p
NOTES ON PRACTICAL ASTRONOMY
AND GEODESY
By L. B. STEWART
SECOND EDITION
Revised and Enlarged
UNIVERSITY OF TORONTO PRESS
1921
Si
tap 3,
COPYRIGHT, CANADA, 1921
BY L. B. STEWART
PREFACE
These Notes form the substance of a course of lectures in
Practical Astronomy and Geodesy given to students of the
Third Year in Civil Engineering of the Faculty of Applied
Science . and Engineering of the University of Toronto.
These lectures are only designed to meet the requirements of
the engineer or land surveyor in his ordinary practice, or
those of the explorer wishing to determine his position in an
unsettled country, and therefore barely touch upon the precise
methods of the geodetic surveyor. The subjects treated,
however, and the methods of treatment, are fundamental;
so that the course serves as a useful introduction for those
who desire to extend their studies to the higher parts of the
subject.
LOUIS B. STEWART.
NOTES ON PRACTICAL ASTRONOMY
AND GEODESY.
PRACTICAL ASTRONOMY.
In these notes it is proposed to set forth in outline the most
useful methods for determining positions and directions on the
surface of the earth. It is assumed that the observer is provided
with an engineer's transit, or a nautical sextant, so that the
methods described are only such as are adapted to the use of
those instruments. More precise methods, necessitating instru-
ments of the highest class, are therefore entirely omitted, or but
briefly referred to.
1. Spherical Co-ordinates. Solution of the Astro-
nomical Triangle.
Determination of the position of a point.
Fig. J
In Fig. 1 CAO and ABO are fixed planes of reference;
is the point of observation. The direction of the line OS
is determined when the angles AOB and BOS are known;
also when the spherical angle ACS and the arc CS are known.
Planes of reference
The planes of reference used in astronomy are those of the
equator, the ecliptic, the meridian, and the horizon.
The plane of the equator is that of the earth's equator.
As the direction of the earth's axis is nearly fixed in space,
being subject only to slow changes of direction due to pre-
cession and nutation, therefore the plane of the equator is,
nearly a fixed plane.
The plane of the ecliptic is the plane of the earth's orbit.
The plane of the observer's meridian is a plane determined
by the earth's axis and the point of observation.
The plane of the horizon is a tangent plane to the earth's
surface i.e., to the surface of standing water at the point
of observation. It is therefore perpendicular to the observer's
plumb line.
The celestial sphere
This is an imaginary sphere of infinite extent, whose centre
is coincident with the centre of the earth. Upon its surface
the heavenly bodies may be assumed to be, as they apparently
are, set like brilliants.
The reference planes above denned are assumed to be
produced to intersect this sphere in great circles. The plane
of the horizon, as above denned, may be assumed to intersect
the sphere in the same circle as that determined by a parallel
plane through the earth's centre, owing to the infinite extent
of the celestial sphere.
Fie. 2
Fig. 2 shews a projection of the celestial sphere on the
plane of the meridian, the reference circles being represented.
Thus
PZHR is the meridian,
EFR the equator, or equinoctial,
HDN the horizon.
The ecliptic is not shewn, but V is a point in which it
intersects the equator.
If S now be the position of a star (by that term denoting
any heavenly body), and secondaries ZSD and PSF to the
horizon and equator respectively be drawn through it, these
arcs, with the meridian PZ, form a spherical triangle PZS,
which, from its frequent use in the solution of astronomical
problems, is termed the astronomical triangle.
Definitions
The circle ZSD is a vertical circle; PSF a declination or
hour circle. P is the celestial pole; Z the zenith. SD is the
altitude of S; ZS its zenith distance; SF its declination; PS its
polar distance; the angle PZS its azimuth; and ZPS its hour
angle. PSZ is generally called the parallactic angle.
As the observer's latitude is the angle between the direction
of the plumb line at the place of observation and the plane
of the equator, it follows that the latitude is the angle ZOE
or the arc ZE. This is also equal to the arc PN.
The following notation will be used:
h denotes the altitude SD of 5.
" denotes the zeiith distance ZS.
8 demotes the declination SF.
p denotes the polar distance PS.
t denotes the hour angle ZPS.
A denotes the azimuth PZS.
C denotes the parallactic angle.
<t> denotes the observer's latitude EZ or PN.
a denotes the right ascension VEF.
Systems of Spherical Co-ordinates.
1st system Altitude and azimuth.
The arcs SD and DN serve to determine the position of 5
with reference to the horizon and the meridian.
A small circle parallel to the horizon is termed an almu-
cantar.
A vertical circle is a great circle perpendicular to the
horizon.
The prime vertical is that vertical circle which passes
through the east and west points of the horizon.
2nd system Declination and hour angle.
The arcs SF and FE determine the position of S with
reference to the equator and the meridian.
A parallel of declination is a small circle parallel to the
equator.
3rd system Declination and right ascension.
The planes of the equator and the ecliptic intersect in a
right line called the line of the equinoxes. This line inter-
sects the sphere in the vernal and autumnal equinoxes.
The vernal equinox is the point through which the sun
passes in going from the south to the north side of the equator;
it is shewn at V, Fig. 2.
The equinoctial colure is the declination circle passing
through the equinoxes. The solstitial colure is the declination
circle passing through the solstices the points of greatest
north and south declination on the ecliptic. It is therefore
at right angles to the equinoctial colure.
The co-ordinates in this system are the arcs SF and FV.
4th system Celestial latitude and longitude.
V
Fig. 3
In Fig. 3 VEAR is the equator, VHAK the ecliptic, VA
the line of the equinoxes, VPA the equinoctial colure, and
EPR the solstitial colure.
The co-ordinates in this system are SG, the latitude of S,
and GV the longitude. These are denoted by /3 and X respec-
tively.
In the first system the co-ordinates change continually
and irregularly on account of the diurnal rotation of the earth.
In the second system the declination is unchanged by that
rotation, and the hour angle changes uniformly with the time.
In the third and fourth systems the co-ordinates are unchanged
by the diurnal rotation.
The third system of co-ordinates is for this reason used in
the construction of ephemerides.
Although unchanged by the diurnal rotation, the co-ordi-
nates of the third and fourth systems are changing continually
though slowly on account of precession and nutation.
Solution of the Astronomical Triangle.
(1) Given the altitude and azimuth of a star, and the latitude
of the place, to find the star's declination and hour angle.
Fie. 4
If we denote the angular points of the astronomical triangle
ZP and Shy AB and C, respectively, then in Fig. 4 we have
given
A=A, 6 = 90-/z, c = 9O-0;
and it is required to find
a = 90 -5, and B = t.
These are given by the first of (1) and (5), Sph. Trig., p. 69
which become
sin 5 = sin h sin 0+cos h cos cos A.
sin A cot t = cos tan h sin cos A.
The first of these may be written
sin 5 = sin h (sin 0+cot h cos cos A)
Then introducing the auxiliary such that
tan = cot h cos A (1)
it becomes
sin 5 = sin h (sin^-fcos < tan 6)
_ sin h sin (0+0) (2)
COS0
The second equation may be written
sin A
tan t =
cos <f> tan h sin cf> cos A
sin A
tan h(cos <j> sin </> cot h cos A)
sin A
tan h(cos sin tan 0)
sin A cos
tan h cos (0+0)
Eliminating tan h by (1) this becomes
tan A sin (3)
tan T = (j. i a\
COs(0 + 0)
Equations (1), (2) and (3) give the solution.
(2) Given the declination and hour angle of a star, and
the latitude of the place, to find the altitude and azimuth of
the star.
