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ON NON-SPHERICAL SURFACES IN OPTICAL INSTRUMENTS 







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Translation of! 



^ber asphafiische Flachen in optischen jinstrumenten | 
Kungliga Ivenska VetenskapsakadeiaieaarHandlingsEF""^ 
Vol. 60, 1919-1920, Number 1, pp. 1-155- 




NATIONAL AERONAUTICS AND SPAC E ABMINISTBATION 



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KUNGLIGA SITSIBKA 71BE11BKSPSAK&DMIS1© HiUSBLINGAR 

[TRM.TIS1S 0? OEE ROYAL SllDISH AOADMY OF SGIENGES] 

¥olume 6O5 1919 - 1920, Number 1, pages 1 - 155 

ON 

NOH-SfflERICiU. SURFACES 
IN OPTICAL INSTRGMEWrS 

by Allvar Gullstrand 
With 10 figures in the text 
Presented 12 February 1919 



StocMiolm 
Almqvist & Wiksells BoJctryokeri-Ao-Bt 



Table of contents 



Chapter 

II - New Methods of Production . 

Convex Surfaces of the Second Order 

The Duplex Method 

Ill <- Determination of Machine Constant .... 
Centric Osculation of Higher Order . . 

Eccentric Osculation ...... 

IV - Calculation with Hon-Spherical Surfaces ; 



• « • 



Page 
1 

23 

23 

32 

68 

69 

98 
136 



V - Examples of the Application of Duplex Surfaces I69 



I - Method 

The attention of researchers has been directed to the utilization 
of non-spherical surfaces in optical instruments since Kepler and Cartesius. 
Extensive practical use among the surfaces belonging to this category was 
first made of the cylindrical surfaces in eyeglasses for correction of 
ocular asti^atism. Now toric surfaces are also being used for this pur- 
pose and the wide distribution of astigmatian of the human eye has brought 
it about that eyeglasses with cylindrical or toric surfaces belong among 
the most occurring optical instruments. 

However, if we restrict ourselves to non-spherical planes of rotation, 
it is no longer possible to speak of a general utilization. This doubt- 
lessly is related to the difficulties both of the production as well as of 
the calculation of such surfaces. The present short history of the latter 
in practical optics is that of the surmounting of these difficulties and 
we can hardly go amiss by predicting that real progress in this field will 
be accomplished only by methods which permit both practical production 
with simple mechanical as well as calculations with elementary mathemat- 
ical means. 

Until recently, such surfaces seem to have served only for the purpose 
of focussing in one point of the axis. From this viewpoint, parabolic mir- 
rors are highly suitable and have been utilized in telescopes, search- 
lights, and microscope condensers. This type presents no difficulties of 
calculation and the technical difficulties in production can be overcome 
as results have proved so that the only remaining question of importance is 
that of cost which does, however, influence the possibility of wide use. 
To an entirely different type belong the surfaces produced through so-called 
local retouching which have been employed principally for large telescope 



objectives and where surfaces ground initially spherically were further 
processed by polishing of given zones. As a test of the effect of such 
retouching, it has been possible only to utilize the efficiency of the ob- 
jective so that the method must be designated as purely empirical. The 
surfaces obtained in this matter are consequently unknown from the mathe- 
matical point of view and cannot be designed by calculation. 

Only the introduction of a third type of non-spherical surface of 
revolution by Ernest Abbe opened the path to the utilization of such sur- 
faces for the correction of deviations of oblique [light] beams and only 

2) 
10 years later was it possible for Kohr to report on a practical success 

of this type by correcting the astigmatism of oblique bundles in cataract 
glasses. Abbe has embodied his respective findings in two patent descrip- 
tions which are dated I899 according to Rohr . In the first of these der- 
scriptionsj he introduces, for the determination of a point on the surface, 
the distance of this point from the osculating sphere belonging to the ver- 
tex. This distance is designated by x and the radius of the osculating 
sphere by r . The distance s is calculated as positive when the point of 
the surface is located within the osculating sphere so that the radius vec- 
tor from the center of curvature of the latter has the value r - s. The 
length of arc of meridian in the osculating sphere from the vertex to the 
point where it is intersected by the prolonged radius vector is designated 

1. ERNST ABBE, Linsensystem mit Correction der Abweichungen schiefer Bue- 
schel, Gesammelte Abhandlungen von Krnst Abbe, Vol II, P. 301, Jena I906. 
— Verfahren, sphaeroidische Flaechen zu pruefen und Abweichangen von der 
forgeschriebenen Gestalt nach Lage und Groesse zu bestimmen, ibid, p.311> 

2. M. V. ROHR, Iber Gullstrandsche Starbrillen mit besonderer Beruecksich- 
tigung der Korrektion von post-operativem Astigmatismus. Report on the 
36th meeting of the Ophthalmologischen Gesselschaft Heidelberg 1910, 
Wiesbaden 1911. 

5. ibid, p. 189 



by him with 1 and the equation of the meridian curve of the "spheroid sur- 
face" written as: 

By this designation, he intends to express the assumption that s is very 
small everywhere in relation to the length of radius r°. Based on this as- 
sumption he developed approximate formulas which take into account only the 
first member of the series. The second patent description treats the 
method of production of spheroid surfaces by stating first that only such 
spheroid surfaces are suitable for optical purposes in which the linear de- 
viation s from the sphere of the vertex remains restricted, within the ef- 
fective area of the soirface to a very small magnitude (a few 1/100 mm). 
The procedure consists in reducing, over given areas, the spherical surface 
by grinding and polishing and by employing an accurate lens as template so 
that it "does offer somewhat greater but no other difficulties than the re- 
presentation of a spherical surface by means of the corresponding spherical 
lens template." These difficulties are predominant in the production of 
the template lens, and in general in the first production of a given spheroid 
surface because only spherical surfaces can be utilized for testing. In 
the preliminary processing of the lens surface, the coordinates of the sur- 
face point are checked by utilizi-Jig a suitable spherometer and compared 
with those of a model sphere with accurately known radius and with the same 
sign as that of the curvature of the vertex of the spheroid surface to be 
reduced and only little different from the radius. It is stated that the 
desired shape of the spheroid surface can be produced in this manner with«- 
out difficulty in all zones with an accuracy of +/- 1 micron, provided the 
model sphere utilized is correspondingly accurate, but that an appreciably 
greater accuracy of the spheroid shape is required in general for optical 
purposes. In order to achieve this accuracy, a second model sphere is 



utilized during the last retouching and the radius of curvature of this 
sphere with opposed sign is numerically approximately equal to that of 
the vertex of the spheroid surface to be tested and where the diameters of 
the interference rings, when placing this model sphere against the sphe- 
roid surface, represents the test object. If the radius of curvature de- 
creases for a convex spheroid surface and/or increases for a concave sur- 
face toward the periphery, the two surfaces then touch at the vertex. If 
this is not the case, then the model sphere must lie on the spheroid surface 
along a strictly circular edge. The procedure makes it possible to produce 
accurately spheroid surfaces of given shape, "except for very small frac- 
tions of a micron." If the desired spheroid shape differs everywhere so 
little from the spherical shape that it can be produced from a sphere of 
suitably selected radius only by polishing over given areas, then the pre- 
paratory shaping by means of the spherometer can be eliminated. 

The Abbe method is of such decisive importance that it requires more 
detailed discussion. Concerning first the practical results, the original 
consideration of Abbe seem to have been realized unchanged only in the al- 
ready mentioned cataract glasses with any great degree of success because 
there are no other known optical instruments in which the divergence of 
oblique beams is corrected by surfaces whose shape is little different from 
the sphere. In any event nothing is known on whether any surfaces of this 
type have had any appreciable practical success. However, the Carl Zeiss 
Company has repeatedly used other non-spherical surfaces for the purposes 
of more accurate focussing in axial points, i.e., in large searchlights, in 
certain microsc ope condensers, etc. Rohr assumed from the beginning the 

1. M. V ROHR, Ueber neuere Bestrebungen in der Konstruktion ophthalmol- 
ogischer Instruments. Report on the 37th Meeting of the Ophthalmolo- 
gischen Gesellschaft Heidelberg I9II1 Wiesbaden 1912, p. 53* 



task of calculating a single lens intended to give a sharp picture of an 
object with relatively wide beams. Calculation leads to the aplanatic 
lens in which the sinus condition is also complied with, at least for a 
given ray inclination. For certain delicate ophthalmological methods of 
examination, such lenses are absolutely required and have found wide distri- 
bution as components of the respective instruments but are also used for 
many other purposes. Although nothing further has been published on the 
production of such surfaces, Rohr seems to indicate that the procedure is 
based on the Abbe method. Theoretically also, very little has been publish- 
ed in this field by the followers of Abbe. The surface equation of Abbe is 

1) 2) 

reiterated both by Koenig and Bohr as well as by Siedentopf but with 

changed designations, opposed sign and also (Siedentopf) changed numerical 
value of the coefficient. Koenig and Rohr derived the formulas which indi- 
cate the influence of Seidel image errors through the first coefficient in 
the series of Abbe. (In these formulas, the coefficient X has the same sign 
and the same value as k above.) As far as nomenclature is concerned the 
designation of "spheroidal'' selected by Abbe is not a happy choice because 
spheroid also encompasses an ellipsoid of rotation. In the first of the 
patent descriptions quoted above, there occurred the expression "spheroidal 
deformation of the original spherical surface" which is understandable in 
any event but has led to the unfortunate designation "deformed surface." 
However, in later publications of Rohr, the expression, non-spherical sur - 
face , is employed for a non-spherical surface of revolution, has been ac- 
cepted elsewhere and is here used in this meaning. 

Other than that, only few theoretical studies on non-spherical sur- 
faces are avail able. In postulating the general problems by applying the 

1. M. V. ROHR, Die Theorie der optischen Instrumente, Vol I, Berlin 190^, 

p. 323 ff. 

2, ibid, p. 25. 



iconic ["eikonal"] concept, both Schwarzschild ^ and Kohlschuetter^ take 
into account the respective differential quotients in the equation of a 
general surface of revolution. Whereas these investigations were restrict- 
ed to the paraxial space and consequently considered only the differential 
quotients of the surface equation at the vertex, I have deduced , on the 
exclusive assumption of a symmetrical plane, general formulas for calcula- 
tion of the divergences of finite inclined beams from which result, for 
general axial -symmetrical systens, some of the asymmetrical values deter- 
mining the decimal points, the inclinations of the two image surfaces and 
the asymmetrical value of the tangential enlargement coefficient as well as 
certain other summation formulas ' which can be used in the calculation of 
optical systems. 

The method of differential geometry employed in these investigations 
differs basically (because of the difference of the task; from the treat- 
ment of the problem to comply with given conditions for each ray in a wide- 
open system. Schwarzschild calculated a system consisting of two mir- 
rors and strictly aplanatic for infinite distance from objective in which 
not only the axial image point is strictly free of aberration but which also 
fulfills the sinus condition along each ray, and was able to represent the 
rectangular coordinates of the surface points for both mirrors without ap- 
proximation as explicit functions of a parameter. The same problem, even 

1. K. SOHWAEZS CHILD, Untersuchungen zur geometrischen Optik. Abhandlungen 
der iSesellsehaft , der Wissenschaften zu Goettigen, Math. phys. Klasse, 
new series, Vol IV, N;o 1-3, Berlin 1905* 

2. ARNOLD KOHLSCHUETTER, Die Bildfehler fuenfter Ordnung optischer Systeme 
abgeleitet auf Grund des iikonalbegriffes nebst Anwendung auf ein astro- 
photo graphisches Objektiv. Diss. Goettingen 1908. 

3. Die reelle optische Abbildung. This series, Vol hi. No, 3, Upsala I906 
h, Tatsachen und iiktionen in der lehre von der optischen Abbildung. Archiv 

fuer Optik, Vol 1, I908 
5. ibid. , No. 2. 



complemented by a requirement for achromatism, was solved by Linnemann 
for refracting surfaces. However, the surface equations can here not be 
simply written down but the respective differential equations must be in- 
tegrated numerically. In connection with such integration, there is also 
indicated the numerical method with which a ray can be followed through 
such a system. 

Let us mention finally that the greater interest in non-spherical 

2) 

surfaces is manifested by the fact Kerber outlines the application of his 

new calculation formulas for deformed rays on non-spherical surfaces and 

Lange derives both the calculation formulas for rays traveling in a me- 

3) 
ridian plane as well as the aberration values in the axis for surfaces 

of revolution of the second order 

This brief review shows the non-spherical surfaces utilized so far 
in practice are ei,ther surfaces of revolution of the second order or Abbe 
surfaces by which I understand surfaces of rotation whose meridian sec- 
tions have the equation indicated above. We here disregard the mathe- 
matically non-definable surfaces produced empirically by local retouching. 
In addition, theoretical treatment has been given to reflecting surfaces 
which are represented by certain transcendant equations and certain sur- 
faces siiitable only for numerical integration of differential equations. 

In this connection, *e want to mention also, among earlier investigations, 

5) 
the Huygen metho d with which it is possible to construct a surface 

1. MARTIN LINNEMANN, Ueber nichtsphaerische Objektive. Diss. Goettingen 1905. 

2. ARTHUR KERBER, Neue Durchrechnungsformeln fuer windschiefe Strahlen. 
Zeitschrift fuer Instrument enkunde. Vol. 53, p 75? 1913* 

3. MAX LANGE, Durchrechnungsformeln fuer die Lichtbrechung an Kegelschnit- 
ten. ibid. Vol ^k, p 273, 191^. 

4. The same, Entwicklung des ersten Gliedes der Aberration endlich geoeff- 
neter Lichtbueschel fuer den Achsenobjektpunkt einer lichtbrechenden 
Rotationsflaeche deren Querschnitt ein Kegelschnitt ist. Ibid. Vol 31, 
p. 3^8, 1911. 

5. CHRISTIAN HUYGENS, Treatise on Light. Rendered into English by SILVANUS 
P. THOMPSON. London 1912, p 116 ff. 



point-by-point and which makes a given light beam homocentric by refrac- 
tion. 

Concerning these last categories, it will be evident that the re- 
spective surfaces can be produced equally well with the Abbe method as the 
Abbe or any other surfaces. However, it is equally evident that they re- 
quire much more complicated calculation and are therefore replaced, in 
practice, by the last named or other surfaces equivalent in this respect. 
This is probably also always possible. Actually, we are not concerned in 
the first instance with a fusion of the rays in a mathematical point since 
diffraction produces the fact that, even in those cases where such fusion 
of rays is accomplished theoretically, a finite extended area takes the 
place of the mathematical point, and in the second instance, it is possi- 
ble to select a surface by including an adequate number of coefficients in 
the Abbe series so that it will osculate a given surface with the desired 
accuracy and will still be maniable for mathematical calculation. In prac- 
tice, we should therefore be able to confine ourselves for the time being 
to calculations with surfaces of the second order and with Abbe surfaces 
and the two categories which complement each other, should be adequate for 
the requirements concerning non-spherical surfaces in optical instruments. 

We might be tempted to conclude from this that all reasonable objec- 
tives in regard to such surfaces have already been achieved or are at least 
easily achievable. This would perhaps be the case if the question of the 
production of such lenses were; not of such decisive importance. At the 
present time, the problems of geometrical optics are being investigated 
in the scientific world by very few researchers who are not connected in 
one way or another with the optical industry. That this relation has not 
had only an immense progress of technical optics as consequence but has 



also appreciably advanced pure geometric optics, we can only gratefully 
confirm in regard to the latter. However, on the other hand, this rela- 
tion has also disadvantages for science because certain advances are not 
always published but are treated more or less as industrial secrets. It 
is obvious that we cannot reproach the respective inventors or discoverers 
for this because otherwise there would certainly have been achieved much 
less success and geometric optics would consequently have been less ad- 
vanced. However, this situation results in regard to the production of 
non-spherical surfaces in the fact that information can be obtained only 
from patent descriptions and generally in a very restricted form. 

As far as the Abbe method is concerned specifically, the latter can 
be utilized unchanged only for surfaces which deviate very little from the 
spherical form whereas this deviation assumes considerable proportions, 
for example, in the aplanatic lenses of Zeiss. If the surfaces are produc- 
ed by a method based on the Abbe theory, the latter must have undergone 
certain modifications. Since nothing has become known of this, however, 
we can only attempt to examine the manner in which this method can be 
modified for such purposes. 

In a simple lens with only one non-spherical surface, and when abso- 
lute absence of aberration in one axial point is required, the sinus condi- 
tion can be complied with in general only for a general beam inclination. 
This is achieved easily by an appropriate deflection [curvature?] of the 
lens whereas, for a given deflection, the coordinates of the points on the 
non-spherical siirface can be determined in any desired number and desired 
accuracy through the Huygens method. In order to do so, it is only neces- 
sary to express the geometrical expression of Huygens trigonometrically. 
As already indicated, this results in two possibilities for the production 



of the lens tanplate by either producing this surface directly or else 
calculating a surface by means of the Abbe equation which will adequately 
closely osculate the former to be utilized in production. In the first 
case, there must be calculated at least as many point coordinates as are 
required for spherometric control and, if further control by means of the 
interference rings is to be used, very complicated calculations must be 
added in order to achieve the accuracy intended by this method and without 
which it would seem rather useless. However, for the calculations of the 
Abbe surface, the coordinates of a small number of surface points are suf- 
ficient. The calculation by which they are determined, also furnishes the 
direction of the normal and the radius of curvature and it is possible — 
as will be explained in detail further below — to calculate the osculating 
surface in various ways, even by utilizing only one single surface point 
but where then the residual amount of aberration of different rays must be 
investigated by theoretical calculation in order to judge whether the cal- 
culated surface osculates the prescribed shape sufficiently accurately. If 
the surface equation has been found in this manner, the calculations neces- 
sary for using the interference rings for control can be effected more eas- 
ily than when the surface can be constructed only point-by-point. As al- 
ready stressed, since we are actually not concerned with a mathematically 
accurate fusion of rays, there seems to be no reason to prefer the more 
difficult calculation of a surface constructed point-by-point to the Abbe 
surface calculated with an adequate number of coefficients. 

The method of control by means of the interference rings cannot be em- 
ployed unchanged when a high deviation from the spherical shape exists. A 
surface of revolution can be regarded as the enclosing surface of spheres 
whose centers are located in the axis and whose radii are equal to the 



10 



normals drawn from the respective axial point to the surface. If then 
the radius of curvature steadily increases from the vertex to the periph- 
ery and the evolute has an apex in the direction of the vertex, then the 
distance of the center of the sphere from the vertex as well as the radius 
of curvature of the meridian section of the surface is greater everywhere 
in the points touched by the generating sphere than the radius of the lat~ 
ter and the sphere is located on the concave side of the surface without 
intersecting the latter. Precisely the opposite takes place if the radius 
of the curvature of the surface steadily decreases from the vertex to the 
periphery when the generated surface always lies within the generating 
sphere without being intersected by the latter. It follows from this that, 
in the production of a lens template, control by means of the interference 
rings created in the contact with spherical surfaces along the latitudes 
of the ground surface is possible only if we grind, for an increasing ra- 
dius of curvature (from the vertex to the periphery), a concave and, for 
a decreasing radius of curvature, a convex lens template. Depending on 
the degree of deviation from the spherical form and depending on the re- 
quired accuracy, the check must then be carried out by means of a greater 
or lesser number of different spherical surfaces. Instead of contact, we 
obtain osculation at a very small angle by grinding an appropriate circu- 
lar edge on the spherical surface. A given latitude or parallel of the 
ground surface corresponds to each radius of sphere. If we now imagine 
the spherical surface divided into two parts by section in such manner 
that the plane surface of section has exactly the diameter of the corre- 
sponding parallel and if we imagine further that this surface of section 
is ground off somewhat but only very little in both parts, then each of 
the two parts can be placed against the ground surface so that the edge 



11 



of the spherical surface coincides with a parallel of the ground surface 
and the two surfaces here intersect at a very small angle. Other possi- 
bilities also exist on which we shall not enter here since we are merely 
concerned with showing that the Abbe method of control by interference 
rings can be modified so that the method can be employed also for surfaces 
whose form differs appreciably from the spherical form. However, it is 
obvious that the method becomes complicated in proportion to the increasing 
degree of deviation and requires a corresponding greater number of spheri- 
cal lens surfaces with accurately known radius. 

For the first shaping of surfaces differing appreciably from the 
spherical form, a shorter procedure is preferable to subsequent zonal 
grinding and polishing. Preliminary grinding by machine is highly suitable 
for this, since grinding is based in general on the contact of two sur- 
faces and a non-spherical surface to bfe produced by employing one single 
grinding surface generally cannot touch the latter simultaneously in more 
than one point so that the machine must obviously be capable to accomplish 
such movement of the two surfaces in relation to each other that the non- 
spherical encompasses the grinding surface. It is of technical advantage 
here when both surfaces move in themselves and when we start from a suit- 
ably selected and spherically ground lens. The latter is consequently 
centered on a rotating axis and the grinding surface best represents a 
surface of revolution which rotates around its own axis. If this surface 
is neither cylindrical nor degenerates into a sphere or a plane, we caji 
then grind only through a single parallel of the grinding surface because 
the mechanical installation would otherwise become very complicated. This 
parallel latitude must then always be located in a plane which contains 
also the axis of revolution of the non-spherical surface and the grinding 



12 



surface must have such a form that no other point of it can touch the 
non-spherical surface. Although in general suitably formed surfaces of 
revolution can be employed as grinding surfaces which are symmetrical to 
an equatorial plane and such symmetry is not even necessaiy , it would seem 
to be sufficiently generalized for the present demonstration if we pro- 
ceed on the assumption that the grinding surface represents a torus and 
that both the cylinder as well as the sphere and the plane are considered 
as special cases of the torus. The toric surface may be regarded as the 
encompassing surface of a sphere whose center moves on a fixed circle, 
the base circle and its axis of revolution therefore intersects the plane 
of the base circle perpendicularly in the center of the latter. 

The requirement for having the grinding parallel and the axis of re- 
volution of the non-spherical surface always located in one and the same 
plane is most easily complied with by having both the axis of revolution 
as well as the base surface of the torus always remaining in one and the 
same fixed plane. If we consider the axis of revolution as fixed, the cen- 
ter point of the toric surface must therefore describe a fixed curve which 
represents a curve parallel to the meridian curve of the non-spherical sur- 
face and will be designated as machine curve . Since the base circle of 
the toric surface must remain in the fixed plane and the axis of revolu- 
tion of the latter therefore must always be perpendicular to this plane, 
the machine curve is consequently described by each point on this axis. 
However, if we grind with a sphere, the orientation of the axis of revolu- 
tion becomes indifferent and the machine curve needs to be described only 
by the center of the sphere. In this manner, it is possible to grind a 
convex surface with a concave spherical calotte. On the other hand, if 
a convex surface is to be ground and a cylinder is substituted for the 



13 



toric surface, then a displacement of the cylinder in the direction of 
its axis is permissible although this axis must then continuously inter- 
sect the fixed plane perpendicular in all positions. If such displace- 
ment takes place, the machine curve is then described by the point of in- 
tersection of the cylinder axis and the fixed plane. Finally, if the non- 
spherical surface has no points of inflection on the meridian curve, the 
cylinder can be replaced by a plane which is articulated around an axis 
lying in or parallel to this plane and intersecting the fixed plane always 
perpendicularly. The inclination of the latter in the different positions 
of the axis of articulation is then determined thus that the line drawn in 
the fixed plane from a fixed point located on the axis of revolution of 
the non-spherical surface to the intersection with the axis of articula- 
tion always represents a normal of the grinding surface. This plane can 
then perform any desired movement in itself and the machine curve describ- 
ed by every point on the axis of articulation represents, depending on 
whether this axis lies or does not lie in the plane, the base curve of 
the meridian curve of the non-spherical surface or the base curve of a 
parallel curve of the latter in relation to the fixed point. Whichever 
of these arrangements is selected, the machine curve now can always be 
constructed point-by-point without difficulty as soon as the equation of 
the non-spherical surface is given and we need in essence only to describe 
only a curve constructed point-by-point by a machine part. 

This problem cannot be solved purely Mnema tic ally but requires em- 
ployment of methods which may be grouped under the designation, template 
method , and are characterized by the utilization of a guiding curve con- 
structed punctually. This curve need not be similar to the machine curve 
but can be produced by any kinematic generation of one curve out of another. 



Ik 



i.e., by circular inversion, by rolling up or off, etc., so that eccentric 
curves and evolutes are also included among the guiding curves in connec- 
tion with it. By employing a guide curve not similar to the machine curve, 
it is also possible to make the errors of the machine curve smaller than 
those of the guide curve. This could be accomplished by pantographic trans- 
fer in which the guide curve would be similar to the machine curve but be 
constructed on a larger scale. Template methods seem to have been employ- 
ed already for a considerable time . 

A particular method of preliminary shaping has been patented by Carl 

2) 
Zeiss . A lens first ground with two spherical surfaces is softened by 

heating so that it adapts itself to a supporting surface also ground spher- 
ically. The process imparts to the upper surface of the lens a non-spher- 
ical form depending on the radii of curvature of the three spherical sur- 
faces and on the thickness of the lens. Whether this method has proved 

itself is not known. In any event, the risk of internal stresses in the 

3) 

lens represents a complication which cannot be disregarded . 

It should be clear from the foregoing that the original Abbe method, 
in order to be a,pplicable to non-spherical surfaces whose shape differs 
appreciably from the spherical form, needs to be modified only so that pre- 
liminary shaping is effected by means of a template method (or possibly by 
heating), that more coefficients are included in the equation, and that a 
larger number o f spherical surfaces with accurately known radius of curvature 

1. See, eg., the German patent XMo. 23369, E. Avril.Schleifmaschine um 
Brillenglaeser nach einem Modell zu schleifen. Zeitschr. fuer Instru- 
ment enkunde, vol k, p. 7^, l884. 

2. D. R. P. No. 212621. Deutsche Mechaniker Zeitung 1910, p. 51. 

3. See the communication of Schott ©t al,# . Der linflmss der Abkuehlung auf 
das optische Verhalten des Glases und die HersTeTlung gepresster Linsen 
in gut gekuehltem Zustande. k^eitschr. fuer Instrumentenkunde, Vol 10, 

/ kl, 1890. 



15 



is employed for control with the interference rings. 

We do see on the other hand that this method is too complicated for 
producing a large selection of non-spherical surfaces but can be employed 
for making a small number of such surfaces which can be used in a suffi- 
cient number for given purposes. To this should be added that this selec- 
tion is determined not only by the shape of the surface but also by the 
radius of curvature of the vertex. For example, if the aplanatic projec- 
tion of a point is desired by a simple lens of given focal distance and 
corresponding lenses with this focal distance are not available, there is 
then required, for the production of the desired lens, a new lens template 
in any event and possibly also a new guide curve. The latter is eliminat- 
ed in two cases; first, when the method of preliminary shaping by heating 
is possible in practice and, second, when the preliminary grinding machine 
is constructed so that the guide curve determines only the shape of the 
surface but not the ratio of magnitude. It is obvious that the complicated 
production and consequent restriction in the selection of surfaces produced 
is a handicap in the general use of the latter in optical instruments. 

However, there is no doubt that a wide field is open to non-spherical 
surfaces as soon as the production of the latter is possible under such 
conditions that the designer of optical equipment can incorporate non-spher- 
ical surfaces without any hesitancy in his design. 

Two objectives here occupy the foregoing. In order to advance essen- 
tially the possibilities of construction in general, it would be desirable 
that there should always be available surfaces of a selected simple type 
in which evidentally, aside from the radius of curvature of the vertex, 
only one coefficient could be freely available. If it were always possible 
to employ only a single such surface in an optical instrument and even if 



16 



this possibility vsere restricted only to convex non-spherical surfaces, 
it will be easy to see that even this would accomplish an appreciable ex- 
tension of the optical means available. We need only remember that, in 
general, a Seidel image error can be corrected through the respective co- 
efficient. The second objective must obviously be to encourage the utili- 
zation of such surfaces which will osculate, with optimum possible accur- 
acy, a surface of any desired prescribed form. The means for this would 
be a simplified production of surfaces of an appropriate tj'^pe with several 
coefficients. 

The possibility of simplified production of non-spherical surfaces is 
influenced to a great extent by the requirements made on the accuracy of 
the shape of the surface. In respect to this, relatively high demands for 
axial symmetry must be fulfilled whereas small deviations in the shape of 
the meridian curve ar^ more easily permissible. If the surface does not 
represent strictly a surface of revolution, this then results in deviations 
on the axis which are most closely similar to those which are created by 
inadequate centering of an optical instrument and are as little permissible 
as these. However, if the shape of the surface to be produced deviates 
appreciably from the spherical form and we obtain a surface, in place of 
the desired surface, which, although it does represent a surface of re- 
volution, has a meridian curve with minor zonal deviations from the prescrib- 
ed form, these will be, in the greater part of the cases, without signifi- 
cance for the practical application of the respective optical instruments. 
This can be best illustrated by an example. The hyperboloid of rotation 
whose meridian curve has a numerical eccentricity which is equal to the value 
of the refraction index of the lens, converges — as is well known — a 
beam of rays of arbitrary width which is parallel in the glass medium, on 



17 



one point in the air medium. To begin with, it is obvious that any devia- 
tions of the meridian curve of the surface from the prescribed hyperbolic 
form which cause no greater divergences of the rays than are still contain- 
ed within the diffraction disk, are completely without significance. How- 
ever, even zones of greater deviation are generally permissible without 
any hesitancy. Such zones do occur in existing optical instruments very 
frequently without interferring with the usefulness of the latter. Even if 
no single point of the surface should lie exactly on the prescribed hyper- 
bola, such a surface, provided that it strictly represents a surface of rev- 
olution and that the meridian curve is free of any unsteadiness and has ap- 
proximately the prescribed hyperbolic form, would constitute an enormous 
step forward from the spherical surface. It follows from this that, although 
we must require a strictly axial -symmetric form and ceuanot permit unsteadi- 
ness on the meridian curve, there is no reason for making excessively high 
demands on the trace of the meridian curve, unless we are concerned with 
surfaces whose shape differs only very little from the spherical form. The 
degree to which the demands may be reduced can only be decided by success 
in practice. Initially, the only path open is then to examine the possi- 
bility of a simplified production of non-spherical surfaces from the point 
of view of this principle. 

If we proceed to this from the modification of the original Abbe method 
outlined just now, it would appear initially that the complicated control 
of the lens template by means of the interference rings can be eliminated 
because the spherometer method seems sufficiently accurate, even though the 
latter possesses only an approximate accuracy, in relation to that indicated 
by Abbe, for surfaces whose shape varies appreciably from the spherical 
form. However, we may well ask whether a lens template is absolutely 



18 



necessary. A thorough simplification of production should be oriented to- 
ward mechanical working methods as much as possible in view of industrial 
operations. In the respective machine, the surface must be capable of be~ 
ing produced so accurately that no further finishing except perhaps pol- 
ishing is necessary. The latter can then be effected by skilled workers 
without any essential change in the fonn of the surface. To what extent 
optical or other control methods will here be required can be determined 
only from experience. Where surfaces are concerned of which a large number 
are to be produced, the lens template will of course again take its proper 
place. 

When such requirements are established, it is obvious that production 
by machine cannot be based on a template method because the production of 
the curve d-line guidance would require such complicated working that the 
intended simplification might become questionable. The respective machine 
curves must therefore be capable of being produced kinematically and only 
such guide curves should be employed which can be produced exclusively by 
machine. It follows from this that the meridian curve of the non-spherical 
surface cannot represent an Abbe curve. On the other hand, since the machine 
curve cannot coincide with this meridian curve because this would require 
grinding with one cusp C'Spitze"] represents a parallel or base curve of 
the latter, it is evident that the equation of the meridian curve of the 
non-spherical surface in general will not be obtained at all. The result- 
ing disadvantage that we must start from the machine curve in the calcula- 
tion of optical syst«is which contain such non-spherical surfaces, is of 
completely minor significance since the additional calculations — as will 
be explained in further detail below -- demand only trigonometric methods. 
However, exceptions to this are all those cases in which it is necessary 



19 



to determine the intersection of the non-spherical surface with a given 
ray but in these cases we are restricted to numerical methods even when 
employing the Abbe equation. 

The disadvantage of having to employ the machine curve in place of 
the meridian curve in the theoretical calculation of the system can be 
eliminated only when the latter curve possesses a parallel or base curve 
which can be generated kinematically and can itself be expressed directly 
through an equation. Theoretically, this is the case for all curves whose 
parallel or base curves represent algebraic curves since any plane alge- 
braic curve can be generated by a system of articulation. However, since 
the machine will be as much more accurate as it is simple, only a very few 
known curves can be. utilized in practice and we cannot expect in this man- 
ner to be able to generate curves with more than one freely available co- 
efficient. On the other hand, we have alreac^ stressed that such curves 
can be expected to play an important role. Among them, curves of the second 
order axe at the top, both because the latter possess base curves which can 
be easily generated kinematically and also because of the possibility to 
carry out theoretical calculation in all cases by employing the most ele- 
mentary means. Since the machine curve represents a base curve, only convex 
surfaces can be obtained with such direct methods but the advantage of the 
simpler calculation is so great that the surfaces of revolution of the sec- 
ond order generally do seem to be most suitable to comply with the purpose 
indicated above, provided that we are successful in producing the latter 
with adequate accuracy and with the numerical eccentricity which can be ar- 
bitrarily prescribed within sufficiently large limits at any desired radius 
of curvature of the vertex. 

In the solution of the other task, i.e., to produce non-spherical 



20 



surfaces which will osculate sufficiently accurately a prescribed surface, 
emphasis must be placed on the possibility of calculating the respective 
coefficients in the equation of the machine curve which we shall designate 
as machine constant , for the sake of brevity, with mathematical means as 
simple as. possible. Above all, the equation of the machine curve must 
therefore be suitable for an osculation of higher order in the vertical 
point. Since it is desirable to have available as large as possible a 
number of machine constants for this purpose and the respective calcula- 
tions for more than three constants are too complicated, we shall concen- 
trate on the problem of obtaining a complete contact of the eighth order, 
as far as possible, of the ground surface with the prescribed surface, 
i.e., a nine-point contact of the meridian curves of the two surfaces in 
the vertical point. For this purpose, we must above all determine the re- 
spective generally valid relations between the differential quotients of 
the curve and those of the parallel or base curve for a vertical point. 
If we have found the machine constants which will produce such a central 
osculation of a given order, it is best to employ the latter not always 
unchanged but to attempt a higher degree of osculation through variations 
of the latter under control by theoretical calculation or by the values 
prescribed for the surface. With very large openings, it may be of ad- 
vantage to base the calculation of the machine constant on the condition 
that the meridian curve of the ground and of the prescribed surface shall 
touch each other in a given point or shall possibly also have the same 
radii of curvature. For such an eccentric osculation of the first and/or 
second order, two and/or three machine constants are required. In order 
to obtain simultaneously either a central osculation of the fourth order 
or an eccentric osculation of the second or two eccentric osculations of 



21 



the first order, four machine constants are required. For the Abbe curve 
which seems specifically adapted to this purpose, the coefficients in such 
conditions can be determined by linear equations. We are therefore con- 
fronted by the task to find a machine curve equally ideal in this respect 
and to have in mind, in any event, in the selection of the machine curve 
the possibility of specifying various conditions in the determination of 
the machine constant. 

This investigation of the various methods for calculating the machine 
constant will be followed by a demonstration of the methods for the theo- 
retical calculation of optical systems containing the respective surfaces 
and followed ultimately by a discussion of the employment of such surfaces 
for certain purposes. 



22 



II - New Methods of Production 
Convex Surfaces of Rotation of the iSecond Order . The base curve of 
a conical section in relation to a focus is, as is well known, a circle 
which has its center in the center of the curve, whose radius is equal to 
the semi-axis and/or the major semi-axis, and which therefore degenerates, 
in the case of a parabola, into the vertical tangent of the latter. This 
is equivalent to the fact that the lines drawn through the different points 
of a circle perpendicularly on the connecting lines of these points with 
a given point, are encompassed by a conical section. It follows from this 
that, by employing circular and straight-line guidance, we are able to 
grind convex surfaces of revolution of the second order with one plane by 
the method outlined above. In Fig. 1, let AB be a crank which is able to 




Fig. 1 



revolve around an axis intersecting the paper plane pearpendicularly in 
A and possessing an axis parallel to the axis in B around which the arm 
BC is able to revolve in turn. It is then merely necessary to allow this 
arm in any position to pass through the fixed point D and to rigidly link 
the grinding surface which intersects the paper plane in the figure 



23 



perpendicularly to the line EF, in such manner that it is located perpen- 
dicularly on the line BD and that the axis B is contained in it. When 
pivoting the crank arm, the plane EF is encoapassed by a straight hyper- 
bolic cylinder in which A represents the center point, AB the semiaxis 
and D a focus of the hyperbola. The axis of rotation of the lens surface 
to be ground must therefore be parallel to the paper plane and contained 
in a plane which intersects the latter perpendicularly in the line AD and 
grinding is effected with the plane grinding downward while the lens is 
raised upward with unchanged axis of rotation. If the plane EF intersects 
the line BD or its prolongation perpendicularly in any other point than B, 
the corresponding parallel surface of the hyperboloid will be ground. 

The convex surface thus ground on the upper face is produced by pivot- 
ing the arm AB so that the axis B comes to lie between A and D in the cen- 
ter position. By a full revolution of the crank, however, the two branches 
of the hyperbola are generated. If the arm AB is then pivoted around a 
center position in which the axis B intersects the extension of the line 
AD so that the plane EF now grinds upward, we then obtain, on the same 
axis of revolution, the same hyperboloid which is now convex on the under 
face and/or the corresponding parallel surface. This is equivalent to the 
fact that the line DC in Fig. 1 is not permitted to run through the point 
E but through that point which is located at the same distance from A as 
B along the extension of the line AD so that then the lower shell of the 
hyperboloid is ground by employing the upper focus. As will be seen im- 
mediately, one and the same angle of traverse of the arm AB corresponds in 
this case to a lesser inclination of the plane EF toward the horizontal. 
To this should be added that the contact point of the plane with the sur- 
face is not located on the same side of the line AD as the axis B (Fig. 1) 



24 



but on the opposite side. For one and the same surface points, both the 
angle of traverse of the arm AB as well as the distance of the contact 
point from the axis B is greater when we employ the more distant instead 
of the nearer focus. 




