Full text of "On non-spherical surfaces in optical instruments"

See other formats

CVJ

EH
EH

/ NASA TT F- 872f.

ON NON-SPHERICAL SURFACES IN OPTICAL INSTRUMENTS

A. Gullstrand ^^/^ /^^^ ^^^ f%

5 fNASA

AD/Vi

U/Mber;

..^'1

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

WASHINSfON J) c

APRIL l96i^

NASA TT F- 8729

msA TT F-8729

KUNGLIGA SITSIBKA 71BE11BKSPSAK&DMIS1© HiUSBLINGAR

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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

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

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

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

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.

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

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)

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

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.

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.,

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.

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

Si2_f

Fig. k.

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

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,

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

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

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.

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,

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

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

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

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^,

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

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

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

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.

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

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

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-

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

H . . — ■-,- —

(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

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-

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

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-

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)

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,

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 * ' . '

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

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

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

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

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

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.

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

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

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

^ 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

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

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

«-2

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

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

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

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)}

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

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-,

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

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

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^

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

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.

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.

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

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

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

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-

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.

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

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-

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

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

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

Through the same treatment of the third equation, we obtain

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

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.

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

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

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?„

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

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'

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

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),

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

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.

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

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

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

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

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

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

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,

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

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

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

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

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

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

in which j

We thus obtain

(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

we set or for 3 and k for k, in

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),

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 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

must make I „ ch.Ea

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,

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

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

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 =

" •= 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.

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

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.

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

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

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 .

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") .

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

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

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 £^>,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

: 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±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

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-)

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

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

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

~ ^^^ °°^ 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

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~

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

•='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)

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

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"

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 ; „ , , " ... ,„.

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

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

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^

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

1 C, = - e* - (e« — 1)* + (e" + 1 - 2Xe'=)«.

it is immediately apparent that C possesses a minimum with negative value

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

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

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., - __

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

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

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-

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 -

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 = - — .

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 \

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

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 ,

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

we further obtain

o =- R^ -h 2 R>* ~ a R<>

\ (J, + 00 =» -^r- it ii^

I Pcos(i3~«,) Ocoa^(fZ.Y)

For the curvature of the machine curve, there is valid in general

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

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'

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,

\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

-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:

-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:

?■ ■

■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 :

c'co

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

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-

-0,087 360
-0,175 29

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

N .

M .

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

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'.

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

«== 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^

(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:

_ 4"

- 8"

-12'

-15*

-18°

-20°

—22°

I -24°,

i —25°

i -26°

i -2C°,8

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

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-

— 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

'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

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