In the spherical triangle, Fig. 4, we have given
a = 90-8, c = 90-<t>, and = r
and b = 90-handA
are required. These are given by the second equations of (1)
and (5), Sph. Trig., which become
sin h = sm 8 sin 0+cos 5 cos < cos t
sin t cot A = cos tan 8 sin <f> cos t
These may be written
sin h = sin 6 (sin < + cos 4> cot 8 cos t)
. sin t
tan A = r? 1 = i Ti ^
tan 5(cos sin <f> cot 5 cos t)
Then substituting cot Q\ = cot 8 cos t (4)
they become
. , sin 5 cos(0i </>) (5)
sin h = r
sin 0i
sin t sin B\
tan .4 =
tan 8 sin (0i 0)
Then eliminating tan 5 from this last by (4) it becomes
. tan t cos 0i
tan A = sin^-0) (6)
0i being given by the equation
tan 8 (7)
tan 0i =
COS T
These two problems serve for the transformation from the
first system of co-ordinates to the second; and conversely.
(3) Given the altitude and declination of a star, and the
latitude of the place, to find the azimuth and hour angle.
In this case we have given
a = 90-5, b = 90-h, and c = 90-<f>
and are required to find
A =A, and B = t
These are given by the first and second of either set of
equations (6), (7) or (8), Sph. Trig. In these equations we
have
s =Ka+&+c) =9O-K0+5-r)
s-a =$(-c+6+c)= Kr+5-0)
s-b = \(a-b+c) =90-i(r+<+5)
sc = (a+b-c) = Kf+0-8)
so that on substituting 5' = |(f+0+5) they become
sin 2 \A =
cos s' sin (5' 5)
(8)
cos <f> sin f
cos 2 \A =
cos(s' f)sin(s' </>)
cos sin f
(9)
tan 2 \A =
cos 5' sin(/ 5)
cos(5' f ) sin (V 0)
(10)
^in 2 ir
sin (^' <)sin(s' 5)
(11)
cos < cos 5
cos 2 \r =
cos (V f) cos s'
cos < cos 5
(12)
+ o-i2 i^
sin(s' <f>) sin(s' 8)
(13)
cos s' cos(.s' f)
(4) Given the altitude, declination, and hour angle of a
star, to find its azimuth, and the latitude of the place.
The data here are
o = 90 -8, b = 90-h, and B = r\
and the required quantities
A=A, and c = 9O-0
These may be found by (3) and the second of (1), Sph.
Trig., which become
. sin r cos 5 (14)
sin A = - 7 v
cos h
sin & = sin 8 sin 0+cos 5 cos <f> cos t
This last becomes (see eq. 5)
. T sin 5 cos (01 0)
sin h = : zr
sin di
Then transposing, we have
fa x\ sin \ sin 6l ( 15 )
COS(0i <$>)= : z
sin 5
0i being given by the eq.
tan 5
tan 0i =
cos r
There may be two solutions of this problem; but the
ambiguity may be removed by first determining <j> and then
A by either of the equations (8), (9) or (10).
(5) Given the declination and azimuth of a star, and the
latitude of the place, to find the hour angle and altitude.
Thus we have
a = 90-5, A =A, and c = 9O-0;
and are required to find
B = t, and b = 90-h.
The first of these is given by the second of (5), Sph. Trig.,
which becomes
sin t cot A = cos 4> tan 5 sin <j> cos t
or sin r cot A-\~ sin (f> cos r = cos <f> tan 5
which may be thus transformed:
cot A (sin r + tan A sin < cos t) =cos tan 5
or, substituting tan 2 = tanVl sin <f> (16)
this becomes
cot A sin(r+0 2 ) . s
; = cos <b tan 5
cos 02
or, transposing
sin(T+0 2 ) =cos 4> tan 8 cos 2 tan A
Then eliminating tan A by (16) we have
sin(r-|-0 2 ) =cot <f> tan 5 sin 0j (17)
Equations (16) and (17) determine r.
We may now find h by applying one of equations (3),
Sph. Trig., to the astronomical triangle, which gives
7 sin t cos 5 (18)
cos h = ; - A
sin A
We may also find h directly from the data by means of
the first of (1), Sph. Trig., which gives
sin 8 = sin h sin 0+cos h cos <j> cos A ;
which may be written
sin 8 = sin </>(sin h-\-cos h cot <j> cos A);
in which substituting
cot 3 = cot (f> cos A
we have
sin 8= sin <(sin & + cos h cot 03)
sin <f> cos(h 63)
(19)
Also . tan <j> (20)
tan 3 = -.
cos A
(6) To find the altitude, hour angle, and azimuth of a
circumpolar star when at elongation, or maximum azimuth.
It is assumed that the latitude of the place is known.
When a star is at elongation the angle C, Fig. 4, is a right
angle, and the solution is given by equations (26), (28) and
(27), Sph. Trig., which become
. 7 sin<*> tan* . A cos 8 (21), (22), (23)
sin h= . r, cos t= r, sin A = .
sin 8 tan 8 cos *
sin
03
cos(h-
-w- Sln
8 sin
sin
03
tan 3
tan 4>
(7) To find the altitude and hour angle of a star when on
the prime vertical.
Here the azimuth A is equal to 90, and it is assumed
that 4> and 8 are given. Then applying equations (26) and
(28), Sph. Trig., we find
tan 5 . , sin 5 (24), (25 j
COS T =
tan $
sin h =
sin <f>
(8) Given the right ascension and declination of a star,
and the obliquity of the ecliptic, to find the latitude and
longitude of the star.
In the triangle PP'S, Fig. 3,
PS = 90 -5 P'S = 90-P
SPP' = 90 + a SP'P = 90-X
PP' = t
and we have by equations (1), (4)and (3), Sph. Trig.,
sin /3 = sin 8 cos e cos 5 sin e sin a
cos /3 sin X = sin 8 sin e+cos 8 cos e sin a
cos 13 cos X = cos 5 cos a
Then substituting
m sin M = sin 5
m cos M = cos 5 sin a
they become
sin /3 = m sin (Jlf e)
cos /3 sin X = m cos(Af c)
These may be written
}
(26)
(27)
(28)
(29;
tan M =
tan 5
sin a
. sin 5 sin(M e)
sin /3 = = ,j
sin M
tan a cos(Af e)
tan X =
(30)
cos M
The quadrant in which M is situated is determined by
equations (27), m being assumed always positive.
(9) Given the latitude and longitude of a star, and the
obliquity of the ecliptic, to find the right ascension and declin-
ation of the star.
As in the last case we have
sin 8 = sin /5 cos e+cos /3 sin e sin X _ "j
cos 5 sin a = sin (3 sin e cos /3 cos e sin X J- (31)
cos 5 cos a = cos /3 cos X ; J
in which substituting
n sin iV = sin /3 ) (32)
w cos N = cos sin Xf
9
they become
sin 8 = n sin(7V+e)
cos 5 sin a = n cos(iV+e)
cos 5 cos a = cos jS cos X
From these we derive
tan
(33)
(34)
(35)
tan N =
sin 5 =
tan a =
sin X
sin j3 sin(7V+e)
sin N
tan X cos(7V+6)
cos N
(36)
It
2. Time.
The sidereal day.
The earth's motion of rotation, as far as can at present be
ascertained, is uniform; though theoretical considerations
point to a possible retardation of its, velocity. If such retarda-
tion exists, its amount must be extremely minute, as up to
the present time none has been detected. The time of ap-
parent rotation of the starry sphere is therefore sensibly
constant, and may consequently be adopted as a unit of time
and be denoted the sidereal day. Owing to the proper
motions of the fixed stars the practical sidereal day is the
time of rotation of the vernal equinox.
Sidereal time.
The sidereal day is assumed to begin at the instant of
upper meridian transit of the vernal equinox, which point
will in future be denoted by and referred to as the point V;
and the sidereal time at any instant is the hour angle of V
at that instant. It is thus equal to the right ascension of
any star which is on the meridian of the observer at that
instant.
The solar day.
A unit of time dependent on the sun is necessary for the
purposes of daily life.
F/g.5
On account of the earth's orbital motion about the sun
the latter body has an apparent motion among the stars,
11
so that it returns to the meridian of a place nearly four
minutes later on any given day than on the previous day, as
shewn by a clock regulated to sidereal time.
This apparent motion of the sun, however, is not uniform.