Fig. 2 



The paraboloid is obtained when the crank is replaced by a straight- 
line guidance and if the axis B is guided perpendicular to itself in a 
plane located perpendicularly to the axis of revolution. The distance of 
the point D from this plane is one-half of the radius of the vertex and, 
if the plane EF intersects the line BD in any other point than B, there 
then results in the same manner the corresponding parallel surface. On 
the other hand, if the crank axis A is located below the point D and if AB 
is greater than AD, then the ground surface is an ellipsoid which has the 
greater axis as axis of revolution and/or the corresponding parallel sur- 
face of such an ellipsoid. By a full revolution of the crank, the complete 
ellipse is generated as the encompassing curve of the line EF. It follows 
from this that it will be possible also to grind ellipsoids of rotation 
with the shorter axis as axis of revolution. In Fig, 2, if A is the fixed 
axis, AB the crank arm, BC the line passing in any position always through 
the fixed point D, and EF the secant of the grinding surface located perpen- 
dicularly on the line BC and rigidly linked to the latter, there will then 



25 



be ground, when pivoting the arm AB around the center position drawn in 
the figure, such an ellipsoid and/or the corresponding parallel surface 
of the latter, under corresponding securing of the grinding surface, if 
the axis of revolution of the surface is located parallel to the paper 
plane and contained in the plane perpendicular to AD which passes through 
the pqint A. If the arm AB is pivoted only to one side, it will be easily 
seen that, for one and the same surface point, both the angle of traverse 
as well as the distance of the contact point of the grinding plane with 
the ellipsoid of axis B is sraallef when AB is brought closer to D during 
pivoting than in the obverse case. Simple reflection will show us that 
the same differences exist also in grinding an ellipsoid with the greater 
axis as axis of revolution by utilizing the closer and/or the more dis- 
tant focus. 

Mechanically , the requirement for having the line BC pass in all 
positions through the point I) is complied with by having the arm BG slide 
in a straight-line guide which can rotate at D around an axis intersect- 
ing the paper plane perpendicularly and having the direction of the line 
BD coincide with that of the straight-line guide. The mechanism is con- 
sequently an oscillating crank mechanism or, in the more exact nomencla- 
ture introduced by Burmester , a centrally rotating and/or a centrally 
oscillating crank-drive depending on whether an ellipsoid or a hyperboloid 
is to be ground. If the crank is replaced by a straight-line guide for 
the purpose of grinding a paraboloid, this then results in a central 
straight-line thrust-drive, according to the same nomenclature. Transla- 
tion into practice initially raises the question whether the arrangement 
outlined at the top of Fig. 1 in which the axes A and D as well as the 
1. L. BURMESTER, Lehrbuch der Kinematik I. Leipzig 1888. 



26 



axis of revolution of the lens surface are fixed, is actually the most ad- 
vantageous. If these components are not fixed, the axes A and D must be 
linked to each other by an arm and the axis of revolution of the lens sur- 
face must be fastened to this arm in such manner that it intersects the 
two axes perpendicularly, fhe mechanism then consists of the two arms AB 
and AD with the articulated axes AB and AD and of two equivalent components 
sliding against each other of which one can revolve around B and the other 
around D, and which provide straight-line guidance in the respective di- 
rection BD. These components, called in kinematics infinite members of 
the specialized plane mechanism, will be designated here simply as thrust- 
components. It is therefore a general condition for the grinding of the 
respective surfaces that the grinding surface is linked perpendicularly to 
the direction of thrust on one of the thrust components whereas the axis 
of revolution of the lens surface is rigidly linked to the opposite arm 
and intersects the articulated axes of the latter perpendicularly. Depend- 
ing on whether the grinding surface passes through the articulated axis of 
the respective thrust component or not , a surface of the second order and/or 
the corresponding parallel surface is ground. Since AD represents the dis- 
tance of a focus from the center point, AB the semi-axis and the ratio AD 
to AB is therefore indicated by the numerical eccentricity, the numerical 
eccentricity is therefore in general equal to the ratio of the length of 
the arm linked to the axis of revolution to the length of the other arm. 

However, it is possible to fix rigidly any one of the four links and 
thus distribute the motion in different ways to the grinding surface and 
the axis of revolution. In Fig. 1, the axis of revolution is fixed whereas 
the plane carries out a compound motion. However, if we make the arm AD 
fixed — which again results in a crank drive — then both the axis of 
revolution as well as the plane effect simple motions of revolution, i.e., 



27 



around the axes A and/or B. On the other hand, if one of the thrust com- 
ponents is made fixed — which produces a thrust-crank drive — either the 
grinding surface remains fixed while the axis of revolution carries out a 
compound motion or else the plane is slid back and forth in the direction 
of its normal whereas the motion of the axis of revolution is a simple ro- 
tation, depending on whether the thrust component containing the axis B or 
that containing the axis D is made fixed. 




Fig. 3 



Except for differences of mechanical design, the same is true for the 
sliding mechanism for the grinding of paraboloids. In Fig. 3j GH represents 
the straight-line trace of the point B which originates when the axis A in 
Fig. 1 is infinitely distant. The mechanism consists of two straight-line 
guides with two articulated axes and has four links of which one has an 
articulated connection with another one and slides in a straight line along 
a third. The connecting line of the axes B and D coincides with the direc- 
tion of one of the straight-line guides. A central straight-line guide is 
therefore linked to an eccentric straight-line guide by two articulated 
axes. It will then be easily seen that, in this mechanism also, two links 
belonging to one and the same straight-line guide are kinematically equivalent. 



28 



This is due to the fact that, if we make the link connected to the axis B 
and belonging to the eccentric straight-line guide fixed, the axis D must 
then describe a path parallel to the line GH and if we then make the axis 
of revolution of the lens surface fixed to the fixed link and the grinding 
surface is made fixed to the link connected with the axis D and belonging 
to the central straight-line guide, we then have identically the same mecha- 
nism. It is consequently a general condition for grinding that the grinding 
surface must be made fixed perpendicularly to the direction of thrust on one 
of the links belonging to the central straight-line guide whereas the axis 
of revolution of the lens surface is connected to the opposite link belong- 
ing to the eccentric straight-line guide in such manner that it intersects 
the articulated axis belonging to the latter perpendicularly and is located 
perpendicularly to the direction of the straight-line guide. In the arrange- 
ment shown in Fig. 3» ttie axis of revolution of the lens surface is made fixed 
whereas the grinding surface executes a compound motion. If the thrust com- 
ponent containing the axis B and belonging to the eccentric straight-line 
guide is made fixed, then the axis of revolution executes a straight-line 
and the plane a rotating motion. On the other hand, we can make the plane 
fixed so that the axis of revolution executes a compound motion and if we 
then finally make fixed the link containing the axis D and belonging to the 
central strai^t-line guide, the plane then executes a straight-line motion 
in the direction of its normal whereas the axis of revolution rotates around 
the articulated axis D. 

For the selection among these various types of machines, the guiding 
viewpoint must be to obtain optimum accuracy of motion and optimum exclu- 
sion of vibrations. It follows from this initially that sliding friction 
should not exist in straight-line guidance. Since this requires a carriage, 



2,9 



there remain only two types if we want to exploit the mechanical advantage 
of a fixed carriage path. If we consider further that the grinding surface 
must perform motion in itself which is accomplished most simply by rotation 
around a normal, we then obtain one each axis of revolution in two different 
links of the mechamism and the requirement for having one of these axes 
fixed simultaneously with the path of carriage travel is complied with only 
by that type in which the grinding surface is made fixed. For other reasons 
also, this type appears to be the mechanically most advantageous, at least 
where we are concerned with optimum applicability for the production of dif- 
ferent surfaces in a small number of pieces. On the assumption that we want 
to grind a convex hyperboloid on the underface, this type is shown in the 
diagram of Fig. k» The vertical axis seen on the left has fixed bearings 
and carries the horizontal grinding disk in which the plane grinding upward 

w 



E 



Si2_f 



w 



Fig. k. 



A 



is indicated by the line EF. fhe axes A, B, D have the same significance 
as in Fig. 1. However, B is here ihe crank axis set in fixed bearings where- 
as A represents only the articulated connection of the crank arm with the 
couple. The other end of the couple is made fast to the axis D and the 
bearings of the latter are carried by the carriage W which moves vertically 
in a fixed path. On the left, the machine component carrying the axis of 



30 



revolution of the lens is linked rigidly to the axis D. If the crank is 
rotated, then the axis of revolution of the lens follows the motion of the 
couple. In order to adjust the machine for different surfaces, it is nec- 
essary to be able to give the crank arm and the couple the corresponding 
length. They must therefore be able to be made fixed to the respective 
axes at any desired point. Consequently, the distance of the lens vertex 
from the axis D must also be capable of being varied correspondingly and 
the path of the carriage must be sufficiently long in order to make possi- 
ble guidance at the different initial positions of the axis D*. Lack of 
space prevents us from going into further details. However, it is clear 
that the three axes must be capable of being very closely approached to 
each other when grinding surfaces with a small radius of vertex and a value 
of eccentricity either low or close to 1 and can therefore not be arranged 
above each other but must be next to each other as indicated in the figure. 
For purposes of adjustment, it is also advantageous to bring them into one 
line which requires them to be centered on each other. When grinding an 
ellipsoid, the axis A- must be located above the axis D so that the full 
length of the crank arm, as drawn in the figure, can be made available. 

For the grinding of a paraboloid, the crank arm and the couple must be 
removed and a roller fixed to the axis B» On this roller then rests a plane 
horizontal in the initiaO. position which is made fixed to the axis D in such 
manner that the shortest vertical distance of the two axes from each other 
is equal to one-half of the prescribed semi -parameter. The same arrangement 
can also be applied to the grinding of ellipsoids and hyperboloids when the 
semi-axis is so large that the corresponding length of the crank arm would 
tend to result in mechanical failure. The plane is then replaced by a cyl- 
inder surface whose axis assumes the prescribed position of the axis A, 



31 



Since the crank arm is replaced in these cases by a "closed linkage", the 
force effective in pivoting must consequently have its point of attack on 
the couple. This is moreover of further advantage if the latter is shorter 
than the crank arm which is the case in the grinding of ellipsoids. 

A machine of similar design can silso be utilized for the grinding of 
ellipsoids which have the shorter asd-s as axis of revolution. It is merely 
necessary for this that, in the initial position outlined in Fig. k, the 
axis of revolution of the lens can be made fixed to the axis D perpendic- 
ularly to the paper plane. The already postulated possibility of varying 
the distance of the lens vertex from the axis D in any of usual applications 
now serves for displacing the axis of revolution perpendicularly to itself 
imtil it is intersected by the extension of the axis A located above D. In 
addition to this, the lens vertex m-ast also be capable of being displaced 
in the new position in the direction of the axis of revolution so that the 
distance of the latter from the axis A can be made equal to the shorter semi- 
axis of the ellipse. If the lens has been fixed correctly to the axis D in 

o 

this manner, it is then only necessary to rotate the couple by 90 , in order 

to obtain the middle position necessary for grinding where the axes assume 
the same position as in Fig. 2, when the latter is turned l80 ["stuerzen"]. 

If we want also to grind parallel surfaces of the second order, the axis 
B must then be capable of being displaced in the vertical direction. The 
vertical distance of the latter from the grinding surface is equal to the 
distance of the ground parallel surface from the surface of the second order* 

The Duplex Method . Since only convex surfaces can be ground in this 
manner, this raises initially the question whether concave surfaces in a 
shape suitable for theoretical calculation can be produced with simple me- 
chanical means. A first answer leads us to the composite curve and/or 



32 



circular conchoid generated by a thrust-crank drive and/or sliding-crank 
drive. The former is described in Fig. h by any desired point of the axis 
of revolution of the lens and the latter in Fig. 1 by any desired point of 
the line BC or its extension. In regard to simple construction, these 
curves therefore leave nothing to be desired and the conchoid has more- 
over already been utilized as machine curve in a patent . Although both 
curves are only of the fourth order but offer even so no advantages in re- 
gard to calculation. In the Abbe curve with only one coefficient, the value 
of the latter results directly from the postulate for an osculation of the 
fourth order in the vertex and we should here desire that the respective 
value of the machine curve results from this postulate in an equally simple 
manner. The mathematical expression for such a condition becomes apparent 
in the following manner. 

When p represents in genersuL the radius of curvature and a the length 

2) 
of arc of a plane curve, I have designated the value 

da* p 

as the flattening value of the curve at the respective point. In the verti- 
cal point, the latter is dependent on the differential quotients of the 
second and fourth order and the postulate of a central osculation of the 
fourth order with a given curve is identical to the problem of determining 
the machine curve at a prescribed radius of curvature and flattening value 
in the vertex. If, in order to facilitate this operation as much as possi- 
ble, the r espective coefficient in the equation of the machine curve is to 

1. D. E. P. Nso 21^107 of G. OSSAKE and A. ¥ERGE. Deutsche Mechaaiker 
Zeitung, 1910, P. 91. 

2. Allgemeine Theorie der mono Chromatis ch en Aberrationen und ihre nSchsten 
Ergebnisse fuer die Ophthalmologie. Nova Acta Reg. Soc. Sc. Ups., Vol. 
20, 1900. 



33 



be directly proportional to the prescribed flattening value, then this 
equation must be in such a form that the one coordinate is represented as 
a function of the other and that the differential quotient of the second 
order aiTist vanish in the vertical point. In a finite curvature of the ver- 
tex, this is possible only with polar coordinates when the radius vector 
is represented as a function of the angle and the starting point is located 
in the center of curvature of the vertex. With an infinite radius of curva- 
ture of the vertex, the condition is fulfilled by the corresponding equation 
in Cartesian coordinates. A brief consideration shows that the respective 
machine curve cannot be generated by a simple plane mechanism. However, 
since these are the mechanisms making possible the simpliest calculations, 
the combination of such mechanisms appears to be the solution most advanta- 
geous from the viewpoint of theoretical calculation. In the polar equation 
R = F(g) of the machine curve, for example, we can make the extension of the 
radius vector by a thrust-crank dependent upon an angle of rotation a in order 
to generate automatically the displacement through a thrust-crank correspond- 
ing to a trigonometric function of this angle from the eccentric angle g. 
The equation then assumes the form 






•^G.tf{a.) /(a)^c.y(p) 



in which R = radius of curvature of vertex and C, c = machine constants, 
o 

The functions f(cr) and 9(®) can be generated by thrust-cranks where the math- 
ematical axes of the crank arm and of the couple in the position correspond- 
ing to the vertical point must coincide with the direction of the straight- 
line guidance. Both angles are calculated from this position and conse- 
quently represent the angles of rotation of the two crank-arms. The value 
of the radius vector of the machine curve is consequently dependent only on 
the absolute magnitude but not on the sign of the angle or and if we set, 



3^ 



for example f (ce) as equal to sin o? , then the value of a, remains un- 

1/7 7? ft n 

changed even with a change of sign of B. For 8 = 0, we then havel — — -=-=«0. 

Ida ap 

and, by differentiating four times, we obtain — 

I df» " d^*^^d^\^ij ' 

from which follows that the flattening value at the vertical point of the 
machine curve is directly proportionate to the constant C. 

This advantage points out the necessity of examining more closely 
whether such curves are actually suitable for satisfying the demands made 
above on curves with several available constants. Since this is actually 
the case, I have selected curves of this type as machine curves under the 
general designation of duplex curves. Further investigation then showed 
that similar curves can be utilized advantageously in certain cases also 
if the differential quotient of the second order of the equation of the 
curve does not vanish in the vertical point. I therefore distinguish be- 
tween two categories, depending on whether this is the case or not, and 
designate the curve represented by the equation above as a true duplex 
curve when the following conditions are complied with. The function cp(8) 
must have a value of zero also for 0=0 but remains unchanged otherwise 
at a change of sign of g. Simultaneously with f (,a) , ex must pass through 
the value of zero by having the differential quotient f (cc) different 
from 0. We merely require that the function cp (#) vanishes together with 
the differential quotient of the first order at cr = 0. In the case of a 
non-focal non-spherical surface, R receives an infinitely large value so 

that the equation is written in the form 

y — Cy(«) '/(«)"^C'1>(«) 

and the same requirements must be made on the different functions as when 

employing polar coordinates. In addition, I am introducing two categories 



35 



of pseu do -duplex curves which are represented by the equation 

j:?^«C,.y{a) + 0,,.I.(P) /(«)-c.?)(p) 

and/or 

y^G,.^{<l) + G^.'Hx) f{<x)^c.'p{x) 

in which are valid the conditions indicated above for functions with the 
same sign ["gleichbezeichnet"] and the same requirements must be made on 
^(0) and/or ^^(x) as on cp(@). In the application of certain mechanisms, 
pseudo-duplex curves can be represented also by equations of the same form 
as true duplex curves, except with this difference that the first differ- 
ential quotient of the function (p(<x) does not vanish at a = 0. 

A surface ground as machine curve by employing a duplex curve will be 
designated in general as a duplex surface . A true duplex surface is there- 
fore characterized only by the fact that its meridian curve has a true du- 
plex curve either among the parallel curves or as base curve in relation 
to the center of curvature of the vertex and the same is true of the pseudo- 
duplex surfaces represented by polar coordinates in relation to the pseudo- 
duplex curve, with this difference that a point on the axis other than the 
center of curvature of the vertex represents the point of the nonnal of 
the respective base curve. A pseudo-duplex curve represented by Cartesian 
coordinates is, if the surface is ground with one plane, actually only a 
curve derived from the respective base curve. 

The duplex machine utilized for the grinding of the surfaces must con- 
tain in any case two different mechanisms, as will be seen from the fore- 
going, which we shall call the A- and B-mechanisms . aince the task of the 
A-mechanism consists in converting the rotation equal to or around the A- 
axis into a straight -line displacement of the machine component carrying 
either the abrading surface or the axis of rotation of the lens, it must 
therefore include straight-line guidance for which a carriage is preferable 



36 



for reasons already discussed. iJepending on vshether polar or Cartesian co- 
ordinates are utilized in the equation of the machine curve, the radius 
vector and/or the Y-axis must indicate the direction of the straight-line 
guidance. Should we desire for technical reasons to establish the path 
of the carriage as fixed, the lens must then be pivoted around the B-axis 
intersecting the axis of rotation of the lens perpendicularly in the cen- 
ter of curvature of the vertex, when utilizing polar coordinates, whereas 
the distance of this axis from the abrading surface is changed by the 
straight-line guidance. Without anticipating the question of whether 
greater advantage lies in establishing the B-axis or the abrading surface 
as fixed, let us assume initially, for the sake of easier comprehension 
of this representation, that the E-axis rotates in fixed bearings. Conse- 
quently, when grinding non- focal surfaces, the X-axis corresponding to the 
straight-line path along which the lens is guided, shall be considered as 
fixed. The abrading surface therefore executes a straight-line movement 
in these cases but another arrangement is preferable for grinding pseudo- 
duplex surfaces whose machine curves are given in Cartesian coordinates. 
If the function Wix) is given a form such that the machine curve is trans- 
formed into a circle at C =0, this function is directly generated most 
simply by rotating one machine component around a B-axis where the form 
y = F (8) can be given to the equation of the machine curve. At a fixed 
path of the carriage, we can obtain fixed bearings for the B-axis only by 
pivoting the abrading surface around the B-axis whereas the lens displaces 
itself in the direction of its axis of rotation. In general, we shall there- 
fore temporarily 'assume that the paths of the carriage of the A-mechanism 
and the B-axis are established as fixed. The B-mechanism in which we may 
understand as included also the function tia) more properly connecting the 



57 



two mechanisms, is intended to automatically generate the rotation around 
the A-axis from that around the B-axis by complying with postulated mathe- 
matical conditions. As will be shown immediately, straight-line guidance 
is not absolutely necessary for this. However, such guidance offers cer- 
tain advantages on the other hand so that it would seem indicated for cer- 
tain purposes to accept the inclusion of a second carriage. On the basis 
of the assumption just formulated, the latter is always given a fixed path. 
In the interest of easier comprehension, let us assume further that, in the 
middle position, the axis of rotation of the lens is vertical and the 
abrading surface is located above the lens surface so that the direction of 
thrust of the two straight-line guidances becomes vertical. 

In the examination of the different machine types applicable under the 
indicated conditions, let us start with the B-mechanism. 

The obvious solution for generating the function cpO) is represented 
by the thrust-crank mechanism , possibly in a specialized form. Let us des- 
ignate the length of the crank arm as A and compute it as positive if the 
axis of articulation of the latter is located below the B-axis as shown in 
Fig. k. Let a + b be the length of the couple and computed as positive if 
the axis of articulation linked to the straight-line guidance is located 
above the other as in the figure. It follows from this that, in the ini- 
tial position, b represents the distance from the B-axis of the axis of 
articulation linked to the straight-line guidance and must be computed as 
positive if the former is located above the latter. If the crank arm is 
rotated around the crank axis until it forms an angle 8 with the direction 
of the straight-line guidance, then Y will represent the angle now includ- 
ed by the couple with this direction. For the determination of the dis- 
placement in height * , we then obtain the two equations 



l + bm(a + b) eosY^a cosg ; (o + b) sinY — a sin^, 

38 



which are to be employed in the form 

U — a 1 1 — cos p — jT (I — COB v) j sin 7 -* & sin [i 

by setting k = ^ + h ' 

For the special cases corresponding to an infinite length of the cou- 
ple and defined by the conditions k = and/or k = 1, it is easy to carry 
out the following trigonometric conversion which is moreover advantageous 
for numerical calculation also in the general case. By taking into account 

that 

i I — cosY = sinYtgVn tpf y. p — f g il » J sin V« (g — r) 

■ cosV«p««08V«t 

we obtain through elimination of k: 

i j_ 2a ain V» p gin '/i (g — y) 
j ' ' cos'V»Y ' 
and this expression assumes, in the case of k = 0, the form 

i /-=o(l — C03]S) 

On the other hand, through the elimination of k and a, we obtain with the 
aid of the relation 

I . sin 3 — sin Y — 2 sin >/» (P — y) cos Vt (3 +. Y) 

the expression 

; M. 2^8'» V«PBin 'Ay . 

"" 0O8Vi{p + Y) ! 

to which is given, in the case of k =1, the form 

^_^ 6(1 — cosp) 
cosg 

Further mention need only be made of the case k = -l corresponding to 

a full rotation ["durchschlagend"] of the thrust-crank drive in which 

i i = 2o(l--cosp) 

The thrust-crank drive which becomes transformed into a sliding-crank 
drive at k = as well as at k = +/--1, can generally be replaced by utiliz- 
ing a closed linkage through a curved-line guidance for which cylindrical 
surfaces are fixed on the carriage and on the crank aim which have their 
axes in the respective axes of articulation and are forced against each 



39 



other. The axis of the cylindrical surface fixed to the carriage therefore 
lies in the initial position at the distance b from the B-axis and this dis- 
tance is considered as positive when the former is located above the lat- 
ter. The axis of the cylindrical surface rigidly linked to the B-axis in 
turn lies at the distance A from the latter and this distance is considered 
as positive if the cylinder axis is located below the B-axis. The radius 
of one of the cylindrical surfaces can be selected arbitrarily so that three 
qualitatively different arrangements are possible in the general case, de- 
pending on whether the line of contact lies between the two cylinder axes 
or beyond one or the other. Of the three arrangements, at least one always 
permits utilization of gravity so that the carriage is carried by the B-axis 
at the line of contact of the cylinder surfaces. Only when the couple is 
so short that a pin with a diameter equal to the length of the couple would 
not be able to support the carriage, would it be impossible for this mechan- 
ical reason, at a positive value of a + b, to utilize the weight of the car- 
riage, but in such cases curved-line-guidance is excluded for other reasons. 

When employing curved-line guidance, it is preferable to substitute a 
roller, able to rotate around its axis, for one of the cylinders, in order 
to eliminate sliding friction. If this is done for the cylinder belonging 
to the B-axis, then the roller must be able to rotate around an axis paral- 
lel to the B-axis which is mounted on a crank arm of variable length. The 
carriage then rests on this roller by means of the surface affixed to the 
carriage which represents, depending on whether k ^ 0, a cylindrical sur- 
face with convex underface, a plane, or a cylinder surface with concave 
underface. It is here assumed that a x so that the middle position repre- 
sents the highest position of the carriage when k/' 1. However, if we de- 
sire to have a stable middle position at k / 1, we need only make a 



40 



negative but in that case the surface, by means of which the carriage rests 
on the roller, represents a cylinder surface with concave underface. The 
question is then whether this form of curved-line guidance is actually an 
advantage when the absolute value of k appreciably exceeds unity. By rea- 
son of the fact that here the angle of inclination of the couple is greater 
in absolute value than that of the crank, it will be better to have the 
force, inducing pivoting, attack at the couple which requires the retention 
of the thrust-crank drive without any change. 

If a very high value of a is prescribed, the roller must be mounted 
on the carriage at the respective height. The surface linked to the B-axis 
on which it rolls, then represents at b I^" 0, a cylinder surface concave at 
the topface, a plane or a cylinder. surface ;cbnvexi at the topface depending 
on whether k^l. In those cases, curved-line guidance must therefore be 
employed also at k ^ 1 but in these cases the value of k differs very 
little from unity. 

Obviously, it is also possible to employ another curved-line guidance 
in place of that corresponding to the crank drive where one of the two 
straight circular cylinders is replaced by a straight cylinder whose funda- 
mental curve has a symmetrical axis. If this were done for both cylinders, 
calculations would become entirely too complicated and rolling would more- 
over not produce adequately positive guidance of motion so that sliding 
friction could not be prevented. The expression for the displacement in 
height of the carriage differs for the utilization of such cylinders depend- 
ing on whether the roller is affixed to the carriage or to the crank arm. 
If the former is the case, the cylinder must be affixed to the crank arm 
so that its axis of symmetry contains the B-axis. If the crank aDn is then 
rotated around this axis, then the axis of the roller affixed to the 



^1 



carriage describes a curve in a plane rigidly linked to the crank arm and 
perpendicular to the B-axis which represents the parallel curve of the 
fundamental curve of the cylinder located at the distance of the radius of 
the roller. If this parallel curve is consequently given by an equation 
r = f(B) where the respective point of the B-axis represents the pole and 
when r represents the value of r at 8=0, we then obtain for the displace- 
ment in height 

and must make b = r in the machine. 

o 

On the other hand, if the roller is affixed to the crank arm and the 
cylinder to the carriage, then the plsme of symmetry of the latter must be 
parallel to the direction of thrust and contain the B-axis. If the crank 
is rotated, then the axis of the roller affixed to the crank arm also de- 
scribes a parallel curve of the fundamental curve of the cylinder in a 
plane perpendicular to the B-axis and rigidly linked to the carriage. The 
equation of this parallel curve shall be given in Cartesian coordinates 
where the X-axis is located in the axis of symmetry of the cylinder and is 
considered as positive upward whereas the I-axis represents the tangent of 
the vertical point and consequently passes through the axis of the roller 
in the initial position. Customarily cc is the length of the crank arm and 
the vertical point of the parallel curve in the initial position is then 
located at the distance a faram the B-axis below the latter. After rotating 
the crank by an amount P, this distance is a cos g + x and we thus obtain 
the displacement in height from the equations 

j Z=»o(l— cosp) — a; y«=0 8inp /(3.'/)=»0. 

On the basis of the conditions postulated, only such cylinders may be 
employed for this curved-line guidance which can be produced exclusively 



42 



by machining. It should be here noted in general that a kinematically pro- 
ducible curve may be employed as guiding curve if, in the machine generat- 
ing the curve, the axis of the abrading cylinder describes the curve whereas 
the roller is given the same diameter as the abrading cylinder when employ- 
ing a cylinder ground in this manner in the B-mechanism. The base curve 
of the cylinder so produced and so employed then is a parallel curve located 
at the same distance both from the guiding curve as well as the kinematical- 
ly produced curve and these two curves are consequently identical. In this 
manner, for example, cylinders can be ground in an eccentric grinding de- 
vice ["Ovalwerk"] which produce elliptic guiding curves in the B-mechanism 
and whose long or short axes can correspond to the initial position as de- 
sired. Moreover, since the method described above for the production of 
surfaces of the second order is also imminently suited to the grinding of 
cylindrical surfaces whose fundamental curves are parallel curves of the 
curves of the second order, conical sections can therefore be used qtiite 
generally as guiding curves in the B-mechajaism. 

It follows from the foregoing that the equation must be given in polar 
coordinates if the guiding curve is to be linked to the crank. The form of 
this equation most suitable for the present purpose is obtained from the 
familiar equation in Cartesian coordinates 

y* -= 2 p (a; + r,) + g (« + re)» 
by substituting 

I a; = — r cos p y = r sin p. 

The value r assumed by the radius vector r at S = therefore repre- 
sents the distance of a vertical point from the pole of the coordinate 
system and p, the radius of curvature at this point, has a positive value 
if the center of curvature is located on the saaie side of the respective 



kj> 



V 2 

vertxcal point as the pole. If q ^ - 1, the equation q = e - 1 then in- 
dicates the numerical eccentricity e. At a negative value of q, we have an 
ellipse whose semi-axis, coinciding and/or perpendicular to the initial 

line of the coordinate system, may be designated as A and/or B. In that 

B^ 
case, q = - —j' so that the shorter semi-axis coincides with the initial line 

i 

of the polar coordinate system at q <f - 1. In solving the equation in re- 
gard to r, the sign of the square root must be selected so that r = r is 
obtained also at 3 = 0. In this manner, we obtain for the displacement of 
the carriage 1 = r - r 

l^ rp(qooB^ — q cos' p + sin* p) + p (oca g -- tt ) 
gcos'P — 8in*p 



in which u represents the positive root of the equation 

I P A P / 

2 
If we here substitute e for u and simultaneously set p - r = a which pro- 

(X 

duces k = -r , then these equations can be expressed in the form 

H . . — ■-,- — 



i- 



(o — r,e«cosp)(l — coap) — |(l— «) 



1 — e'ooB*^ 
I ' tt^*sil — Bin»p{fc»-*e»(i— 4)«} 

from which it becomes directly apparent that the expression valid for the 

crank mechanism is obtained at e = and can also be used at q<;f - 1, pro- 

2 
vided we substitute q + 1 for e . 

If a focus of the conical section represents the pole of the coordinate 

systems, we then have r = = — " — from which results •= ?- = +e and b = + — . 

olj+e l-k~--e 

In that case, u = 1 and we obtain 

I 
N_ a(l-cosg) 

I I ± e cos p 

I - ■ . 

where we must utilize the upper or the lower sign, depending on whether the 
pole coincides with the focus closer or more distant in relation to the 
vertical point* 



Of other special cases, we need only note that we have also ? = at 
r = as well as that the case corresponding to the condition $» for 
the machine curve, is characterized by c? = k = and consequently r = — a p 
which gives us u = 1 + e sin 3 so that the expression for / becomes simpler. 
The denominator becomes equal to zero then when indicates the direction 

of asymptote of a hyperbola or is equal to zero in the parabolic equation. 

"? 

In the latter case, the value of C has the form ~ and the corresponding 

differentiation produces the value 0. 

Inversely, if the guiding curve is affixed to the carriage, then its 

equation in the indicated coordinate system is 



and we obtain 



:_|(i_l/.+,,ta.p5;), 



in which p is also positive when the center of curvature is located above 

a 
the vertical point. If we set k = — , this produces 

j /»ajl-c08p + ^(l~„)|, 
in which u represents the positive root of the equation 

1 u*^l + k*q Bin* ^ 
At q =: -1, there results the expression valid for the crank mechanism and 
we obtain for the parabola 

/, I, o fc sin* S\ 

Except for curves of the second order, presently known curves can hardly 
be used to advantage in this manner because the calculations become too com- 
plicated. If this is not to be the caise, then the equation of the curve 
must be able to assume either the form r = fCB) as pole for any desired axial 
point or the form x = f(y) as X-axis for the axis of symmetry. However, 



1^5 



these conditions are satisfied by the pseudo-duplex curves represented in 
polar and/or Cartesian coordinates so that they are highly suitable as 
guiding curves and consequently the respective cylinder can be ground in 
the machine itself by employing abrading cylinder of the same diameter as 
that of the roller. 

If the diameter of the roller is changed when utilizing a guiding curve, 
then the fundamental curve of the cylinder represents the parallel curve lo- 
cated at the correspondingly changed distance. If the diameter of the roller 
increases appreciably and if sliding friction is permitted, then the roller 
can be replaced by a cylindrical segment affixed to the respective machine 
component. As long as the radius of the cylinder surface is finite, the for- 
mulas deduced above are valid without change. However, this is no longer the 
case, if the latter surface becomes transformed into a plane so that the guid- 
ing curve is infinitely distant. Consequently, a plane displaceable in the 
direction of its normal is held in this case in contact with a straight cyl- 
inder which is able to rotate around an axLs perpendicular to the fundamental 
plane and, if the fundamental curve of the cylinder is a circle, the mecha- 
nism represents an eccentric and is therefore designated, for any desired 
form of the fundamental curve, as eccentric mechanism . If r = f(8) repre- 
sents the equation of the base curve of the fundamental cylinder in relation 
to the respective points of the axis of rotation, then the radius of vector 
is equal to the distance of the axis from the plane, from which it follows 
that the displacement of the plane corresponding to an angle of rotation 

is equal to r - r in which r customarily indicates the value of r at P = 0» 
^ o o 

In the present demonstration, the most advantageous general form of the 
equation of the base curve results in the following manner. In a rectangular, 
plane coordinate system whose X- and/or Y-axis coincide with a normal of the 



kG 



curve and/or with the tangent in the respective point of the curve, the 
magnitudes 9 N M are to be defined by the equations 

I cot cp ■== T^ N "s -X— M = x+ N coa^ 
I ^ ax sm f ^ 

in which that value of f corresponding to the cotangent is to be selected 
which turns into zero when the point of the curve is guided along the curve 
to the initial point. N consequently represents the length of the normal 
and M the sum of subnormal and abscissa whereas the angle 9 is formed by 
the normal with the X-axis. If this axis is a perfect normal, e.g., the 
axis of symmetry of the meridian curve of a surface of revolution, these 
three magnitudes then have the properties of intrinsic coordinates. 

In order to find the equation r = f(8) of the base curve in relation 
to a point on the X-axis, we draw both the tangent as well as the normal 
through any desired point of the curve and plot the vertical to both from 
the given axial point whose abscissa is equal to r . Projection to the 
normal produces 

I Nmmr+{M—r,)eoBf 
and consequently, for the displacement c &f the plane in the eccentric 

mechanism, 

I = iv^__ M coatp — r, (1 — cos <p) , 

in which q? represents the eccentric angle &, Since the base curve of the 

parallel curve, in relation to the point vertically below the point of the 

normal, represents in general a conchoid with the base curve of the original 

curve as base, / remains unchanged if a parallel curve of the given curve 

is employed instead of the latter. If we permit this curve to pass through 

the point of the normal, then r N M assume the values zero and/or N = N - r 
^ o '00 

and M » M - r so that we obtain 
o o 

/ = iVj — M, cos (p 
^7 



These values of the displacement of the plane remain valid without 

change for the displacement of the carriage if the plane is affixed to 

the latter and the eccentric is affixed to the crank in which r conse- 

o 

quently has a positive value if the line of contact is located above the 

B-axis. On the other hand, if the plane is affixed to the crank and the 

eccentric is affixed to the carriage, we then must divide the particular 

value of t by cos g, in order to obtain the displacement of the carriage 

from the formulas, and r must be considered as positive in the latter 

o 

when the line of contact is located below the B-axis. If we should assiuae 
that the carriage is fixed and that the B-axis can be displaced in a verti- 
cal direction, there would then take place, with a rotation of the plane 
around this axis, a displacement of the latter in the direction of its 

normal in the amount of which would correspond to a vertical displace- 

t 

ment of the B-axis in the amount •— — =•, 

cos p 

For the same reasons as for curved-line guidance, only curves of the 

second order and duplex curves will be available for the eccentric mechai- 

nism. For the first of these curves, we obtain, by differentiation of the 

equation 

;y* — 2pa! + qx* 

2 2 
the value of the subnormal p + qx whose square is p + qy . Consequently, 

we have 

and elimination of x and y results in 

Vl — c*8in*y Q ' 

where e at q -^ - 1, is as everywhere in this demonstration, only an abbre- 
viated designation for q + 1. The result is 

N r- M cos ^ = -' (cos y — 1^1 — e* sin* y), 



and the value 



Z«-(r, + ^) (i^cosp)+ J{l-r«), 



in which u represents the positive root of the equation 

1 «*■=■!— c*sin»p 

l -- 

i.e., the value ■ _ -g consequently furnishes the displacement of the car- 
riage, depending on whether the eccentric is affixed to the crank or to the 
carriage . 

At q SB 0, the expression for i contains a number of the form —. How- 
ever, from the above calculation, we obtain directly 

and can therefore write the expression for the parabolic eccentric in the 

form - _ ^ _._ , ,, ^_ . 

The general expression which turns into the expression valid for a 
cylinder with a circular base at q = - 1 and assumes a particularly simple 
form when the B-axis passes through the center of the conical section which 
corresponds to the condition r + — = 0, can be written, as q x' ~ li i^^ ^^^ 



form 



i/-.a,jl>«co8^.--~(l— cosY)J sinY-±«8inp 



in which 



\a, P + g^» 7, ^ P + 9U 

9 p 

and therefore coincides, at k^ = +e, i.e., r(l + e) = p which indicates 

that the B-axis passes through a focus, with the esqjression valid for the 

crank mechanism. In fact, it is possible to grind with the crank mechanism 



if.9 



whose basis represents a conical section, when the abrading plane is af- 
fixed to the carriage and the ground cylinder to the crank. It thus be- 
comes clear that the movement of the carriage must be precisely that gen- 
erated by the crank mechanism when the cylinder acts as eccentric on the 
plane . 

The eccentric mechanism has the advantage that it may eliminate a 
carriage in the B-mechanism under certain conditions. If we arrange the 
horizontal A-axis running in fixed bearings perpendicular to the B-axis, 
then a cylinder surface rigidly linked to the latter with its axis parallel 
to it, can rest directly on an eccentric rigidly linked to the B-axis so 
that the plane affixed to the carriage is replaced by the always horizontal 
tangential plane common to the crossed cylinders . This arrangement conse- 
quently requires the A-axis to be perpendicular to the B-axis but still 
retains the disadvantage that the form of the function tia) cannot be as 
freely selected as when straight-line guidance is given. 

As far as the function fCor) is concerned in general, the latter must 
represent above all such a specialization of the simple plane mechanism 
that calctilation becomes as simple as possible and positive guidance as 
accurate as possible. From the former viewpoint, a more complicated func- 
tion is to be preferred only then when real advantages are gained by the 
introduction of a new machine constant. This is the case in a generaliza- 
tion of the two most simple functions, i.e., sin a and tg cc, but elsewhere 
the introduction of a new constant results only in very complicated calcula- 
tions. I shall therefore discuss here only the two main types of the B- 
mechanism derived in this manner. 