The earth moves in an ellipse, of which the sun occupies
one of the foci, and its angular velocity about the sun varies
inversely as the square of its radius vector; the angular
velocity of the sun on the ecliptic therefore varies in the
same manner. An inequality in the lengths of the solar days
results from this; but a further irregularity is due to the
obliquity of the ecliptic; for, even if the sun's motion on the
ecliptic were uniform, its motion in right ascension would
not be so.
F/g. 6
This is illustrated in Fig. 6, which is a projection on a
plane perpendicular to the earth's axis. Pi and P% are two
consecutive positions of the earth in which the sun is on a
given meridian. The earth in the interval has performed a
complete rotation on its axis plus the angle M'PiM" ' , which
equals PiSP 2 , which is the angle through which the pro-
jection of the radius vector has revolved during the interval.
This angle varies from day to day, owing to the causes above
mentioned, viz., the eccentricity of the earth's orbit and the
obliquity of the ecliptic. The solar day, being equal in length
to the time of an absolute rotation of the earth on its axis
plus the variable angle P1SP2, is therefore variable in length.
The angle P1SP2 is clearly the motion of the sun in right
ascension in the solar day.
To obtain an invariable unit of time dependent upon the
sun astronomers invented a fictitious sun, called the mean
sun, and denoted by S in Fig. 7, which is assumed to move
at a uniform rate on the equator and to return to the vernal
equinox at the same instant as another fictitious sun Si,
12
assumed to move at a uniform rate on the ecliptic. Sy i*
also assumed to pass through perigee, and therefore apogee,
at the same instant as the true sun.
Fig. 7
The relative positions of the three suns at different times
of the year are shewn in Fig. 7. There, the points VBXA
shew the positions of the sun when the earth is at correspond-
ing points in Fig. 5.
Solar time.
Apparent solar time at any instant is the hour angle of
the true sun at that instant.
Mean solar time is the hour angle of the mean sun.
Apparent noon is the instant when the sun is on the meridian
of a place. Mean noon is the instant when the mean, sun is
on the meridian.
The equation of time is the difference between apparent
13
and mean solar time ; or, it is the difference of right ascen.si.yn
of the true and mean suns.
Tracing out the relative positions of the three suns in
Fig. 7 shews that the equation of time changes its algebraic
sign four times in the year, about April loth, June 14th,
Aug. 31st, and Dec. 24th. It therefore has four maximum
values.
Civil and astronomical time. 6-t*f+-9*<~ /?&& .
The civil day begins at the instant of lower meridian
transit of the mean sun, or at midnight; while the astro-
nomical day of the same date begins at upper meridian
transit 12 h. later.
Time at different meridians.
F/g.8
At any instant at two places in different longitudes, the
hour angles of the sun, or of V, differ by an amount equal
to their difference of longitude; consequently the difference!
between the local times of the two places, either solar cr
sidereal, is equal to their difference of longitude.
This as shewn by Fig. 8. Thus if PA and PB are the
meridians of two places, S and V the mean sun and the
vernal equinox, respectively; then the M.T, at A is the
angle APS , and the sidereal time the angle APV. The
corresponding times at B exceed these by the angle APB
(denoted by L).
Standard time.
For convenience, since 1883 the time used ac any place in
N. America, instead of being the local time of the place, is
theoretically the time which differs by the nearest who!e
number of hours from Greenwich time. This is called st.n-
14
The change in the astronomical day.
Since the beginning of the year 1925 the astronomical day
has been discontinued, or made coincident with the civil day.
This change necessitates certain modifications in the method
of converting sidereal to mean time.
Equations (37) and (38) remain unchanged in form, thus:
Q = (T+L)(l+k') + V -L (37)
T=(0+L)(l-k)+M-L (38)
In the former of these equations T-\-L denotes the G.M.T.
corresponding to the given time, and reckoned from midnight,
and V the G.S.T. at the previous Greenwich O h .
In the latter equation, if 0+L > 24* it must be diminished
by 24 /f before being reduced to the equivalent mean time inter-
val. The value of the quantity M must be that given in the
ephemeris for the immediately preceding Greenwich sidereal
noon. Thus, if, when M is taken for the given date, the
quantity
(e+L)(i-&)+M
is less than L, then M must be taken for the next following
date ; but, if that quantity is greater than 24 h -\-L, then M must
be taken for the previous date. For intermediate values M
must be taken for the given date.
dard erne. Thus, the time which differs by 5 h from Gr.
time 2s used at all points whose longitudes lie between i*
30 m and 5 h 30 m W. The following standard times ?ve
used in N America:
Atlantic, differing by 4 h from Gr time;
Eastern, differing by 5 h from Gr. time;
Central, differing by 6 h from Gr. time;
Mountain, differing by 7 h frorc Gr. time;
Pacific, differing by 8 h from Gr. time;
Yukon, differing by 9 h from Gr. time.
Relation between the lengths of the solar and sidereal units of time.
The tropical year is the interval of time between two cdb
secutive passages of the mean sun thiough the niran vern^
equinox
Fig. 9
In Fig. 9 let 5 and S' be the positions of the mean sui
relatively to 7 at the instants of two consecutive 1 ransiti
over the meridian of some place. Then it is evident that the
mean solar day is equal to the sidereal day plus the motion
of the mean sun in right ascension in one mean solar day.
(See also Fig. 6.) Therefore if
D the length of the solar day, and
D' = the length of the sidereal day, and
n =the number of mean solar days in 1 tropical year;
then
1 tropical year = rZ>
rv{D'+D')
n
= (-r l)D'.
15
2
But 1 tropical year = 365.24222 mean solar days
.'. 1 tropical year = 366.24222 sidereal days,
Tf then
M any interval of time expressed in mean solar days,
eS = the same interval expressed in sidereal days;
M 365.24222 , ,
TS^ 366^4222 ==1 -^ aSSUme;
. S 366.24222 , . ,,
*** M '365.24222 = 1+ *
in which = 0.00273043
'=0.00273791
Also
24 h M.S.T. = 24 h 03 m 56 s .555 Sid. T.
24 Sid. T. = 23 56 04 .091 M.S.T.
The cor. version of a given interval of M.S.T. into
the corresponding interval of Sid. T., or conversely, is
best effected by means of tables given in the Nautical
Almanac.
To convert the mean time at a given meridian into the corre-
sponding sidereal time.
Let T the given local M.T.;
G -the corresponding Sid. T ;
L =the longitude of the place;
V, =the Gr Sid. T. at the previous Gr. mean noon.
Then
e = (r4-L) (l+k') + V -L (37)
V is taken from the ephemeris. Instead of using the
factor k' the reduction of T-k-L to the equivalent sidereal
interval is made by means of tables given in the N. A
To convert the sidereal firz> at a given meridian into the
corresponding mean time.
Using e same notation, and in addition denoting by
M the mean time at Gr. of the previous Gr sidereal noon,
we have
r-(e-fL) {l-k)+M-L (38)
Here again the tables of the N A. are used instead of the
factor k.
The value of M, to be taken from the N. A., is to be that
for the date of the transit of V immediately preceding the
given time. Thus if
(9-j-L) (l-k)+M>2&
then the. value of M must be taken for the previous date.
16
To convert the apparent solar time at a given meridian into
the corresponding sidereal time.
This may be done by first reducing to M.T. by applying
the equation of time to be taken from the N. A. and then
reducing to sidereal time by the method given above.
A more convenient method, however, is to interpolate
from the N. A. the value of the sun's right ascension at^ the
Gr. instant corresponding to the given time. Then if in Fig. 8
So represents the true sun it is clear that if / = the hour angle
of the sun, or the apparent time, then
= ;+a (39)
To determine the hour angle of a heavenly body at a given
time.
If in (39) it is assumed that the hour angle / may have
any value up to 24 h , then that equation is general and
applies to every case and any heavenly body. It may be
necessary in some cases to deduct 24 h from t -\- a. Transpos-
ing we have
t = Q-a (40)
Here it may be necessary in some cases to increase by
24 h to render subtraction possible.
The hour angle denoted by r, found by solving the astro-
nomical triangle the parts of that triangle being limited
to values less than 180 , being given, we have
/ = r if west
= 24 h -r if east.