One of these, the general sinus mechanism is characterized by the func- 
tion 

/ (a) = sina — tga)(l — cosa) 

50 



which is generated in the following manner. On a horizontal plsme rigidly- 
linked to the B-carriage, there rests a cylinder rigidly linked to the A- 
axis with its axis parallel to this axis and which can therefore be re- 
placed by a roller, in order to prevent sliding friction if the possibility 
exists of momnting the supporting plane at any desired height on the car» 
riage. fhe distance of the A-axis from the vertical plane in which the 
cylinder axis and/or the roller is located in the initial position, is 
designated with E whereas u) represents the angle formed with the horizontal 
by the plane drawn through the A-axis and the cylinder axis in the initial 
position and which is to be considered as positive upward. If the carriage 
is displaced upward by the distance ^ when the A-axis rotates around the 
angle a, then this plane assui^s a position in which it forms the angle 
a + w with the horizontal. The height of the cylinder axis above the hori- 
zontal plane passing through the A-axis is the product of the distance of 

the two axes from each other and the sinus of the respective angles. Since, 

^o 

according to the definition, this distance is , the vertical displace- 

cos U) -^ 

ment of the carriage is indicated by the equation 

1 1 ^ « (sin (a + (o) — sin <o) 
008(0 * ' . ' 

1 

which furnishes the expression above when fCof) = ~--, 

'^^ 
o 

In the general tangential mechanism which is characterized by the func- 
tion 

I /(a)-='tg(a + M) — tgo) 

a cylinder is affixed to the carriage with the cylinder axis being parallel 
to the A-axis and on which rests a plane rigidly linked to the axis and 
parallel to the plane passing through the two axes. The plane parallel to 
the latter forms the angle u) with the horizontal in the initial position 
whereas E represents the distance of the A-axis from the plane in which 



51 



the axis of the cylinder moves. 

In the general sinus mechanism, crossed cylinders may be utilized. 
It is here merely necessary for the A-axis perpendicular to the B-axis to 
be at a height sufficient for allowing absolute large negative values ©f 
tt). This angle is varied by changing the distance of the axes of the crossed 
cylinders from each other. If both cylinders have circular fundamental 
curves, this can be accomplished by changing the diameter either of only 
one or of both cylinders. However, if the eccentric represents a special 
cylinder, the vertex of the latter above the B-axis can -- as will be ex- 
plained in detail further below — be selected arbitrarily so that even in 
this case o) can be varied by changing either one or both of the cylinders. 

However, even with the general tangential mechanism, the B-carriage 
can be eliminated under certain conditions if we utilize a toric eccentric . 
If the toric surface is regarded as the enclosing surface of a sphere whose 
center moves on the fundamental circle, then the distance of the A-axis 
from the plane perpendicular to the latter and rigidly linked to it which 
lies on the eccentric, must be equal to the radius of the sphere. The 
plane passing through the A-axis and the horizontal tangent of the funda- 
mental circle then forms the angle or + U) with the horizontal plane. In 
the expression for the upward; displacement 

i if — 0|(l — COSj^) 

a is then the distance of the center of the torus f2x>m the B-axis and the 
angle u) results out of the diameter of the fundamental circle. However, 
this method has various disadvantages as compared to the crossed cylinders. 
In the first instance, the utilization of special cylinders is excluded 
and, in the second instance, the production is more difficult and, thirdly, 
mounting on the B-axis is more complicated because it is necessary to be 



52 



able to determine the distance E of the A-axis from the plane of the funda- 
mental circle with marLmum possible accuracy. 

Since it appears from the foregoing that the function 9( P) — as far 

as it has been the subject of investigation — ■ can be represented by an 

1 

expression of the form — in which a has the dimension of length whereas the 

function itself, except for the respective trigonometrical functions of 8, 
contains only the coefficients k and q, so that the constant occurring in 

the equation f(<y) a c • cpO) representing the B~mechanism results through 

1 

elimination with the equation f(ff) = ^ which furnishes c = |r-. If we uti- 

o o 

lize a duplex curve as guiding curve or as fundamental curve of an eccentric 

(in which the cylinder must be ground with a plane in the latter case), the 
fimction 9(B) and the constant c is to be formed in a similar manner • 

In a B-meohanism as described above, we therefore have available at 
least one and at the most four machine constants — disregarding the uti- 
lization of duplex curves which' permit a greater number* The effective 
variability of the mechanism corresponding to these constants depends both 
on the mathematical means making possible the determination of the constants 
in a given case -- which will be disc^^ssed further below — and also on the 
possibility of adapting one and the same machine to the different functions. 
In this respect, the carriage offers a great advantage because it allows 
not only the utilization of the crank but also makes possible the setting 
of the various angles (b in a mechanically more advantageous manner. For 
example, if the crank as shown in the diagram of Fig. k is arranged between 
the end of the B-axis and the carriage, the A-axis can be located parallel 
to the latter sufficiently high to permit any desired variation of m in the 
sinus and/or the tangential mechanism. Within the limits controlling the 
construction or design of the machine, it is therefore possible, if the B- 
mechanism has a carriage, to have available three constants c (» k without 



55 



utiliasing special cylinders nor does the utilization of any desired special 
cylinder encounter any difficulty so that, at the cost of such a cylinder, 
we would also have available the constant e and/or the constants of the 
duplex curve. Without a carriage, we are restricted to toric eccentrics 
and/or crossed cylinders but here the limits of variation of the angle U3 
arfe restricted for mechanical reasons since it is difficult to provide for 
a distance E sufficiently small to permit large absolute values of m and/or 
or + (B. Because, when utilizing crossed cylinders, the appropriate width of 
the eccentric is equal to the difference of the maximum and the minimum 
value of |— £- — ^ (« + <!>) ^ |.jjg eccentric must be wider than when utilizing a 

1 cos <i) 

carriage and this is a decisive factor particularly for special eccentrics. 
On the assumption that the absolute value of a + ta required is not too high , 
however, it is possible to have available the constants c and u) even without 
a carriage and the utilization of a special cylinder together with a sinus 
mechanism then makes possible additionally the utilization of the constants 
k and e and/or of the constants of a duplex curve. The cylinder lying on 
the eccentric can also be replaced by a sphere whose center must then move 
in the plane containing the B-axis and perpendicular to the A-axis . In the 
initial position, the distance of the center of the sphere fi*om the B-axis 
is equal to the magnitude b occurring in the equation of the creink mechanism 
whereas the distance of the former from the plane passing through the A-axis 
and parallel to the B-axis determines the angle u). With fixed A-axis, the 
number of constants is consequently not increased and k varies instead 
through mathematical interrelations with w. However, if the A-axis were 
capable of being displaced upward, then, even without a carriage, the three 
coefficients cask would be available within certain limits without the uti- 
lization of a special eccentric. 



3k 



A merely theoretical interest is represented by the fact that we can 
here also replace the eccentric by a sphere which creates a simple spatial 
mechanism with closed linkage in which the spheres can be replaced by a 
rod which is linked, corresponding to the centers of the spheres, through 
universal joints with the crank arm and/or with a crank arm starting out 
from the A-axis . 

If the equation of the machine curve is given in Cartesian coordinates, 
then the function cp(x) occurs in the B-mechanism. If this then concerns an 
actual duplex curve, i.e., a non-focal surface, then the lens must be dis- 
placed back and forth on a horizontal path witn unchanged direction of the 
axis of revolution. Although such displacement can be converted into the 
required vertical displacement of the B-carriage by a sliding cross-head 
driven, it is preferable to introduce a B-axis in order to exploit the exist- 
ing possibilities and to generate the rotation around the latter in the most 
simple manner by displacement of the lens. This then furnishes the function 

cp(x) from the function 9(B) by means of one of the two equations sin S = — ~ 

o 
and/or tg 3 ss ~ so that this case does not result in any change of the 

B-mechanism. The same is true for the pseudo-duplex curves in Cartesian co- 
ordinates where it is best (as already mentioned) to utilize a B-axis in the 
production and where the dependence of the angle P on the coordinate x is 
formulated in the function ^(x) so that cp(x) is replaced simply by a function 
q)(g). However, since the displacement effected by the A-mechanism is to act 
on the machine component carrying the axis of revolution of the lens, it is 
best to locate both the A-axLs as well as the carriage of the B-mechanism 
below the B-axis since this causes only changes of detail in this mechanism. 

In the A-mechanism which is intended to convert the angle of rotation 
a into the prescribed straight-line movement, only the function cp(a) occurs 

55 



which miist vanish together with the differential quotient of the first order 
at ff = 0. It will be evident that these conditions are satisfied by the 
different functions cp(8) and that consequently the corresponding mechanism 
can also be applied to the A-mechanism. However, it will be equally evident 
especially here that great advantages for the determination of the machine 
constants could be procurred by an appropriate function. Actually, there 
exists the function cp(Qr) , practically ideal from the mathematical viewpoint, 
which does, however, require the utilization of special cylinders in an ec- 
centric mechanism. If the fundamental curve of the cylinder of the eccentric 
is a circular involute of any desired order, then we have available amy de- 
sired number of machine constants and the latter are determined by a system 
of linear equations for various given problems. On the other hand, such 
cylinders can be produced by purely mechanical means where mechanical diffi- 
culties occur only when very small radii of evolutes and/or cusps are in- 
volved. The abrading tool ["fraize"] and/or the abrading roller can run in 
fixed bearings but the "band" must start from a fixed point located on the 
extension of the axis and the cylinder to be ground must be coupled rigidly 
to the evolmte cylinder utilized. Both cylinders are best mounted on a 
common axis whose bearings are supported^ by a carriage which can be displaced 
in a direction along this axis and perpendicular to the axis of the abrading 
surface . The development can then take place on the one and the grinding ©n 
the other side of the carriage. In order to be able to generate involute 
radii as small as possible , the plane containing the axis should be parallel 
to the path of the carriage. In dimensioning the length of the band, it 
should be remembered that the curve described by one point of the axis of 
the abrading surface represents the parallel curve, located at the distance 
of the radius of this surface, of the ground curve. The accuracy with which 



56 



the length of the band can be adjusted can be determined only by practical 
experience. As far as the involutes of the first order are concerned, an 
exact length of the band is not necessary since these involutes are identi- 
cal with their parallel curves so that an error in the length of the band 
is corrected by utilization of the correct point of the involute. 

The equations defining M and N as indicated above, furnish in the form 

ii^T — -J~ M" * + y cot © 

by taking into account that if p represents the radius of curvature in gen- 
eral we have 

dy — p cos ipc^'f dx = p sin ^d^ 

in accordance with the given definitions, the value 

iV — i/ cos y — sin ^ I p cos yd© — cos y | p sin <fd'i 
for the displacement of the eccentric. Let 

! «-o 
be the equation of a circular involute of the order m in which consequently 

a represents the radius of curvature of the curve at the point cp = but 
that otherwise oi represents the radius of curvature corresponding to the 
same point of the n-th successive evolute and in which the symbolic designa- 
tion zero [l] is equal to unity. We therefore have 



p COS 'fcZo =- 2 — M ffi" COS >^d'f 
«-o ■ 'J 



in addition to the analog expression for the other integral. The familiar 
reduction formula 

j I if» cos y c?y »= 9" sin y + n'f "-' cos 9 — ■«(« — 1) I y""^ cos. 9 d'f 

results in both for even-number n 



57 



^ U» COS fdf = ± a„ sin <p (l — |y + |j— •• • ± ^) =F 

j / tps to* (»"-i \ 

I Ta„co8y(cp-|! + ^---T(^:^). 

where the upper and/or lower signs are to be used, depending on whether n 
is or is not divisible by four without residue, and also for odd-nxuaber n 

j. ^ j r cos ^-tZcp = ± a„ sin y (cp - 1* + |J ±~{j± 

, /, ■ 9^ 9* <P"~* \ 

ri- Oil COS 'i>l — i-4.1 ... J I 1 

in which the upper and/or lower signs are to be used, depending on whether 

n divided by four furnishes a residue of 1 or 3* If we now define, through 

the equations 

I Ch = a„ — c»+2 + a„+ i 

which all terminate in the member a and/or a , . the m + 1 magnitudes 

m ffi - X 

c , the sum of the integral can then be given the form 



«— m— 1 



2 ^'; / r cos cpcZ^ = sin © 2 «»!? + °°« ?'^<'»+i Jf-*" 



in which the integration constant is determined by the fact that the sum 
must vanish at q) = 0. 

In the same manner, the reduction formula 

jffl" sin (pdf = — 9" cos tp + n<p"~^ sin cp — n(n — 1) j <P-"~^ sin cp dtp 

furnishes for even-number n 

fejrsin9cZ9==-F«»cos9(l-|-; + f-;--...±£^)T 

and for odd-number n 

,^jV8in9dcp=Ta„cosy(9-g + U-...±|^j± 

j ■ ' • ■ 

±«„sincp(9-| + |;_...±_^). 
58 



where the signs in these equations mtist be applied acaording to the rules 
indicated above. Summation results in 

1 ""'" a C '^~'" a" n-?n-l „ 

I 2 ,77 >" 8^'" ?^'f = - COS f 2 <="-, + sin ?> I]c„+, ^ + c„. 
!»-o '-^ «-o ^- »-o "• 

where the constant of integration is also determined by the fact that the 

sura must vanish at cp = 0. We further obtain 

ten 



/r- n-m „ 

P COS y(Zcp T— COS o I p sin '^df ==2 *'» ~ 
•^ «-0 



sin >ii i [J uus (pciw 1 — coa © i o sin maoi == > fi.. j c cos O c sin tp 



and this e3q)ression, if cp is replaced by a, indicates the amount of displace- 
ment in the straight-line guidance of the A-mechanism if a cylinder with 
the respective circular involute as fundamental curve is used as eccentric 
in the latter. As was demonstrated above, the magnitudes M N here have a 
fixed relation to the parallel curve passing through the A-axis of the funda- 
mental curve of the cylinder so that a represents the distance of the center 

o 

of curvature of the fundamentstL curve from the A-axis and is considered pos- 
itive when this point is located below the axis. In concordance with this, 
the eccentric must be affixed to the axis so that the normal deteiroined by 
the value a is intersected by the axis in the point determined precisely 
by this value and, at cy s 0, is perpendicular to the plane on which the ec- 
centric acts. If we set the displacement as equal to c cp(cr) , we then obtain 

o 



(p(a) = 1 — 003 a + i, (a — sin a) + Y ^» *. , 



«-2 

c 

in which the numbers k = — represent the available machine constants con- 

o 
tained in this function. 

The practical importance of this eccentric mechanism will be shown 

only by experience. As will be shown further below, machine constants are 

generally available, in a number adequate for most cases, without the 



59 



utilization of special cylinders and it therefore seems probable that cir- 
cular involutes of a higher order than the first order will need to be em- 
ployed only in relatively infrequent exceptional cases. 

In those cases where machine constants are not required in the function 
cp(a) , this function is most simply made equal to 1 - cos or and generated by 
a standard eccentric. It should be noted that, if a is obtained by the 
sinus mechanism, the bearings of the A-axLs can then be mounted on the car- 
i^iage so that the eccentric acts downward on a fixed plane. However, in 
the csilculation of the angle cu, the axis of the cylinder of the eccentric 
then takes the place of the A-axis . The motion of the eccentric and the 
machine components coupled to the latter can be decomposed into rotation 
around the axis of the cylinder and a horizontal displacement of this axis. 
When utilizing the general sinus mechanism, however, horizontal displacement 
has no influence on the angle a because the respective machine component 
with another cylinder rests on an horizontal plane also. The process is 
therefore the same as if the axis of the eccentric were fixed. This ar- 
rangement can be utilized for the purpose of enlarging the numerical value 
of a negative angle U3 at an equally high position of the A-axis » 

As a compound duplex machine , I designate a machine in which a straight- 
line motion is generated through summation of the effects of two or more 
individual mechanisms. In a general case, such a summation must take place 
in the production of the pseudo-duplex curves given in polar coordinates 
where the extension of the radius vector of the machine curve is equal to 
the sum of the displacement effected by the functions ^(B) and: cpCof). If 
*|/(B) = 9(B) « it is only necessary to mount the bearings of the A-axLs on the 
carriage of the B-mechanism. The rotation of the latter can then be effected 



60 



arbitrarily by the general sinas or tangential mechanism by forcing the 
cylinder and/or the respective plane rigidly coupled to the latter, in an 
upward direction against a fixed horizontal plane and/or fixed cylinder. 
If the A-Bie chanism consists of a standard eccentric, the bearings of the 
A-axis can be mounted, in accordance with the method jtist described, also 
on the carriage of the A-mechanism in which the eccentidc acts downward on 
a horizontal plane coupled to the carriage of the B-mechanism, provided, 
however, that the sinus mechanism is used. If the crank mechanism is not 
utilized in the function cp(a), the A-carriage can be located above the B- 
carriage so that both carriages could run in one and the same path if this 
were a mechanical advantage. If both the A- and the B-mechanism consist 
of standard eccentrics and the sinus mechanism is employed, one of the car- 
riages can even be eliminated by utilizing the method of the crossed cylin- 
ders. The A-axis must then be perpendicular to the B-axis and the A-ec cen- 
tric directed downward then rests directly on the B-eccentric directed up- 
ward. 

If we select, in order to have available one more machine constant, 
different functions for iff iB) and <?)(?), the arrangement just described must 
be modified so that the rotation of the A-axis is no longer effected by 
forcing a machine component coupled to this axis against a fixed machine 
component and the latter must be set in motion instead through a special 
B-mechanism. The displacement effected by this second B-mechanism may then 
take place either in horizontal or in vertical direction. In the first 
case, the sinus mechanism must be employed by mounting a vertical plane 
parallel to the B-axis on the carriage displaceable in a horizontal direc- 
tion and on which acts the cylinder parallel to the B-axis which is coupled 
to the A-axis parallel with the ssune axis. The second B-mechanism in this 



61 



case directly produces the function q)(B). However, if the straight-line 
guidance of this mechanism is vertical — when we can also employ the 
tangential mechanism -«- we obtain this function from the difference of the 
displacements effected through the two B-mechanisms. 

In the case of cp(a) = 1 - cos a and cp(8) = ^(B) and where fiat) repre- 
sents the sinus mechanism, the pseudo-duplex curve represented in polar co- 
ordinates can also be generated in a standard duplex machine. The equation 
of the latter can be written in this case in the form 



I "~g— ^ =" G {cos S — cos (8 -i- «)) sin a — tg «> ( 1 — cos a^ — c ?> (0) 



which is demonstrated in the following manner. By elimination of sin a, we 

obtain 

I cos (3 + a) = cos S COS a — sin 5{cf{^) + tg a)(l — cos a)} 

i 

and therefore \E — Ji„ _,, » , . s,^ ,,, , . ^ ,„,^ 

— -—5 =» C{(cos 8 + sin 5 tg o>)(i -rcos. a.) + c sm Sf(^)}, 



from which restilts , by talking into account that 

1 cos8 + sin5tg<o-S2lfciL) 

j ' cos (0 

the equation 



— o—=. Cod-cos a) + 0/f(P) 

-"0 



in which 

I /^ _^cos(5— £o) ' ^ 

00 = 5 J. n „ /-, . 

I cos (O Oi = cC SI 



sin 8 



If the machine constants C , C^ , c, ao are given, we therefore obtain 6 and 
C from the equations 

,cotS = ^_tg<o C = -^. 
^» c sin 8 



62 



The standard eccentric serves for generating the function cos 5 - 
COB ( 6 + a) where we need only provide that, in the initial position at 
ff = 0, the plane containing the A-axis and the axis of the cylinder of 
the eccentric forms the angle S with the vertical plane and that this angle 
is considered positive in the same direction as or. If or represents the 
distance of the two axes from each other, then C = 15-, 

O 

This method makes it possible to grind non-focal surfaces with the 
standard duplex machine where the curvature of the vertex of the machine 

curve can be made equal to zero. Condition for this is that the value of 

d R 

— ^ for B = becomes equal to S which is satisfied when |Ciyo(p) = l . 

dF ° ■ - 

The importance of the compound duplex machines lies in the fact that 

these can impart properties to the machine curve which can otherwise not be 
achieved, except by utilizing special cylinders. For example, if we re- 
quire of a machine curve that the radius vector at a given finite inclina- 
tion against the axis shall have the same value as on the axis, this would 
be possible in a standard duplex machine without special cylinder only by 
a full revolution of the A-axis. This could be achieved only by making 
the functions f(Qr) equal to 01 and by generating it through rolling off 
which would be, however, a disadvantage both in mechanical as well as in 
mathematical respect. However, if a straight-line motion in the machine 
is composed of two separate such movements, then the range of the latter 
also includes machine curves of this type. 

If such compounding takes place in the A- and/or B-mechanism of a du- 
plex machine, the latter will be designated as an A- and/or B- triplex machine 
If D represents in general a function of the form 

Z> = <|.(s) + A;<p(Y) /(y) = c,<p(£) 

in which are valid for the respective separate functions the same conditions 



63 



established above for functions of equal sign, a triplex machine is conse- 
quently defined by the fact that the function cpCcf) and/or cp(g) is equal to 
D in which or and/or P assumes the place of e. The corresponding mechaniceO. 
characteristic is the existence of a third axis, the G-axis , which is ro- 
tated by the amount of the angle y. What has been said above on the com- 
pound duplex machine is valid also for the compound A- and/or B-mechanism» 
It follows from this that, when employing the sinus mechamism for f (y) s *^e 
G-axis can run in fixed bearings if iJTCe) = cp(e) and cp(Y) = 1 - cos y, but 
that, if these conditions are not satisfied, the bearings must be mounted 
on the carriage of the respective mechanism. To enter on the details of 
the arrangements mechanically possible would lead us too far afield here 
because actually many different types offer themselves. Let us stress mere- 
ly that generally one of the three straight-line guidances can be replaced 
by the method of the crossed cylinders. An A-triplex machine can even be 
built with only one carriage. A simple but efficient B-triplex machine is 
obtained from a standard duplex machine with two carriages where the bear- 
ings of the G-axis are mounted at the top of the B-carriage and the cylinder 
rigidly coupled to the AVaxis rests on an eccentric coupled to the G-axis „ 
Although it is here necessary that fCor) represents the sinus mechanism, this 
is not a disadvantage as will be shown further below in the determination 
of the machine constants. The same is true of the condition generally in- 
herent in this arrangement that a + w may not sissume too great a numerical 
value . 

Since the function D represents a pseudo-duplex curve, it is evident 
that a triplex machine can be replaced by a duplex machine if we employ in 
the latter a corresponding duplex cylinder for curved-line guidance. This 
designation is intended to signify in general a cylinder whose fundamental 



Sk 



curve represents the meridian curve of a duplex surface and which can there- 
fore be ground in a duplex machine the fundamental curve of the cylinder must 
therefore be in this case the parallel curve of a pseudo-duplex curve; the 
duplex cylinder atost be mounted, depending on whether this curve is given in 
polar or in Cartesian coordinates, on the respective axis and/or carriage; 
and the diameter of the cylinder rolling on the latter must be equal to that 
of the abrading cylinder which was utilized in the production of the duplex 
cylinder. That the duplex cylinder in operation must further assume exact- 
ly the same position in relation to the respective axis as in the production 
of the B-axis , will be automatically evident. The triplex machine, however, 
can also be replaced by a duplex machine if a corresponding duplex cylinder 
is utilized in the latter as eccentric. This eccentric must then have been 
ground with a plane and the fundamental curve may represent, if ^(e) in the 
function D is made equal to 1 - cos e, the base curve of a true duplex curve 
in relation to the center of curvature of the vertex. For example, if we 
desire to generate the machine curve of an A-triplex machine 



i?o 



. Gil — cos a + ki fi'i)} /(•;) = c, ?> (a) /(a) = c f{^) 



in a duplex machine, we first grind a duplex cylinder with one plane by 
titilizing the machine curve 



In this equation, the functions cp(a) , f (a) , cpO) must here be the same as 
<f(Y)» f (y) 1 ^(<^) in the former and c, must have one and the same value in 
both equations whereas it is sufficient otheirwise that the product CE in 
the second equation is equal to the product Ck. B in the first. The cyl- 
inder thus ground is then incorporated in the A-mechanism as eccentric by 
making the distance of the A-axis from the center of curvature at the vertex 
equal to the product GR formed from the coefficient of the first equation^ 

65 



That the action of the eccentrie here corresponds to this equation will be 
seen from the equation given above on page k7 for the displacement Z, Since 
consequently the radius of curvature of the vertex of the duplex eccentric 
can be selected freely, we are able to influence the distance of the vertex 
from the A-axis , Whether we should prefer the employment of the duplex cyl- 
inder as eccentric or for the purpose of curved-line guidance will depend 
on given circumstamces. The eccentric has the disadvantage of sliding fric" 
tion which can be eliminated through curved-line guidance but operates more 
satisfactorily than a steeply rising guidance curve because of more favor- 
able transmission of power. 

It will be evident that duplex cylinders can be ground also with ma- 
chines in which such cylinders are employed and this would correspond to the 
utilization of a compound machine with four axes and this procedure can be 
continued ad infinitum. As we intend to demonstrate further below, the 
same purpose can be accomplished in this manner as with involute eccentrics 
of higher order and we obtsdn with both methods the same number of machine 
constants available in the same manner if the same number of special cyl- 
inders is ground. 

By this demonstration of the different possibilities of application of 
the duplex method, I intended to give the necessary review for being able 
to investigate the methods of determination of machine constants. A selec- 
tion among the individual types or a detailed evaluation of the latter can 
be effected only from a knowledge of these methods. Here we desire to 
stress in this respect only that, the simpler the A-mechanism, the easier 
it will be to grind lenses of one and the same type with different curva- 
ture of the vertex. If the A-mechanism consists of a standard eccentric, 
we then need change only the latter (and the abrading surface correspond- 
ingly) to convert to another curvature of the vertex. From the same point 



66 



of view, grinding with one plane is preferable for convex surfaces whose 
meridian curves do not have points of inflection because we then need only 
change the eccentric for a change in curvature of the vertex. 

In standard eccentrics, this change signifies only the adjustment of 
the prescribed distance of the cylinder axis from the machine axis. This 
change cam be accomplished by a change in the length of a crank arm where 
the cylinder can rotate around its axis in order to eliminate sliding fric- 
tion and may have any desired diameter. 



67 



Ill - Determination of Machine Constants 
Depending on the objective to be accomplished by the introduction of 
a non-spherical surface in an optical instrument, we will require different 
expressions for the demands made on this surface. If we are concerned 
merely with the correction of a Seidel image error, only the flattening 
value of the meridian curve in the vertex is prescribed. In other cases, 
e.g., surfaces of the second order or Cartesian ovals, the equation of the 
meridian curve is given in Cartesian coordinates. If we speak generally of 
an osculation of the order zero when the meridian curve of the gjround 6ur~ 
face intersects that of the prescribed surface in a given point, we can 
then express the problem as the intention of deriving a centric osculation 
of prescribed order and a number of eccentric osculations of also prescribed 
order. As will be explained in further detail below, it is theoretically 
possible to satisfy any desired one of these prescriptions by employing 
circular-involute eccentrics or duplex eccentrics of higher order. However, 
in practice, the objective is always achieved with a restricted number of 
machine constants. Even when the meridian curve of the non -spherical sur- 
face can be constructed only punctually, the problem is given the same ex- 
pression. The direction of the normal and the radius of curvature general- 
ly results from the calculation for punctual construction but can, if this 
should not be the case, be determined with any desired accuracy by numerical 
methods and the same is true of the differential quotients of higher order 
in the vertex so that even in this case a central osculation of higher order 
can be prescribed. When employing a restricted number of machine constants, 
the problem must be given a special formulation. Depending on whether this 
formulation is restricted to the centric or an eccentric osculation, entire- 
ly different methods of calculation result so that it seems indicated to 
treat these problems separately. 

68 



Centric Oscnlation of Higher Order . In order to be able to employ 
the maximum possible number of machine constants for centric osculation, 
the simplest possible coordinate system must be selected because the com- 
plexity of calculation increases in proportion to the increase in order of 
osculation. For this reason, we prefer the polar coordinate system forming 
the basis of the typical duplex curve and whose pole is located in the cen- 
ter of curvature of the vertex. Because the machine curve whose equation 
has the form K = f(S) with the condition R" s for B = 0, represents a 
parallel curve of the meridian curve of the ground surface and/or the base 
curve of the latter in relation to the center of curvature of the vertex, 
the differential quotients of higher order of this equation must be deter- 
mined from the corresponding differential quotients of the equation of the 
meridian curve of the prescribed surface. If the equation of the meridian 
curve was given in the form r = f(9) and if r" = at 8 » in the latter 
which corresponds to the condition that the center of curvature of the ver- 
tex should represent the pole of the coordinate system, then the problem 
would be restricted to calculating the corresponding differential quotients 
of a parallel curve and/or the respective base curve from the differential 
quotients of this equation valid for the vertex. However, since this is 
not the case in general, we therefore must determine first an equation 

——. , — « = 4, 6, 8 . . . 

in which r represents the value assun^d by r at 9 s and which must be 
equal to the radius of curvature of the vertex. In this equation, the 

highest value of n indicates the prescribed order of osculation and the cor- 

IV VI 
responding differential quotients which we will designate as r r ... for 

the sake of brevity, mtist be Calculated from the data prescribed for the 

non»spherical surface. If the latter is constructed punctually, there is 



69 



then known in any event the radius of curvature of the vertex and r is thus 

o 

determined. The coordinates of ^ - 1 or, if the flattening value has been 
determined, those of ■^ - 2 points are expressed in r& from which we obtain 
a corresponding number of linear equations, by substitution of these values 
in the above equation, and the differential quotients result from these. 
Compensation calculation by utilizing the coordinates of a greater number 
of points would be of no advantage because the curve represented by the equa- 
tion is not identical with the curve ground but has only a contact of the 
n-th order with the latter so that a correction, when required, is carried 
out only in the machine constants. We can proceed in the same manner when 
the equation of the meridian curve of the prescribed surface is given in 
such a form that the latter cannot be expressed by Cartesian coordinates. 
For example, this would be the case with a transcendant equation in polar 

coordinates if the second differential quotient in the vertex had a finite 

IV 
value. The direct calculation of the differential quotients r «.« from 

those of such an equation could be effected but would doubtlessly be without 
practical significance. 

This calculation assumes the following form when the equation of the 
meridian curve is given in Cartesian coordinates. Let such an equation in 
the coordinates If^ be referenced to a coordinate system whose Y-axis coin- 
cides with the axis of symmetry of the curve and is considered positive in 
the direction from vertex to the center of curvature of the vertex whereas 
the X-axis represents the tangent of the vertex. The differential quotients 
determined from the latter shall be designated as lv]'-/j" •/]'"•/]"' , ., Through the 

substitutions 

i V] = »•„ — y cos S = rsinO, 

in which r has the value of the radius of the curvature of the vertex, and 
o 



70 



through subsequent successive differentiation, it is now possible to deter- 

IV 
mine the differential quotients r ..* corresponding to the vertex but it 

is preferable to make this substitution calculation, so~to-speak , once and 

for all by deriving the formulas through which the differential quotients 

r ... are obtained directly from the values of \'i"'q^ ... in the vertex. 

For this purpose , the substitution equations are to be differentiated by 

considering one of the variables 9 or § as independent. For greater ease 

in calculation, let 9 be considered an independent variable. Since the Y- 

axis represents an axis of symmetry of the curve, all differential quotients 

l-j— in the vertex are equal to zero when n represents an odd-number figure, 

and the successive differentiation of the first equation shows that this is 

d r 
the case under the sauae condition also for the differential quotients — — » 

In concordance with this , the differential quotients — — ^ vainish for even- 

number n which becomes apparent also in the successive differentiations of 

the second order. The differential equations of odd-number order therefore 

vanish for the first equation of substitution and those of the even-number 

order for the second of these equations. The equations obtained during the 

first step for 9=0 

furnish 

from which follows r" = because r is equal to the radius of curva- 

o 

ture and M" is equal to the reciprocal value of the latter. From further 
differentiations of the second equation, we obtain, if we set 0=0 after 

the differentiation 

;#5 == r^d^ sin 6 d'^'i = r^d^ sin 6 + Sd^rd sin 6 
d'S = r^d'' sin 8 + 25d*rd^ sin 6 + Id^rd ain 6. 



71 



and therefore have 

In the same manner, further differentiation of the first equation for 

9 s produce 

«?* vj == — ti* r — ro d* cos 6 

rfoyj = —(?«,•— 15d*rd«cosG — r„d«cose 

d'^ri =. - d<'r'-28d0rd> cos 6 — 70d*rd* cos - f„<f« cos , 



in which 






as well as 



d»cos6 = -d0' d*cosG = rfO* J«cosO--d;e« d«cosO=rfe8 



must be set. Since d r = r d9 , etc., we need only substitute the values 

3 

of dCd |... in order to obtain the formulas. In the form most favorable for 

numerical application, the latter are 



fVi + 20?-'^ -qvir", + IB-q^Vrl 

rv"i + 28rv« + 329riv_280^ « _v,vm,.« + 56YjVir«_316riv,. 



Because the flattening value; <J>^yJy 3y)"^ « ^® must make 



riv= rj4) 



if the former is prescribed. I have also determined the relations of the 

d^ 1 
differential quotients to the flattening values of higher order — r ■- and 

.6 dff P 

— ^ — through differentiation of the general equation for the radius of curva- 

da 

ture both in Cartesian and in polar coordinates. Comparison of the values 

so obtained shows that no error is contained in the above calculation. 



72 



In this way, we therefore always know the differential quotients of an 
equation r = f(0) representing the meridian curve of the prescribed surface 
and in which r" = for the vertexo In order; to obtain from them the cor- 
responding differential quotients of the equation R = f ( g) of the machine 
curve, let us assume first that this curve represents a parallel curve of 
the letter located at the distance a from the meridian curve and where this 
distance is considered as positive when the radius of curvature of the ma- 
chine curve is greater in the vertex than that of the meridian curve. If 
^ is the angle formed by a normal common to the two curves with the symmetry 
axis and if Kr are the radius vectors of the points of the curve determined 
by this normal, we then obtain through projection of the latter both to the 
normal and the tangent the two equations 

I iZcosCp — 9) = rcos(0 — y) + c i?sm(p— ^)-=rsin {0 — 9) 

and obtain moreover the familiar relations 

ji?'cos(p — y) = i2sin(p — y) r'co8(6 — ?>)=rsin(0 — y). 

These equations are now to be differentiated by treating as an inde- 
pendent variable. The first differentiation of the second and of the fourth 
equation furnishes dS s dq) = dB from which is obtained E" = by two differ- 
entiations of the first equation. The second equation is now no longer 
necessary. Because of the symmetry, the derivations of odd-number order of 
the first and those of the even-number order of the last two equations vanish, 
Successive differentiations of the last equation produce 

I d^r' = r^cP sin (6 — <p) = r, (d»e — d'y) 
I d^r> = r.d'' sin (6 — ?>) = r, (d^e — d^y), ; 

where 

I dV = r^^^d^^ d^r' =• rVi^^Qs + lOfivde'd'9 

so that this results in 

73 



Through the same treatment of the third equation, we obtain 






K 



d^\ 



so that we now know, because d^ can be changed for d8, all derivations of 
@ and cp from 6 necessary in further calculations. The successive deriva- 
tives of the first equation are 

■ d'^E + i?od« cos (p — <p) -= dV + ud" COS (6 — y) 
d^B + 5od« cos (p — <p) = d^r + r^d^ cos (6 — y) , 



where 



and 



d« cos (P — 9) = — 10 (d^f)* d« cos (6 — ip) «- — 10 (<Z»e — d'f)* 

fl!»cos(p — ?)) — ■— 56 d»y<Z*?> . c?«cos(e — <p) = — 56{d!»6 — d»y)(d*e--d'>y) 






so that we obtain the respective formulas through substituting of the values 
derived above. These can be given the symmetrical form 

«" + 10^ = rV« + 10'— 



iSy^ + 56 






+ 280 



i?„ 



jtm 



HI 



rvin + 56 



,.vi fVr 



-iv» 



+ 280 ~ 
ro rl 



from which we see that the magnitudes occurring in them generally represent 
invariants for any desired parallel curves . The latter must therefore geo- 
metrically characterize the common evolute. That this is actually the case 
is shown when we deduce, from the flattening values through corresponding 

differentiations, the values — - which represent the radii of curvature of 

, n ^ 

dcp 

the successive evolutes. I have checked the above formulas through these 
values. 

7h 



If we substitute another value for R in the equation R - R = f (B) , 

o ^ o 

then the equation represents a conchoid with the parallel curve as base. 
Since now the base curve represents such a conchoid with the infinitely dis- 
tant parallel curve as base, we obtain the differential quotients of the 
base curve simply by setting R = oo in the above formulas. The same re- 
sult is produced by differentiation of the easily verified equations 

5 « r cos (6 - y) B'-^r' cos (e - y) -. r sin (6 - 9), 
where R represents the radius vector in the equation R = f (8) of the base 
curve and 6 = cp. 

Although the formulas consequently can be applied in the above form 
also for the base curve, they can be written, by utilizing the distance of 

= R - r , in the form 
o o 



which is both more convenient for numerical application and can also be 
used with non-focal surfaces. In this case, the equation r = f (9) of the 
meridian curve must be replaced by an equation t{^,Y\) = and the equation 
R=f (9)of the parallel curve either permanently or provisionally, depending 
on the method utilized for grinding, by an equation f (xy) = from which 
follows a = 19 - y , according to the definition of the coordinate system 
given above (page 70). The formulas thus derived result, for r = cp and/or 



R - cp in 

o ^ 






and in the same relations for the parallel curve. The formulas above are 
first divided by r and/or r and/or r and the last values substituted sub- 
sequently. Since all members in the equations so created have finite 



75 



values, it is possible to set r = E in the denominators which produces 
the formulas 

\ yvin = yjviii _ 28 o 7jJV(tjVI + yVi) 



The latter are obtained also through differentiation of the equations 

•q — y = a COB f ■ x — S^asiny j/' cos ^ -> sin ^, 

of which the last represents the definition of the angle cp formed by the 
normal with the Y-axis, and the other two can be obtained through projec- 
tion of the distance a on the coordinate axes. 

In order to grind non-focal surfaces with a standard duplex machine 
without horizontal straight-line guidance, it is necessary to know the 
differential quotients of the machine curve for the equation in the form 
R = f (0). These are obtained from the differential quotients of the 
equation f (xy) = valid for the respective parallel curve by differen- 
tiating, in the same manner as above (page 79), the equations 

I 1/ = ^0 — ■R ops p « =. jB sin p 

where the distance B of the pole of the coordinate system from the vertex 
of the curve is selected freely. The simplification conditioned by y" = 
is counteracted by the occurrence, in the values of d y and/or d x, a mem- 



ber 



)ccurs. 