The hour angle of the sun may be found by equation (40)
if the sidereal time is given. If the mean time is given, it
may be reduced to apparent time by applying the equation
of time, thus finding the required hour angle.
Reduction of time to arc ; and conversely.
These reductions may be made by means of the following
numerical relations:
l h =15 l = 4 m
l m = 15' l' = 4 s
I s =15"
17
3. Determination of Time by Observation.
Correction and rate of a chronometer.
As the term implies the correction of a chronometer is
the amount that must be added to the chronometer time to
give the true time.
The rate of a chronometer is the amount it loses in a unit
of time.
Thus, if
7\ and !T 2 = the true times at given instants;
T\ and r' 2 = the chronometer time at those instants;
A7"i and A2" 2 = the chronometer corrections;
57"= the chronometer rate.
Then LT X =T X -TA (41)
AT 2 = T 2 -T 2 'f
Ar 2 -Ar x (42)
7Y-7Y
These equations give the corrections and rate with their
proper algebraic signs. The rate is thus given in terms of the
chronometer interval.
1st method By transits.
(a) Meridian transits.
A transit instrument having been adjusted in the meridian,
the time of transit of any heavenly body across the vert,
wire may be observed by a chronometer whose correction is
to be found.
If the chronometer is regulated to sidereal time the true
sidereal time of transit is at once given by the right ascension
of the body, whence the chronometer correction at once
follows by (41). If regulated to mean time, the sidereal
time of transit of the body must be reduced to mean time.
If the sun is observed, the time of transit of each limb
should be noted and the mean taken; thus finding the time
of transit of the centre. If only one limb can be observed,
then the observed time must be corrected by the "time of
semi-diameter passing the meridian", which may be taken
from the N. A., or computed by the equation
5 (43)
/=- =- sec 5 v '
15
in which 5 is the angular semi-diameter of the sun.
If the correction of a M.T. chronometer is to be found
by a transit of the sun, the true M.T. of transit may at once
be found by applying the equation of time to the apparent
time of transit (fi 1 .
18
(b) Transits across any vert, circle of known azimuth.
In this case the latitude of the place and the declination
of the heavenly body must be known; then the hour angle
may be computed by means of (16) and (17), which may be
written
tan = tan A sin </>
sin(r+0) =cot </> tan 5 sin 6
The sidereal time then follows by (39), or the M.T. by apply-
ing the equation of time as already shewn.
The rate of a chronometer may be found by observing two
consecutive transits of a star across the same vert, circle.
The true interval between the transits is
24 h Sid. T. or 23 h 56 m 04 s .09 M.T.
(c) Transits across the vertical circle of Polaris.
This method will be described under Azimuth.
2nd method By a single altitude.
The method of observing an altitude of a heavenly body is
described below, p. 65 et seq.
Corrections to be applied to an Observed Altitude.
(a) Refraction.
The ray of light that reaches an observer from a star, in
traversing the earth's atmosphere is continually bent down-
wards from a rectilinear path by the increasing refractive
power of the air with density as the surface of the earth is
approached. In consequence, the apparent direction of a
F/g. /O
star is that of a tangent to the curved path of the ray at the
point where it reaches the observer. This is illustrated in
Fig. 10.
19
An observed altitude must then be diminished by an
amount equal to the angle between the final direction of the
ray and the straight line drawn to the star, as appears in the
figure. The magnitude of r decreases as the altitude increases,
and its value is best found from tables. These contain
corrections depending upon the readings of the barometer
and thermometer. An approximate value of r may be found
by the equation
r = bl".l tan f
or a closer approximation by the formula
9836
r= 460+7 tanr
in which
b =the barometer reading in inches; and
/ = the temperature of the air in degrees F.
(See Field Astronomy for Engineers, by Prof. G. C. Comstock).
(b) Semi-diameter.
In observing the sun or moon the altitude of its upper or
lower limb is observed. To find the altitude of its centre a
correction for semi-diameter must be applied. This may be
found in the N. A.
(c) Parallax.
As the centre of the celestial sphere is coincident with that
of the earth, if the directions of a heavenly body from that
point and from a point on the earth's surface differ sensibly,
F/G //
then a correction must be applied to any observed co-ordinate
to reduce it to the centre of the earth. This is only necessary
with members of the solar system.
20
In Fig. 11 .S is the centre of the heavenly body observed,
the centre of the earth, A the point of observation. The
triangle A SO gives
sin p = sin C
p being the parallax in altitude. If f' = 90 the resulting
value of p is the horizontal parallax. Denoting it by ir we
have
sin 7r =
a
sin p = sin w sin $"' ; (44)
or very nearly
P = t sin f ' = it cos &' (45)
This gives the correction for parallax with sufficient accuracy
for any body except the moon.
(d) Dip of the horizon.
At sea the altitude of a heavenly body is measured with a
sextant from the sea horizon, the observer standing on the
deck of a ship. A correction must therefore be applied to
the observed angle on account of the dip of the visible below
the true horizon.
Fte. /2
F/G./3
In Fig. 12 we have from PI. Geom.
AB J(2a + h)h V2ah
tan D' =
a
a
a
nearly
or
D' =
V
2)i
a
This gives the dip uncorrected for refraction; but, as shewn
in Fig. 13,. the ray of light which reaches the observer from
the horizon follows a curved path, so that the apparent dip
21
D is less than D' . The mean value of the ratio of D to D'
is .9216 : 1, so that
D = .9216 -i/
2h_
a
or in seconds of arc
.9216 /IT
D =
\"
sinl' v a
Substituting a mean value of a in feet, this becomes
D = 58". 82 JT (46)
h being in feet.
The rule known to navigators: "Take the square root of
the height of the eye above sea level in feet and call the result
minutes", is thus very approximately correct.
Having applied the necessary corrections to the observed
altitude, the reduction may be made by either of the equa-
tions (11), (12) or (13). If a number of observations are to
be reduced an equation derived as follows is more convenient:
Taking the equation
cos f = sin 5 sin 0+cos g cos cos t
it may be written
1 versin f = sin 5 sin 0+cos 5 cos $(1 versin r)
= cos(0 5) cos cos 5 versin t
= 1 versin (0 5) cos cos 5 versin t
versin f versin (0 5) (47)
versin r = r
cos cos 5
This requires the use of tables of natural and logarithmic
versins. In the absence of such a table the following form of
the equation may be used
. , cos(0-5)-cos f (48)
Sin^ f r = r
2 cos cos 5
Example. The following observations were taken with a
sextant and artificial horizon on Aug. 1, 1892, at a place
in latitude 52 31' 04", and approximate longitude 7 h 50 m W.;
to find the watch correction.
2 -alt. O Watch
52 11' 30" 7 h 21 m 29 s A.M.
10 22 54
30 24 27
30 25 28
00 28 18
10 29 52
10 30 54
Index error =+20".
52
38
53
05
53
24
54
16
54
44
55
03
First find the approximate Gr. M.T., thus:
Mean of extreme times = 7 h 26 m ll 8
Ast. time, July 31 =19 26 11
Long = 7 50
Gr. M.T., Aug. 1 = 3 16 11
For this time we take from the N. A.
8 =+17 48' 56"
S= 15 48
E= 6 ra 03 s .6
Reduction of first observation:
Obs'd. 2-alt. =52
Index error
h'
r
P
h
f
Eq. (13) s'
s'<f>
s'-S
log sin (s' 4>)
log sin (s' 8)
log cos s'
log cos 0' f)
11' 30"
+20
2)52
= 26
11 50
05 55
1 52
26
04 03
15 48
25
48 15
8
= 25
= 64
48 23
11 37
67
15' 48"
.5
14
44 44
.5
49
26 52
.5
3
04 11
.5
9.405738
: 9.880708
9.587143
; 9.999377
9.286446
9.586520
log tan 2 \t
= 9.699926
log tan \t
= 9.849963
h
= 35 17' 38".9
T
= 70 35 17 .8
= 4 h 42 m 21 s .2
23
'. App. Time
= 7 17 38
.8
E
6 03
.6
Mean Time
= 7 23 42
.4
Watch
= 7 21 29
AT
= + 2 13
.4
Having reduced the remaining observations the complete
results are as follows:
AT
+2 m 13 s
.4
16
.0
13
.2
15
.0
15
.3
14
.3
15
.2
Mean =+2 14
.6
Another example will be found on p. 43.
To find the effect of errors in the data on the time computed
from an observed altitude.
Taking the equation (see Fig. 4) :
cos b cos a cos c sin a sin c cos B =
and differentiating by means of the expression
%dB= M- da+-^-db+ # dc
dB da db dc
we find
sin a sin c sin BdB =
(sin a cos c cos a sin c cos B)da
sin bdb-\-{cos a sin c-f-sin a cos c cos B)dc
= sin b cos Cda sin M&+sin & cos ylrfc
by applying equations (4), Sph. Trig. Then substituting in
the left-hand number
sin a sin B = sin b sin ^4
we have
JT> cos CV/a d& </c
dB = ; t ; j +
sin c sin ^4 sin c sin ^4 sin c tan A
Then introducing the astronomical co-ordinates, and re-
membering that
da=d8 db=dh dc=d(j)
we have finally
, cos Cd8 dh d(j> (49)
cos <t> sin A cos < sin A cos < tan .4
24
The errors being small may be regarded as differentials,
so that (49) gives the effect of errors in 8, h, and <f> upon the
resulting hour angle r. It shews moreover that the effect of
those errors is least when A and C are both large, or when the
star observed is near the prime vertical.
3rd method By equal altitudes of a heavenly body.
Method of observation with a transit or sextant.
Equal altitudes of a heavenly body on opposite sides of
the meridian correspond, generally speaking, to equal hour
angles. This is the case of a fixed star, whose change of
declination between the two positions may be neglected.
The mean of the times of equal altitudes is then the time of
meridian transit. The method is therefore an indirect one
for observing the time of meridian transit.
In the case of the sun, however, allowance must be made
for the change of declination in the interval between the two
observations. An expression for the correction to be applied
to the mean of the observed times is derived as follows:
F/e. 14
Fig. 14 shews a projection of the celestial sphere on the
plane of the horizon. S\ and 52 are the two positions of the
sun's centre at the instants of the two observations; S\ the
position it would have occupied if there had been no change
of declination. The two triangles PZSi and PZS't are then
equal in all respects. It is therefore required to find the
change of hour angle resulting from a small change of declin-
ation. Taking the equation
cos f sin 5 sin cos 5 cos <t> cos r
25
we find by differentiation
dr cos 5 sin sin 5 cos # cos t
d8 cos 5 cos <j> sin t
tan <f> tan 8
If we now write
this becomes
sin r tan t
dr=-2AT d8 = 2A8
-2Ar =(^-^Y2A5
\ sin t tan r /
or in seconds of time
. AS /tan </> tan 5\
15 \sin r tan t/
(50)
This is the "equation of equal altitudes."
In this equation
AT = the correction to be applied to the mean of the
observed times to find the time of meridian transit;
A5 =half the change in the sun's declination in the
interval between the observations, positive if the
sun is moving north.
t may be assumed equal to half the elapsed interval between
the observations. Attention must be paid to the algebraic
sign of 8; it is positive if north.
The advantages of this method are that the absolute
altitudes need not be known; and small errors in 4> and 8
have no appreciable effect.
To find the time of rising or setting of a heavenly
body.
Take the equation
sin A = sin 8 sin #+cos 5 cos <f> cos t;
which may be written
cos r = sin h sec <f> sec 5 tan <j> tan 5 (51)
In the case of the sun, when its upper limb is just visible
in the horizon it is in reality below the horizon by the amount
of the refraction, 34' approximately; and its centre is below
the limb by the amount of the semi-diameter, which may
be taken as 16'; parallax may be neglected. Therefore
}i = 50', and sin 50' = 0.0145 ; .". the above equation becomes
cos t= 0.0145 sec <t> sec 5 tan <f> tan 8 (52)
The time of rising of the moon's centre is usually computed.
In this case the effect of parallax is important. Assuming its
amount as 57', the altitude of the moon's centre when it is
26
apparently in the horizon = 57' -34' = 23'. Also sin 23' =
0.0067; so that (51) becomes
cos t = 0.0067 sec <f> sec 8 tan <f> tan 8 (53)
Having computed the hour angle, the time readily follows.
Construction of sun dials.
The horizontal dial and the prime vertical dial only will
be considered.
In any form of dial the edge of the gnomon which casts the
shadow must be parallel to the earth's axis, as the position
of the shadow cast upon any plane is then independent of
the sun's declination
Fig. 15 shews the construction of the horizontal dial.
The edge of the gnomon if produced will intersect the celestial
F/0 /6
sphere in the pole P, Fig. 16. PON is the meridian plane,
NOL a horizontal plane, and POL a plane through the
sun's centre. LON (denoted by a) is the angle which an
hour line, corresponding to a given hour angle t, makes with
the noon line. The triangle PLN then gives
tan a = sin cf> tan t (54)
27
The construction for a prime vertical dial is shewn in
Fig. 17. OPZ' is the meridian plane; OMZ' that of the
Fig. 17
prime vertical; and OP'M a plane through the sun's centre.
is the required angle corresponding to the hour angle t.
The triangle P'MZ' gives
tan /3 = cos tan t (55)
A sun dial gives apparent solar time.
fi
4. Determination of Latitude by Observation.
As shewn on p. 3, the latitude of a place is equal to the
altitude of the pole, or the declination of the zenith, i.e., to
either arc PN or EZ, Fig. 2.
1st method By meridian altitudes or zenith distances.
Fte/8
If the altitude or zenith distance of a heavenly body be
observed when crossing the meridian, and the necessary
corrections be applied, the latitude at once follows by one
of the following equations, depending upon the position of
the body. For the star
So. . . .0 = f + 5 (5 being negative) (56)
S 3 ....<j>=5-{ = h-p
S A . . . .0 = 18O-5-f = fc+p
If S 3 and S 4 are the positions of the same star observed at
both culminations, then by taking the mean
h+h' p-p' (57)
* = ~2 2T
the accented letters belonging to lower culmination.
If S\ and 53 are two stars observed at nearly equal zenith
distances, we have by taking the mean of the first and third
of (56)
5 + 5' r-f (58)
</,= ~2~ + ^r
the accented letters belonging to the north star. This formula
is the basis of Talcott's method of determining latitude, the
observed quantity being the difference of zenith distance of
the two stars, which are selected so that that difference is
small enough to be measured by a filar micrometer placed in
the focus of a telescope. Details of method outlined.
29
If the direction of the meridian is not known the maximum
altitude of the heavenly body may be observed. If that
body is the sun the maximum altitude differs slightly from
the meridian altitude, owing to its rapidly changing declin-
ation. The resulting error is entirely negligible, especially
if instruments of only moderate precision are used; its
value is given by the expression
()' ^^ or [5.54861, (^(tan,-tan 8 )
in which A5 is the hourly change in the declination expressed
in seconds. The correction is always subtractive.
Example. On July 10, 1914, the meridian altitude of
the sun's upper limb was observed (Cir. L) to be:
68 11' 30".
To find the index error of the transit used the following
V.C.R's were taken on a terrestrial point:
Cir. L 034'30"
Cir. R 31
Diff.
I.E.
Obs'd alt.
I. E.
h'
r
3 30
1 45
= 68
11' 30"
1 45
= 68
09 45
23
68
09 22
15 46
67
53 36
3
= 67
= 22
= 22
53 39
06 21
17 48
= 44
24 09
h
f
5
2nd method By an altitude observed out of the meridian,
the time being known.
30
To the observed altitude the necessary corrections must
be applied, and the hour angle derived from the observed
time. The latitude then follows by means of (15)
, i. sin h sin
COS(d> d)= :
v J sin 8
6 being found by the equation
a tan 5
tan =
cos r
To find the effect of errors in the data we have by trans-
posing (49)
d<t>= cos C sec Add + sec Adh-\- cos "<j> tan Adr (59)
This equation shews that the effect of errors in the data is
least when A is small and C large, though the second con-
dition is unimportant, as the error in the declination is always
small in comparison with the other errors. These conditions
are fulfilled, however, by observing a close circumpolar star
near elongation.