- (2) d'Rd^-^ cos g and/or \\^d?BdP^ sin p oc 

The result is 

i^ -X -S'^^ - y"^ /?J + 5 .Bo ^R^'- — y'^ni — 55 y'^K + 61 ii, 

iiviii „ _ j,viu ii- — 140 j^vi jg« _ 3486 j/»v jJJ + 280 y'V R\ + 1385 iJ, . 



After the respective differential quotients of the machine curve have 
thus become known, we must first determine the relation of the latter to 
the machine constants through differentiation of the machine curve. Let us 



76 



here employ the abbreviated designation 

I K ' I 

\. I n n 

so that d R = R d in general. The differential quotients and/or 

o ° ^ , n ' ,_n 

IV • IV ^°' ^^ 

shall be designated as G'...e ... and/or « ...of ... Differentiation pro- 
duces, for or = 8 = 



d*s 



:3e"a"« 



d«£ 



15 6"a"aiv+ i5s"'a"» 



I ^ = 28 e" a" a^i + 35 g" aiv + 210 e'" a"* ai^ + 105 eiVa"*. 

In order to obtain lower numbers for the coefficients and because of 
the form of the differential quotients of the function representing the 
crank mechanism, let us employ the abbreviated designations 



!St = 



3i?„ 



« 






e 



ii;vm_2tffv 
. 105 i?iv 



so that we obtain thus 



2t = e"a"« 



^ 1/aiv s"'a" , \ 



= J_ /i*^ J. Satv ^ 30 £'" aiv 15 e^v «"« ' 
45 U" 



^ --1." + ,»2 + g» +— .-fp- 9j 



There are consequently required six successive differentiations of the func- 
tion cp(8). For the crank mechanism, this is (cf. p. 39) 

I y (P) =. 1 — cos p — r (1 — cos y) ' sin V r= ^ sin p. 

Because of the symmetry, the derivations of odd-number order of the first 
and those of the even-number order of the second equation vanish for 8=0. 
We thus have 

. » = 2, 4, 6 



tZ" cos Y 

dn y (P) d» COS p + j^ ^ 

\ d» sin Y ==-id**-sin-P • 



» = 3, 3, 5, 



where 



d*008Y — — dv* d^eos'{x=ad'(* — 4,dyd'i 

d« cos 7 •=» — ti Y« + 20 cZ •/» d« y — 10 (d» y)* — 6 <iY d'^t 



11 



and 

\d sin Y = d Y d?Bin^'=— <?•/» + dr^ y 

! d'>e.m.'(^dt — \Od'i*d?'{ + d'"{ 

whereas the corresponding first member of these equations occurs only in 
the derivations of cos 6 and sin 3. Initially, the second equation pro- 
duces 

U-f^kd^ d^'[ = —k{l—k*)d^^- 

1 d»Y-=i*{l— /fc'){l — 9ifc'')dp^ 



and from this results, if j 

\t = k{\ -V k) 

[■: 

is set for the sake of brevity 

f (p) - 1 _ & .p>v(p) -(!-&) (3 <~ 1) 

^vi (p) « (1 _ ;[.) (45 jfcsf _ 16 ^ + 1) 

In differentiation of the equation of the B-mechanism, we further have 
|d«/{a)=./'(a)a"d;p« d* /(«)="{/'(«) e^ + 3 /" (a) a"«>rfp* 

! d6 j (a) - {/' (a) avi + 15 /" (a) a"a" + 16 /'" («) a"»}dp« 

and thus Obtain | ^„ ^ ^p^^g) ^^f^^ ..J'i^ 

!«^_y^MP) i5,;v/"(«) i5^».r(«). 

In the equation of the sinus mechanism 

j /(a) == sin a — tg<o (1 — cos a) 

^^ lyW-l /"(a)«-tg«> r(«) — 1. 

and in that of the tangential mechanism 

j /(a)=-tg(a + <*) — tgw 

there is produced in turn 

!' ^ ' cos*a> ' C08*W ' COS*M 

In order to examine first what can be accomplished without the utili- 
zation of special eccentrics, the function representing the crank mechanism' 



78 



is also to be employed for cp(Qr) where, however, in order to prevent confu- 
sion, the designation tC is to be utilized for the coefficient otherwise 
represented by k and a number T^ corresponding to t shall be defined by the 
equation I 

1 T «= %* + X 



f"(a) a 

is first to be eliminated „.> .a from the value for 



In substituting the now deduced values in the equation f or \(5. , there 

yi 

The equation so 



f'ia) 



Q?' 



created 



4.Z— 4. K - = 4. X_ici 20 i i£-i . -— J- ^S- fin /y"» / v*/ 



produces, if the value 



.IV 



ZB-1 



obtained from the equation for j ^j at s'" i is substituted. 



{ «VI „XV* Mil /„\ 

P;;r + 5-JJ5- — 9- 180fe«< + 225«« — 90«(1 + 20 — 60a"»Li^, 

a a . / (a) 



so that finally, by taking into consideration that 



sIV 



8"=.C(i-.X) . ^„ 



3t— 1 

there result for a machine with two crank mechanism the formulas 






in wjiich 



.„ c(l-k) 
f («) 



3I«0(l-x)a"« 
S-.4&»/ + 6«»-2«(2< + l) +«"«' (t-|-|^^) 



and the products C(l-1^ and/or c(l-k) at 'yC = 1 and/or k = 1 have finite 
values. With abbreviated designations 



C,= (S + 2« — 5«8 <7,= 0, +2S3« 

we obtain, for the sinus mechanism , 

j 91 = Gc^\\—i/.){\ — hY • «" - c(l — 1) ' 
|s3==i + a"tgw C,+ 4<(5B — fc«) = a"«(t + 1) 

and, for the tangential mechanism , 

J9I = Gc* cos* » (1 — x) (1 — &)• a" = c co8» w (I — A) 
i <8 = «_2a"tgw ■ Ot + 2P{l— *') = a"*(T-3), 



79 



It 2 2 

where the member 8a tg (u has been eliminated from the last equation. 

To begin with, as far as an oscillation of the fourth order is con- 
cerned, it will be seen from the formulas that the latter can always be 
obtained under the assumption that the product C(l->0 bas the same sign as 
J5l as soon as anyone of the four and/or five machine constants can be var- 
ied. The greatest possible simplification of the methods results when 
both U) as well as the coefficients 7< and k are made equal to zero and the 
sinus mechanism is employed. The theoretical calculations are simplified 
in this manner as much as possible; the two crank mechanisms are represent- 
ed by eccentrics; and the carriage of the B-mechanism can be replaced by 

2 
crossed cylinders. Since only the product Cc is determined through the 

prescribed osculation, there are available an infinite number of solutions 
among which we can select the best through variation of c and through 
mathematical checking. On the other hand, if we assign once and for all 
to c a given value, we obtain a standard non-spherical surface with only 
one coefficient which can be produced so easily that we can always count 
on the possibility of procuring such surfaces. The disadvantage that con- 
cave surfaces of the second order cannot be produced, is compensated by 
this method in the simplest possible manner. In order for these surfaces 
to actually have the properties of such standard surfaces, however, it is 
necessary that the distance of the parallel curve, selected as machine 
curve, from the meridian curve of the surface is in a certain ratio to the 
radius of curvature of the vertex. In order to grind such a surface with 
prescribed curvature of vertex and prescribed flattening value, it is 
therefore only necessary to give the corresponding values to the diameter 
of the abrading surface and to the height of the A-eccentric and to adjust 
the required distance of the abrading surface from the B-axis. On the 



80 



other hand, it is evident that we can also select, in order to accomplish 
in special cases as much as possible with the simple machine, different 
parallel curves and/or the base curve as machine curve. 

In this manner, it is also possible to achieve osculation of the sixth 
order with the simplest machine in certain cases. Since ^ = in the 
machine curve, the condition for this is that a parallel curve characterized 
by this equation can be employed for grinding. That any curve has such a 
parallel curve is shown by the equation 



ijvi=rvi4. 



r.J?, 



o-"* 



IV IV 

which is linear in a for R = -5r and E = a + r as well as for any 

— '^ o o •' 

value of E . Whether an osculation of the sixth order can be obtained 
with this machine is therefore based on whether the value of a obtained 
through this equation is mechanically applicable or not. 

However, if the angle U) is added as machine constant, then the oscula- 
tion of the sixth order at any desired machine curve results from two lin- 
ear equations regardless of whether we employ the sinus or the tangential 
mechanism. The two equations 

j SJ = ctg(o and/or |S3 = — csin2a) 
show that c or 03 can be selected as desired in the sinus mechanism whereas 
this is not necessarily the case for c in the tangential mechanism. We can 
therefore employ the simplest machine if the latter is equipped for a finite 
angle ou for which only the A~axis needs to be shifted higher. In order to 
avoid the mechanically disadvantageous negative values of cr" , we need have 
available only two different cylinders which produce one positive and one 
negative value each of o) when resting on the B-eccentric and which are to 
be employed correspondingly depending on whether!-© -^0. Osculation of the 
sixth order is then accomplished by simply giving the two eccentrics the 
corresponding values. 



81 



If the B-mechanism has a carriage so that the coefficient k is avail- 
able, the respective osculation can be achieved also at u) = when the 
equation \ S3 = t produces a real and mechanically employable value of k. 
The former is the case whenj55> - 0.25 and this condition is satisfied by 
infinitely many parallel curves among which selection is made in order also 
to achieve the latter. 

Of the equations representing the complete osculation of the eighth 
order, one is quadratic in a". Since 7^+1 cannot be made negative but 
has a minimum value of + 0.75 at a real value of "tt 

I 0. + 4< (S3— /<;*)> 

is a necessary condition in the sinus mechanism. That the latter cannot be 
satisfied in general by appropriate selection of the coefficient k will be 

apparent without detailed discussion. For example, if both Cp as well as 

i 

is are negative, t must have a negative value in order to satisfy the condi- 
tion but here the absolute value of this magnitude may not exceed 0.25 so 
that satisfaction of the condition at a sufficiently large absolute value 
of Cp is impossible. It is therefore not possible to achieve osculation 
of the eighth order at any desired machine curve with the sinus mechanism . 

In the tangential mechanism, the necessary condition is 

! Ci + 2l={l — F) 



T— 3 



>0, 



and requires a negative value of C at >€ = k = 0. However, if C^ 0, the 
numerator can be made negative at "^ = by a sufficiently large value of k 
and, on the other hand, the denominator can be made positive at k = 
through a sufficiently large value of -4^ so that the condition can always be 
satisfied through different means. Consequently, osculation of eighth order 
at any desired machine curve is always possible with the tangential mecha - 
nism if the machine contains a variable crank mechanism . 

It follows from this that the simplest machine does not correspond in 



82 



general to the requirements made by osculation of eighth order but that a 

carriage is necessary in the B-mechanism. Since this mechanism permits 

the choice of sinus or tangential mechanism from case to case, the variable 

crank mechanism is necessary only in the case C. ^ ^ C, for mathematical 

t p 

reasons because we can mak.e If^ = k = in all other cases which corresponds 
to the utilization of eccentrics both in the A- as well as in the B-mecha- 
nism. Contrary to this, it may be advantageous for mechanical reasons to 
employ the crank mechanism also in other cases, in order to influence the 
value of the other machine constants. 

A crank mechanism is more difficult to incorporate in the A-mechanism 
than in the B-mechanism. To this should be added that its purpose in the 
first of these mechanisms would be to eventually make f~ 3>C where X? i^^ 
order not to obtain excessively large values for a", could be rarely small- 
er than 1.5 which corresponds to "2^- 3 = 0.75- Since the force in the A- 
mechanism must attack at the crank arm but the length of the couple could 
represent, in accordance with this, at the most two-thirds of the length 
of this arm, such a crank mechanism would be mechanically very disadvan- 
tageous. However, the force in the B-mechanism can attack the couple so 
that there is no hesitancy in regard to high values of k from this point of 
view. Such values have the disadvantage, however, of reducing the maximum 
value of the angle 8 and consequently the maximum diameter of the ground 
surface. However, since a value of k = 2 already is 2k (1-k ) = -2k and 
further permits 8=30, this disadvantage need not be anticipated except 
in very infrequent cases. It would therefore be generally preferable to 
make "K = 0, i.e., to employ an eccentric in the A-mechanism -- at least as 
concerns centric osculation. 

With C, > or C, <f 0, we must therefore employ the sinus and/or 
3 t 



83 



tangential mechanism and can make k = or, if this procures mechanical 

advantages, give this coefficient a suitable value. At C > /^ C , on 

t J 

the other hand, k must first receive such a value that cc" assumes a suit- 
able real value in the tangential mechanism. In all cases, there then re- 
sult, when the value of csr" has been determined, the machine constants Cc«3 
from linear equations. If it appears to be of advantage, the latter can 
subsequently still be influenced through variations of k. To this should 
be added, moreover, that it will be possible in many cases to favorably 
influence the values of the coefficient also through suitable selection of 
the parallel curve. For example, if the parallel curve j^S = can be util- 
ized in practice, we have C^ = C. = |S> where a real value of Qf" is obtain- 
ed, either in the sinus or in the tangential mechanism, also for k = 0. 
If B ^- 0.25, we can make O) = from which results 1 ^. = t and 

so that we obtain again a real value of or" either in the sinus or in the 
tangential mechanism. In contrast to this, we would have to solve an equa- 
tion of the fourth order if we were to assign a given finite value to m. 

Although consequently the utilization of special cylinders to achieve 
centric osculation of the eighth order would seem to be superfluous, the 
latter are indispensable either in certain special cases to b e treated 
further below or else furnish certain advantages also in the general case. 
Initially, there arises the question whether a curved-line guidance and/or 
an eccentric would not be able to accomplish a more exact positive guidance 
than the crank mechanism and it is further possible, with the employment 
of special cylinders, to make do in all cases, even with the simplest 
machine, without a carriage in the B-mechanism. Condition for this is 
evidently that the equation for/ ^ is linear in a machine constant intro- 
duced by such a cylinder. The latter cannot be contained in s'" because 



84 



IV 
Of contains either no machine constant when the constant [?] contained in 

£'" would be determined already by,'® or else there results a quadratic 

equation. On the other hand it will seem obvious that both e as well as 

VI 
a may contain the respective constant. It follows from this that when 

the special cylinder is utilized in the A-mechanism, differentiation four 
times of the respective function is sufficient whereas the B-mechanism 
would require one of six times. When employing curves of the second order 
for curved-line guidance, more complicated expressions result for the dif- 
ferential quotients than by the employment of them and/or their parallel 
curves in an eccentric mechanism. Curved-line guidance should therefore be 
considered only for the A-mechanism but the eccentric also for the B-mech- 
anism. This last combination has the advantage of making the B-carriage 
superfluous to which should be added that one and the same cylinder is 
utilized for the grinding of surfaces with different curvature of vertex. 
If we are merely concerned to replace the crank mechanism by an ec- 
centric in those cases where osculation of the eighth order requires a 
value of k ?£: and where c and k are consequently given and necessarily 

j ?>(P) — 1-— cos p~v(l — cosy) sinY — ^sinp 
we must set e = jk| and k^ = k in the formulas deduced above (cf. p. k9) 

from which is obtained 

Since a, = cE and Q?" = c (l-k) , these expressions can also be written in 

the form i^ a"(l + k) r^ a.^ 

I^„°° k E^^'k 

It follows from this that, at > k ">-l, p has the opposite sign from or", 
so that the eccentric must act downward at a positive value of or" but is 
able to act upward otherwise and that the axis of the eccentric must be 



85 



located below or above the vertex of the curve at positive a", depending 
on whether k has a positive or a negative value. However, as will be seen 
from the foregoing, since k does not need to have a negative value in the 
cases where this constant cannot be made equal to zero, we can always em- 
ploy an eccentric acting upward and whose axis is located below the vertex 
of the curve. Whether we will employ the curve of the second order or a 
parallel curve of the latter will be decided on the basis of mechanical 
considerations. However, the magnitudes p and r occurring in the formu- 
las always refer to the conical section itself. 

For the successive differentiations, the following form of the equa- 
tion is best suitable. We set 

where a then represents the distance of the center of curvature of the ver- 
tex from the axis of the eccentric and is considered as positive when the 
former is located below the latter. This results in 

j (l-e')y(p) ^ |i + ^'(^-^) j (1-C08 fi)-|(l-M) u'^l-e'Bin*^. 

2 
in which e may also have a negative value. In the values of the differen- 

^- -, ^- ^ d cp(B) J., J • J.- 4. d cos 8 •, d u 
tial quotxents — ■^ — - no other derivatxons occur except and 

^ d0^ dS" dS" 

where those of odd-number order vanish because of the symmetry. - The second 

equation produces 

d*U' e»id sin p)« d*u + 3 (d»«)» «= — 4 e«d sin pd» sin p 

\ d^u + 15 d*ud^u-^—e^{Qd sin pd" sin p + 10(d» sin p)»}, 

and, when substituting these values in the derivations of the first equation, 

the right side becomes divisible by 1-e without residue. We obtain in 



this manner 



|y"(P)-l >iv(p)_3|^_i 



,vr(p)»l|£.*_l|iVL 



86 



We will see instantly that the equation forj^ is linear in e_ and that, 

k 
after this magnitude has been determined, the equation for IS is linear in 

2 IV 
e . I'e obtain first, for e'" = 0, by replacing a with 3/^ - 1, 

a" 



and, after substitution of these values, for the sinus mechanism 

2I = 0c'(l-x) S3 = ^ + ctgfl> 



k 



4e» 



C7.+ ^{«-e*)-=c''(r+.l) 
and/or for the tangential mechanism 

I St = (7c*oo3*(o(l — %) a" — ccos*» 

|i8 = J-2a"tga>. C^ + ^(J-2e«)=a"«(t-3), 

if a B-eccentric is employed whose fundamental curve represents a conical 

2 
section or the parallel curve of the former . A negative value of e corre- 
sponds to an ellipse whose shorter axis is vertical in the zero position 
and whose semi-axes are obtained in the manner indicated. If k has a nega- 
tive value, this is the case also either with o? or p in which a" is nega- 
tive, i.e., the eccentric acts downward. The latter is not possible in the 
simplest machine and the former is not advantageous. However, a negative 
value of k is not necessary to make the sinus mechanism applicable also 
for the cases C^ ^ at ^ = 0. If we write this last equation 

i O, + 453(<8-c tg w) _-c»-« \- , 

1. K 

it is then evident, after c has received a suitable positive value, that 
we can select u) so that k ^ 0. Only in those cases when in this manner a 
mechanically disadvantageous value of w would result, would it be preferable 



87 



to apply the eccentric and/or a curved-line guidance in the A-mechanism. 
In the equations deduced for a machine with two crank-mechanisms, 1 - x 
need then be replaced only by q)"(Qf) and '£^ by — > iTT — \^ + l}- However, 
since here a crank mechanism in the B-mechanism is not excluded, it is pre- 
ferable to designate the coefficient k occurring in the equation of the ec- 
centric and/or the curve-line guidance by k . If an A-eccentric is utiliz- 

^^ ^^ which the fundamental curve represents a conical section or the paral - 

e2 
lei curve of the former , we must then set J|< = and replace t^ by r— in the 

^e 

equations valid for a machine with two crank mechanisms. The osculation 
of the eighth order can then in all cases be obtained with the sinus mech- 
anism, without the necessity for giving k a negative value. It follows 
from this that the simplest machine, under the conditions specified above, 
for the osculation of the sixth order always makes possible the osculation 
of the eighth order when employing such an eccentric. 

This is also true of the corresponding curved-line guidances , although 
the expressions then do not become as simple. For a cylinder linked with 
the axis, the equations deduced above Cp. ^3) must be differentiated, by 



setting 1= acp(8) in the fo 



rm 



(1 - c« cos* p)V(P) = (I - w COS p)(l - C08 P) - ^(1 — «) «» - 1 — t; sin« p, 



in which j 



We thus obtain 



nS 



(l-eW(B=l-» + ^^ 



where the derivations of u have the already deduced form. After the re- 
spective values have been inserted, the right side of the equations becomes 

2 
divisible without residue both by 1 - e and by 1 - k and this results, when 



88 



we set or for 3 and k for k, in 

e 

and these values must then apply in the manner indicated above in the 
equations valid for a machine with two crank-mechanisms. 



If the cylinder is fixed to the carriage, the corresponding equations 

|?{P)=.l- 

produced in the same manner 



|?>{p) = l — cosp ^^i-(l_M) m' =. 1 + Pg 8in« p 



^>'-'-*- l(^,^-)-«+*-+i^ 



and k can be established in both cases by taking into account mechanical 
advantages. 

The application of evolvent eccentrics to obtain osculation of the 
eighth order will hardly be practical in the general case. An evolvent 
of the first order would offer no advantages and an evolvent of the second 
order does presuppose the grinding of two special cylinders. However, 
with the utilization of the latter, j9r, S, S would then be determined re- 
spectively by e" , s"^', G so that the three variable machine constants 
would all be obtained through the shape of the A-eccentric. Theoretically 
ideal, this method of making the simplest machine applicable to all cases 
has the only disadvantage of being expensive. 

On the other hand, duplex cylinders can be employed to considerable 
advantage because they can be made in the machine itself. The equation of 
the machine curve is that of the corresponding triplex curve when utilizing 
a duplex cylinder. If the cylinder is to perform as a B-eccentric, we then 
have the general equation 

I fc^- 09(a)- i(o.)^cD 
l> = 'MP) + i,»{Y) /(7) = c,«p(p), 



89 



in which we will make 

j ?(«)-= 1 — cos a 4i(P) = y{p)=^l — cosp 9 (y) — !-- COSY 

I /(a) = sin oi — tg (d(1 — cos a) /('/) = sin f — tg (o, (1 — cos if) 

since this concerns the application of the simplest machine. Differentia- 
tion produces initially 



J----..vm_„,„.£^, 



in which ^g^'* * * ^® designated by D" ... and in which 

Since oo, occurs only in the value 

7'^-c,(3c,tg«,-l) 

it is apparent already at this stage of the calculation that the equation 

for linearis in the new machine constant is tg cb . Calculation takes 

place in the same manner as above for a machine with two crank-mechanisms 

and produces 

! ¥= (7c* « =. c tg 0) +^ 

i 

I 0, + 4 k,c,*iS8 — c, tg w,) = c«, 

and these formulas are consequently valid for a machine with sinus mechanism 
in which a dUplex cylinder ground in the same machine is utilized as B-ec- 
centric and where the A-mechanism is represented by a standard eccentric. 
It is obvious that the simplest machine is sufficient in all cases, provided 
only one given positive and one given negative value of «) and/or ®-, is pos- 
sible because we then have available also the sign of the three coefficients 
c, c, , k^ . The cylinder must be ground with one plane and the radius of 

curvature of the vertex p can be selected arbitrarily. If C U) c represents 

o '^ o o o 

the machine constants to be utilized in the grinding of the latter, then we 



90 



must make I „ ch.Ea 

Po 

and, in the utilization of the duplex cylinder, the distance of the center 
of curvature of the vertex from the B-axis equal to cE and in that case, 
when this magnitude is positive, this point will be located below the axis. 

If a duplex cylinder is utilized in the A-mechanism, then the machine 
curve has the equation 

I :^-^ = C{?(a) + i,y{'r)> /(T) = c,9(a) /(«)== cy(p), 

r<> - -.... -.. ^ - ' 2 

d Y 
and we will here assume the same simple functions as above. Since — ^ = 0, 

dS 

there occurs, as will be seen from the formulas deducted above (p. 77) , a 

j^VIII 

number containing y only in the value for -^ , i.e. 

I H 

in which 



In the value forj S , this number is divided by 315 Cc and consequently 

2 2 
equal to k^ , c, , c . This produces therefore the equations 

which furnish the machine constants in the easiest manner. That the value 

of k^ becomes negative at C '^ 0, produces no disadvantage in the A-mech- 
1 s 

anism. Since UJ, actually only occurs in the derivations of higher than the 
eighth order, this angle can therefore be made equal to zero when grinding 
the cylinder. In the case that the latter is to be used as eccentric, the 
detailed instructions for its production have already been given above 
(p. 65). However, if the duplex cylinder is to be utilized for the purpose 
of curved-line guidance, then the machine curve in the production of the 
latter must be a pseudo-duplex curve. If we write the equation of this 



curve as 



Po 



91 



J^ 



we then have 



PoO^-^E.G C,^h,0, c^c, 



and can arbitrarily select p . When utilizing the cylinder, the roller 
must have the same diameter as in the production of the abrading cylinder 
and the distance of the roller axis from the A-axis in the initial position 
must be equal to p . Since cpCor) = 1 - cos a the cylinder can therefore be 
ground with the simplest duplex machine in accordance with the instructions 
given above (p. 63) 



Among the special cases , we encounter first the case r" =0 which can 

be realized for convex surfaces not intersected by the evolute. In regard 

to optical surfaces, no need appears to exist so far which would lead to 

this special case. However, it may be of advantage, in the production of 

duplex eccentrics, to have the possibility, in arbitrary selection of the 

radius of curvature of the vertex, to make the latter equal to zero. Thus 

E =0 and we have to use A = —=— in the above calculations instead of 
o 3 

The first equation of the machine curve is multiplied by R and the product 

E C receives a finite value but the calculations remain unchanged otherwise, 
o 

As can be seen from the equations (p. 7^)1 7 = R ^^ tiie equation 

of the fundamental curve of the cylinder ground with one plane whereas r 

VIII 
and r have infinitely large values. However, this is without importance 

since only the machine curve is of influence in the production as well as 

in the utilization of the eccentric. 

The case =0 permits only one solution at r ^ 0. Since E =0 

VI VI 
and R = r both for the base curve as well as for each parallel curve, 

VI 
we must have «" = or e" = 0. If the former ia the case, then also R = 

VI VIII 

whereas in the latter case both E as well as E can have finite 



92 



values at e"'7i^0. The resulting condition £"'7^0 at e" = can be complied 
with by an evolvent eccentric of the first order if the A-axis passes 
through that point of the evolute circle in which the tangent is in the 
initial position perpendicular to the plane linked to the A-carriage and 
resting on the eccentric. We then have 

I tp(a) = a. — sin a 

and consequently, for or = 



s 



" •= civ 



siv = e"' = C, 



where C represents the ratio of the radius of the evolute circle to B . 



We then obtain first 



^„ = 15 £"'«"" S| - 210 s"'a"«a»v 



and from this 



for the simplest machine. 

In the case 7 s r =0, this is true also for the parallel curve 

and for the base curve and for them also E = r . Osculation of the 

IV IV 

eighth order requires either e" = e*" = at e -^ or cr" = at ff 92^ 0» 

Such a condition can be complied with only by a special cylinder whose evo- 
lute edge coincides with the respective machine axis. If the fundamental 
curve of the cylinder is a 'conical section and/or the parallel curve of 

one, we then must make, when using the cylinder as eccentric, a = and 

2 
q>(g) = — from which we obtain 
P 

! ,„, e*(l — cos S) — 1+« - , . . .„ 

1 ^(P) = —- r» «« = 1 — e« sin* p 

and, if the conical section is to be utilized as a guidance curve linked 

to the axis, we obtain in the same manner 

,a. — e* cos S(l — cos 8) — ! + « „ 

p(P) „ i^,.^oJl - «• - 1 + «* ««• P. 



93 



However, if the guidance curve is to be rigidly linked to the carriage, 
we must make a = p which res tilts in 

I ?(?)=. 1— COS p + i~ M»«-l + j8m'p 



In all three cases, we obtain 

i 

2 
for g = where e can also have a negative value. If the cylinder is 

IV 2 

utilized as eccentiic in the B-mechanism, we then have or = 5ce and ob- 



tain 

1 ijvni 



315^0 



Cc'eS 



for the simplest machine whereas, if it functions in one way or another in 

IV 2 

the A-mechanism, e = 5Ge and we then obtain 



315^0 

The former manner of application has the advantage among others that any 

given cylinder possibly on hand can be utilized which depends in the latter 

VIII 
on the sign of R , 

A duplex cylinder will also comply with the condition when the machine 

curve of the surface to be ground is given the form 

\ 5^»C?(a) /(a) = c,9(T) /W-cyCp) 

where o? can be interchanged with y» This results in 

I 315iio ^*'' *' • 

for the simplest machine regardless of whether the cylinder is utilized in 
the A- or in the B-mechanism. Here also exists the same difference between 
these methods of application. 



9^ 



In regard to osculation of the eighth order, we now need only treat 
the non-focal surfaces . In the machine curve 

I t/='Of{a) /{a) = c9(a;) 



it will be simplest to replace the function ^{x) by a function cp( @) with 

the addition of an equation x = a f(0). We then need merely form the dif- 

,n 
ferential quotients f— ^ which can then be used in the above calculations 

dB jy 
instead of the values B ... so that the calculations remain unchanged in 

all details. The simplest values are obtained when f(P) = sin B, Differ- 
entiation first results in 

dij, =. yiydx* d'^y = y^^dafi + 20 y^'^dx^d^x 
dOy^ymidx» + sey'^dx^d^x + 2S0 y^dx»{d'^x)* + 56yivda:»d»a;, 

from which result the values 

We can then write, as equation of the machine curve 

||-=0®(a) /(a) = cv.(p) 



and makel2l=»;; — •■^4 » after which all deductions receive identically the 
i 3ao"P 

same form as for standard surfaces. 

This can be accomplished also when utilizing a pseudo-duplex curve as 
machine curve. The above deductions are actually valid generally on condi- 
tions tha t J e, Sr, ^, g is defined by the equations 






95 



The equation of a pseudo- duplex curve can be expressed in polar coordinates 
by 

and the latter can consequently be generally utilized for obtaining an os- 
culation of the eighth order, provided G^ and the machine constants possibly 
contained in the function l''(@) can be selected arbitrarily. With standard 
surfaces, however, the method would become only more complicated by this. 
However, when we are concerned with non-focal surfaces, it offers a means 
by which we can avoid the otherwise necessary horizontal straight-line guid- 
ance. In that case, we must have C^{p"(3) = 1 for the vertex and it is pre- 
ferable to make ^(3) = 1 - cos which corresponds to the utilization of 
a standard eccentric. Since consequently C^ = 1, we obtain 

ff -g™ — 21^iv-20.R, 

The necessary compound duplex machine required for production in the 
general case can be constructed according to the type of the simplest ma- 
chine and makes possible, if it is equipped with two B-eccentrics, the ap- 
plication of special cylinders both in the A- and in the B-mechanism. How- 
ever, if we make cp(^) = ^B), in order to eliminate the second B-eccentric, 
we may possibly need to use special cylinders in the A~mechanism. Finally, 

at G !!> 0, a standard duplex machine can be utilized for grinding if we 
s 

make 9(0?) = 1 - cos 01 and the angle 6 is determined by the method indicated 
above (p, 62). 

Centric osculation of the eighth order can consequently be obtained in 
all cases. If the B-mechanism represents a variable crank-mechanism with 



96 



straight -line guidance, special cylinders will then be required only when 
the surface to be ground has a contact of at least the fourth order with 
a sphere. The simplest machine characterized by A-eccentrics and crossed 
cylinders is always adequate, although special cylinders may be necessary 
in certain cases. A circular evolvent eccentric, however, only of the first 
order, is required only when the surface has a contact of the fourth but 
not of the sixth order with a sphere. In all other cases, the special cyl - 
inders can be ground arbitrarily either in the simplest duplex machine or 
in the machine described for the grinding of surfaces of the second degree . 
Non-focal surfaces can be ground, without the horizontal straight-line 
guidance, by iising a compound duplex machine and, in certain cases, even 
with the simplest machine . 



97 



Eccentric Osculation 
If it has been prescribed that the meridian curve of the ground surface 
shall pass through a point given in relation to the vertical apex and at a 
finite distance from the latter and that the normal shall have a given in- 
clination toward the axis in this point, then an eccentric osculation of 
the first order has been prescribed which passes over into one of the second 
order if the radius of curvature in the given point is also given, le then 
need first to determine the respective values valid for the machine curve. 
If the latter constitutes a parallel curve and if the equation of the meri- 
dian curve of the surfaces is given in Cartesian coordinates where the X- 
axis is to coincide with the axis of symmetry and the Y-axis with the verti- 
cal tangent, then the magnitudes 9 N M are to be determined through the 
equations defining them as indicated above (p. k?) 



'^^ . dy Bvaf . . ^ 



For the radiiis of curvature p, we have 



II . d'^x 

I - =. cos' tp s—i ' 

If, on the other hand, the equation of the meridian curve of the surface 
exists in polar coordinates in which the coordinate system is to be deter- 
mined through the relations 

I a;=r, — rcosO y^rsine 

to the rectangular coordinate system defined above, we then obtain for the 

equation of curve r = f(9), the same magnitudes from the equations 

r' - ■ 

tg (8 — y) = - iV sin tp •= f sin 6 

(M — r,) sin <p «= r Bin (e - 9) i = cos' (9^ (^, + 2r'» - rr") . 



98 



The corresponding values N M p of the parallel curve located at the 

Sr £l 3. 

distance a are N + a, etc. From these, we obtain the coordinates E0 and 
the differential quotients R'E" of the equation R = fO) of the machine 
curve through the above relations by beginning with the equation 

1 ^' Na costs — Ma +ja. 

If the machine curve is to be represented in Cartesian coordinates, then 

the relations first indicated above produce the coordinates and differential 

quotients at the corresponding point of the former. 

If a base curve R = f(3) is to be utilized as machine curve, we have 

B = cf and obtain R R' R" from the equations 

j iJ-iV — (iff — iJ,) cos 9 iJ' = (J/ — iZJsin? iJ"~p — ii, 

of which the first is obtained by projection of the radius vector on the 
normal whereas the other two are derived most simply from those valid for 
a parallel curve. Since the base curve represents a conchoid with the in- 
finitely distant parallel curve as base, the differential quotients R' R" 
have the same values as for this curve so that we need only to make a = C30 
and 3 = 9 in the respective equations. If E represents the infinitely 
large radius vector of the parallel curve, we then have 

i JS' cos (P — 9) = i?„ sin {^ — f) = (31 - So) sin?, 

where M - R has the same value for the infinitely distant parallel curve 

as for the fundamental curve. Since further the radius of curvature of 

the parallel curve is R - R + p, there then results generally for such a 

a 

curve 

This esqjression is valid for any desired parallel curve if R represents 
the radius vector of the conchoid of the latter defined through the rela- 
tion R = R - a. The right side is now to be converted into a fraction 
a 



99 



and the denominator and the numerator are then to "be divided by R where- 

upon we can make -5— = and 3 - cp = 0. The numerator here contains the 

a 
member 

i?„{cos-''(p - cp) [1 + 2 tg* (P — ?)] - 1>, 

which is, however, equal to zero because E sin ( - cp) has a finite value 
so that the simple relation above results. 

After we have thus determined the respective magnitudes of the machine 
curves, the elimination necessary for the determination of the machine con- 
stants can be perfoitnedo In the equation of the true duplex curve 



i-J^^Ctp{«.) /(a)-c?(P) 



,^Jl^,_. _ 

E = P + a and/or E = P , depending on whether the latter represents a 
00 00 

parallel curve or the base curve of the meridian curve of the ground suf- 
face. For the sake of brevity, let us set 

I R-R.^-K ?>(p) = J5 

Two successive differentiations produce 

i ^ = (7y'{a)a' /'(«)«' = cJ5' 

■"0 . . 

:^ = e{<p"(a)<x" + ?'(«)«"} /"(«)«" + /'(a)a"-cfi", 
from which we obtain first, through elimination of C and c, 






and then, through elimination a» and 0?" 



where 






?(«)/' (a) /'(a)\?'{a) />) 



100 



If the function B is known, U and V are thus determined by B, R' and the 
three machine constants C c o) can be utilized for obtaining the eccentric 
osculation of the second order if it is possible to calculate the angles 
OB and or from the values of U and V and if these angles receive mechanically- 
applicable values, The next task is consequently to investigate these func- 
tions for the different mechanisms. Since an eccentric is to be preferred 
in the A-mechanism simply for mechanical reasons and since the calculations 
here would obviously be purposeless when utilizating special eccentrics, it 
shall be here assumed that a standard eccentric represents the A-mechanism» 
For the sinus mechanism , we therefore have 

I y(a) = i— cosa /(a) — sina — tga>(l — cosa). 
By taking into account that 



1 — cos*« 



8in« = i=^ cosa- sin «tga>»52i{fL+^ 



we obtain 



.- . 1 — cosa/, , cos(a + w)\ 

i /(a) « ; 1 + 1 ■ '-] I 

:" ' smot \ cosw / 



whereas the expression 



,, . sin (a + w) ^ 
/(a) == — '- — tg w 



COSO) 



is easier for the differentiation. The latter produces 

U'(a) -. sin a /'(«) ». 5^i?L±i^ 

j COSO) 

j ?"(«) =. COS a /"(a) Bm(o. + o,) 



COSO) 



and consequently 



S_iiz M— - «» cot a + tg (a + w) » -; — - , --; — :' 

I f' (a) /' (a) '-"''« ^ "8 v« T / gjj^ ^ gQ3 (,^ .^ ^j 

so that there finally results 



cos (a + (o) 1 + cos « 



101 



Since 



U—1 



= cosa — sinatgw 



the values of the angles a and CB result from the equations 



^^^tr(ir-i)-7 ,g,^,,ta- 



Since both the absolute value of co as well as that of ff + uo must be smaller 

than ■#, it follows from the value U that we must get U > 1 from which 

follows in turn V / 0. The condition 1 > cos or ^ - A where X <C It 

can then be written in the form 

2V>U(U—l)>V{l — i»)>0 

At a high negative value of cb, an angle of Qf /■ -^ ±s actually mechanically 

possible, although it is obviously more advantageous if we can make A = 0« 

Since further the absolute magnitude of the angle (B may not exceed a certain 

value u) for technical reasons, there is added to this also the condition 
m ' 

i 

i £^>,1 + 008 (fl«, 

from which it will be clearly seen that a carriage in the B-mechanism per- 
mits smaller values of U than the method of the crossed cylinders. 

For the tangent mechanism with standard A-eccentric, there is valid 
i 9(a) =. 1 — cos a /(a) = tg (a + <o) — tg (o. 