Hence the method by means of the pole star.
As the altitude of this star never differs much from the
latitude, the method consists in computing a correction to
apply to the former to give the latter. An expression for this
correction is derived as follows:
Taking the equation
sin h = sm < sin 5+cos 4> cos 5 cos t
and substituting in it
(j) = h-}-x
8 = 90-p
we have
sin /* = sin(/z+x)cos p-\-cos(h-\-x)s'm p cos t
Then expanding the sin and cos of h+x, and again expanding
the sin and cos of x and p and neglecting the powers of their
circular measures above the second, we have
sin h = \ sin h ( 1 -~- ) + x cos h > ( 1 ~ )
+ ) cos fe I 1 J x sin h Kp cos r
x 2 . p 2
= sin h sin h-\-x cos h ~ sin h + p cos f cos h
z
px cos t sin //.
Whence
x cos h= p cos h cos T-\-\(x 2J rp"--\-2px cos t) sin h
or x= p cos T-\-\{x 2 -\p 2 -\-2px cos t) tan h.
31
3
Assuming now as a first approximation
x= p COS T,
and substituting in the right-hand member, we have
x= -p cos r + i( 2 cos 2 r+p 2 -2p 2 cos 2 t) tan h
= p cos r + 2 sin' t tan h
or in seconds of arc
x= p cos T-\-hp 2 sin 1" sin 2 t tan /z
We have then finally
<f> = h-p cos t + | 2 sin 1" sin 2 r tan h (60)
The effect of the omission of the smaller terms in the
above expansions can never amount to 0".5.
Example. The following observations of Polaris were
taken on June 14, 1904, with a small transit:
Cir. V. C. R. Watch
R. 45 44' 14 h 50 04 s
L. 45 43 53 46
R. 45 45 57 10
L. 45 44 59 44
The watch was regulated to sid. time, and its correction was
20 s . The star's co-ordinates were:
a = ih 24 m 26 s
8 =88 47' 27" (."./) = 4353").
The mean of the first and second observations being taken,
and that of the third and fourth, the reduction is made as
follows:
Eq. (60) r = 14 h 51 m 55 s = 14 h 58 m 27 s
AT = -20 = -20
6
a
t
T
V
r
h
log p
log COS T
= 14 51 35
= 1 24 26
= 13 27 09
= 10 32 51
= 158 12' 45"
= 45 43 30
56
= 45 42 34
= 3.638789
= 9.967813
log 1st term = 3.606602w
=
14 58 07
1 24 26
=
13 33 41
10 26 19
156 34' 45"
=
45 44 30
56
=
45 43 34
=
3.638789
9.962659w
=
3.601448w
32
log 0.5
log 2
log sin 1"
log sin 2 r
log tan h
log 2nd term =
h
1st term
2nd term
<t>
1.698970
7.277578
6.685575
9.139134
10.010756
=
0.812013
45
1
42'
07
34"
22
6
=
46
50
02
= 1.698970
= 7.277578
= 6.685575
= 9.198634
= 10.011009
= 0.871766
= 45 43' 34"
= 1 06 34
7
46 50 15
Mean = 46 50' 08"
Circum-meridian Altitudes.
If a number of altitudes of a star be observed in quick
succession when near the meridian, each will differ by but a
small amount from the meridian altitude. A correction may
then be computed for each altitude which, when applied to
it will give a value of the meridian altitude. The mean of
these resulting values having been taken the latitude then
follows by means of one of the equations (56).
To find an expression for this correction we return to the
equation
sin h = s\n <$> sin 5 + cos cos 5 cos t,
which is transformed as follows :
sin A = sin (/> sin 5+cos <f> cos 5(12 sin-^r)
= cos(</> 5) cos <f> cos 8 . 2 sin 2 f t
= cos f cos <j> cos 8 . 2 sin 2 ^ t
= sin h cos 4> cos 8 . 2 sin 2 | r
by (56), f being the meridian zenith distance and h the
meridian altitude. If we now write
h = h y
we have sin h = s\n{h ^y)=sin h y cos h
by expanding and discarding powers of y above the first.
Substituting in the above expression for sin h, it becomes
sin h y cos h = sin h cos < cos 8 . 2 sin 2 \ r
or cos d> cos 8
y =
2 sin 2 h t
cos h
or in seconds of arc
cos </> cos 5 2 sin 2 \ t
cos h sin 1"
This gives the required correction.
(61)
33
If higher powers of small quantities be retained in the
above expansions the expression for y becomes
y = y'~ ^^tan * y* + 25"J*(i+3 tan \) y' - (61')
in which y' =
6
cos 4> cos 5 2 sin 2 \ t
cos h ' sin 1
//
and log^- =6.3845449, log S -^- = 12.5929985.
Then h = h-\-y
In applying this method stars giving values of h approach-
ing 90 5 must be avoided. If = 45 a and h = 60 the second
term of (61') will not exceed 1" for values of t less than 13 m ,
and the third term for values less than 33 m .
The value of log m is given in Table III.
Example. The following observations were taken with a
sextant and artificial horizon on Sept. 2, 1893:
2 -alt. _ Watch
89 59' 15" ll h 53 m 36 9
90 00 15 56 37
90 00 45 59 28
89 59 15 12 03 57
89 58 30 05 46
89 57 30 07 11
89 55 15 09 13
Index error = 0; watch correction = 8 s .
An approximate value of the latitude is found by regarding
the maximum observed altitude as the meridian altitude, as
follows :
Max. 2 -alt. =90 00' 45"
Eccentric error +2 00
Obs'd alt.
r
90
= 45
02 45
01 22
58
45
00 24
15 54
45
16 18
6
p
h =45 16 24
34
f =44 43 36
5 = 7 37 54
(approx.) = 52 21 30
The hour angles corresponding to the observed times may
be found by first finding the watch time of culmination, thus
App. time of culm'n = 12 h 00 m 00 s
E = -21
M.T. =11 59 39
AT = 08
Watch time of culm'n =11 59 47
From this follow the hour angles tabulated below. The
corrected zenith distances are also found as above. We then
proceed as follows:
log cos 4> =9.785843
log cos 8 =9.996136
log cos h =9.847403
9.781979
9.934576
logra =1.87545
log (h -h) =1.81003
h -h =64".57
The remaining corrections are computed in a similar
manner, and are tabulated below.
f t h h f
44 44' 21" 6 m 11 s 1' 05" 44 43' 16"
43 51 3 10 17 34
43 36 19 00 36
44 21 4 10 29 52
44 43 5 59 1 00 43
45 13 7 24 1 32 41
46 21 9 26 2 30 51
Mean =44 43 39
6 = 7 37 54
<f> =52 21 33
35
3rd method By two observed altitudes of a star, or the
altitudes of two stars, and the elapsed time between the
observations.
In addition to the latitude this method also serves to
determine the time and azimuth.
F/g/9
Let S\ and S 2 be the positions of the star or stars at tht
instants of observation. The first step in the reduction is
to determine the difference of hour angle S1PS2. If the sun
is observed twice, this angle is equal to the elapsed interval
of apparent time between the observations, though usually
the effect of the change in the equation of time may be
neglected. If one fixed star has been observed the angle
S1PS2 is equal to the elapsed sidereal interval between the
observations. If two stars are observed at the times T\ and
r 2 , the right ascensions being ai and a 2 , then
S.PS, = ( ai - a,) - (TV- r 2 ) (62)
Si being the more easterly star. The interval 7\ T 2 must be
in sidereal time.
Then, PS\ and PS? being known, the triangle S1PS2 may
be solved, finding S1S2 and PS1S2. The three sides of the
triangle ZS1S2 are now known, so that it mav be solved,
finding the angle ZSiS 2 . Then PSiZ = ZS 1 S 2 -PS 1 S 2 . The
triangle PZS is finally solved, finding PZ the co-latitude.
Completing the solution gives also the hour angle ZPS and
the azimuth PZS.
This method is further developed in works on navigation,
in which graphical solutions are given.
4th method By transits of stars across the prime vertical.
A star whose declination lies between the limits and <
will cross the prime vertical above the horizon twice in its
diurnal course.
The times of transit across the p. v. may be observed by
means of a transit adjusted in the p. v. If S\ and S 2 are the
two positions of a star at the instants of observation, then
the elapsed sidereal interval between the observations is
equal to the angle SiPS 2 , and half that interval is the hour
F/G.20
angle of the star at either observation. Transposing eq. (24)
we have
tan 4>
tan 8
(63)
COS r
by which the latitude may be found.
This method is little used with small instruments, but
when applied to the astronomical transit instrument it is
one of the most precise methods known for determining
latitude.
5th method -By observations of stars at elongation.
If two circumpolar stars be selected, whose times of elonga-
tion, one east and the other west of the meridian, are not
widely different, we have for the two stars, applying eq. (23)
cos 5i A cos <5 2 (64)
whence
sin A\= sin A 2 =
cos 4>
sin A 1 cos 5i
sin A2 cos 8 2
cos
From this by composition and division
sin ^4i+sin A2 cos 81 -f- cos 5 2
sin A\ sin A 2 cos 81 cos 82
or
tan %(A!+A 2 )
= - cot \ (5i + 5 2 ) cot \ (5i-5 2 );
tan %(Ai-A 2 )
from which finally
tan %(Ai Ai) = -tan ^(^4 1 +i4 2 )tan |(5i + 5 2 )tan J(*i k) (65)
From this may be found the difference of the azimuths of
the two stars when their sum is known. The sum of the
azimuths may be observed by poirting the telescope of a
transit to each star in turn, when at elongation, noting the
37
readings of the horizontal circle and taking their difference.
From the sum and difference of A\ and A* their separate
values may be found. The latitude then follows by either
equation
cos 5i cos 5 2 (66)
cos 4> =
sin^4i sin^lj
This method was due to Prof. J. S. Corti.
The best stars for observation are those having large
azimuths when at elongation, or whose declinations do not
greatly exceed the latitude. Their elongations then occur
at high altitudes, and therefore this principle must not be
pushed to an extreme, as the effect of an unknown inclina-
tion error of the horizontal axis of the transit increases
rapidly with the altitude.
5. Determination of Azimuth by Observation.
1st method By meridian transits.
The time of meridian transit of any star may be computed
as shewn on pp. 11 and 16. If the correction of a chrono-
meter be known, the chronometer time of transit may be
found. By directing the sight line of a well adjusted transit
to the star at that instant, it will thus be placed in the meri-
dian plane, and a meridian line may then be established on
the ground; or by horizontal circle readings when pointing
to the star and a mark, the azimuth of the latter may be
determined.
It is clear that a slow-moving circumpolar star is best ior
this observation, as then the effect of an error in the computed
time of transit is a minimum. The rate of change of azimuth
of a star when crossing the meridian is given by the relation
A A 1 - A cos 5 (67.)
AA = lo.Ar -r 7- - rr
sin \4> 8)
(see eq. 75) AA being expressed in arc and At in time. In
the case of the pole star over 2 m are required for a change
of azimuth of I', when crossing the meridian.
2nd method By transits across any vertical c'~Je, the
latitude being known.
Having computed the hour angle from the observed time,
the data of the problem are r, 8, and <i>, and the azimuth of the
star may be computed by means of (6) and (7), or
tan 5 . tan t cos 6
tan 9 = , tan A - n r-
cos t sin 'yd <p)
The same considerations as in the last method lead to the
choice of a close circumpolar star for this observation. The
equation from which (6) was derived may be placed i \ more
convenient forms. Thus it may be written,
tan A =
cos tan 5 sin <f> cos
then multiplying the right-hand member through by sec j>
cot 5, this becomes
. sec <j> cot 8 sin r (68)
tan A ~z - .
1 tan <f> cot 5 ccs r
This form is convenient when subtraction log's are available.
(See Manual of Survey of Dominion Land*.)
39
Again, the above equation (68) may be written
. tan A '
tan A =
1 m
in which
tan A ' = sec </> cot 5 sin r (a)
and m=ta.n </> cot 5 cos r
= tan A' sin 4> cot r (&)
Taking logarithms and expanding, we have
log tan A =log tan A' log (1 m)
= log tan yl'+/x (m-\-\ m 2 -\-\ m 3 -\-)
= log tan A' +ii m-\-\ /j, m--\-l /x w 3 + (69)
H being the modulus of the common system of logarithms-
We also have
log M =1.6377843
log |/i = 1.3367543
log \n =1.1606631
The algebraic sign of m is the same as that of cot t. The
third term of (69) is seldom required.
In taking the observation the procedure is as follows:
Point to the reference point and note H.C.R.
Then point to the star, note time and H.C.R.
Then reverse instrument and again point to the star and
note time and H.C.R.
Then point to the reference point and note H.C.R.
The means of the H.C.R's on the star and reference point
are then taken, increasing or diminishing one in each case by
180; and also the mean of the times of pointing to the star,
from which the hour angle is derived.
Having computed the azimuth by (68) or (69), let:
A s denote the azimuth of the star reckoned from the north
in the direction ESW;
A p that of the reference point.
R s the H.C.R. on pointing to the star
Rp that on pointing to the reference point.
Then A p -A, =R P -R s
or A p =A S +R P -R s (70)
40
Example. The following observations were taken in
Aug., 1904, at a place in latitude 46 54':
PL obs'd. Cir. H.C.R. Watch
R-P. R. 178 14'.5
Polaris R. 57.5 15 h 55 m 08 s
Polaris L. 181 02 .5 16 01 05
R.P. L. 358 14 .5
The watch correction was found by observing the meridian
transit of a Scorpii, as follows:
Watch time of transit = 16 h 23 m 00 s
R't ascension of star =16 23 34
Watch corr'n = +34
From the N.A.
a (of Polaris) = l h 25 m 03 s
8 =88 47' 28"
.'. p =4352"
The computation then proceeds as follows:
Mean of obs'd times = 15 h 58 m 06 s
Watch corr'n = -f 34
Eq. (40) Sid. time =15 58 40
a = 1 25 03
1 =14 33 37
T = 9 26 23
= 141 35' 45"
Eq. (68) log sec = 10.165405 log tan <j> = 10.028825
log cot 8 = 8.324328 log cot 8 = 8.324328
log sin r= 9.793235 log cos r= 9.894122w
8.282968 8.247275w
Subt. log= 0.007610
log tan 4= 8.275358
A= 1 04' 48"
Eq. (70) A s = 1 04' 48"
R P =178 14 30
179 19 18
Rs = 1 00 00
A p -178 19' 18"
41
The computation by (69) is as follows:
Eq. (a)
Eq. (6)
iog sec
log cot 8
log sin t
= 10.165405
= 8.324328
= 9.793235
log tan A'
log sin
log cot r
= 8.282968
= 9.863419
= 10.100887
8.282968
log m
= 2.247274
log n
= 1.637784
log ^ w
= 3.885058
- 0.007674,6
log m 2
= 4.494548
log | m
= 1.336754
a
log ^ m w 2
= 5". 83 1302..
..0.000067,8
log tan A
= 1 04' 48".
8.283035,8
8.275361,2
A
07
This method may be used to advantage in finding the
variation of a compass. An explorer's instrumental equipment
may consist of a sextant and a compass. With the former
instrument an observation of the sun for time may be taken.
If the compass bearing of the sun's limb then be taken, the
true azimuth of that body may be computed in terms of
t 8 and 0, which, compared with the magnetic azimuth, will
give the variation. The quantity 6 1 sec h must be added to
or subtracted from the azimuth of the sun's centre to obtain
the azimuth of the limb, h is given by (5) and need only be
known approximately.