Through a small conversion and through differentiation, there results 

{ /fa)==~ ^- /7a\ = — — 1— /"/ ^ 2 tg (g + (o) 

I ' ^ ' cos w cos (a + (0) / w •=' ^Q^i (a + w) ' ^"' ■°' cos* (a + w) ' 

where, by taking into account the identity 

• 4. ■/ ■ , \ COS <^ . , 

i tg (a + w) =. -; ; -1. oot a 

j **^ ' sin a cos (a +-<o) 

there is obtained the value 



and finally 



to' (a) /' (a) sm a cos (a + w) 



„ ,, , .cos(a + w) „ 3Z7cosa „ 

Z7 — (1 + cos a) i V ■= r— 2 

* cos CO 1 + cos a 



102 



Since further 



I + cos a 



cos a — sin a tg w 



a and u) can be determined from the equations 

1 V + 2 IT 

cosa« - tgw=.cota---; ~ 

au— y —2 ° sina(l+c08a) 

The value of U shows that the condition that neither ff + O) nor o) may 

reach the amount of 4 is identical with the condition that U must have a 

finite positive value. The value of V shows in turn that cos c? has the 

same sign as V + 2 and that we have 3U > V + 2 even when V + 2 )> 0. The 

condition cos or ^ - A receives, through the value of V, the form 

and the condition 1 ^ cos a y- " \ can consequently be written in the 
form , - 

The simultaneous condition U >■ is mathematically adequate with finite 
U but must be carried out for technical reasons. 

Since the possibility of achieving eccentric osculation is dependent 
not only on the functions U and V but also on the function B, there still 
remains to investigate the latter where special cylinders are to be excluded 
initially, le then have available the crank mechanism for which 

; JJ — 1 — cosp — T-{1 — posv) sinv = &8inp 



and the coefficient k can be selected arbitrarily. For this, it is best 
to eliminate k so that the angle y assumes the role of the latter. Dif- 
ferentiation initially produces 

Bin'['(' 



.B' = SmP; g=-*- cos YV'^fc 003,1!, 



103 



and we obtain from the last equation 



so that there results 



7 —tgYCOtS l—'T '='-7—- —> 

** ' ' sm p COS f 



! cos Y 

Differentiation once more produces 

I B" = co8Ycos(g-v)a -V ) + sin (IS -7) sin v/ 
f cos* Y ' 

and this expression assumes the form 

„„ sin (S — y) /■,,.. ^ 

^ " 8inpco3»Y ^°"^ ^^ - ''^ +,«'" "^ ^S'[ COS p} 

through insertion of the above values. By utilizing the value derived al- 
ready above (p. 39) 

I B ^s»"V»PsinVi(p — y) 
! cos Ys Y 

we obtain 



}g'_ cosVaYco8Vg(p— y) ^ C03(p — y) + sin Ytg YOGS p 

jS sin V«P COSY -A' "^ sin p cosy 

and then, by utilizing the identity 

I oos*ViY-i±^2iX 

the expressions 

! ^^SSmU- + i\ + *|I f^tgY + oot|5{l + tg»Y). 

: B 2 \cosY I 2 B 

which are suitable for investigation of the limits of variation of these 
values. For this purpose, the angle y must be treated as a variable param- 
eter at a given value of P. Differentiation in regard to the latter pro- 
duces 

i i. IK] BinYCOtVgp+ 1 _8_ IB^\ I-l-2cotgtgY 

\Sx\BI^ 2co8«y SyU'/"' cos«y • 



lOif 



^*j^>I3>0i *® have|cot V3P>1, and it follows from this that not only 

B" B* 

g7 but also ~ assume infinitely large positive values if y approaches one 

of the values + ^. If the differential quotients are made equal to zero 

where respectively 

we then obtain the respective minima 



8int--tgV,p fcgY»_i|- 



iMin. 



of which the former is always positive whereas the latter has a negative 
value of >■ arc tg 2. 
The function 

X=.2|^-|;^ = cotV3p(3^ + l)-ootI3{l+tg»Y) 

to be utilized immediately is symmetrical in regard to the value Y = and 
assumes an infinitely large negative value when \y\ approaches the value 
^. The differential quotient 



! 3^ == 52^*81 (COS 7 - 2 cot P tg V. P) 

therefore results in a relative minimum at y = and two symmetrical maxima 

which correspond to the condition 

I cosY=r— tg*V«P = 2cotptgV*P 

By taking into account the identities 



1 ^- COtVsp — tgV«P xi, o -1 

jcot S «= f-J-- — 2_lj: = cot Vs 3 r— s 

j "^ 2 ' "^ sinp 

there results for the relative minimum X 

o 

: Xo-cotVi,p + ^. 

and this value is consequently always positive. For the maxima X , we ob- 
tain initially 

^ir».=cotv^p + 3^(cotv.?-^) 



105 



and then, by application of the second of the above values of cos y 

Through the relation 

j* sm p 

there results the value of the difference 

A.m — Ao ■=• 2 * 

We are now able to investigate the conditions under which an eccentric 
osculation of the first and/or second order can be accomplished with a 
standard A-eccentric and B-crank mechanism. For the sake of brevity, we 
shall utilize here, and in the following examinations, the abbreviated 
designations 

\ K ^--W ^-B «=f 

with the express assumption ■^ /■ ?> /^ so that the relations deduced above 
receive the form 

As far as the eccentric osculation of the first order is concerned, 
it is immediately clear that | SK> represents a condition necessary in all 
cases since neither U nor m can have a negative value. If the tangential 
mechanism is utilized, this condition is adequate also mathematically where- 
as we must require for technical reasons that! 2)? shall not exceed a certain 
minimum value dependent on the construction of the machine. If the sinus 
mechanism is utilized, then we must have 

I iK > OTmin. (1 + cos W^) 

The same conditions are valid also for machines in which od = 0. For the 

sinus mechanism, we here need only make cos (» =1 and the postulate 
' " m 

2 /" U /" resulting for the tangential mechanism can always be complied 



106 



with because m can assume an arbitrarily large value. However, if the 
crank mechanism is replaced by a standard B-eccentric where m = cot )^ 3 , 
the mathematical conditions for the sinus- and/or tangential mechanism 
then read 

|2)l — 2cot»Ap>0 and/or j SJ^ — ^ cot Vip < o, 

and it is clear from the latter that even the eccentric osculation of the 
first order requires the possibility of the application of a finite angle 
0) if a standard eccentric is utilized in the B-mechanism, in order to be 
successful in the border cases where the above differences are small » 

For eccentric osculation of the second order there is valid, under 
application of the sinus mechanism, 

■^2V>U{U~l)>V(l-\')>0 

i 2m{S« — »)>aK(SOl — m)>m(9l — n)(l — )J')>0. 
Necessary conditions are therefore 

but compliance with the latter is not adequate since m and n are dependent 
on each other, even when we disregard that the technical conditions must be 
carried somewhat further. The respective equation resulting through elimi- 
nation of cos y from the formulas deduced above would hardly lend itself, 
however, to a representation of the necessary and adequate conditions. 
The condition 



3J7>2F + 4> 



1— X* 
valid for the tangential mechanism is written as 



i 1 — A* 



107 



Of these two inequalities, the first can always be satisfied because 2m - n 
can receive any arbitrary value located between the positive value X and 
-Co derived above. The necessary and adequate mathematical conditions are 
therefore 

I m>o iifi>-x„-J^,- 

In order to achieve eccentric osculation of the first and/or the second 
order in both cases in which the first and/or both conditions are not ful- 
filled, compound machines or special cylinders must be utilized. In the 
investigation, we shall consider first the mechanically advantageous appli- 
cation of these cylinders in the B-mechanism. The following method is ap- 
plicable both to duplex cylinders as well as for general cylinders of the 

second order. Let the function B be written in the general form 

iB^E + kfif) f('s)'^e,F 

in which E and F are functions of 3, Since B must be symmetrical around 

zero value, this must therefore necessarily be the case also with E and 

either with cpCy) or with F. Through differentiation and elimination of k 

and c^ in the same manner as above (p. 100) , there results 

\ B' — E' F'/ B"-E" ^.i^rr 
\ B—E "F ' B'—E''"F''^F^" 

in which U- and V^ represent the same function of Y as U and V of or. In 

order to make out of these two equations , containing the three magnitudes 



B, B' , B" , one equation in which only m and n occur, let us first subtract 

E' ./ E" 
~ and/or gy-: 

[ We— BE' iljj _E[^ B"E' — B' E" F' „ F" E" 

\E(B — E)^F' E E'(B' — E)"~¥^''^r'~W' 



whereupon the last of these equations is multiplied by the first and divided 
by the third. The equation so obtained is to be written in the form 



F '^V'-Wj 



F' E 



108 



-„r 



where the function T thus introduced is defined by the equation 



r-!lMflM 



^' EF' 



The solution of the problem of eccentric osculation of the second order 
is essentially dependent, in those cases where the simple duplex machine 
is inadequate, on the properties of this function which are different for 
the different types of special cylinders and/or compound machines. 

When an eccentric is concerned in which the fundamental curve repre- 
sents a conical section or the parallel curve of one , we have, as demon- 
strated above (p. ^9) » 

I £=.1 — cos^ + &(!—«) tt« = 1 — (y + 1) sin* ?. 

At q + 1 !1^ 0, this equation can be written in the general form 

- - . -. . I 

i JB^E^+ktpii) f('()-~c;F 

I 
where 

i JE? -= 1 _ cos p y (7) -» 1 — cos Y 7fl = — 



P + gu 



and, at q + 1 < 0, we need insert here only the hyperbolic functions 
So« Y ©in Y and/or I ]/ — {q -r I) in place of cos y sin y e. This produces ini- 
tially 



F' E" 

Y = ^7- -=• cot p 



E' 



E±F _ , , 1 F IF" E"\ 1 ' 

'EF''~^+^i^: f'[f^-w}^-^-^s'^'--^^/ 

At q + 1 ^ 0, the functions U, , V^ have the values which were determined 
above for U V with the sinus mechanism, and in which we set te = 0: 



COSY 



i + cos Y . cos* Y 



At q + 1 <f 0, the differentiation of the hyperbolic functions produces the 



109 



same values where we need to replace cos y byl ^o9'[ . This produces 

i »^tgp(»— cotp) ■ a , , \/l TT^\ 

I m-cotV2,3 "'/ ^U + V U"^^/' 
in which u = cos y and/or Im = 6o« Y » depending on whether u ^ 1. 

It follows from the deduction (p. 49) that u must have a positive 
value. OtherwdLse N and p would have different signs which would correspond 
in the hyperbola to transition from one branch to the other and in the 
ellipse to an angle 3 /" ^. When y == Oi c^ =0 and the fundamental curve 
represents a circle if the conditions m = cot )^ B and n = cot P are satis- 
fied. If this is not the case, then k assumes an infinitely large value 
and the infinitely distant parallel curve of the fundamental curve of the 
cylinder is a conical section. Since such cylinders cannot be ground with 
the machine described, this case is mechanically, although not mathemati- 
cally, excluded. Since we must have u ^t*!^ 1, this results in the conditions 



I ^>^ IV2tgPcotV*P, 

in which the latter has meauiing only if the value of T obtained by m and 
n does not assume the form — , which is necessary and adequate in this form. 
For eccentric osculation of the first order, we need only solve the 

equation 

i tgp(m5 — ainp) „ 
i 5-sinptgVisP *" ' 

depending on whether TL or B has been selected arbitrarily. If this is the 
case with U^ , then B assumes a real value at any desired value of m. The 
eccentric osculation of the first order can therefore always be accomplish- 
ed at an arbitrary value ofjJOT regardless of whether the sinus or the tan- 
gential mechanism is utilized. We can here make O) = and select a arbi- 
trarily after which C is determined by K. If E is then also selected 
arbitrarily which determines c, and Y, then there results from the equation 



110 



above B, from which the values of c and k are obtained through which are 
determined in turn the magnitudes p r characterizing the cylinder in addi~ 
tion to e. 

When I a)? = and consequently when m = 0, we find from the equation for 
J" that the product m n must have a finite value and therefore must be 
1 91 = 00 which does not indicate any singularity, however, but merely means 
that E" has a finite value in the machine curve. The consequence of K = 
at R' jz^ would be C = and is consequently impossible to realize by a 
special cylinder in the B-mechanism. The casej 932 = oo is also mechanically 
excluded as a finite value of K and of cos (or + c«) and/or cos od in the sinus 
and/or tangential mechanism because, with cos y = 0, the point of contact 
of the fundamental curve of the cylinder with the plane would lie on the 
asymptote and the eccentric would consequently have to be infinitely large. 
Concerning osculation of the first order, there is consequently applicable 
the only condition thatIS)? may not have an infinitely large value. To this 
should be added merely that the case cos (a + w) =0 can be realized with 
the sinus mechanism and that therefore this condition is strictly valid 
only in regard to the osculation of the first order obtainable through the 
special cylinder. 

In examining the possibility of achieving eccentric osculation of the 
second degree:, we have for the sinus mechanism 

U ■ I + cos a \ 

and therefore obtain 

'\ ^ ^ m — ucot'h? 

It will be seen immediately that a large positive value of JT can always 
be obtained provided ; £02 is large enough in order to make the numerator 



111 



equal to zero by a mechanically appropriate value of U. However, if this 
is not the case , then osculation of the second order cannot always be 
achieved. If UP is designated with [?] and/or x y, then the equation 
for 1 represents a right-sided hyperbola in which the asymptotes are 
parallel to the coordinate axes but of which only a part is available be- 
cause X cannot assume arbitrary values. If then the numerator cannot be 
made equal to zero , then this is the case only with one part of one branch 
and y reaches the highest value either at x = 1 + cos m or at x = +00. If 
the corresponding values of y are designated by i ^ and/or i^jo ^^^ examina- 
tion is restricted to the mathematical possibility, i.e., only U > 1 is 
required, we then have 

and the mathematical condition states that, when! 2)J< cot V« P » either P^ or 

- 2. 

Lnn must be greater than ». 

fj" ° cos p 

LZg^ 3 3«cosa ,2 
/ 1 + cos a ^ Z7 

is valid for the tangential mechanism and consequently 



r- 



oiii. o /co 4.D 3 3)1 COS a , 2\ 

9)i tg S m — cot P — r— + ff) 

. "^ \ ' 1 + cos a U 



m—.Ucotys^j 

Since U can here assume arbitrarily small positive values and since 

r2 ts: B 
, if U decreases toward zero, approaches the value &— » osculation of 

the second order is always possible in the general case. The special case 
|2W = 0, however, requires the product j 2)^91 and consequently also =- to have 
a finite negative value which is true moreover for the sinus mechanism but 
is contained in the condition formulated for that case. In this special 
case, the normal of the machine curve passes through the center of curva- 
ture of the vertex which presupposes a cusp in the evolute and the condition 



112 



means that the radius vector has a maximum and/or minimum, depending on 
whether the latter is larger or smaller than the radius of curvature of 
the vertex. A different behavior would indicate that a second point E' = 
existed between the given point and the vertex and/or coincided with the 
first point at B' = K" =0 but such a complicated singularity is there- 
fore completely excluded, except when K = 0, 

After we have thus obtained a mechanically applicable value of IP 
through appropriate selection of U and or in this manner, a positive value 
for u results from the quadratic equation 



lT=('Ui)(I-,JL-) 



u 



after which the other machine constants and the magnitudes determining the 

cylinder can be obtained in the same manner as above for osculation of the 

2 
first order. Difficulty results only when e =1 results where the funda- 
mental curve of the cylinder represents a parabola or the parallel curve 
of one. However, it is apparent from the equation 

deduced above (p. ^9) that we can set 

I J5 = 1 ^ COS p + & sill p fcg p 

By differentiation and elimination of k, we obtain the equation 



I mB — sin p 1 + cos*p 

! B — sin p tg Va p '°° sin p cos p ' 

which furnishes the value of B and the other machine constants subsequently 
result in the customary manner. If we determine the function J after 
differentiating once more in the manner indicated above, then the latter 
must receive the same value as when we set u = cos in the general ex- 
pression which is actually the case. 



113 



When utilizing duplex cylinders as eccentrics in the B-mechanism, 
the simplest circumstances exist if the latter are ground in a standard 
duplex machine by utilizing a standard B-eccentric. The equation of the 
machine curve obtained by utilizing such a cylinder is the same as the 
one just examined with the difference that 

and that f(Y) represents the general sinus or tangential mechanism with 

the angle of inclination (i)^ . There consequently results in general 

I ?wtg'/iifi{n---oot g) ^ p ^ UiyL 
I »t — cot Vs ^ ^1 "^ 

and, if the duplex cylinder belongs to the sinus type , i.e., has been 
ground by utilizing the sinus mechanism, 

1 i- "=» r— • 

1 1 + COS Y 

We therefore obtain the values irriFoD from the analogous values indicated 

above by multiplying the latter with tg J^ 8 cot B and the mathematical postu- 
re'""""'"" T 

late is i->^r~7"" • With these differences, everything that has been said 

! 1 + cos Y 

above on the special cylinders already examined, is valid for these cyl- 
inders also, regardless of whether the sinus or the tangential mechanism 
is utilized with them. If P is determined in the customary manner, Y can 
be selected arbitrarily after which m can be determined from the resultant 
value for U. in the same manner as OD from U. If we desire to have ou = 
in those cases where 1 is sufficiently large, we then have 

i cos'v cos Y 

i 

If the duplex cylinder is of the tangential type , there then results 

U, 13 U, cos Y 



The equation 



r C/, ( 3 U, cos Y g\ 

■ " U, — lh + cost j 



sUi' tg Vs Y cot Y - ^; (r + 2) + r = 



114 



resulting from this has the discriminant 

i 
I (r + 2)« — 12ri.gVjYcotY, 

which can be written in the form 



r 4- 2(1 — 2cosy) \'' 12 cosy (2— cos y) 
1 + cosy / (1 + cosy)* 



The roots are therefore real at cos y ^ and both have positive, with 
r ]> 0, and opposed signs at P <r so that, at arbitrary values of 1 
and cos Y» always at least one real positive U is obtained. It follows 
from this that, with the application of such a cylinder, eccentric oscula- 
tion of the second order can always be achieved and not even the special 
case mentioned above is excluded which is equivalent to the fact that the 
evolute of the machine curve may have two cusps. Whenj 2}i -■ oo , however, we 
cannot make cos y = which would require an infinitely large eccentric 
but would have to utilize the sinus mechanism and make cos (c? + ou) =0. 
Whether in such a case an osculation of the second order can actually be 
achieved is not worthwhile of investigation because we will always attempt 
actually to avoid this which is always possible by selecting a pseudo-du- 
plex curve as machine curve. The duplex cylinder of the taingential type 
therefore represents the best means in the B-mechanism. Unforttmately, it 
is inconvenient to make u) = a priori because we must then solve a bi- 
quadratic equation in cos y. 

When utilizing the special cylinders in the A-mechanism , the investi- 
gation can be carried out by the same method. In the equation 

\il^^C.D /(«)-c?{p), 

j -^0 

in which D is a compound function of a which is represented in the same 
manner as B above by the equations 

iZ> = S+J;?i(y), ih)^c,F 



115 



E and F are therefore functions of a and the same postulates must be made 
on the latter and/or on the function cpCy) as above. The derivations of 

D E F from or are designated by D' ... whereas B' R" represent the customary 

D' D" 

derivations from 3, If m n then also signify the quotients ^ and/or ^, 

then the formxilas for the function i deduced above remain valid unchanged 
and we need only express m n ±n\ fSffSl U ^^^ ®» By differentiation and elimi- 
nation of C and/or c, we obtain 



a' 



If now the functions U and V are formed in the customary manner from the 
function f(a) and the derivations of the latter in addition to the function 
tp(a) = 1 - cos <x and the derivations of the latter, there then results 

^ '■ % (P) ?" ^(p) "^ f(p) (^ - ^*S V* a cot a). 

from which results in turn 



7n 



~ ^^^ °°^ V* " cot Vn a 



by setting 



«i' 



'^ru~ ^^^ —n'~m'V) + cot a 






for the sake of brevity. The general expression for X then assumes, for 

E = 1 - cos a, the form 

rp_j_F i? cot Via ^ — ■„i-fn'v 
! "^F'' m'U m-m'U 

F 
where the value tg a and/or tg )^ a must be applied for ^Tf depending on 

whether the cylinder has been ground in the machine serving for the produc- 
tion of surfaces of the second order or in the duplex machine. In the first 

case, the postulate !r> .exists in the latter if the cylinder of the sinus 

cos a 



116 



type iSir>-- whereas 1 can have an arbitrary value if the latter is 

j 1+cosy' 

of the tangential type. If (^(0) = 1 - cos 3, then 'jm'=.cot 'AP and i??/--cotl3 
are valid. We see that in this case also sm adequately large positive 
value of can be obtained with certainty only when, atj 9.1i > cot '/a p $ the 

numerator can receive an adequately small and either positive or negative 
value as needed through appropriate selection of U. If this is not the 
case, success will not be certain when the tangential mechanism is utilized 
for f(Qr). In this regard, the utilization of the special cylinder in the 
A-mechanism is therefore a disadvantage as compared to the utilization in 
the B-mechanism. However, the first procedure is also always successful 
if the special cylinder represents a duplex cylinder of the tangential type. 

The investigation so far has therefore produced the result that it is 
necessary to be able to utilize the tangential mechanism in order to achieve 
in all cases eccentric osculation of the second order. This would be tanta- 
mount to saying that the simplest machine will not be adequate in all cases. 
However, this is still possible, if we take the trouble of grinding two 
special cylinders, and in two ways. 

The simultaneous utilization of special cylinders in the A- and B-mecha - 

isms corresponds to the utilization of a function B in place of q5( g) where 

B' B" 

ffl' = •=— and n' = rr-r» As was demonstrated above, m' can here have an arbi- 

trary value so that it is always possible to obtain an adequately large 
positive value for the function i of the A-mechanism. 

The same purpose can also be realized by special cylinders of a com - 
bined type . This designation is intended to comprise the different types 
of cylinders which can be ground in the duplex machine when utilizing a spe- 
cial cylinder. Of the multiple possible types, we shall mention here only 
two which will have to be considered in the following. For the sake of 
simplicity, we shall here assume that all cylinders are ground by utilizing 



117 



a plane. As demonstrated above (p. 65), the result is the same when this 

is not the case but production is somewhat more coraplicatedo 

If we write the equation of the machine curve of a duplex cylinder 

ground with a plane 

j 
j K^ - a, (1 - cos aj /{«J «= c. (1 — 00s ?>), 

and if this cylinder is used as eccentric where a^ represents the distance 
of the respective axis from the center of curvature of the vertex, then 
the displacement in height corresponding to the angle of inclination 9 is 

|«3(1 — oosy) + ic,. 
If the cylinder is utilized in the A-mechanism for the grinding of a new 
cylinder, this produces an A-triplex cylinder and, when this procedure is 
repeated as many times as desired, an A-multiplex cylinder * We then need 
each time to replace 1^ by the angle a applied to the respective grinding 
process. The machine curve obtained through utilization of a duplex cyl- 
inder in an A-mechanism can therefore be represented by the equation 

j r, -^ ^, ! I - cos ^ j -(- a.Jl -- mn r,.X j(a,) =^ *, n - cm 0.,) /f^^) . ,.j^ ,,^, ov * 

and, for the m-th grinding, the machine curve in general is represented by 
the ffl + 1 equation 

I -^ = ^1 «».(1 — COS a„) /(«„) «. c„{l — cos «„+,) 

where the designation a ^ is applied for P. If now a standard eccentric 
and/or a duplex, a triplex, etc., eccentric is designated as eccentric of 
the first) second, etc., order , then the last equations represent the ma- 
chine curve if an A-multiplex eccentric of the order m is used in the A- 
mechanism. However, such an eccentric can also be used in the B-mechanism 
where we shall designate the angle a applied to the production of the ec- 
centric by y, le now obtain without further difficulty the same m + 1 

equations 

^L^C[l- cos a) Jeo/(a) =• 2 ««(! " f^^ 'f«) /("^"^ = ''"^^ ~ ''"^ *^"+'' ' 

-^0 «-i ' 



* Translator's note ; Photocopy of original text supplied is illegible, in part. 

118 



where a represents the distance of the B-axis from the center of cunrature 

of the vertex of the curve of the eccentric and B is designated by y • 

It is now evident that, if the coefficients c as isell as C in the last 

method of application are selected arbitrarily, an A-multiplex cylinder of 

the order m will always offer m machine constants a which are obtained 

n 

through linear equations when as many conditions are prescribed and that 
consequently, when only the simplest machine is available, an eccentric os- 
culation of the second order can also always be obtained by an A-triplex cyl- 
inder, regardless of whether the latter is utilized in the A-mechanism or, 

with K 3!fc , in the B-mecheinism. In the former case, the values of a'oi^* are 
^^ ' ' n n 

determined through differentiation twice of the three last equations after 

which the first equation represents three linear equations for the desired 

machine constants a , in addition to those obtained from it by twice differ~ 

n 

entiating, and in the latter case, the procedure is analogous by obtaining 
i^ and ^^ from B-R". 

In addition to the A-multiplex cylinders , the B~triplex cylinder is 
also of benefit in the following calculations. If we write the equation 
for the grinding of the duplex cylinder as 



|-K^. = ?i(l — cosf.) ^o/(y.) = Z',{1— cosy), 

we then obtain for the second grinding, if the cylinder is utilized in the 
B-mechanism and b- represents the distance of the B-axis from the center of 
curvature of the vertex of the curve of the eccentric, 

I Zj = a,(l — COST,) EJit,) ==bjil — cos (p) + a^il—ooa 'u) ^o/(Y.) = &t(l — cos ?) 

and, if the triplex cylinder so produced is utilized in turn in the B-mechanism 

I JS"s«=0,(l— COSfa) ^o/iTa) = &s(l —COS P) + 0,(1— -008 7,) 

■2?o/{T2) = 62(1-008 p) + a.(l-oos Y,) ^o/(Y,)=-&i(I -COS p). 

and these equations can also be written in the form 

5- = 0(l — cosa) /(a)=c,B, JSj- 1 — cos p + fc,(l — cos y») 

lit 

/(Y») = C2-B, J?i»l-cosp + *i(l-co8Y,) /(Yi)-Cj(l-C08p) 

119 



In the same manner, for the utilization of the B-triplex in the A- 
mechanism, there results 

i ^3 *= 03(1 — COS a) + a, (1 — cos f,) -So/CVj) = 6,(1 — cos a) + a, (1 — cos f,) 
jB'o/(7i) «« 6,(1 — 008 «) /(«)«=* 0,(1—- cos P) 



and 



K 



•='C .Dj D, ■= 1 — cos a + Aj(l — cos 7,) /(if,) =■ CjDi 



iPj^l — cos a + *,(! — COSY,) /(7,) = Cj(l — cos a) /(a) = c(l — cos p). 

It is obvious at this point of the investigation, that it is necessary 
as a rule to have available a large number of machine constants in order to 
be able to apply three of them for achieving the prescribed osculation of 
the second order. However, it should be noted here that the problem can be 
appreciably simplified in many cases through appropriate selection of the 
machine curve so that a smaller number of means will be sufficient. It is 
also possible in other cases to achieve a better osculation of the curve 
and/or satisfaction of an additional condition by variation of the constants, 
so-to-speak, in excess. In this regard, we intend to investigate in the near 
future what means are necessary in the different cases to achieve a centric 
osculation of the fourth order simultaneously with an eccentric osculation 
of the second order . 

The foregoing will immediately show that this problem may be solved in 
all cases by an A~quadruplex eccentric even when only the simplest machine is 
available. If such an eccentric is utilized in the A-mechanism, we then have 

and the other three constants a result by means of three linear equations 
from the values K R' E" . We shall show by example further below that the 
solution can also be obtained in certain cases through the selection of an 



120 



appropriate parallel curve as machine curve by utilizing only the three 
constants C c o) of the simplest machine. However, if the machine curve 
is prescribed, we can generally achieve our objective with simpler means 
than a quadruples cylinder, although a special cylinder and/or a compound 
machine is always necessary in order to obtain the fourth constant. 

If we utilize a special cylinder in the B-mechanism, then the equa- 
tion 



IS. 



■0 



utilized above is valid and we have generally 

where all differential quotients are valid for 8=0. If C c are elimi- 
nated in this expression by means of the equations above, there then re- 
sults . , ■_____■ 

In order to make possible further eliminations, B" may not contain any con- 
stant characterizing the special cylinder which would be the case if a cyl- 
inder of the second order were used. The function B must therefore repre- 
sent a duplex cylinder so that we obtain, when E = 1 - cos 8, B" s 1. If 
<p(a) = 1 - cos a and we state, for the sake of brevity 



8^]/_^=]/. 



2R,% V 2R^J 



there then results 



5=^ 



^ /(a) 



•sin y« a /'„(«) 

When utilizing the sinus mechanism , as demonstrated above (p. 100), 
we have 



„ , 1 — cosa/, , cos (g + td) \ 
A(«) = 1 /(«) - sin« V ^ cos« i ' 



121, 



from which we obtain 



5= 8^ 



(U—1) cos Vs. a 
whereas we have, for the tangential mechanism , 

l/o(«)-»-~- /(a) = ^1^ 

j cos* (0 ' V ' cos to cos (a + (0) 

and obtain 

ij5=.ii££!lV*a 

m and n are determined in the usual manner so that then also B' and B" are 
known functions of U and oi and the equations 

^' — sing F[ JS"— cosp F" F'. 

B — ain^tg'l.^'^F^' B'-sin^"^ ¥ "^ F ^' 

decide whether the problem can be solved in this manner or not. The cir- 
cumstances are here the same as in the problem of eccentric osculation of 
the second order in the standard machine without the utilization of special 
eccentrics, except for the difference that variation of U and a affords more 
possibilities. A complete discussion would lead us too far but it is ob- 
vious that in those cases where a duplex cylinder does not accomplish its 
purpose, this is achieved if F represents a duplex cylinder of the tan- 
gential type which is expressed by the equations 

rF^B, + k,f(r(^ f(^J^e„F, 

if f(Yv) represents the respective function. The cylinder is then a B- 

triplex cylinder produced by using a duplex cylinder of the tangential type 

for the grinding process. By stating 

IF' , F" , 

the coefficients can be determined in the manner indicated above through 

the respective function i- ^ where not only U and or but also U^ and ¥. can 

be: freely selected in the equations above. If f (Yt ) represents the tangential 



122 



mechanism, then i can have an arbitrary value as demonstrated above. How- 
ever, if only the simplest machine is available so that f (Y-, ) must represent 
the sinus mechanism, then there is valid for the equation 



the necessary condition 



"^ i; 



I 1 + cos Y; 



Although the values of m' and n' can now be varied through variation of Uo? 
U'V, it is not possible to prove without a very complicated examination 
that this condition can be satisfied in all cases. However, in cases where 
this is impossible, we must, in order to still be able to employ the sim- 
plest machine, utilize the quadruplex cylinder. 

The simultaneous centric osculation of the fourth and of the second 
order therefore requires at the most a triplex and/or quadruplex cylinder 
depending on whether the tangential mechanism is available or not and can 

always be achieved by these means provided that JS has a finite real value. 

IV 
If K and R have different signs, then this must be also the case 
o 

with «p(a) and cp"(ci?) as will be seen from the general expression for B 
above but this then requires a special cylinder in the A-mechanism. This 
is an expression for the fact that the machine curve, if the A-mechanism 
consists of a standard eccentric, cannot intersect the circle of curvature 
of the vertex because the radius vector cannot have a smaller value than 

the one corresponding to the zero position of the machine. In contrast to 

IV 
this, the case B =0 can be treated by a special cylinder in the B-mech- 



anism by making B" = which produces the equations 

GficC) /{a) = ctp(Y) ■ /(t) = c,?(P) 



o 

i2« 



123 



If 9(y) is designated with B and if B'B" represent the derivations from P, 
there are then determined only the quotients m = B'/B and n = B"/B' through 
the eccentric osculation of the second order but since B represents a stand- 
ard duplex function, U-, and V are determined in turn through these quotients. 



It will be evident that 






and the problem therefore is, after calculation beyond the function f (ff) , 
exactly the same as the standard problem of osculation of the second order 
calculated beyond K/R . The result therefore is that the same means are 
adequate in the case E = as in the general case. We need only make 
b = for the respective B-triplex cylinder. 

The method of treating the problem is exactly the same if the special 
cylinder is utilized in the A-mechanism which constitutes the only possi- 
bility if K and R have different signs. Through the value of | St , D is 
then ftiade into a known function of U and a which is then the case also for 
D'D" through the values of m n. Only the value of D is different. Through 
elimination of C and c from the equations 

there results in general 

ii) = ^o//(«)9;:(p)\* 

If the cylinder represents a duplex cylinder and if 9(&) = 1 - cos B, we 

then have D" = cp"(3) = 1 and obtain, through insertion of the respective 
o ^o 

values of f(a) and t'(a), for the sinus mechanism 

and/or, for the tangential mechanism, 

12 K sin^ a cos* Va a 



D' 



B^ U' sin^ p tg» Vs p 



12k 



Since D may also have a negative value, a point of intersection of the 
machine curve with its circle of curvature of the vertex located between 

the point of the eccentric osculation and the vertical point, is possible. 

IV 
At R = 0, we need only state D" = which produces the same equations as 

above where only a and y change places. Accordingly, we must make a, = 

when utilizing the G-triplex cylinder in the A-mechanism. The utilization 

of a corresponding special cylinder in the A-mechanism therefore makes it 

possible in all cases to obtain a simultaneous centric osculation of the 

fourth and of the second order. 



The special case K = has been left aside so far in order to be able 
to discuss it in context. That it requires a compound machine and/or spe- 
cial cylinder has alreacfy been stressed. At E' =^ 0, there exists a point 
of intersection of the machine curve with its circle of curvature of the 
vertex and, in accordance with the foregoing, the A-mechanism must be a 
compound mechanism and/or contain a special cylinder. D is then equal to 
zero and the determination of the machine constants is performed in the cus- 
tomary manner. If only an eccentric osculation of the second order is pre- 
scribed, the general expression for i can be utilized since the latter is 
valid also at M =oo . It follows from this that a duplex cylinder is al- 
ways adequate, as in the general case, if it is of the tangential type but 
that a triplex cylinder may be required in the opposite case. On the other 
hand, if a simultaneotis centric osculation of the fourth order is prescrib- 
ed, then C is eliminated from the value ! 2t by means of the equation 

which results in 

iilo^ J7 tg V2 a sin* p tg Va p \/', (a)/ 

for cp(3) = 1 - cos B. The value of n then also makes D" into a known 



125 



function of U and a after which calculation is continued in the customary 
manner. If E'-' is also equal to zero, the procedure remains unchanged in 
spite of this. 

When K = E' =0, the machine curve has a point of contact with its 
circle of curvature of the vertex and there consequently follows D is D* = 

for E = 1 - cos a as 

1 F' 
|C0tVaa>J-.?7,. 

It is evident from this that the duplex cylinder must be of the tangential 
type at F = 1 - cos at and that, if we have only the simplest machine avail- 
able, eccentric osculation of the second order can be achieved only by a 
triplex cylinder in the A-mechanism and/or special cylinders in both mech- 
anisms. In the first case and if a B-triplex cylinder is utilized, F 
constitutes a compound function and, in the latter case, we can make or' = 
and arbitrarily select D' so that in both C finally results from the value 
of E". If a simultaneous centric osculation of the fourth order is pre- 
scribed, C is eliminated from the value of | 2f by means of the equation 

after which the determination of the machine constants takes place in the 
customary manner. The case where the machine curve is to have, in the ec- 
centric point, a contact of the second order with the circle of curvature 
of the vertex and, in the vertex, a contact of the fourth order with the 

prescribed curve, can be treated in the same manner. We obtain D = D' = 

2 
D" = and select C and c so that the product C c receives the prescribed 

value. 

Under certain conditions, the case K = E' =0 permits the utilization 

of the special cylinder in the B-mechanism whereas the A-mechanism consists 

of a standard eccentric. Since the machine curve is here located in its 



126 



entirety on one and the same side of this circle of curvature of the vertex, 
it will be apparent that the centric escalation of the fourth order can be 

obtained simultaneously with an eccentric osculation of the second order 

IV 
only if the radius vector has a minimum, at a posxtxve value of R , and 

a maximum, at a negative value of the latter, in the point of eccentric os- 
culation which is equivalent to the fact that E" must have the same sign as 

IV 
R or must also be equal to zero. We obtain a = and consequently also 

B = from which results k after y has been selected arbitrarily. If we 

then make o) = 0, B' is also known and we obtain the value of a\ after c 

has been selected arbitrarily, after which C results from the equation 

1 S" 



II 



6'a's 



so that the eccentric osculation of the second order is always possible. 
If the centric osculation of the fourth order is also prescribed, C is then 
eliminated from the value of | 21 by means of this last equation and the equa- 
tion , cB' 

which results, by taking into account that f'(a?) = f'(of), 

so that the problem can always be solved, provided that the condition just 

stated is satisfied. 

IV 
The case of R = at K = also offers no difficulties since we only 
o 

need to utilize duplex cylinders both in the A- and in the B-mechanisms. 

In the equations , 

I f^ = jO,I> /(a) =.6.5 

there then must be ; „ , , " ... ,„. 

I 

so that B" = and consequently also ; Sf = 0. From the values for E" and E' , 

T 
there results in the customary manner the equation for the function .- 



127 



inherent in the A-eccentric so that we need only set D = in further cal- 
culation. 



It appears from the above that it is always possible to employ four 
machine constants in such manner that an eccentric osculation of the second 
order and simultaneously a centric osculation of the fourth order is obtain- 
ed. The question is now under what conditions the same number of machine 
constants can also be made available in another manner and this confronts 
us initially with the problem to obtain an eccentric osculation of the 
first order in two different points or, abbreviated, two eccentric oscula - 
tions of the first order . That this can be solved directly in certain 
cases without the utilization of a quadruplex cylinder is demonstrated in 
the following manner. When utilizing a duplex cylinder in the B-mechanism, 

let the machine curve be represented by the equations 

\ j^ " "' " " ■■■ ' 

where 

i5 = i5' + ^^(Y) /(y) = c,J' F-1— cosp 

If there are now given two points located on the machine curve which we 
shall assume to be defined by the values F , K^ , R' and/or F^ , K_ , E' , we 
are then required to determine four machine constants through elimination 
from the equations obtained through these values. In connection with the 
method utilized above, these are offered by the coefficients c, k, c, and 
the angle a), contained in the function fCy). We therefore arbitrarily 
select not only the constants Cu) but also the functions cp(Qf) and f(c?) by 
taking into account the conditions valid for the latter which makes the 
four products cB^ , cB' , cBp, cB'^ known. Through elimination of k, we 
obtain in the customary manner, by application of the functions U^ Up 



128 



formed from the functions cpCy) f(Y)> 

B,-FrK'^^ -K^^^'^i:"'' g^^zt;--^' 



l^^z^^Z^.u, fr=4^»|^.ir. 5c=Z._^). 



J? J; r T 

From these equations is subtracted on both sides _1 and/or _2 and/or -^ ; 

Fi F2 F2 

i -B't-Fi — i?t.g", „; ,^, .. B\Fj — BjF'^ _,, .J., . 

! jBt J, - -g, J'. _ -F,? (T.) - -F^yCY.) 

The fourth of these six equations is multiplied with the third and divided 
by the sixth whereas the fifth is divided directly by the sixth. We obtain 

in this manner 

I B\F,-B,F\ _ F\f{'iMU,-l) B\F,-bJ>, F'M'[,)iU, - 1) 
I B,F, -B,F, F, 'f(Y,) - F.fif,) B,F, - B,F, = h'H'U) -lxf{ri^_ 

and also , through elimination of c^ , 

The magnitudes on the left side in these three equations are known where- 
as the three unknown angles Y-i Yp**^-. occur on the right side. If f(Y) repre- 
sents the sinus mechanism, all equations are linear in tg co when 

I ^ = cos Y„ — • sin '[„ tg «, fi'in) = siii u — tg w, (I — cos y„) ^(t..) — 1 -- cos -[„ 

The elimination of this magnitude produces, by utilizing the abbreviated 

designation , 

iA - -B'.^x-^.^'. • . _ B',F,-B, F\ 
' ^' B,F, - B^F, ^» "= B,F,-BJ\ 

the two equations 

I A^F, cos (y, - Ya) = cos Y, {A,F, — 4, JT, - F\) + ^.i^-, + J", 

I A:,F, cos (Y, - Y,) = cos -[, (A,F,- A,F, — F',) 4 ^^i?'. + F',. 

which lead to a bi-quadratic equation in cos Y-i °r cos Yo* -^^ ^^^ latter 
furnishes a real and mechanically applicable value of the angle, we obtain 
the respective machine constants from this value throu^ linear equations. 
The variation of the values of C and u) furnish the means for influencing 



129 



the bi-quadratic equation. Evidently, the method is not easy and not 
generally applicable but will be in appropriate cases of benefit if a 
mechanical simplification is to be achieved at the expense of increased 
calculations. If the latter does not lead to the objective, it is only 
necessary to grind the respective quadruplex cylinder but we can also em- 
ploy the following method. 

The machine curve generated through an A-triplex machine with two 
evolvent eccentrics of the first order is represented by the equations 

R-— T? 

—~i «- Co{l — COS a) + Gtia — sin a) + Cjil~ cos y) + C,(y — sin y) 

/(Y)=c,{l~cosa) /(a) = c(l — cosp) 

i. . , .. . . 

if standard eccentrics are utilized otherwise. If here the coefficients 
cc, and the angles cuu)^ contained in the function f(a) fCy) are selected 
arbitrarily, then (srs'a" yy'Y" ^^e known at a given value of B and we obtain, 
exactly as with a quadruplex cylinder, four machine constants, i.e., C , 
which result from any arbitrarily formulated prescriptions out of four 
linear equations. The curve can then be made to pass through four given 
points and thus achieve four osculations of the order zero or else combine 
a centric osculation of the fourth order with an eccentric osculation of 
the first order or another osculation of the order zero, etc. If mechani- 
cally unsuitable values should result from the solution of the four linear 
equations, we then have adequate means for influencing the latter through 
the variation of the four arbitrarily selected constants. Whether this 
method or that of the quadruplex cylinder is preferable, will be decided 
probably only by practical experience. Serious mechanical difficulties 
do not seem to exist, in any event, with the A-triplex machine, especially 
not when a B-carriage exists so that the A-axis can be arranged parallel 
to the B-axis. 



130 



It follows from the foregoing that a pseudo- duplex curve is indis- 
pensible only when the radius vector of the machine curve constructed in 
the center of the curvature of the vertex touches the machine curve in one 
point. We must then select another point as the pole of the coordinate 
system from which a tangent to the utilized section of the machine curve 
cannot be drawn. The treatment of the equations 

I ^-^'^C.fW + G.m /(a) = c«p(p) 

i ' _ _^ 

is the same as above, except that it is possible to influence the equations 

through appropriate selection of the pole of the coordinate system. This 

selection determines R and C^ is then obtained from the radius of curva- 

o 1 

ture of the vertex of the machine curve and the form of the function ufO) . 

If p represents this radius, there are then valid the relations 

1, ■ ■ ■ ■ ■ ■ ■ 

1 p ~Bl~' bT '^''^^'^'. 

For the sake of mechanical simplification, it is preferable to make >|^(B) = 

1 - cos B so that U/"(B) = 1 and the first equation can be written in the 

I ^-(.-f)a-cosw = <,.„., 

from which results in the most simple manner the influence of the variation 

tf 

of E . Since the left side of this equation corresponds to the value ^r— 
o Rq 

in the customary method, it will be apparent also that the special case 
corresponding to the case K = can always be avoided from which it follows 
an eccentric osculation of the second order can always be achieved by em- 
ploying a special cylinder in the B-mechanism. 

In order to clearly represent the influence of the variation of E on 
the possibility of obtaining a simultaneous eccentric osculation of the 
fourth order, it is necessary initially to derive the expression for the 
flattening value $ at R"=^0. Differentiation twice of the generally 



131 



valid equation 

^- - cos" (p — 2:)(7?' + 2 72'* - RR") 

initially produces 

1 "i i2* 
^cl^-+ — d'i? =. 2 iJcZ'7? + 4 diJ'« - Rd?R" - R"d'R -B{R'-~ RR") (cZp - d's)\ 

for P = cp = 0. If P represents the independent variable and d a the element 



of arc of the machine curve, then 

P 



d*-=-<i>dc!\ da = pd'f^2ia^^ 



and we obtain 



^'*^-(l; + ^«) + 3^J(i + ^.). 



by applying the relations above. Bince further 
there then results 



i ^«'<''^'W(^)*=^JO + C?.)- 



R'o^ 



3 

The right side of this equation is a polynomial of the third degree in 
R and shall be designated as F^ (R ). By taking into account that the value 
of R inherent in the point of eccentric osculation is also a function of S , 
the first equation of the machine curve can be written in the form 

Through elimination of C , we obtain the condition for employing a standard 



eccentric in the A~mechanism: 



>o. 



i lARo) 

Since there is always at least one real value of R which makes F^ (R ) =0 

"' o 1 o 

at F,' (R ) 9^ and since a real value of F^(R ) corresponds to any value of 
1 o c. o 

R , this condition can always be satisfied in any case, provided F^CR ) does 
not pass through zero simultaneously with F (R ) and in the opposite direc- 
tion. In those cases where the standard machine requires a special cylinder 
in the A-mechanism for obtaining a simultaneous centric and eccentric oscu- 
lation of the fourth and/or second order, we can therefore, if this method 



132 



produces mechanically applicable values, utilize the cylinder in the B-mech- 
anism by grinding the surface with a compound machine. 

In non-focal surfaces also, the method of the determination of machine 
constants remains unchanged for the various problematics, regardless of 
whether they are ground by employing a horizontal straight-line guidance or 
with a pseudo-duplex curve as machine curve. In the latter case, when the 
machine curve is less different from a straight-line than from the corre- 
sponding Pascal convolute ["Schnecke"] , it may be of advantage to set \F(P) 

= o - 1 so that we obtain 

cos B I u , V 

since C^ = 1 and W^ (3) = 5« We need only to remember here that the form of 
the function cp(6) is also influenced because of the construction of the com- 
pound machine. 



The principal results in regard to the means required for achieving ec- 
centric osculation can be summarized as follows. 

Eccentric osculation of the first order requires, at a finite value 
of K and a finite positive value of | 23? » only a standard machine with- 
out special cylinder. However, if the tangential mechanism is not 
available, j S)Z may not be smaller than a certain value depending on 
in that case. In all other cases, a cylinder of the second degree or 
a duplex cylinder in the A- or in the B-mechanism is adequate. Only 
in the case of K = at R' :?^ 0, is it necessary to employ such a cy- 
linder in the A-mechanism. 

At a finite value of K and a finite positive value of jSJJ , eccen- 
tric osculation of the second order can also be achieved with the 
standard machine without special cylinder, provided iSJ exceeds a cer- 
tain negative value depending on S and the tangential mechanism with 



133 



variable B-crank mechanism is employed. If the latter is not the case, 
the conditions are more complicated. In all cases, a duplex cylinder 
of the tangential type is adequate. Only when the machine curve inter- 
sects its circle of curvature of the vertex in the point of osculation 
must the latter be employed in the A-mechanism. Utilization of a cy- 
linder of the second degree or of a duplex cylinder of the sinus type 
in the B-mechanism also achieves the objective, except in certain 
special cases, provided grinding is affected with the tangential mech- 
anism. If no other than the simplest machine without tangential mech- 
anism is available, an A-triplex cylinder is adequate in every case. 
The simultaneous centric and eccentric osculation of the fourth 
and/or second order requires a special cylinder, when the machine curve 
is prescribed, and can be achieved in all cases either by means of 
duplex cylinders in both mechanisms or by means of a B-triplex cylinder 
if the latter has been ground by employing a duplex cylinder of the 
tangential type. If no other than the simplest machine is available, 
an A-quadruplex cylinder is adequate in each case. 

Two simultaneous eccentric osculations of the first order can be 
obtained in certain cases by means of a duplex cylinder in the B-mech- 
anism. In all other cases, an A-quadruplex eccentric or an A-triplex 
machine with two evolvent eccentrics of the first order is adequate. 
By employing the corresponding triplex machine, the order number 
of an eccentric can be reduced by one unity. 

Any desired number m of machine constants are available in linear equa- 
tions for different problematics if m - 1 and/or m - 2 special cylinders are 
ground, depending on whether the duplex or the triplex machine is utilized. 
The A-multiplex cylinder of the order m and/or m - 1 thus obtained can be 



I3h 



employed in the B-mechanism if the machine curve does not intersect its 
circle of curvature of the vertex which is not the case with the respec- 
tive evolvent eccentric of the order m - 1 and/or m - 2. In contrast to 
this, the evolvent eccentric of the order m - 2 can be replaced in the 
A-triplex machine by two such eccentrics if the sum of the order numbers 
of the latter remains m - 2. 



155 



IV. Calculation with Non-Spherical Surfaces 
In the application of the method for determination of the machine 
constants demonstrated in the preceding section, difficulties may result 
because the systems of equations expounded may not furnish sufficiently 
exact values if these equations are utilized in precisely the form given. 
Before we enter on a discussion of the methods of mathematical calculation 
of optical systems with surfaces of the second degree or duplex surfaces, 
we shall therefore discuss the respective modifications of the equations 
for some of the simpler cases and indicate by means of a few examples that 
calculation on the basis of the methods indicated above are not really too 
complicated. 

In the simplest case where it is merely intended to eliminate a Seidel 
image error, surfaces of the second degree offer such appreciable simplifi- 
cations, if the system is to be; computed mathematically, that they should 
be selected whenever possible. The Seidel formulas, in the form which I 
have given them, directly furnish the flattening value # of the surface, and 
for a surface of revolution of the second degree whose meridian curve has 
the radius of curvature of vertex p and the numerical eccentricity e, there 
is valid the relation — 

At a positive value of the product p §, the meridian curve therefore 
represents an ellipse whose shorter axis coincides with the axis of revolu- 
tion. The general equation . ■ — . 

furnishes, if the X-axis coincides with the axis of revolution and y there- 
fore represents the independent variable, for x = y = 0: 



Ipd^X'^dy^ ■pd^x+Sqid'xY'^'O, 



136 



from which results, because 



in the vertex of a curve symmetrical around the X-axis, the expression 

which represents, at a negative value of the product p$, the relation 
given above of the flattening value to the eccentricity but represents, 
in the contrary case, by means of the relation q = - -rj the relation of 
the semi-axis B perpendicular on the axis of revolution to the same axis 
A coinciding with the latter in which B >' A. From any arbitrary value of 
$, we consequently obtain in this manner the absolute constants of the 
corresponding conical sections when the radius of curvature of the vertex 
is prescribed. 

In most cases, the present purpose will be achieved by substituting 
the corresponding surface of the second degree for a convex spherical sur- 
face and the former can then be ground directly as such a surface by the 
method described above. However, in the cases where a concave non-spheri- 
cal surface is to be preferred for special reasons or where, e.g., a bi- 
concave lens is required, a duplex surface must be utilized. From the 
mechanical viewpoint, the greatest advantage then lies in selecting a 
standard surface as simple as possible in accordance with the proposal 
demonstrated above. However, if the system is to be calculated, it may 
be, especially in the cases where the diaphram cannot be incorporated in 
a medium bonded by the non-spherical surface, of great advantage to have 
the surface osculate a surface of the second degree as accurately as pos- 
sible. Since we can obtain the intersection of a given ray with such a 
surface through solving an equation of the second degree whereas the for- 
mer can be determined only by more complicated calculation for duplex 



137 



surfaces, we can initially calculate with the surface of the second de- 
gree and subsequently utilize the values obtained either directly or as 
first approximate values depending on the degree of accuracy required. 
Under certain circumstances, it may also be advantageous to utilize a 
duplex surface which has optimum possible osculation with a prescribed con- 
vex surface of the second degree. 

If we now want to calculate a duplex surface in consequence of these 
reflections and this surface has a centric osculation of the eighth order 
with a surface of the second degree , let us first correspondingly differ- 
entiate the equation of the second degree of the curve just utilized. At 
n y 1, we have for the vertex: 

in which ^ - 

! dHx*)-^15d*xd*x dHx^)^28d<>xd^x + i5[d^xy 

so that we obtain, after exchange of the variables: 

By substitution of these values of the formulas deducted above (p. 72) 

we obtain the differential quotients in the polar coordinate system r" = 

0: .-..,^^- ---- - - - - ----- - — -,-....- -..- 

\r^ = Spe' 7-vi = — 15pe«(3e* + 1) r^i" = 63pe»(25c* + 25e^ + 1) 

and for a parallel curve (p. 7^) if we set 



X=:.itnP' 



- V p p* 

For the magnitudes (p. 77) to be applied to the calculation of the 
machine constants, we therefore obtain 



138 



S = 5e* + 5e» — 8Xe«(3e8 + 1) + 24X«e* 
I C, = 3.e* — 4Xe«{e» + l) + 4X«c* 
! C< = 2e* + 3e* — 4Xe*(3e« + l) + 12X*e*. 

Since p 0, the parallel curve lies on the convex side of the conical 

section in the case 1 > X > whereas, at A ^ 1, the latter is located 

on the concave side beyond the center of curvature of the vertex and, at 

A\0, between the latter point and the vertex. The case /\ = 1 corresponds 

to an infinitely distant parallel curve. Since the base curve represents 

a conchoid with this curve as basis, we obtain the values for the latter 

if A = 1 is inserted in the values of |53G C C, whereas R = p for the 

I St o - 

base curve and, in concordance herewith, A = must be inserted in the value 

of /?r - 

For the simplest machine without crank mechanism and without carriage, 

we have c = c . From the expression above, we obtain the value C = 3e 
s ^ ' s 

and/or C = - e for A = and/or /\ = 1 . If we set c = 0, this produces 
two real values of A of which one consequently corresponds to a parallel 
curve located on the convex side. Since the roots of the quadratic equa- 
tion in A both have positive or opposed signs, depending on whether e is 
or is not positive, the other parallel curve determined by the condition 
C„ = lies on the concave side and, in the first case, beyond the center 
of curvature of the vertex and, in the second case, between this point and 

the surface. If we write the value of C as 

s 

1 C, = - e* - (e« — 1)* + (e" + 1 - 2Xe'=)«. 

it is immediately apparent that C possesses a minimum with negative value 

s 

but no maximum which is true also of the cases concerning ellipsoids of 
rotation with the shorter axis as axis of revolution in which e <C 0. 
Furthermore, since {5J = ctg «) for this type of machine and c is always 



139 



made positive, it follows from the value of 133 that u) has the same or 
the opposed sign of e , depending on whether A ^0,5 whereas u) = when 

A has precisely this value. 

2 
For the case e ^ 0, there initially follows from this in regard to 



convex surfaces that the latter can be ground in general either with a cyl- 
inder or with a concave semisphere since C is positive both at a very 

s 

small and at a very large value of A . The former method will be prefer- 
able for mechanical reasons as soon as the cylinder would not have to have 



too small a diameter which is the case when the ratio of numerical eccen- 

2 
tricity to parameter is large. The larger e is, the more closely will 

the two values of A which correspond to C =0, approach the values 
and 1. If therefore the utilization of the cylinder at large numerical 
eccentricity is impossible for mechanical reasons, we can then always grind 
with the concave semisphere of relatively large radius where, however, a 
limit for the opening of the ground surface is established by the fact 
that the latter may not have a radius of curvature larger than that of the 
cup. The concave surfaces, on the other hand, can always be ground by 
means of a small sphere. Although this produces a value of c which is as 
much larger as numerical eccentricity is greater but, since u) has a nega- 
tive value which permits a larger angle a from the mechanical viewpoint, 

this counteracts a narrowing of the opening. 

2 
At e *C 0, the convex surface can in general be ground with a plane 

where c = "V - e^ and tg lo = - c. However, at a large value of je | , mechan- 
iC-al difficulties: are created by the magnitude of (ti)( which can be elimi- 
nated, however, through grinding with a concave spherical surface. Be- 
cause the value tg w = - 1 corresponds to the case A =°^ > it is possible 
to obtain, in the cases occurring in practice, a mechanically applicable 



140 



value of u) through an adequately large value of A where, hovsever, the 

large value of c narrows the opening through the dimension of the angle 

a + w. Since a parallel surface is located between the surface and the 

center of curvature of the vertex of the latter for which C =0 for this 

s 

surface, a numerically adequately large negative value of X produces a 
real value of c so that concave surfaces can consequently be ground by 
the application of an arbitrarily selected value of this machine constant. 
It should merely be remembered here that too large a value of the positive 
angle w is obtained at too small a value of c and that, on the other hand, 
when c is too large, the opening is narrowed both by the large value of a 
+ oj as well as through the condition that the non-spherical surface may not 
have a radius of curvature smaller than that of the abrading surface. 

With this review, I have merely intended to show that the calculations 
in the determination of the machine constants concerning centric osculation 
of the eighth order are relatively simple. When other surfaces than those 
of the second degree are concerned, the latter can then obviously not be 
represented so simply and clearly that the entire category of the surfaces 
can be treated at once. However, if a given surface is concerned, the dis- 
cussion above can be carried out in the same manner and the calculations 
reaain the same as soon as the differential quotients have been derived. 

That such a derivation in other cases also does not necessarily re- 
quire endless calculations will now be shown by the example of the Carte - 
sian oval . If ss' designate the distances, considered as positive in the 
direction of motion of the light, of the points of focus and/or image from 
the vertex of the curve and nn' the indices of refraction, then the opti- 
cal length from the point of focus to the point of image is equal to Ans 
where the designation of the optical invariant 



l4l 



If q q' represent the distances of the same point, considered as positive in 

the same way, from an arbitrary point of the curve, then the condition that 

the optical length between the two points shall be the same on a ray of 

like incidence as on the axis, 

Ang = Ans, 

and we obtain, through the relation 

and through the similar relation valid for q', the equation of the curve 
in Cartesian coordinates. In the successive differentiations, y shall be 
treated as an independent variable and x"x ... and/or q"q shall desig- 
nate the respective derivations of x and/or q. Differentiation two times 
results in I 

for X = y = where q = s, i.e., - __ 

i 

1 «(2" + »")'= 1 . 



This equation is multiplied with — and subtracted from the similar 
equation valid for the image medium where we obtain 

f ^, . ' 

by considering that /^nq" = 0. Since x" represents the reciprocal value 

of the radius of curvature of the vertex r, the above equation furnishes 

the relation of this radius to the distances s s' and we obtain 



» 1 1 

In the subsequent differentiations, there result for V^ 1 the equa- 
tions 

where | j^iv=.3a;"» X''^ =^ ISx'"" x" 



lk2 



and where the binomial quotients of the even-number members are applied 

unchanged but those of the odd-number number members after division by 2, 

Hi) 
and the magnitudes Q are formed in the same manner. These equations 

are treated in the same manner as above, i.e., multiplied initially with 

— and subtracted from the similar equations valid for the image medium. 

. II iJ 
The equation so obtained produces, together with the equation A q 

= 0, the differential quotients x and q which also furnishes us 
the value for X "*" and Q "^ '^ . After these, we immediately ob- 
tain the differential quotient of the desired order of the equation of 
curve X = f(y) where we need only know the respective binomial quotients. 

A useful subsequent and small transformation of the values is best 
illustrated by the example >*= 2. The equation 



initially furnishes 

s s 

but receives, by the substitution of A— = x" and by taking into account 

3 

that n q" is an optic invariant, the form 

I An ns 

from which is obtained j 

In the same way, there further results 

An ns 

and 

. s 

■ - An ns 

After this, the calculations in the determination of the machine con - 
stants for an eccentric osculation shall be discussed by means of an 



1^3 



example . If a magnitude important for the calculation is obtained as the 
difference of two approximately equal magnitudes, there is then necessary 
in general a transformation in order to achieve sufficient accuracy, and 
in certain cases it is preferable to employ series development for this. 
Since it would be much too complicated to enter here on a large number of 
the necessary calculation methods:, we shall select an example where such 
transformations occur at different points. If we are concerned with im- 
proving a duplex surface without utilization of a crank mechanism by the 
introduction of an evolvent eccentric of the first order into the A-mech- 
anism, c and cu can be selected freely and C and k^ can then be determined 
through the condition of an eccentric osculation of the first order. The 
best curve is determined in this manner by variation of c and «). If c 
receives here a low value, we must transform, especially if ^ is low at 
the same time, both the equations representing the general sinus mechanism 
as well as those representing the general tangential mechanism. In the 

former mechanism, we obtain a from the formula 

i 8in{ct + (o) — sinw ,, .. 

1 ^ = c(l — cosS). 

cos 0) 

Let the magnitude on the right side be designated as h. If we substitute 

2 B & 

1 - cos P by 2 sin p or by sinBtg 2» ^ can then be calculated with any 

desired accuracy. However, in order to obtain an accurate value for cr 

at small h and when us) is not small at the same time, it is best to use the 

value resulting from the equation above as a first approximate value a 

in the equation ^ 

I sina, =7j + 2tgwsin''^ 

highly suitable for iteration, and thus obtain a better value of «_ which 
can be made as accurate as desired by repetition of the procedure. 

The smallest values of h do not occur in the calculation of the 



Ikk 



machine constants but in the trigonometric pursuit of a ray encountering 
the non-spherical surface in the proximity of the vertex. In spite of 
this, they shall be treated in this connection. It vsill be apparent that 
the method above is successful but, with a very small h, a sufficiently 
accurate value will be obtained more quickly with a series. The original 

equation can be written as a quadratic equation in sin or = x in the form 

i , x-laio cot CO, ,., ■ 

1 a; = 7t + ~-^— + — ~ {X — h)* 

The first three differentiations furnish 

dx-^dh 
d'x — tgiodx'' 
d^x= Stg<adxd*x, 

and, after the fourth differentiation, we obtain, in consideration of the 

fact that 

. , , cotw 
tg 0) + cot W ' 



COS* 0) 



d^x = 4 tg 0) dxd^x ■\ s — (d*x)- 

" cos* to ^ ' 

I d^x •='5tg (A dxd^x-\ ^-r~d^xd^x 

cos'' w 

i 

j d^x='6tg(adxd^x + ^^^ll5d*xd^x+10{d»x)^] 
i cos* CO ^ ^ ' ■' 

etc. , where we need only apply the binomial coefficients in the manner al- 
ready indicated. By taking into consideration the numbers up to and in- 
cluding the sixth order, we find the series 

In the tangent mechanism, the equation 

1 tg(a + to) — tg(o = 7i, 

where the tangent of the sum of the angles is expressed by the tangent of 

the two angles, is brought into the form 

, „ h cus* to 

i tga = -— 



1 + h sin to cos to 



which permits calculation of o? vsith any desired accuracy. 
The equations 

G (l—ooH a) + G 7c, {a — Bin a) ^^i 



C sin « + C/c, (1 — cos a) 



li' 



serving for the calculation of the machine constants Ck^ produce, for the 
determinant D which constitutes after solution of the latter the numerator 
of the values obtained, the expression 

1 2? = 2 (1 — cos a) — a sin a , 

which cannot be calculated sufficiently accurately in the usual manner 
when the angle 0? is small. However, by means of the two series 



there results 



asina=2~~i~+6~-. 



Df=2 4 — 4-6 



4! 6! ' "8! 

which makes it possible to achieve any desired accuracy. 



After this brief explanation of the calculations required for the de- 
termination of the machine constants, we shall now turn to the actual 
methods of theoretical calculation . 

In the mathematical examination of the design calculations for an 
optical instrument, that is, in re-calculation of the latter, the emphasis 
lies in general on the trigonometric pursuit of individual rays. This is 
done in connection with a still prevailing concept of the predominant im- 
portance of the cross-section of a bundle of rays for the optical projection. 



146 



However, since the latter is effected through the fusion of rays which af- 
fects in turn in general only the closely adjacent rays, the cross-sec- 
tions of the caustic surfaces in many cases have so large an importance 
for the sharpness of the image that the cross-sections of the bundles of 
rays become of secondary importance. The simplest way of convincing our- 
selves of this is to project the image of the glowing filament of an elec- 
tric bulb on a screen by means of a simple bi-convex lens with a large 
aperture. If we then adjust to the smallest circle of diffusion, then 
the image does not give any idea of the object whereas the latter can be 
clearly seen as soon as the distance is adequately enlarged in order to 
bring the cusp of the evolute on the screen. The large circle of diffu- 
sion so created mainly has the effect of a veil. However, if the lens is 
tilted so that the caustic surface receives a less favorable form, there 
occurs an appreciable deterioration of the image, although the magnitude 
of the diffusion figure does not appreciably increase with the most favor- 
able adjustment. These simple experiments teach us without any possibil- 
ity of error that, for non-homocentric bundles of rays, the cross-section 
of the caustic surface plays the essential and that of the bundle of rays 
a minor role. Since the greater part of the optical images further should 
fall in the last analysis on the retina of the eye or on the photographic 
plate and since in either one or the other case primarily differences of 
intensity are decisive, this is a further circumstance contributing to 
make the veil created by the circle of diffusion relatively without effect. 

However, that this conclusion is not immediately applicable to all 
possible optical images can be seen simply by the fact that a whole cate- 
gory of such images is not offered either to the eye or to the photographic 
plate. This is especially the case with the images of the pupils and/or 



lk7 



the apertures of optical instruments, regardless of whether we are con- 
cerned with projecting an opening entirely within or entirely without 
another. For example, this last problem is formulated by the conditions 
of the non-reflecting ophthalmoscopy. In these cases, it is evident that 
the smallest circle of diffusion maintains its former place of predomi- 
nance. 

However, disregarding these and similar cases, since the extent and 
form of the caustic surface plays a decisive role, it is obvious that, in 
the general case, re-calculation is not primarily intended for the trigo- 
nometric pursuit of the largest possible number of rays but is capable of 
giving the most reliable icnowledge of the projection through the determina- 
tion of the properties of the caustic surfaces in the close proximity of 
a lesser number of t rigonometrically pursued rays. This is equivalent to 
saying that the [Abbe] Laws of Imagery of Higher Order must be employed. 
This is, of course, done in regard to the laws of the first order in 
general where we are concerned with the projection of the image of an ec- 
centrically located focal point through an axial -symmetric system by cal- 
culating the tangential and sagittal image point on the main ray passing 
through the center of the diaphram. However, in order to obtain more de- 
tailed knowledge of the caustic surface, these image points must be cal- 
culated also for other rays starting from the same object point or else 
we must, by applying the laws of the second order, determine the asymmet- 
rical values of the bundle of rays parallel to the main ray. In any case, 
it is preferable to calculate the transverse asymmetrical value which, in 
systems of revolution, does not require the knowledge of the differential 
quotients of the third order of the equation of the meridian curve of the 
non-spherical surface. It will therefore be evident that the 



lifS 



trigonometric pursuit of rays which do not intersect the axis, so-called 
"warped" rays, is practically of no value without the time-consuming ef- 
fort of the calculation of the image points on the latter and has a higher 
value with this calculation than the calculation of the transverse asym- 
metry only then when several warped rays are made the basis. The advan- 
tage which is afforded through this extremely difficult effort of calcula- 
tion would, however, be able to make itself felt only with the very 
largest apertures from which follows that the trigonometric pursuit of 
warped rays should be utilized only in the most infrequent cases. We shall 
therefore indicate the pertinent point, only for the sake of completeness, 
the method by which the intersection of a given warped ray with a non- 
spherical surface is obtained. 

What has been said here on the application of the laws of higher order 
is valid not only for the caustic surfaces but — all things being equal — 
also for the other magnitudes determining the image. For example, if the 
abberration of a given ray is corrected for one axial point and the sinus 
condition is thus satisfied, we then obtain, by the investigation of the 
sinus relation parallel to other rays, adequate understanding of the re- 
spective relations but must, parallel to these rays, calculate also the 
tangential image point in addition to the respective coefficient of en- 
largement as will be explained further by means of the example below. The 
same is true also for distortion. As far as the image surfaces are concern- 
ed, we obtain, by application of the laws of second order, the tangients 
corresponding to the respective image points but can obviously replace this 
calculation by the determination of a larger number of image points. As 
disclosed by this review, an exact representation of the methods of re- 
calculation must also take into consideration the laws of the second order. 



149 



In the derivation of the formula, I have retained as far as possible 
the designations of the distances and angles utilized previously for 
spherical surfaces and considered the latter positive in accordance with 
the usual or customary method. The axis of revolution therefore repre- 
sents the X-axis of the coordinate system whose starting point coincides 
vsith the vertex of the surface and the distances on the latter are calcu- 
lated as positive with a refraction in the direction of the motion of 
light. The incident and/or refracted ray intersects the axis in a point 
whose distance from the vertex of the surface is s and/or s' and forms 
the angle u and/or u' with the axis. The coordinates of the intersection 
of the two rays with the meridian curve of the surface are xy and the sign 
of the angles is determined through the relations tg u = — ^ and juj <" rr 

S — X c. 

as well as the similar ones valid for the image medium. This consequently 
excludes from consideration the case where the projection of the motion of 
light taking place on a warped ray on the axis of the former would be op- 
posite to that of the latter. In the point of the curve xy is drawn the 
normal which forms the angle cp with the axis and has the length N between 
the point of the curve and the intersection with the axis whereas M repre- 
sents the distance of the intersection from the vertex. The sign of this 
last magnitude is therefore given and the sign of N shall be the same. 
The case M = is consequently excluded from consideration. Through the 

two equations :• - __ _.— . 

y^Nmi's yi# — a; = iVeos«) 

cp is definitely determined so that, e.g., at M (M - X) -C 0, a value of 
|'fl>9 is obtained. The sign of the angle of incidence and/or refraction 
in i or i' is determined by the conditions 

sin i _ s — M ... 3t 



150 



as well as through the similar ones valid for the image medium. Due to 

these observations, the equations 9=u+i=u' +i' always produce a 

value of j.»j<s. The distance of the intersection of the incident and/or 

refracted ray with the axis from the curve point xy is designated by q 

and/ or q'. Due to the conditions noted in regard to the angles u u' , the 

relations i 

i g sin u = y q cos u=s — x 

as well as the similar ones valid for the image medium signify that q q" 
have respectively the same sign as s s' and are therefore calculated as 
positive in the direction of the motion of light. The same is true of the 
distances pp' of the tangential focal points corresponding to an axial 
point located in any desired medium from the point xy. The tangential 
and/or sagittal coefficients of an enlargement in the projection of this 
axial point in the present object and/or picture medium are XiXuXlXit* ^^ 
the same manner, the tangential and/or sagittal focal distances U c t' c' 
and coefficients of enlargement K,K„K'K,', . correspond to an extra-axial 
point located in any desired medium. The indices of refraction are desig- 
nated by n n' , The sign of the latter is positive or negative depending on 
whether the light in the respective medium moves or moves not in the di- 
rection of the positive X-axis. In a reflection, we therefore must set 
n ' = -n but the distances on the axis are calculated as positive for both 
media in one and the same direction.. In certain cases, it is preferable 
to make both indices of refraction negative. For example, if we are con- 
cerned with a lens reflecting from the back, the positive direction of the 
X-axis is permitted to coincide with the direction of the motion of light 
in the object medium, in order to avoid a change of sign during the calcu- 
lation, and must make then both indices of refraction negative for the 



151 



refraction arriving after reflection has taken place. However, for the 
sake of simplicity, it is preferable in the present representation to era- 
ploy positive indices of refraction at single refraction which is equiva- 
lent to the fact that the distances in the direction of the motion of 
light are calculated as positive. Furthermore, p is to design the radius 
of curvature in the vertex and p,p„ the tangential and/or sagittal radius 
of curvature in point xy where consequently p„ = N. The sign is deter- 
mined by the fact that a radius of curvature is defined as the distance of 
the center of curvature from the point at the surface where the positive 
direction on the normal is determined by the sign of M. Finally, D is to 
designate the power of refraction in the vertex and D,!),, the tangential 
and/or sagittal power of refraction in the point xy. 

Since the determination of the point of incidence xy and its respec- 
tive magnitudes MNcpp takes place by different methods depending on whether 
a surface of the second degree or a duplex surface is utilized, whereas 
the calculations, after these magnitudes are known, are carried out in 
both cases in one and the same manner, it seems therefore indicated to dis- 
cuss the determination of these magnitudes only later and to initially as- 
sume them here as known. Moreover, we know the indices of refraction and 
the magnitudes su characterizing the incident ray and, where the applica- 
tion of the laws of higher order is concerned, at least the distance of 
q - p and the coefficients of enlargement XiXm* These correspond to an 
axial point which is located in the respective medium on the respective 
ray but may otherwise have a different significance. If we are treating 
with an optical system which is intended to project an axial point as 
sharply as possible, they then correspond to this point which therefore 
constitutes a focal point. However, if the optical system is to be 



152 



^— fj— ^ and/or j— i = — If = — i . The 



employed for the projection of extra-axial points, these same magnitudes 
correspond to the diaphragm center whereas, in regard to the focal 
point, the distances q - T^ and q ~ C ^s well as the coefficients of en- 
largement K,K„ are known. If one or the other point is located in the 
respective first medium, we then have to set q - p = and/or ^= £ and 
X, = Xii = 1 and/or K, = K„ = 1. If here one of the points is infinitely 
distant, it is simplest to set 
coefficients of enlargement belonging to the second medium then no longer 
represent figures but have the dimension of a length. However, if an in- 
finitely large value of a focal distance in the respective first medium 
corresponds to a focal point or diaphram center located at a finite dis- 

y 

tance, then we always know the magnitude of the form •^'. Finally, at in- 
finitely large s, the respective focal distances are determined through 
values of the form p + x. 

Since cf and u are known, the equation An sin i = is sufficient 
in order to make known all angles. We then obtain s* from the equation 

sm w 

which furnishes, however, an inaccurate value if s' is very small in re- 
lation to M. In those cases, the equations 



sm u 



Aq sin M = l. . As = Aq cos u 



furnish a sufficiently exact value. Both the former are also used for the 
determination of q qS after which the last can be employed as control. 

If qq' are eliminated from the latter, it can be written in the form 

i \ . _ -^ Ef^ 'f sin At 
* ~~ sin u sin n' 

which is more convenient in those cases where we are concerned exclusively 
with the trigonometric pursuit of a ray. The sagittal coefficient of en- 
largement corresponding to the axial point and the sagittal power or 



153 



refraction produce the formulas 

j A • /^ r, \ ^ An cos i. 71 n' sin A i 
I A n x„ sm « = D,, = A - = — ^^— = _____-. , 

where, in the last expression very appropriate for trigonometric calcula- 
tion, I = n sin i has been set according to the procedure of Abbe. For 
the tangential projection corresponding to the axial point, we have 

^ — r~ ==— — == A cos t cos t' a12^^I/.j^q 
P Pi p 

and as control , 

If the axial point represents the diaphra^ center and if we desire 
-- with a narrow diaphragm — to employ only the laws of the first order 
to the projection of the extra-axial points, we then need not pay any at- 
tention in the individual surfaces to the latter but can calculate them 
in a manner to be indicated below by application of the complete system. 
However, if the laws of the second order are to be taken into account, 
then the values TC and the respective magnification coefficients must be 
known everywhere. We have, for T'K[, similar formulas as for p'xl» and 

further , - - ^^ - 

I Af = A, A^ = 

as well as for control 



A'JiJtUJ^O A,... 



I . ■= — P " ~{c— 5)sin?t'~^• 

When u = 0, the value of q' is derived from the equation 

I sin u 

but otherwise the formulas can be. utilized unchanged. With q) = and y 

=5it 0, M and H have infinitely large values where M - N = x. N sin cp has 

to be replaced by y so that we obtain D„ = and 

i A cos » cos i' = ^-^ = _ "i^l^ifL^ . 



15^ 



whereas the formulas remain valid unchanged otherwise. 

If the entire system has been recalculated in this manner, me ob- 
tain the tangential power of refraction,'©, of the entire system parallel 
to the given ray through the formula 

where the magnification coefficients on the left side belong to the first 
and/or last medium. In the same manner, the value of the sagittal power 
of refraction JS),, is produced and, if we have also determined the magni- 
fication coefficient corresponding to an extra-axial object point, the 
formula is valid also with the application of the latter which enables 
us to check the entire calculation. The deduction has been given else- 
where. 

If we now wish to project any desired object point located on the 
known ray, then the formulas 



\ t — p ^"^■'^' t — p 

and the similar ones for sagittal projection are valid. 

In the formulas deduced by me for the application of the laws of im - 

2) 

agery of the second order , the designations and signs have been selected 

everywhere so that they correspond to the definitions given above. The 
formulas can therefore be applied directly with the magnitudes obtained 
in recalculation and it remains only to briefly discuss the asymmetrical 
values 

i • o a p, do Pi, 



1. See, e.g., Handbuch der Physiologischen Optik by H. v. HELMHOLTZ , 3d 
ed. , Hamburg and Leipzig 1909. 

2. Die reele optische Abbildung. Ci. "Handlingar" , Vol, kl , No. 3, 1906, 



155 



occurring in them. Here da is the element of arc of the meridian curve of 
the non-spherical surface and, in accordance with the given definitions, 
there are valid without restriction the relations 

I ' sm <f costo 

With surfaces of revolution, Vif is determined through the differential quo- 
tients of the first and second order and we thus have p„ = N. If a paral- 
lel curve of the meridian curve of the non-spherical surface is drawn 
through the intersection of the normal with the axis, then the element of 
arc of this parallel curve is equal to 

(p, — i\r)(Z^==sincprfiI/"= ig'fdN , 



which results in 






The value of U is to be determined separately below for the surfaces of the 
second degree and for the duplex surfaces. Here we shall merely draw at- 
tention to the fact that the magnitudes occurring in the asymmetrical val- 
ues are algebraic, and not absolute, magnitudes so that the asymmetrical 
values with equal absolute ordinate magnitude change sign both with a 
change of sign of the ordinate as well as with inversion of the curve with 
the ordinate as axis. The latter is true also of the abberation values in- 
troduced by me into the Seidel formulas 

f do"^ {J, da^ [J,,' 

for which # = 3^ in systems of revolution. The value of # is obtained from 

the differential quotient of the fourth order of the meridian curve of the 

3 . rlV 
non-spherical surface where p s = - — . 

o 

Concerning the calculations so far not discussed for surfaces of the 
second degree , the values of the intrinsic coordinates Ml cp have been 



156 



determined already above (p. k7) from the equation 



tgf--'^- 



Q + qx 

I if =• p + e*a; iV* =• p» + e^yi 

aince, at e •<r - 1, only that part of the curve for which M does not 
pass through zero is taken into consideration and, at e ^1, only one 
branch of the hyperbola is appropriate; the relations noted there and im- 
mediately above agree completely and N receives the same sign as p when 
extracting the root. The relation valid for conical sections 

j . i.= PL. ■ 

I I', N^ 

directly produces \ 

i 

It has already been shown above that p § = -Je . 

In those cases where the axial point of intersection of the oblique 
rays is located in one of the two media separated by the surface of the 
second degree, we can select the point of incidence on the surface and then 
need apply only the formulas indicated. If this condition is not satisfied, 
the intersection of the conical section with a given ray passing in the 
plane of the latter is determined in the following manner. If the incident 
ray, as usual, is determined by the magnitudes su, we then have the quad- 
ratic equation 

2pa;+ qx^^{s — x)''ig^ti., 

whose roots are i ~.=,4l^— 

* B 



157 



and where ;. . „ ^ 

At B < 0, the roots have different signs and the conical section there- 
fore represents a hyperbola, the two branches of which are intersected 
by the incident ray. Since here \c] > \ aI and since x must have the same 
sign as p, the lower sign must be utilized for C at positive p and vice 
versa. If B = 0, then the roots at C >• are real with the same sign as 
A. If here the product pA is negative, we then have a hyperbola in which 
the other branch is intersected by the ray at two points. In the contra- 
ry case, we must give C the sign opposed to p, in order to obtain the nu- 
merically smaller root. Imaginary roots correspond to the case A = at 
B >!>■ 0. At B = 0, the one intersection of the ray with the conical section 
moves into infinity so that a hyperbola is given and the ray is parallel 
to the asymptote. For the other intersection, there results 

1 «« tg^M 

and the intersection belongs to the other branch unless pA > 0. Finally, 
the case C = is excluded through the conditions jij <^ JI, This pro- 
duces the following rules which make it possible to turn over calculation 
to a routine mathematician which is an important factor in practical cal- 
culations. In order for the ray to intersect the surface, we must have 



C^ > and either B < or else pA > at B ^ 0. The sign of C is to b 



e 



opposed to that of p. However, this last rule has an exception in such 
cases as may occur, e.g., in dark- field illumination where the numerically 
larger root must be selected among two roots with the same sign. 

In order to finally obtain the intersection of a sxirface of revolu - 
tion of the second degree with a ray not intersecting the axis , the equa- 
tion of the surface is formed in the three-axis coordinate system in which 



158 



the Z-axis perpendicularly intersects the already employed X Y-plane by 

2 2 2 

substituting y + z for y in the equation of the conical section. Re- 
gardless of which characteristics are utilized for the ray, it is always 
easy to form two equations y = x, z = Xp where the right side contains 
only X. If these equations are squared and suffloiated, we then obtain a 
quadratic equation in x which is to be treated according to a similar 
scheme as above. 



In calculation with duplex surfaces , the magnitudes xyMN cp p Pj are 
to be employed always only for the meridian curve of the surface itself 
whereas the machine curve is designated through the respective equation 
R = f O) as above. Since a passage of R' through infinity is excluded, 
one point of the meridian curve of the surface is definitely determined 
through the value of 3, This angle is to be calculated as positive in the 
same direction as cp and must also pass through zero simultaneously with 
this last angle. In grinding of concave surfaces in which the radii of 
curvature increase toward the periphery, there may be required, at a large 
opening, a value ofj ippnalso at a moderate value of 9 and can be realiz- 
ed mechanically also under certain conditions. Signs and magnitudes of R 
are indicated in grinding for a parallel curve and/or base curve through 
the relation R a p (1 * o) and/or R = p. In order to obtain an even ap- 
proximate idea of the form of the duplex surface, it is preferable to cal- 
culate a number of point coordinates in which we start from freely select- 
ed values of 3 and base ourselves preferably on equal intervals in order 
to facilitate interpolation which may possibly occur later. For the deter- 
mination of the point coordinates, it is only necessary to know the radius 
vector and the first derivation. The equations 



159 



\m-Cfm /'(a)a' = c(?'{p) 

produce the values of a and cv' corresponding to the respective value of B 
which are inserted in the equations 

In those cases where the axial point decisive for recalculation is locat- 
ed in one of the two media separated by the duplex surface, the ray can be 
determined by the selection of the value of 3. In order to obtain the tan- 
gential radius of curvature and the direct asymmetry of curvature, we then 
need the derivations of the second and/or third order and must employ for 

this 

|/'(a)a" + /"(a)a'«==c(p"(p) Rn ^ j^^q^^, i^^^^,, _^ (p"(a)a'2] 

I /'(a) «'" + 3 /" (a) a' a" + /'" (a) «'» = c f ((3) 

! R'" ^R,C [?' (a) «'" + 3 w" (a) a' a" + tp'" (a) a'»] 

It would obviously lead us too far to indicate here the formulas for the 
various machine types but it may be pertinent also to demonstrate by an 
example that the calculations are not too complicated. For this, we shall 
select the most important case in practical application of the general si- 
nus mechanism with standard eccentrics both in the A- and in the B-mechan- 
ism by taking into account the general principle that the formulas should 
make as little demand as possible on the routine calculation. Consequent- 
ly, we shall also indicate here and there control formulas for early dis- 
closure of any eventual errors of calculation. The machine constants C c c« 
and the magnitudes R B are therefore known. We first determine a by means 

of the equation , -- 

j sin (a + 0)) = sin w + 2 c cos w sin* -^ 

after which the equation i g 

; sin a = c sin p tg | + 2 tg w sin^ ? 



160 



is utilized for control and, at small a, for determination of an accurate 

srmore, if we set 
,, _ Ho Cc sin « sin p cos w 



value in the manner indicated above. Furthermore, if we set K = R - fi , 

o' 



K^2Ii,0sm'^ R' 



cos (a + w) 



and for control 



Subsequently, 



and as control 



J7=l + 



cos (0 



?„.. p. 



^HTI^h:^ . 7i' = i?„OCrsinatg?cot| 



r 
y^'MLz^ ]i"^B' cot ^ + R'V cot I 



2cos2 



a' = c sin ,3 (£7-1) ' «" == «' cot p + a'-" tg (a -(- to) 

i?" ■= 7?a C a" sin a w-^, C a'^ cos a. 



The magnitudes U V occurring here are the same which were examined in de- 
tail during the determination of the machine constants. A confusion of 
the former with the similarly designated asymmetrical value need not be 
anticipated because the respective magnitudes do not occur in one and the 
same stage of the calculation. The elimination of or"' produces, for any 
desired mechanisms. 



from which we can derive for the present case, in which the quotient of 

the derivation of the third order into that of the first order is equal 

to -1 for all functions, the two formulas 

Li?'" + B'= ^''"-^'!'^~ ^^ = 3 Foot I (J?"-a'i?' cot a) 
I sma i 

without difficulty. 

Further calculation will differ, depending on whether the machine 
curve represents a parallel curve or the base curve. In the first case, 
we have 



I6l 



tg(p— ?) = 






sin o 
i^ cos ?) =. 2 p sin* I + 2 p sin *^^ sin ^-? _ K cos (5. 



iar=i/ — iV^cos?)=-2i3 sin* ^ + 2 oo sin *!' 

2 "- 2 

and, among these formulas, the last offers not only an excellent control 



but produces the more accurate value at small x. By employing the designa- 
tions 

i 

P - 1 + sin' (s ~ ©) _ ;?!l£22!Il=i?) 



we further obtain 



o =- R^ -h 2 R>* ~ a R<> 



R 



\ (J, + 00 =» -^r- it ii^ 

I Pcos(i3~«,) Ocoa^(fZ.Y) 



For the curvature of the machine curve, there is valid in general 

1 



in which 



If the designation 



p, + po /j* 



i L^^R* + B'* 



dL_ R'jR-i'R" ) 



d p p, + p o 



is introduced, we then have, since 

fp, + po) cos (P — a) (Z '^ = iJ <i p Q = 



(Pf + po)" S|3 






for the asymmetrical value 



V = /f'L±_Po\8 Q cos ff i — B)' 

\ P. 7 ' ye '"•■ 



Differentiation initially produces 

! = JL^ ^G dL 

and we obtain, by employing this expression, the formula system 



162 



R 



9^ 



cos (p — tp) 

Q ■ GdL E'(R+R") 

B (R'" + R'} ■ _ R> (R + R") (RIt _ R Jitn 

V< jrr ' 



^ I dG 2RR' + SR'R"—RR"' 

^'"^T^Tpr : Zi — - — 



j:' 



/v« 



j,_(eipf)=,c,_3^,.,,_(pi£_»)V. + .,5,,, 



If the machine curve represents the base curve , the formulas are 
basically simple: 



'f '■■='? 



M -^ o 4- -^ 



iJ' 



a; = ilf — iV cos P == 2 p sin* | + R' sin p — Z cos[5 



iV = ^ + -»' cot (3 



sm p 



y = N sin p 



.0, =-i?" -1- i,' 



U'-' 



i?"' + 7v" 



In those cases also where the respective axial point is not located 
in one of the media separated by the non-spherical surface, we can, if the 
system contains only one non-spherical surface, determine a surface point 
by the selection of the angle B in order then to search for the ray which 
passes in the respective field through the axial point and on which the 
surface point determined by 3 is located. V'^hether this method is given 
preference or whether we search directly for the intersection of the sur- 
face with a given ray, will depend in most cases on how many surfaces exist 
between the axial point and the non-spherical surface. The latter method 
must be employed in all cases where the optical instrument contains more 
than one non-spherical surface. 

In order to find a ray which passes through a given point of the 



163 



non-spherical surface and through a given axial point in another medium , 
we attempt to determine, if no other ray passing through the respective 
axial point is already known, initially a rough approximation value by 
employing the equations valid on the axis where we project the two points 
in the media where the other point is located. For the sake of simplicity, 
it shall be assumed here that the light moves in the direction from the 
axial point toward the surface point and the respective media are consid- 
ered in calculation as the first and last medium of an optical system 
where the respective magnitudes are designated by us... u's'.... The ax- 
ial point is therefore located at the distance s from the first surface, 
and the distance of the point conjugate on the axis in the last medium of 
the latter shall be s' and the magnification coefficient — . If the axial 
point represents a real diaphragm center, we have consequently s <^ 0. The 
vertex of the first and/or last surface represents the starting point of 
the coordinate system in the respective medium. The coordinates of the 
surface points shall be x'y'. Through the laws valid on the axis, we de- 
termine the point x conjugate to the axial point x'O and the magnifica- 
tion coefficient corresponding to these points through which the ordinate 
y is determined. The equation 

produces a first approximate value u which determines a ray passing 
through the axial point. The latter is pursued through the optical system 

V ' 

by eventually calculating also the magnitudes q' - p' and — '. In this last 
medium, the ray does not pass through the surface point in the general 
case. However, we can obviously vary the value of u until this is the 
case, and it is merely a question of saving of time and labor whether we 
want or do not want to employ the Newton method in the manner to be described 



16^ 



below. If x'y, are the coordinates of the intersection of the refracted 
ray with the ordinate of the given surface point, we then have 

and obtain by differentiation 

cos* M 

If we further plot a circle through the intersection of the ray with 
the axis and the center of the circle coincides with the tangential focal 
point corresponding to the given axial point, it will then become immediate- 
ly evident that { 

1 sin u' ds' == {p' — q') du' 

I 

and we obtain, by application of the fundamental equation: 

d u._ n' x'l cos^ u' 

where subsequently, if the angles u are measured in degrees and if E rep- 
resents the length of the radius of the circle measured in degrees, the 
equation ' , 

produces a better approximation value which can be made as accurate as 
desired by repetition of the procedure. 

In the practical application of this method, it should be considered 
that the calculation of the tangential focal points and of the correspond- 
ing magnification coefficients sometimes requires a greater effort than 
the repetition of the calculation with another ray. We can therefore set, 
if the given surface point is not located in the proximity of the caustic 
curve corresponding to the given axial point, p' - q' = and employ for 
Xj the value x' valid on the axis which makes calculation extremely simple. 
At a large value of u^, it is preferable, however, to employ the approxima- 
tion value 

!p'-2' = 2(5'-s'„) 



165 



in the first calculation and this value results, for u' = 0, through dif- 



ferentiation two times of the above equation for ^— , . 



ds' 
du' 



If w/e are concerned with directly finding t he intersection of the du - 
plex surface with a given ray , then this ray shall be detennined by the 
values s u where the vertex of the surface represents the starting point of 
the coordinate system. Such a problem presupposes that a number of surface 
points are already known through previous calculation of the coordinates xy 
in accordance with certain values of 8. The two points through which the 
ray passes are determined by means of a drawing or by employing the equa- 
tion of the ray 



y = {s — x) tgu 

by comparing the resulting ordinate values with the given ordinates after 
insertion of the given abscissa values. In the first calculation, it is 
best to employ quadratic interpolation and to select first the three point 
most closely located to the ray for this purpose. if these points are 
characterized by the magnitudes P x y (n = 1, 2, 3) , we form in the cus- 
tomary manner the interpolation equation 

in which c is obtained by substituting Xp8p for xP. After the correspond- 
ing equation for y has been determined in the same manner, we obtain, by 
substitution of the values of x and y in the above equation of the ray, a 
quadratic equation in P which, when solved, produces the first approxima- 
tion value. If the initially known surface points were not separated top 
far, then this value will be so accurate that it will be no longer neces- 
sary to employ the quadratic interpolation. In the continuation, we can 
now either carry out linear interpolation in this same manner or else em- 
ploy the Newton method. In the latter case, it becomes necessary to 



166 



calculate, in addition to the magnitudes MN cp necessary for the determina- 
tion of the coordinates xy, also the radius of curvature p, . The respec- 
tive differential quotients are obtained, if the machine curve represents 
a parallel curve of the meridian curve of the non-spherical surface, from 

the easily derived relations 

Itdf, =. (p, + p o) cos (p — ^) (Z ,f, 

i 

\dx=^ p, sin ?(Z ?) . d7j = p, cos 9 d f. 

vshereas dB = dcp if the machine curve represents the base curve. If vue 
designate the magnitudes determining the first approximation value by B 
X y , we then obtain the values of x and/or y occurring in the differences 
X - X and/or y - y by elimination from the equations of the ray and of 
the tangent. This is equivalent to saying that the value 

and the value of y formed in the same manner are substituted in the equa- 
tion of the ray. The closer the approximation value utilized, the more 
closely will this operation coincide with the linear interpolation. 

The intersection of a duplex surface with a ray not intersecting the 
axis is determined in a similar manner. If the equations of the ray have 
been brought into the form y = X z = Xp where the right side contains only 
X, this produces, through quadrature and summation of these equations, the 
equation of a hyperboloid of revolution of one sheet, whose secant with 
the non-spherical surface represents a circle which must contain the desir- 
ed intersection. It follows from this that the X-coordinate of this inter- 
section is the same as the abscissa of the intersection of the meridian 
curve contained in the XY plane with the hyperbola 

which therefore in this calculation takes the place of the ray in the former. 
The next consequence of this is that the method of quadratic interpolation 



167 



leads to an equation of the fourth degree in P. Whether we want to solve 
the latter or will prefer repeated linear interpolation, is best decided 
(when the machine curve represents a parallel curve) on the basis of the 
skill of the routine mathematician. However, if the machine curve repre- 
sents the base curve, it is preferable in all cases to repeat linear inter- 
polation and/or the Newton method as in the above procedure. The actual 
difference then consists in the fact that a quadratic equation is to be 
solved for each new approximation value. However, this can be avoided by 
neglecting the member of the second order in - B but it is necessary 
to examine in each case whether this method is more rapid. 



168 



V. Examples of the Application of Duplex Surfaces 
It is now merely necessary to show by a few examples that mechanical- 
ly applicable values for the machine constants can be obtained for cases 
occurring in practice. For the sake of easier comprehension, we shall con- 
sider here only the simplest forms of the duplex curve which will also af- 
ford us an evaluation of the capacity of the simplest machine. Non-spher- 
ical surfaces have so far been used mainly in two different fields, i.e., 
first for the purpose of better fusion of rays in one axial point as in 
non-planate ophthalmoscope lenses and second for improvement of the projec- 
tion of extra-axial points as in the non-spherical cataract glasses. To 
the extent in which technology is becoming more familiar with the utiliza- 
tion of such surfaces, it is probable that both purposes can eventually be 
reached in one and the same optical instrument, perhaps by the application 
of two non-spherical surfaces where the effect can be distributed both to 
the axial as well as to the extra-axial projection on both surfaces. For 
the time being, it would seem more appropriate to treat each of the two 
purposes separately and we shall therefore discuss examples for the two 
main types which are characterized by these different purposes separately. 
Duplex Surfaces Eliminating Aberration . 

Where improvement of the fusion of rays in one axial point is concern- 
ed, there is no point, as has already been explained above, to try for a 
homocentric bundle of rays in the mathematical sense in practical ececution 
and it is preferable to utilize a surface through which such a fusion of 
rays is obtained that the latter is practically not differentiated from a 
homocentric bimdle in regard to the purpose of the respective instrument. 
A non-spherical surface which satisfies this condition shall be here desig- 
nated generally as aberration- eliminating . Since the concept of 



169 



non-planateness also implies the satisfaction of the sinus condition and 
since in general, merely through the change from a spherical to a non- 
spherical surface, the sinus condition in the elimination of aberration 
cannot be satisfied, there then do not exist, if we cling to this concept 
of non-planateness, any non-planatizing surfaces. The expression ''aber- 
ration-eliminating" selected for this reason comprises as a special case 
freedom from aberration, i.e., the incident bundle of rays is free of aber- 
ration. Aberration-free surfaces are therefore the surfaces of revolution 
whose meridian curves represent Cartesian ovals and we know that these con- 
vert for certain cases into a curve of the second order. In agreement with 
the discussion above, such duplex surfaces shall be designated as aberra- 
tion-free duplex surfaces which can be utilized with an accuracy sufficient 
in practice instead of the exact aberration-free surfaces. There is no 
doubt that the hyperboloid, once it becomes more easily accessible, will 
play the most important role among the aberration-free surfaces because 
the utilization of two plane-hyperbolic lenses with water cooling between 
them as condenser would constitute an extraordinary increase of the effi- 
ciency of the projection apparatus. For this reason and since it is not 
excluded that the respective aberration- free duplex surface, e.g., in in- 
stitutes where a duplex machine exists, may be preferred to the hyperbo- 
loid for mechanical or economical reasons, this surface is selected here 
as the first example. This has the further advantage that, because of the 
simple equation of the hyperbola, a detailed comparison of the two surfaces 
will require much less effort. 

We shall first search for a duplex surface which can replace the so - 
called non-planate hyperboloid . 

By a correction of aberration, there is usually understood, in the 



170 



literature of geometrical optics, that state where a peripheral ray pass- 
es through the axial image point. The aberration on the axis is here 
generally not corrected in that the respective aberration value is dif- 
ferent from zero and the intermediate rays also intersect the axis in 
other points. We then speak of zones of aberration. As will be easily 
understandable, the evolute of the meridian curve of the Fresnel zone of 
the refracted bundle of rays has in these cases a cusp corresponding to 
a ray passing between the axis and the given ray and the point in which 
the given ray touches the evolute is located, in relation to the cusp, 
on the opposite side both of the axis as well as of the focal plane. 
This is evident, if the designations pMKcpp are applied to the meridian 

curve of the Fresnel zone, from the differential quotient 

l.aitifdMI =:{p, — N)df 



already deduced above. In order for a ray to pass through the focal point, 

there must be, as on the axis, M = p and this is possible only when a point 

dM 

-7— = 0, i.e., p, = N is located in between which corresponds to an inter- 
section of the evolute with the axis. The latter is possible in turn only 
when the evolute has a cusp between it and the axial focal point and, af- 
ter the evolute has intersected the axis, the latter must also intersect 
the focal plane in order to be touched by a ray passing through the axial 
cusp ifj[2|<^ for the latter. We know sufficiently from experience that 
such a correction is in practice entirely adequate if the opening of the 
optical system is not too large. For larger openings and when high demands 
are made on this system, the usual alternate consists in making two differ- 
ent rays intersect the axial focal point . We find in the same manner that 
in this case the evolute has a second cusp located on the same side of the 
axis and of the focal plane as the point of contact with the first ray and 



171 



then in turn intersects the axis and the focal plane so that the point of 
contact with the second ray is located on the same side of the axis and 
of the focal plane as the first cusp. It was stressed above that the cross- 
section of the caustic surface has an essential and that of the bundle of 
rays a secondary influence on the quality of the image. It follows from 
this that the existence of the cusps on the evolute of the meridian curve 
of the Fresnel zone of the bundle of rays represents precisely the nature 
of the correction of the aberration and that one cusp corresponds to the 
usual but two to greater demands. To this needs to be added only that it 
is not indifferent on what part of the evolute the cusp is located. If 
only one cusp exists, then the effect of the latter is obviously much 
less when it is located in the immediate proximity of the axial focal 
point or in the most peripheral part of the evolute as when it has a medium 
position and the same is true — all things being equal — also for the 
existence of two cusps. 

There are infinitely many surfaces which refract, at prescribed ver- 
tex and radius of curvature of the latter, a given ray issuing from a given 
object point so that it passes through the axial image point in the second 
medium. The optical length from the object point to the axial image point 
changes with the position of the intersection of the surface with the in- 
cident ray. Only for the aberration-free surface and for those surfaces 
which have a contact of the first order with the former at the intersection 
with the incident ray, is the optical length on the refracted ray equal to 
that on the axis. That this behavior must represent an advantage may be ex- 
pected a priori and is determined in the following manner. 

If we construct an evolvent intersecting a symmetrical evolute which 
has no other cusps than the axial, we then find that the evolvent is divided 



172 



in three parts through two cusps touching the evolute. From the inter- 
section with the axis, the evolvent proceeds initially with the concave 
side turned toward the cusp until it encounters the corresponding branch 
of the evolute where it turns back with the formation of a cusp and then 
intersects, in proceeding further, the other branch of the evolute with 
the convex side turned toward the cusp. The evolute which touches the 
evolvent in the cusp of the latter therefore has a cusp between the 
points of contact and it will be easily seen that this constitutes a gen- 
erally valid rule for the evolute of the meridian curve of the Fresnel 
zone under the conditions determined for optical instruments. The same is 
true also of the circumstance that two branches joining in a cusp — 
branches either of the evolute or of the evolvent • — turn the convex side 
toward each other. If now, on a ray with finite inclination and passing 
through the axial focal point, the optical length from the object point 
is the same as on the axis, then the meridian curve of the evolvent of 
the evolute of the Fresnel zone passing through the axial cusp of 
the evolute must intersect itself in the same cusp and have the respec- 
tive ray as its normal under these circumstances. However, this is only 
possible if the evolvent has two cusps on both sides which is equivalent 
to three points of contact with the evolute and therefore requires two 
pairs of symmetrical evolute cusps in addition to the axial cusp. Since 
further the point of contact of the given ray with the evolute must be 
located on the same side of the axis and of the plane perpendicular to 
the axis, and passing through the axial image point, as the evolute cusp 
corresponding to the lesser ray inclination, there consequently must also 
exist a ray with lesser inclination which passes through the axial focal 
point. It follows from this that an eccentric osculation of the first 



173 



order with an aberration" free surface on each side of the axis produces 
irwo evolute cusps and two rays passing through the axial focal point . 

In an osculation of the second order , the evolute must also return 
to the axial focal point and here touch the given ray which produces at 
this point in the corresponding evolvent a third cusp which conditions 
in turn a third double-sided evolute cusp . In general, with increasing 
ray inclination, the lateral aberration of a ray changes sign when the 
latter passes through the axial focal point. However, since this is not 
the case when the evolute passes simultaneously through the focal point, 
we can therefore, in an osculation of the second order, regard the given 
ray geometrically as two coinciding rays passing through the axial focal 
point. If a centric osculation of the fourth order is added to this, then 
the centric cusp of the evolute is touched in five points by the axis, or 
it is possible to create, through variation of the respective constants, 
on the evolute one more double-sided cusp in the proximity of the axial 
cusp. 

Based on these findings, it was evident that a relatively large open- 
ing could be made the basis of the calculations without any risk of fail- 
ure. I therefore selected a priori that point of the hyperbola as point 
of osculation for which the ordinate has the same value as the radius of 
curvature of the vertex. If this value is assumed as equal to 1 and if 
a refraction index of 1.53 is selected, then 



r a; = 0,30526 2/ " 1 y = 33°,i69 

! J/ =1,9253 iV= 1,8278 p,«= 6,1064 

for this point and 

I p=l '-r- = — e*-= — 2,3 4 00. 

for the vertex. 



174 



The first experiment with an osculation of the first order already 
led to a relatively satisfactory result. This first calculation was con- 
cerned with the simplest duplex machine with crossed cylinders and with 
an evolvent eccentric of the first order in the A-mechanism at u) = so 

that the equation of the machine curve could be written in the form 
I i2 — i2„== Co(l — cos a) + Cj (a — sin a) sin a -= c{^ — cos [5) 

The values c = 1, o = 0.25 were selected so that the radius of curvature 
of the abrading cylinder then constituted one- fourth of the radius of ver- 
tex of the surface. This produced 

I Co = 1,046 32 C, = — 2,252 24. 

A plane-convex lens provided with this duplex surface and turned with 
the plane surface to the light showed, for light with a parallel incidence, 
the following lateral aberration of the various rays: 



p 


n 


-10° 


— 0,000 057 


-20° 


- 0,005 428 


-30* 


- 0,007 502 


-40° 


+ 0,005 107 


-50° 


+ 0,016 07 


-60° 


- 5,076 03 



Here f\ is the ordinate of the intersection of the refracted ray with 
the focal plane and the sign of the ordinate is referred to a positive 
value of the ordinate of the intersection with the surface as will be seen 
from the sign of the angles 3. Therefore 

I P = 54°,397 a = 24°,o08 p,=,d,i700. 

for the point of osculation. 

In order for a centric osculation of the fourth order to exist, it 

2. 
would be necessary that 00=6*^^ j-t ig therefore shown by the value of 

c that the surface is located in the proximity of the axis between the 
o 

hyperboloid and the sphere osculating the latter in the vertex. In agree- 
ment with this, the aberration of the rays passing in the proximity of 



175 



the axis is also positive. The first change of sign of the latter corre- 
sponds to the ray intersecting the axial focal point which must be locat- 
ed between the axis and the ray refracted in the point of osculation. 
The relative high value of the aberration of the most peripheral ray re- 
sults from the difference of the radii of curvature. 

In order to judge the value of such a duplex surface, it is not suf- 
ficient, however, to know the difference from the hyperboloid but we must 
also take into account the difference from the sphere. In a spherical 
surface, the occurrence of total reflection conditions a maximum of the 
ordinate of the surface point at a value of O.6536 and the corresponding 
value of /| is -I.8128. The ordinate of the surface point corresponds in 
the duplex surface to a value of B <! 40 . 

The favorable result justified the expectation that the experiment 
with an eccentric osculation of the first order at tu = and by employing 
a standard eccentric in the A-mechanism would also be successful. In the 

equation of the machine curve 

I li— li, = c„(l — cos a) sin a - c(l — cos [i) 

there is continually written c for the product H C because the mechanical 
applicability of the corresponding A-eccentric results immediately from 
this value. Certain experiences in the preceding calculations made it 
probable that a larger radius of curvature of the abrading cylinder would 
be of advantage so that a value of o = 2 was selected. There resulted 

i Co = 0.989 IB • C"==l,ezi2 C»Co = 2,5098; 

I ! 

and, for the point of osculation, 

IP = 42°,683 a = 25°,3io' f.,== 7,107. 

Consequently, the hyperboloid lies here both in the proximity of the 
axis as well as in the proximity of the points of osculation between the 
duplex surface and the sphere osculating the two surfaces in the vertex. 



176 



Under the same supposition as above, the lateral aberration ^ was calcu- 
lated for the various refracted rays. These values are listed below to- 
gether with the ordinates y of the surface points in which the rays are 



refracted: 



?■ ■ 


y 


■1 


10° 


-0,178 21 


-0,000 01 


20° 


-0,377 40 


• -0,002 57 


30° 


— 0,015 57* 


-0,000 21 


40° 


-0,010 02 


+0,005 42 


45°. 


— l.OSOOG 


-0,013 08 



Here also the double change of sign manifests that a ray exists be- 
tween the axis and the ray refracted in the point of osculation which pass- 
es through the axial focal point and that, correspondingly, two pairs of 
symmetrical cusps are found on the evolute of the meridian curve of the 
Fresnel zone of the refracted bundle of rays. As will be seen from the 
values demonstrated, the correction is so satisfactory that the possibility 
of differentiating this duplex surface from the hyperboloid in practical 
application is very minor. 

Partly in order to show that it is possible to obtain even much bet- 
ter results and partly in order to demonstrate the manifold applicability 
of the duplex method, we have carried out further calculations of machine 
constants. 

Initially, there was obtained an eccentric osculation of the second 

order through application of the general sinus mechanism where we therefore 

, , ^ J. sin (cy + «)) - sin lU-.j^.^, .,- t, ,. -. 

had to set xnstead of sxn ot m tne last equatxon above 

cos «) ^ 

of the machine curve. The first experiment was effected with o = 0.25 and, 
since this was satisfactory, o was then varied for the purpose of achieving 
a simultaneous eccentric osculation of the fourth order. The main results 
are listed in the table below and in the sequence in which the calculations 



177 



were carried out : 






(U 


c'co 


c 


0,25 


-31°,700 


2,3029 


. 2,252 


1,0 


- 14°,323 


2,4045 


1,207 


0,0 


- •24'',S57 


2,3810 


1,834 


0.4 


-29°,048 


' 2,3503 


2,094 


0,35 


- 29°,904 


2,3359 


2,148 


0,37 


-29*,063 


2,3447 


' 2,1320 



Since c c must have the value of 2.3^09 in order for us to obtain 
o 

the osculation of the fourth order in the vertex, there would be no point 
in carrying the calculations out further. If this value has been reached 
approximately, then the correction may be as satisfactory as at mathemati- 
cal equality which can be decided only through the time-consuming com- 
parative examination of the lateral aberration of the various rays. This 
is specifically the case when the special cusp characterizing the centric 
osculation of the fourth order on the evolute on the meridian curve of 
the Fresnel zone of the refracted bundle of rays has become decomposed in- 
to three cusps which is made manifest by the sign of the lateral aberra- 
tion of the rays passing nearest to the axis. Based on the machine con- 
stants ; -, .- __ __. ._ --.- - 



I = 0,37 JCo I ==0,61564 C=2,1328 



(0 



29°.603 



we therefore first calculate the lateral aberration of the various rays in 
the manner indicated above. The values are listed in the following table 
together with the respective ordinates of the surface points: 



lo- 


y 

-0,087 360 
-0,175 29 


1 





15" 


-0,264 26 


0,000 07 


20" 


-0,354 50 


0,000 25 


25° 


-0,446 45 


0,000 41' 


30° 


—0,540 11 


0,000 47 


35° ; 
40° 


-0,635 80 
-0,733 05 


0,000 28 
-0,000 24 


45° 


-0,835 14 


-0,000 46 


00° 
-55°, 
60° • 


— 0,040 14 

, . -1,050 34 

-1,16700 


-0,000 07. 

. -0,000 50 

-0,008 85 



178 



For the calculations, we employed seven-place logarithms until we 
obtained a value at a certain place where the accuracy of this value was 
such that greater accuracy could not be obtained through continuation 
with a higher number of places than five. The values obtained for the 
lateral aberration of the two rays with the smallest inclination were 
positive but so small that the sign was uncertain, in view of the ntimber 
of places employed, and are therefore designated in the table with 0. 

As is shown by tne table, the lateral aberration is everywhere so 
small that a greater correction, even though mathematically possible in 
all probability, would most likely be physically not noticeable. The 
change of sign of the lateral aberration indicates that, in addition to 
the ray refracted in the point of osculation, another ray passes through 
the axial focal point which is possible only when three cusps exist on 
the evolute of the meridian curve of the Fresnel zone of the refracted 

bundle of rays between the point of contact with the first ray and the 

2 2 
axial focal point. Since moreover c c ">■ e , an also infinitely small 

negative value of y must correspond to an infinitely small negative value 
of /I or, in other words, the cusp of the axial evolute has become decom- 
posed into three cusps so that the evolute has a total of not less than 
nine cusps. 

In order to demonstrate the degree of similarity of the duplex sur- 
face with the hyperboloid, I have calculated the table shown further be- 
low. The value of the abscissa belonging to each value of 8 was substi- 
tuted in the equation of the hyperboloid whereupon the corresponding 
values of ycpNMp, belonging to the hyperboloid were calculated. These 
values are designated in the table by H whereas D indicates the values 
belonging to the duplex surface at the same abscissa. The numerical 



179 



calculations were carried out by routine mathematicians in my laboratory 
and a sufficient number of control formulas guaranty the accuracy of the 
results. However, since I have neither verified myself that there is not 
an error of the last place in a logarithm nor performed a calculation of 
the degree of accuracy which would represent an extremely time-consuming 
effort, it is probably not possible to determine with certainty from the 
differences of the ordinates that the meridian curves of the two surfaces 
intersect in accordance with the four changes of sign of the curves in 
four points which could have been decided only through more detailed cal- 
culations. However, we may conclude, because of the great number of cal- 
culations, that the differences are not greater than the maxima indicated 
in the table. Since the distance of the two surfaces from each other (if 
they touch in the vertex) is approximately equal to the difference of the 
ordinate multiplied by sin cp, the greatest difference amounts to about 
1/10,000 of the radius of curvature of the vertex. In a plane-convex lens 
with a radius of curvature of the vertex of 10 cm and with a diameter of 
21 cm, the largest deviation of the form of the duplex surface from that 
of the hyperboloid would therefore be indicated by a distance of about 
0.01 mm and, if this lens is struck on the plane surface by axial-parallel 
light, the maximum lateral aberration of a ray amounts to 0.06 mm. Here 
the angles P and or have, in accordance with the point of osculation, the 
value of 52.7^9 and/or 43.3^9 so that the machine constants are entirely 
satisfactory from the mechanical viewpoint. 

However, duplex surfaces can also be obtained in another manner in 
these surfaces which are exceptionally appropriate for replacing the hyper- 
boloid. For example, the duplex surface first described above which was 
calculated for the application of an evolvent eccentric in the A-mechanism 



180 



Table for comparison of the non-planate iiyperboloid 
with a duplex surface. 



« y 


to 


N . 


M . 


P' 


0,003 79 


D 0,087 37 
H 0,087 17 


4°,967 70 
4°,956 97 


1,008 90 
1,008 86 


1,008 02 
1,008 87 


1,026 03 
1,026 81 


0,015 20 < 


D 0,175 20 
H 0,175 22 


9°,748 18 
9°,743 80 


1,035 31 1 l,0;i5 57 
1,035 55 ,1 l,on5 57 


1,100 55 
1,109 72 


0,034 14 -I 


D 0,264 26 
H 0,264 20 


U°,184 33 
14°,181 25 


1,078 41 
1,078 65 


1,07!) 07 
l,i..9 91 


1,253 59 
1,255 00 


0,0 CO 44 


D 0,354 56 
H 0,354 65 


IS^IOS 5 
18',162 4 


1,137 09 
1,137 74 


1,110 83 
1,141 48 


1,400 71 
1,472 72 


0,093 70 


D 0,446 45 
H 0,446 44 


21°,647 8 
2r,632 3 


1,210 19 
{,21103 


1,218 62 . 
1,210 48 


1,773 OS 
1,776 04 


0,133 80 < 


D 0,540 11 
H 0,540 21 


24°,619 5 
24*,607 


1,290 48 
1,297 40 


1,312 53 
. 1,313 42 


2,186 04 
2,183 65 


0,180 42 I 


D 0,635 80 
H 0,635 99 


27°,119 6 
27°,116 8 


1.394 78. 

1.395 30 


1,42184 
1,422 35 


2,729 9 
2,710 5 


0,233 04 < 


D 0,733 95 
H 0,734 08 


29°,211 
29°,219 


1,503 92 
1,503 80 


1,546 72 
1,545 52 


3,418 1 
3,400 9 


0,291 75 < 


D 0,833 14 
H 0,835 24 


30',8G8 ■ 
30° ,979 


1,623 01 
1,622 70 


1 ,683 40 
1,082 95 


4,263 8 
4,272 


0,356 78 


D 0,940 14 
H 0,940 35 


32°,455 
32°,459 


1.751 92 

1.752 12 


1,835 00 
1,835 18 


5,328 4 
5,378 8 


0,428 58 ] 


r> 1,050 34 
B 1,050 46 


33%704 
33°,707 


1,892 82 • 
1,892 91 


2,003 25 
2,003 25 


- 6,955 9 
6,782 6 


0,508 50 < 


D 1,167 9 
H 1,107 8 


34°,666 
34°,774 


2,053 3 
2,047 6 


2,197 4 
2,190 3 


11,143 3 
8,584 



at U3 = 0, can be improved through variation of the machine constants o 
and c until a centric osculation of the fourth and an eccentric oscula- 
tion of the second order can be obtained with any desired degree of ac- 
curacy. Without including a table of all calculations, I shall indicate 
here only the final result. If vue set o = O.65 and c = I.8325, there 
then result 

! Ca==0,096 998 Ci = — 0,687 384 C^Cj ^= 2,34055 p, ■== 0, 1 058, } 



l8l 



and the values 

!P = 49°,722 a = 40'',375, 

correspond to the point of osculation so that the machine constants are 
entirely satisfactory from the mechanical viewpoint also in this case. 
Moreover, the duplex surface calculated above for application of a 

standard A-eccentric for ^ = 0, can be improved also through reducing the 

2 
value of o where both the product c c as well as the radius of curvature 

■^ o 

on the point of osculation decreases, we thus obtain for o = 1.35: 

i C = 0,745 95 C, = 4,363 33 C*Co = 2,4279 p, = 6,244, 

where the values for the osculation P and/or a are ^5.065 and/or 12.655 . 
Although the height of the eccentric in the A~mechanism is large, it is 
still mechanically applicable. If we continue further on this path, it 
is possible to come theoretically even closer to the optical ideal but c 
then rapidly rises to mechanically disadvantageous values. For example, 
at = 1.25, there results the high value of c = 10. 883 which is presum- 
ably applicable only at a very small radius of curvature of the vertex. 
However, with this we obtain 

I C*Co = 2,39C4 p,— 6,039. 

If we are able to freely select the diameter of the abrading cylinder, 
we can then obtain duplex surfaces in various ways which will replace the 
hyperboloid. However, this raises the question of whether this is also 
possible if we are to grind with a certain given surface and especially 
if this surface represents a plane because the latter offers particular 
mechanical advantages as was explained above (p. 66). 

If the evolvent eccentric is utilized in the A-mechanism, there then 
result unsuitable values for the base curve as machine curve. The latter 
can be improved by the application of the general crank mechanism as B- 
raechanism but, since this produces an unnecessarily complicated device. 



182 



we investigated only the case K = 1 which did show better results. How- 
ever, since the latter did not seem sufficiently satisfactory to me, we 
turned again to the general sinus mechanism. The figures in the table 
of machine constants previously given (p. I78) made it seem probable that 
an attempt with k = would produce no advantage. For k = 1 , i.e., by 

employing the machine curve 

inn /, V sin{a + (i>) — sin (I) /I ,\ 

jB — i?B =" Co ( 1 — cos a) - i '- =» c g — 1 

I » » ^ \ COS w \cos fi / 

as base curve, however, there resulted for the eccentric osculation of the 
second order 

! Cj==0,2ai!38 = 2,793 00 (0 = 8°,9008 C^Co ■== 2,3 I afi , 

whereas 

i p == ffl = 33°. 109 a = 34°,S92 

in accordance with the point of osculation. The machine constants are 
therefore excellent from the mechanical viewpoint and the small difference 
e - c c signifies that the surface can replace in practice the hyperbo- 
loid from the optical viewpoint. It is moreover possible to obtain the 
simultaneous centric osculation of the fourth order through variation of 
k. If such a machine is constructed only for the grinding of these duplex 
surfaces, then a carriage is not necessary in the B-mechanism because 
the crank mechanism, as explained above (p. 5^), can be replaced by a 
closed linkage consisting of a sphere integrated with the A-axis and rest- 
ing on the cylinder linked to the B-axis where this cylinder degenerates 
into a plane at k = 1. When grinding such duplex surfaces with a differ- 
ent radius of curvature of the vertex, we need only change the distance of 
the abrading plane from the B-axis and the A-eccentric accordingly. It 
will be difficult to decide a priori whether a duplex surface produced in 
this or some other manner will be able to displace the directly ground 



183 



hyperboloid on the market through mechanical and economical considera- 
tions. 

If I have thus shown that there exist at least four different duplex 
surfaces capable of being produced with simple means which can replace the 
non-planate hyperboloid, I have simultaneously also demonstrated the multi- 
fold applicability of the duplex method. The difference of the different 
machine curves is illustrated most easily by a comparison of the parallel 
curve of the hyperbola at a small value of o with a base curve which pos- 
sesses a point of inflection within the section which is utilized. 

After this, let us examine as the next example an aberration- eliminat - 
ing duplex surface with a point of inflection. The exact aberration-elim- 
inating surface along each ray can be constructed geometrically point-by- 
point in the following manner according to the method sketched by Huygens. 
To a given optical system of revolution of m - 1 surfaces, a last surface 
is added for which the radius of curvature of the vertex and the locus of 
the vertex are preselected and whose form must be such that a given axial 
point located in the first medium is projected homocentrically through the 
entire system which thus consists of m surfaces. A necessary and adequate 
condition for this is that the optical length on each ray has one and the 
same value. The optical length existing along an arbitrarily selected ray 
between the surface with the order number V and that with the order num- 
ber V+ 1 can be indicated in the form 

For the first and/or last medium, the members - n., q, and/or + n'q' are add- 

' ' 1^1 m^m 

ed. After summation, we obtain, by employing the invariant designation, 
the condition of freedom of aberration represented by the optical length 
in the form 



l8k 



if s s' represent the paraxial lengths of intercept. 

Since the locus of the vertex and the radius of curvature of the ver- 
tex of the last surface are given, the right side is known. We then begin 
by laying off on the ray refracted in the next to the last surface, and 
therefore belonging to the next to the last medium, such a distance that 
the optical length from the object point to the end point of this section 
is equal to the optical length on the axis. In Fig. 5, let ED be the ray 
refracted on the surface m - 1; E = the point belonging to this surface in 
which the refraction takes place; = the vertex of the last surface; and 
A = the image point . 




Fig. 5. 
If the optical length on the oblique ray from the object point to the 
point E is designated by L, then 

i m—l «i— 2 



and we lay off the section 



\ ^^^^'^«>)-i-^ 



on the ray ED. The optical length from the object point to the point B 
located in the next to the last medium is then the same as to the point A 
located in the last medium. If we now draw the normal AF on the ray ED 



from this point and draw a circle with the radius EG = AF 



ffl 



m 



with B 



185 



as its center, then the intersection C of the tangent of this circle 
passing through A with the ray ED represents the respective point of the 
last surface and CA is consequently the ray refracted in the latter. Be- 
cause of the similar triangles, we have n • CB = n ' • CA and, since the 
optical length from the object point to the point C is equal to 2^^^ns 
- n • CB, this length to the point A is the same on the refracted ray as 
on the axis . 

In the case where the image point is infinitely distant and all rays 
must therefore issue axial-parallel from the last surface, the optical 
length is measured as far as an arbitrary plane intersecting the axis per- 
pendicular in the last medium. However, in this case, the above geometri- 
cal construction fails us but the corresponding trigonometric calculation 
presents no difficulty. 

In order to determine trigonometrically the point C as well as the 
direction of the normal and the tangential radius of curvature in the 
former, we need to determine first, when concerned with the surface m - 
1, the coordinates x,y, of the point B referenced to as starting point 
of the coordinate system. If EB is here designated as X and calculated 
in the manner already indicated, we then have, if xy designate the coor- 
dinates of the point Ji referenced to the vertex of the surface m - 1 as 

starting point, - _ ._ 

\ x, = x + XG03u' — d t/, = y — Xsmu'. 

i 

where d represents the distance of the point from the starting point 

of the coordinate system. In the calculation of the last surface, let 

us temporarily designate the angles BCA and BAC with 6 and/or £. We 

then obtain 

i sin s : sin (s + {)•)=.»':» = sin i : sin *' . 



186 



since moreover 6 = u' - u = i - i* , as is evident from the figure, this re- 
sults in 



and from this 



sin i sin (e + ■&) = sin e sin (i — &) 
I tgt = — tge. 



and this last equation is unique under the condition |ij < ^, Since there- 
fore the angle OAB is in the figure equal tou'-i=u-i',we obtain i' 

from the equation , 

} tg(M_i') = _lL_ 
I ■ *"*» 

and then successively and in the customary manner i cp and u'. Furthermore: 



t' — ^£r" *') ^'" " 



y, sm V 



sin {i — %') sin (/ - i') sin (« — f) ' 

where the first of these expressions is the simpler one and is employed ad- 
vantageously as soon as s has not too high a value. The last expression 
provided for thir case results from the application of the triangle ABC 

y : 

where AB = — : 7 — • r-rr* Through the usual formulas 

sm (u - X • ) _ '^ 

sm 9 sin (5 

I 

«== Jf— iVcoao y^'iV'sin'f 

we know the magnitudes necessary for the calculation of an osculation of 
the first order and those for the construction of the surface point. In 
order to obtain also the radius of curvature, the value of p is desired 
either in the usual manner, if s is not too large, from the difference q 
- p known from the preceding surface or else from the expression 



from which p, is derived from the formula 



I ^ UGoa^i ^ nn'smAi 



in which we must set p'= qV 



187 




Fig. 6. 
The case of an infinitely distant image point is illustrated most 
simply by Fig. 6 where 0, E and C have the same significance as in Fig. 5 
and ED therefore represents the ray refracted in the next to the last sur- 
face. A is an arbitrarily selected axial point in the last medium and the 
optical length from the object point to the former is equal to the optical 
length to the point B located in the next to the last medium. CG is the 
ray refracted in the last surface which intersects the line AG perpendic- 
ular to the axis in the point G. This then gives n • CB = n ' • CG and 

^ m m 

proves in the same manner as above that the angle BGC = -i. Since u' = 0, 
we have i - i' equal to -u and consequently 

fi sin » = n' sin (i + m) , 



fi-om which results 



itgt = - 



»i' sin u 



n — n' cos M 

If OA is designated by E and x.,y, are the coordinates of the point B as 

above, we then obtain the coordinates of the surface point C by means of 

the formulas r. . ■ 

I y = y, + (jS' — r,) tgz = (s— a;) tg«, 

after which all required values are determined through the usual formulas. 
Where we are concerned only with calculation but not with explanation by 
means of a figure, it is simplest to make E = 0. 



188 



In order to apply the application of this method to calculation of 
an aberration-eliminating duplex surface, let us select an example which 
makes high demands on the duplex method. If we wish to satisfy, in re- 
verse projection at the same magnitude by a simple lens, also the sinus 
condition for the peripheral ray when one surface is spherical, then the 
other surface calculated in the above manner receives a point of inflection 
of the meridian curve which will be located within the optically effective 
part at an adequately large opening. In order to produce such a lens, 
we must construct, in the general case and after experimentally effected 
deflection, the respective point of the non-spherical surface and the ray 
refracted in the former in the manner just described until that value of 
the deflection has been found which satisfied the sinus condition. How- 
ever, these calculations may be replaced, if the lens has a sharp periph- 
ery and the peripheral ray is selected for the elimination of the sinus 
condition, in the following manner by solving a cubic equation. That it 
is best, even when the lens cannot be optically exploited all the way to 
the sharp periphery, to select the peripheral ray for the satisfaction of 
the sinus condition will be seen from the following. Similarily acting 
lenses are in use for other magnification as "non-planate ophthalmoscope 
lenses" in the methods of non-reflecting ophthalmoscopy developed by me. 
Since the designation "non-planate" has already been introduced for similar 
lenses, .although it should actually only designate lens in which both the 
aberration is eliminated and the sinus condition is satisfied for any 
arbitrary inclination of the ray, it should be employed here also. We 
are therefore concerned with the construction of a duplex lens with one 
spherical surface non-planate for the magnification coef ficient-1 . 

In order to find the deflection corresponding to an arbitrary 



189 



magnification coefficient by means of the cubic equation, we can proceed 
in the following manner. For the ray refracted at the sharp periphery of 
the lens, we select q, and u, which is equivalent to the choice of scale 
and lens aperture. Through the sinus condition, q' and u'^ result from 
the prescribed magnification coefficient. The thickness of the lens is 
determined by the condition that the optical length on the peripheral ray 
shall be the same as on the axis. If the lengths of intercept otherwise 
designated by s are designated instead with S and if n is the refraction 
index of the lens, then this condition is written 

—gi + q'i'='—Si + nd + S',, 

whereas we have on the other side 

— Qi COS Ml + g'j cos u'i'^—Si-i-d-i- S'^ 

We obtain from this for the thickness of lens 



d = g'a (^ ~~ pos u'n)—- gi (I -T- ooa Ml) 
» — 1 



The two equations 



UvX^ 



S, 



(v-1.2) 



result in 



<S"i __ /g, sin Ui 



because of the satisfaction of the sinus condition. On the other hand, 
we have, when using 

a='q'i~g^ — nd 



tor abbreviation 



which furnishes 



\S\: 



S'i^St + a, 
<S, (sin «, — sin m'j) — a sin «', 



The paraxial projection in the first surface signifies 

n _»— 1 1 



190 



and if xy represent the coordinates of the intersection of the XY-plane 
with the periphery of the plane in the standard coordinate system whose 
starting point is shifted to the intersection of the first spherical sur- 
face with the axis, we then have 

ar^jS, — jicostt, 1/ =» jj sm ttj (Pi — «)* + J^* = p! 

and obtain, through elimination of S ' S^ p and y from the last five equa- 
tions, the cubic equation 

j a;«n(l— ifc) + a»{7tg, cosw,(l— /«) — d{2« — 1) — wifca} + 

ij. a;<7, {nqi sin'ttj (1 — k) — 2d cos«i(n— 1)> + gj am'Ui{nq^GOSUi{l — k)~d—^ 

sin U ' P 

in which k was substituted for —. If k = -1 and if we designate 

sm U]_ ^ 

q' = -q-, and/or u' = -u by q and/or u, then this equation can be brought 
into the form 

i ^ [ « *~* 1 \ 

•"^■^"131";^^^^— **5'(3«— !) + «?*(» — !)(» + » cos«+ 2cosw) — ff»sin*M (» + !) = 

For n = 1.53 and u = 26.6 , there then results — = 0.2^572 and, if the 
absolute value of the radius of curvature of the vertex of the second sur- 
face is selected as unit, 

|p, = l,8290 d=l,3765 Pa = — 1 . Si^ — 2,2347. 

Based on these values, I have calculated the coordinates of the re- 
spective points on the exact aberration-eliminating surface for a differ- 
ent inclination of ray and listed them in the following table: 



M, 

_ 4" 

- 8" 

-12' 

-15* 

-18° 

-20° 

—22° 

I -24°, 

i —25° 

i -26° 

i -2C°,8 



X 


V 


0,015 05- 


0,177 08 


0,0U1 02 


0,3SS 03 


0,130 SO 


0,531 58 


0,206 76 


0,605 29 


0,270 39 


0,804 50 


0,324 40 ° 


0,904 00 


0,382 33 


1,015 38 


0,444 97 


1,153 32 


0,478 72 


1,245 72 


0,514 32 


■ 1,38182 


0,532 22 


1,544 12 



191 



By lasing these values, Fig. 7 has been drawn in order to give a pic- 
ture of the form of the lens and of the demands made on the duplex sur- 
face to be calculated. Strokes indicate the principal paraxial points 
and crosses the center of curvature of the spherical as well as the center 
of curvature of the vertex of the non-spherical surface. At the periphery 
of the lens, this produces for the latter surface at negative u^ 

[ Jf ■= — 2269,038 N = — 2268,500 « = _ 0°,0390 p, « + 0,83522 . 

It is here evident that the surface 
normal is nearly parallel to the axis and 
the degree of curvature is negative because 
the tangential but not the sagittal radius 
of curvature has changed signs. The corre- 
sponding duplex surface can consequently 
still be ground wiith a cylinder but, since 
o may not exceed the absolute value of •^' , 
the possibility of satisfying a prescribed 
condition through variation of this magni- 
tude is correspondingly limited a priori. 
To this should be added, as shown by the 
figure, that the radius vector at the pe- 
riphery of the lens is very appreciably prolonged and intersects the tan- 
gential plane of the surface at a relatively small angle so that high 
values of the differential quotients in the polar equation of the machine 
curve result. If I utilize in spite of this, the above values valid for 
a point of the periphery of the lens for the osculation of the duplex 
surface, then this is not done merely in order to calculate a practically 
advantageous form of lens (which will be discussed further below) but in 
order to select an example which makes high demand on the duplex method. 




Fig. 7 



192 



VUe will therefore have to investigate whether it is possible to obtain 
with simple technical means an osculation of the first and/or second 
order on the periphery of the lens and perhaps also a simultaneous centric 
osculation of the fourth order. In order to satisfy this last postulate, 
we need — = 2.0651 at a negative radius of curvature of the vertex as is 
shown through the respective Seidel formula in the form indicated by me. 

The first attempt proved that a rather satisfactory result is obtain- 
ed already with the simple sinus mechanism, i.e. , at U) = 0. With a posi- 
tive radius of curvature of the vertex, there resulted for = 0.6, at 
eccentric osculation of the first order, 

j p, = — 0,83893 C*Co = 1,8548, 

and the eccentric osculation of the second order can be achieved through 
a minor reduction of o at any desired degree of accuracy as proved by cal- 
culations which are not given here. However, the value of c c does de- 
viate from the desired value but must be considered as relatively favor- 
able since only three machine constants are available for the satisfaction 
of four conditions. The value of a corresponding to the point of oscula- 
tion is 75.957 which has to be regarded as a disadvantage from the tech- 
nical viewpoint. However, if we consider that, as will be seen from Fig. 
7, the speed of the A-carriage at very large 3 must be very high in rela- 
tion to the speed of rotation around the B-axis, we can predict that it 
will be scarcely possible to ever avoid this disadvantage. The latter has 
as necessary result that the 'A-axis must be parallel to the B-axis and 
that therefore a carriage is necessary also in the B-mechanism. Since the 
speed of rotation around the A-axis is also very large in the proximity 
of the point of osculation in relation to the speed of the, B-carriage , it 
will probably be preferable to grind at least the most peripheral part of 



193 



the surface only in the direction from the periphery toward the center. 
When employing the general sinus mechanism for obtaining an eccen- 
tric osculation of the second order, there then results for o = O.65, at 
positive radius of curvature of the vertex, 

|= — 8°,0954 C==2,7I76 Co = 0,283 38- C*Co = 2,OU29, 



and calculations here not included show that the simultaneous centric os- 
culation of the fourth order can be achieved through a minor reduction of 

with any desired degree of accuracy. However, the minor difference 

jc c j - I — signifies that we cannot decide a priori whether the exact 
centric osculation of the fourth order is actually more advantageous since 
this could only be confirmed by a complicated calculation. For the oscula- 
tion point, a + CD = 76.^83 so that consequently the remarks above on the 
type of machine and grinding are valid without change also in regard to 
this machine curve. 

The maximum value of 8 is 5^«1066 . For the values indicated below, 

1 have calculated the lateral aberration of the rays refracted in the lens 

with this duplex surface where the direction of the rays was reversed in 

order to make the calculations less complicated which is without importance 

for projection at natural magnitude. The values so obtained are listed 

below : 

I P - 'Q ' 

j 10° 0,000 041 



j 20° 0,002 588 j 

30° -0,002 979 

1 40° -0,008 69G 

jSO" —0,007 070 



Since the optical length on the ray refracted in the point of oscula- 
tion is the same as on the axis for a surface constructed by this method, 
the meridian curve of the evolute of the refracted bundle of rays must in 
this case also have three double-sided cusps and a second ray must also 



19^ 



pass through the axial focal point. In agreement with this, the figures 
above indicate the respective change of sign. 

In order to find out whether it would be possible with other simple 
machine curves to avoid the large angle Of + co, a series of calculations 
were carried out from which it resulted that, although the eccentric oscu- 
lation of the second order can be achieved in various ways, there are al- 
ways obtained large values either for ou or o? + u) as soon as a simultane- 
ous centric osculation of the fourth order was obtained. Since, among the 
various duplex surfaces determined in this manner, the one investigated 
above appears to offer the greatest advantages, this shall not be dis- 
cussed further here. 

As indicated above, this lens was not calculated in view of any sort 
of practical purpose. Whether it can serve such a purpose, we shall not 
here decide. Since the principal focus is 1.^6713, i.e., insignificantly 
less than the thickness whereas the diameter exceeds twice the principal 
focus, it should satisfy high demands on the aperture. However, because 
of its thickness, it can probably be employed only in those cases where 
object and image can be sufficiently approached to the respective lens 
surface so as to permit a low absolute size of the lens. Moreover, the 
presence of the point of inflection on the meridian curve of the non- 
spherical surface conditions less favorable reflection on the lens sur- 
faces. It is therefore a moot question whether a bi-hyperbolic lens 
and/or a combination of two plane-hyperbolic lenses may not be preferable! 
in spite of the double non-spherical surface. Entirely disregarding an 
eventual achromatization which should be as easily possible in one as 
in the other case, there is an important circumstance in regard to the 
sinus condition which speaks very much in favor of the bi-hyperbolic lens. 



195 



As well as the sinus condition is understood, the effect capable of 

being achieved by the satisfaction of the former along a given ray still 

seems to be misunderstood. The equation valid in a system of revolution 

for two arbitrary media 

I Anx,; sin « == o 

signifies, if it is applied to object and image media, that the sagittal 
magnification coefficient in the projection of an axial point is independ- 
ent of the inclination of the ray at a constant ratio of the sinus. If 
this equation is differentiated and divided through the fundamental equa- 
tion 

Anxidu = 

also valid for two arbitrary media, there then results 

i A /sin u dy,, y,, \ 

from which we obtain,^ by employing the abbreviated designations 

» »= .«, «' = w'„. V, = '^ y, = 5^ 



for m surfaces, the equation 



au' COS u' ■ ^* 



Since __1 represents the actual tangential magnification in the 

cos u' 
projection of axial-perpendicular planes into each other for the immediate 

proximity of a finite inclined ray passing through the axial image point, 
the last equation therefore shows that the actual tangential magnification 
is also independent of the inclination of the ray at a constant ratio of 
the sinus. However, the results are valid only on the assumption that the 
aberration is eliminated and the sinus condition satisfied along each ray. 
The effect can be expressed most simply by saying that, in the projection 
of an infinitely small axial-perpendicular object surface on an axial- 
perpendicular plane by employing an infinitely small diaphragm, the image 



196 



is independent of any arbitrary eccentricity of the diaphragm or, in other 
words, the aberration is esliminated along each ray not only for the axial 
point but also for a point located in the same axial-perpendicular plane 
infinitely close to the axis where this last point is projected in the same 
axial-perpendicular plane as the axial point . 

However, what do we accomplish by satisfying the sinus condition for 
a given ray with eliminated aberration? As will be immediately apparent 
from the above equations, we gain by this only that the sagittal magnifica- 
tion along this ray is the same as along the axis whereas the satisfaction 
of another condition is necessary in order to achieve the same result for 
the tangential magnification. If consequently an infinitely small diaphragm 
is decentered so that the respective ray passes through the center of the 
former, there is then projected an infinitely small axial-perpendicular 
surface anamorphotically on the axial-perpendicular plane and, for an object 
point infinitely close to the axis, the aberration is eliminated only along 
those rays intersecting that line which is perpendicular in the diaphragm 
center to the plane passing through the latter and the axis so that these 
rays therefore constitute an infinitely small part of the infinitely thin 
bundle of rays delimited through the diaphragm. 

It follows from this that the mere indication of the sinus ratios cor- 
responding to the various inclinations of the ray is not suitable, as seems 
to be the general viewpoint, for giving a comprehension of the effect of 
satisfying the sinus condition along a given ray. In order to derive this 
comprehension, we must also know the coefficients of the tangential magni- 
fication corresponding to the various inclinations of ray for axial-perpendie- 
ular object and image planes. If the paraxial magnification coefficient is 
indicated briefly by x? we must then determine, in accordance with the 



197 



various inclinations of ray, not only the value ofj t:=^— . — f,^^* also 



X- 



that of 



1 cos u 



X 



71' sm u 



, in order to be able to judge the effect of satis fyinj 



X cos u' 
the sinus condition for the ray passing through the periphery of the lens. 

These values have been calculated by me, by employing the exact aberration- 
eliminating surface, for those inclinations of ray for which the coordinates 
of the surface points given above were determined, and have utilized these 
values for the construction of the curves in fig. 8. The inclinations of 




Fig. 8 
ray listed in the table above are plotted as abscissae and the corresponding 
magnification ratios as ordinates. The flatter curve represents the sagittal 
and the steeply decreasing curve the tangential magnification ratio on the 
basis of the above indication, and the ordinates of the end points of the 
former therefore represent the unit of the ordinate scale. 

If the siniis condition were satisfied along each ray, the two curves 
would then coincide in a straight line parallel to the axis of the abscissa. 
However, in the cases where, as is the case here, this condition is satis- 
fied only for a given inclination of ray, we know a priori nothing on the 
ordinate of the curve of the tangential magnification ratio corresponding 
to the former. Inversely, it follows from the last of the equations given 



198 



above that the two curves intersect in that point where the tangent of the 
curve of the sagittal magnification ratio is parallel to the axis of the 
abscissa. Consequently, there must also exist, if the sinus condition is 
satisfied for two different rays, a second point of intersection of the 
two curves. If we let the two rays approach each other infinitely closely, 
then the common ordinate in the second point of intersection has the value 
of one. Only when this is the case, can the purpose attempted through 
satisfaction of the sinus condition be considered as achieved. 

It now follows initially, and specifically for the present case, that 
it is better to satisfy the sinus condition for the peripheral ray than for 
an intermediate ray. Corresponding to the peripheral ray, the tangential 
magnification on the axial-perpendicular plane is in any event only about 
1/7 of the sagittal magnification which corresponds to a high degree of 
anamorphotic projection by an infinitely small diaphragm along the margin 
["randstehend"] . Moreover, the unfavorable trace of the curve for the tan- 
gential magnification seems to very inuch indicate that the utilization of 
two non-spherical surfaces will be advantageous where, because of symmetry, 
the sinus condition is eliminated along each ray. At least, such a lens 
should be preferable where one projection is concerned. However, if we are 
confronted only by the task of making all rays issuing from a small light 
source pass through a narrow aperture, the lens with one non-spherical sur- 
face will be able to hold its own, imless mirror images and required thick- 
ness of lens produce disadvantages. 

Finally, let us merely stress in regard to the aberration-eliminating 
surfaces that such a surface does not necessarily need to represent the 
first or last surface of the system but can be located at any desired locus 
but that then the simple construction utilized here must be replaced by very 
complicated calculations. 

199 



Image-Planating Duplex Surfaces . If, in a given optical system of 
revolution in which only the locus of the vertex and the radius of curva- 
ture of the vertex is prescribed for its last surface, an optimum satis- 
factory projection of a certain axial-perpendicular plane on an also axial- 
perpendicular plane is required with a narrow diaphragm in the prescribed 
locus, this objective can in msmy cases be achieved by giving the last 
surface a suitable form. If this form of the surface accomplishes that 
the two image planes corresponding to the object plane intersect in the 
axial -perpendicular plane passing through the axial image point, the sur- 
face will then be designated as anastigmatically image-planating . If the 
ratio of the axial distance of the anastigmatic image point to the axial 
distance of the object point is equal to the axial magnification coefficient, 
then the non-spherical surface is orthoscopically and anastigmatically image - 
planating . However, if we achieve for the prescribed inclination of ray 
only that the sagittal image surface and the axial-perpendicular image 
plane intersect, the surface will even then be designated as image-planating , 
provided the tangential image surface intersects this plane at some other 
point . 

It should be noted in regard to these definitions that, like an aberra- 
tion-eliminating surface, an image-planating surface can also occupy any 
desired place in the optical system but that I have found a simple method of 
construction only for the case where this surface is the last and/or first 
surface of the system. It should also be kept in mind that a surface image- 
planating in accordance with the above definition, for example orthoscopi- 
cally and anastigmatically, is not necessarily applicable in practice. Since 
the definition takes into consideration only a certain inclination of ray, 
it is therefore conceivable that values not possible in practice are obtained 



200 



for small inclinations of ray where, for instance, the meridian curve of 
the image surfaces may even have infinitely distant points. After cal- 
culation of an image~planating surface, we must therefore examine in each 
case the practical applicability of the latter through the behavior of the 
image surfaces between the ray forming the basis of calculation and the 
axis. It will finally also be apparent that, on the basis of these defini- 
tions, an image planating surface is characterized, depending on whether 
the image planation is or is not anastigmatic , by the respective magnitudes 
determining eccentric osculation of the second and/or first order. 

These magnitudes can be found through a geometrical construction in 
the following manner. In fig. 9, let A be the vertex of the last surface, 
AB = the axis, CD = the incident principal ray which intersects the axis in 
the point B, and d be the sagittal focal point corresponding to the given 
object point in the next to the last medium. In addition, let E be the 
point conjugate to the axial object point in the last medium and F be the 
point in the axial -perpendicular plane passing through E in which the given 
extra-axial object point in the last medium is to be projected. The dis- 
tance FE is therefore determined in the usual manner through the condition 
of orthoscopy but is selected freely when we must forego orthoscopy. We 
then draw the line DFG which intersects the axis at G, plot the normal GH 
from G on the ray CP and draw a circle with a radius GK = -, • GH with G 




201 



as its center. The point of intersection I of the tangent of the circle 
passing through F with the ray CB is the desired surface point and IG the 
respective surface normal. That the refracted ray will pass through F 
follows from the construction where the angle GIH and/or GIK represent an- 
gles of incidence and/or refraction and it also follows from the construc- 
tion that F is the sagittal focal point in the last medium because this 
point must be located on the line DG. Since the sagittal projection is 
independent of the tangential radius of curvature of the surface at the 
point I, the non-spherical surface acts in regard to it like a sphere with 
the radius GI . Finally — by trigonometry, however — the tangential radius 
of curvature at the point I is determined by the condition that the tan- 
gential focal point corresponding to the given object point in the last 
medium shall also be located in the F. If special reasons make it neces- 
sary to forego anastigmatic image-planation , the value of the tangentiail 
radius of curvature can be varied in order to obtain the optimum tangential 
image surface by experimentation. 

This construction furnishes the trigonometric formulas in which we 
shall designate, for A as starting point, the coordinates of the points G 
and F by a b and/or a'b' the radius of circle GK with e and the angle BGD 
with o and employ otherwise the usual designations utilized for recalcula- 
tion. In addition to the radius of curvature of the vertex, the indices 
of refraction and the characteristics of the incident ray, we therefore 
know a'b' as well as q - ^ and q - t". We initially have 

; a'^s — {g--c) COS u 6 =■ (j — c) sin « 
and then obtain M and o by means of the equation 

6 b' 



202 



from which results e u' and s' out of the equation 

n{s — M) sin u _ 6' sin (o + v!) 



e = -^ -, - == - ---.;-• ••' = {s' — M) sin «' 

sin o ' 



After we have then determined i' and N by utilizing the relations 

I »' — % cos («' — MJ ■'*^ ~ sin »' 
we also know q) and obtain in the usual manner p,. In the case of C = o° 
and a* =00, the same system of formulas can also be employed with the 
corresponding easily effected modification. In the former case, the angle 
IDG = and consequently o = ~u and, in the latter, the angle IFG = and 
consequently o s -u' where u' is determined by the condition of orthoscopy 
and/or selected freely so that o is known a priori in both cases. 

As an example of the application of this method, we shall calculate 
an orthoscopically and anastigmatically image-planating duplex surface on 
the supposition of a simple lens with an anterior stop and infinitely dis - 
tant object . In order to find out first how a plane-convex lens turned 
with the plane surface to the light, behaves under these circumstances, 
let us base our calculation on the lens characterized by the scheme p,(d/n)pp 
through the numerical values «> (0.5/1.53) - 1 with an anterior stop, the 
center of which is s = - O.25 at the inclination of ray u' = 59 . Here 
a' = 1.88679 and the condition of orthoscopy produces b' = » 5«1^05 which 
results, on the basis of the calculation just indicated, in the values 

\M — — 2,7566 N <=■ — 2,2660 9==10°,00U0 p, = — 3,6925 

A brief reflection shows us that no practically applicable lens can be con- 

N 



— I < JM|, the tangent of the meridian 



structed with these values. Since 

cos 

curve erected in the respective surface point intersects the axis in a point 
located on the concave side of the surface from which follows that the merid- 
ian curve must have two points of inflection between the respective point 
and the vertex. For those rays which enter the non-spherical surface at 



203 



the points of inflection, the tangential image point lies in infinity, 
however. Because of these points of inflection, the corresponding duplex 
surface is therefore practically inapplicable although it can be produced 
by the utilization of compound machines or special cylinders. Since a re- 
duction of the angle of inclination to kk will not eliminate this defect 
as further calculation shows, we have no other choice than either to fore- 
go orthoscopy or else to attempt deflection of the lens. Calculations 
continued in the former direction for the greater inclination of ray indi- 
cated above show that the defect is not yet eliminated at b' = - 2.5 where- 
as, at b' = - 2 there occurs the opposite effect iNJ ^ iM), Among the 
intermediate values, there are several which permit a simple duplex curve 
as machine curve, e.g., at b' = - 2.25: 

I Jlf= — 1,430 88 ^= — 1,400 35 ?> = 26°,8506 fj, = — l,40a 56. 

At o = 0.25, these values result for the tangential mechanism in the 
machine constants 

j Co = 0,067 396 C= 6,443 34 tO = — 43°,100, 

which are mechanically very satisfactory in spite of the unusually high 
value of c since a + m does not reach 15 in accordance with the given 
point. That this makes it possible to achieve anastigmatic image-planation 
by means of the plane-convex lens while foregoing orthoscopy is hardly sub- 
ject to doubt. However, since there would be no practical purpose in exam- 
ining the image surfaces of the various possible lens forms and to compare 
them with each other which would require extremely complicated calculations, 
it will be sufficient here to have indicated the numerical values above as 
an example of the applicability of the method. 

In the attempts for the deflection of the lens, we may now consider 
the objective of not only satisfying the condition of orthoscopy but of 
simultaneously making possible also the application of the mechanically 



204 



more advantageous sinus mechanism. These attempts were turned over by me 
to Mr. B. Lindblad who worked for some time in my laboratory. On the as- 
sumption of a concave anterior surface, diaphragm distance and lens thick- 
ness were determined as }i and/or 1/5 of the radius of curvature of this 
surface and the radius of the second surface varied experimentally by basing 
the calculations on an angle of inclination of u. = - 36.^8 . This produced 
a favorable lens form - 1(0.2/1,53) - 0.255 with which we obtained, at 
s^ = 0.25 and u = - 36.^8 

i/-= — 0,32472 J\r=- — 0,31641 ^ = _-46°,418 . p, = _0,30000 



At o =0.25 and cp = 0.255* these values furnished for the sinus mechanism 

\ Ca>= 0,009 04 C= 2,1400 W = — 50%206, 

and these machine constants must be regarded as very favorable. At higher 
values of o, there result numerically greater values of w and the latter 
reached amounts of - 51.^29° and/or - 58.087° at o = 1 and/or o = 2. The 
possibility of influencing the form of the surface through variation of o 
is therefore restricted a priori and is further limited by the fact that 
the product c c cannot receive a value greater by more than k%. 

Because of the more uniform distribution of the eccentric action on 
the A- and B-mechanism, the parallel curve determined by o = 1 was selected 
as machine curve which has the machine consteints, 

: Co =0,11109 = 1,7208 W = — 51°,420 

for p = 0.255. By applying these constants, we first determined those 

o o 

points of the duplex surface which correspond to the values = 5 , 10 , 

... 60° after which those rays were determined which pass through the dia- 
phragm center in the first medium and intersect the duplex surface in these 
points after refraction in the first surface and we finally calculated the 

focal points on these rays. Fig. 10 shows the trace of the meridian curves 



205 



of the image surfaces constructed in this manner. (The less curved trace 
represents the sagittal image surface.) The coordinates of the curve 

points calculated are grouped in the table 
following below ishere the rays are identified 
by the determined values of u. and I,)?, 
and/or §„ )|„ represent the coordinates of the 
tangential and/or sagittal focal points if 
the starting point is shifted to the axial 
image point. 

In addition to the coordinates of the 
focal points, the table contains in the last 
column the numerical values Q which afford 
a review of the so-called zones of distortion. 
If >2 represents the ordinate of the point of 
intersection of a ray refracted in the lens 
with the axial -perpendicular plane passing 
Fig. 10 through the axial focal point and if D is the 

power of refraction of the lens, i.e., the reciprocal value of the principal 

focal length of the latter, then''/] = =r- is the condition of absence of 

distortion. However, since the latter is satisfied only for a given ray, 
the number Q = -YJD*cot u, furnishes for the other rays the ratio of the 
real ordinate to the ordinate which corresponds to complete absence of dis- 
tortion. At u.^ = - 36.^8 , I, and |„ pass through zero and Q passes through 
one whereas /? = /'^ = 0.^369. The principal focal length of the lens is 
0.5909. 




206 



Table 
for evaluating the efficiency of an orthoscopically 
and anas tigmati call y image -pi an a ting duplex surface. 



- 3°,2103 

- 6°,4e]8 

- 9*,7780 

- 13°,1804 

- 16°,6743 

- 20'',245C . 

- 23°,8801 

• 27°,5683 

- 3r,2856 

• 35',02S3 

- 38',7714 

- 42°,5093 



s, 


'i- 


0,0060 . 


0,0333 


0,0247 


0,0684 


0,0400 


0,1003 


0,0742 


0,1480 


0,0939 


0,1027 


0,1040 


•0,2401 


0,1028 


0,2882 


0,0880 


0,3360 


0,0601 


0,3811 


0,0183 


0,4220 


0,0337 


0,4580 


0,0970 


0,4871 



5.1 

0,0019 
0,0070 
0,0130 
0,0225 
0,0281 
0,0322 
0,0322 
0,0293 
0,0207 
0,0008 
• 0,0125 
■ 0,0372 



■'t« 
0,0332 

0,0077 

0,1043 

0,1439 

0,1857 

0,2303 

0,2704 

0,3242 

0,3719 

0,4100 

0,4644 

0,5062 



y 

1,0023 
1,0082 
I.OICS 
1 ,0200 
1,0320 
1,0367 
1,0303 
1,0320 
1,0224 
1,0073 
0,9866 
0,0601 



These values only represent the result of the first attempt of deflec- 
tion and could probably be improved through further experimentation. If 
the unit is made equal to 10 cm, the lens is then suitable for an eyepiece 
because the diaphragm center utilized in the calculation can coincide with 
the center of revolution of the eye. At a focal length of about 6 cm, the 
object plane could have a diameter of at least 9 cm and the eye piece wo\ild 
therefore give a full-scale reproduction of corresponding wide-angle photo- 
graphs. However, whether the construction of such gm eye piece would be 
advantageous without achromatization shall not be decided here. 

Among the possibilities of application of the duplex surfaces, let us 
here briefly call attention to the non-focal lenses . A thin lens with in- 
finitely distant principal focuses and with spherical surfaces has a very 
minor action on an optical system. However, if one surface is non-spheri- 
cal, then only the action on the paraxial projection remains irrevelant 
whereas -- depending on the locus of the diaphragm ~- the aberration on the 
axis or the properties of the extra-axial projection or both are changed. 



207 



Such a lens can be incorporated without difficulty as a non~ focal comple - 
mentary lens in most optical instruments and offers an additional means of 
correction by deflection. The characteristics of optical projection here 
require in most of the optical instruments utilized so far a lens which 
is thicker at the periphery than in the center so that the latter can 
really be thin as far as the paraxial projection is concerned. In cases 
where a plane is preferable to the spherical surface, we need only utilize 
a non-focal duplex surface. 

Practical preliminary experimentation on small machines had been com- 
pleted by me in the first half of 191^. The findings showed that neither 
the method of grinding surfaces of the second degree nor the duplex method 
encounters serious difficulties so that an agreement with a large foreign 
company had been reached in July 191^ for immediate construction of a du- 
plex machine for actual practical operation which was to be ready within 
six months. World War I has been the reason why practical technical experi- 
ence has not kept step with theoretical developments. 



208