The equations then are:
tan 5 . 7 sin 6 cos (0 0)
tan = sin h = -.^ '-
cos r sin
. tan r cos A . ,
tan A = . ,_ ^ AA=S sec h
sin (0 0)
The best time for this observation is when the sun is near
the prime vertical.
3rd method By an observed altitude.
The method of observation is described on p. 65 et seq.
The data are h 8 and 0, and the reduction is made by one
of the equations (8), (9) or (10).
Example. The following observations of the sun for
azimuth and time were taken on July 30, 1914, at a place
in latitude 44 24' 09", and approximate longitude 5 h 18 m 15 s W.:
42
Pt. obs'd. Cir. H.C.R. V.C.R. Watch
R.P. R. 23 26'.5
_| R. 219 22 2826'.5 4 h 56 m 39 s p.m.
|o~ L. 40 34 27 23 .5 59 47.5
R.P. L. 203 26
The reduction is as follows:
To find the azimuth:
Mean of V.C.R's. =27 55' 00"
r 1 49
P
h
r
4>
8
s'
s'-4>
s'-8
s'-Z
log cos s'
log sin (V 5)
log cos(>'-f)
log sin (s' <t>)
27 53 11
8
= 27 53 19
= 62 06 41
= 44 24 09
= 18 34 07
= 62 32 28
= 18 08 19
= 43 58 21
= 25 47
= 9.663807
9.841555
9.999988
9.493203
9.505362
9.493191
To find the time:
log sin (>'-</>) =9.493203
logsin(5'-5) =9.841555
log cos s' =9.663807
logcosO'-f) =9.999988
log tan \t
h
T
A.T.
E
M.T.
Stand. T
Watch
AT
9.334758
9.633795
logtan^r =9.670963
= 9.835481
= 34 23' 54"
= 68 47 48
= 4 h 35 m ll s .2
= +6 16 .0
= 4 41
18
27 .2
15
= 4 59
= 4 58
42 .2
13 .2
= +1
29 .0
log tan 2 \A
= 10.012171
log tan \A
= 10.006085
\A
= 45 24' 05"
A
= 90 48 10
A s
= 269 11 50
Rp
= 23 26 15
292 38 05
Rs
= 219 58
A,
= 72 40 05
43
Example. The following observations were taken with a
small transit in Sept., 1899, to determine azimuth, time and
latitude.
PL obs'd. Cir. H.C.R. V.C.R. Watch
R 157 10' 27 21' 7 h 43 m ll s p.m.
L 337 58 27 06 46 23
R 158 42 26 46 .5 49 35
L 339 37 26 19 53 24
R 160 22 25 59 .5 56 43
L 341 16 25 33 8 00 29
R 225 45
L
Arcturus
Arcturus
Arcturus
Arcturus
Arcturus
Arcturus
R.P.
Altair
Altair
Altair
Altair
Altair
Altair
R
L
R
L
R
34 23
34 25
34 27 .5
34 29
34 30
34 32
16 55
19 11
21 18
23 00
24 45
27 19
A mean time watch was used. Arcturus was to the west of
the meridian, and Altair near the meridian and east of it.
The approximate meridian altitude of Altair was observed
to be
34 39' 30"
whence a value of the latitude for reducing the azimuth
observations was found as follows:
h' = 34 39' 30"
r = 1 23
h
= 34
38
07
.<-
= 55
21
53
5
= +8
36
23
$
= 63
58
16
The apparent places of the two stars were:
a 5
Arcturus 14 h ll m 05 s +19 42' 24"
Altair 19 45 55 +8 36 23
The reduction of the first azimuth observation is as follows:
V.C.R.,Cir.i? =27 27'
V.C.R., Cir. L =27 06
Mean
r
h
= 27
16
30"
=
1
51
= 27
14
39
= 62
45
21
44
Eq.
(10)
(70)
s'
s'<f>
s'-h
*'-r
log cos s'
log si n(Y 5)
log cos^ f)
log sin(V <j>)
log tan 2 \A
log tan \A
u
A
A s
Rp
Rs
Ap
= 73 13' 00".5
= 9 14 44 .5
= 53 30 36 .5
= 10 27 39 .5
= 9.460524
= 9.905235
= 9.992721
= 9.205930
9.365759
9.198651
Eq.
= 10.167108
= 10.083554
= 50 28' 40".5
= 100 57 21 .0
= 259 02 39
= 225 45
484 47 39
= 157 34
= 327 13 39
Reducing the remaining azimuth observations in a similar
manner, and taking the mean, the result is
A p =327 13' 31"
In order to reduce the latitude observations it is necessary
to find the hour angle of Altair corresponding to each of the
observed times. This may be done by computing the hour
angle of Arcturus from the observations of that star, and
combining it with the difference of right ascension of the two
stars. Thus:
Eq. (13) logsin(>'-0) =9.205930
logsin(s'-5) =9.905235
log cos 5' =9.460524
logcos(s'-r) =9.992721
9.111165
9.453245
log tan 2 A' =9.657920
45
log tan \r =9.828960
\t =33 59' 54"
t =67 59 48
= 4 h 31 m 59 s .2
Reducing the remaining observations in the same way, the
hour angles are:
4 h 31 m 59 s .2
38 45 .1
45 54 .4
Mean = 4 38 52 .9
The difference of r.a. of the two stars is
5 h 34 m 50 s ;
therefore the hour angle of Altair
= 4 h 38 m 53 s
-5 34 50
= -55 57
(the star being east of the meridian) at an instant equal to
the mean of the observed times, or
7 h 51 m 37 s .5
Then as the change of hour angle of a star is equal to the
change in the sidereal time, the hour angle of Altair at the
time of the first latitude observation is found as follows:
Observed time, 1st obs'n = 8 h 16 m 55 s
Mean of times of az. obs'ns = 7 51 37 .5
Diff. = 25 17 .5
Equivalent sid. interval = 25 21 .7
Hour angle at mean of times = 55 57
Hour angle at 1st lat. obs'n. = 30 35
The hour angles of Altair are thus found to be
-30 m 35 s
28 19
26 12
24 29
22 44
20 10
The latitude observations are now reduced as follows:
Eq. (61) h' =34 23' 00'
r 1 24
\"
= 34 21 36
46
f
resu
= 63 c
Its ;
1 58
ire:
'40"
59
28
28
53
46
= 55 38 24
8
ho
= 63 58 16
= 8 36 23
= 34 38 07
log COS
log cos 5
= 9.642291
= 9.995082
log cos h
= 9.915287
log m
9.637373
9.722086
= 3.263353
logy
y=mvt
r
= 2.985439
= 16' 07"
= 55 38 24
5
= 55 22 17
= 8 36 23
= 63 58 40
The complete latitude i
=
Mean =
= 63
58
42
The inclusion of the small term in the expression for
increases this result by less than 1".
The effect of the error of 26" in the value of the latitude
used in the computation of A is found by the formula
d A = - d(j)
cos tan r
to be about 21". 5.
4th method By an observation of a circumpolar star at
elongation.
The azimuth and hour angle of the star may be found by
(22) and (23). From the former the time of elongation may
be computed.
47
4
Description of method of taking the observation.
In the case of the pole star, assuming a = l h 26 m , 5 = 88 50',
we find t = 5 h 58 m 20 s , and .'. G = 7 h 24 m 20 s , the sidereal time
of western elongation. This may be used to compute ap-
proximately the time of either elongation at any time of the
year.
5th method By transits of stars across the vertical circle
of Polaris.
From the observed times of transit of two stars across the
same vertical circle, the azimuth of that circle may be com-
puted.
F/g.21
To find the azimuth: In Fig. 21, S\ is the position of Polaris
at the time of transit and 6 1 that of an equatorial star. SZS\
is then the vertical circle of the instrument, and PZ the
meridian. The angle SPS\ (denoted by A) differs from the
difference of r.a. of the two stars by the sidereal interval
between their transits, or
A = (a 1 -a)-(r 1 -D (71)
T\ and T being the observed times of transit of Polaris and
the other star, respectively, ai and a their right ascensions.
In computing A the subtractions should be algebraic; A will
then be affected by the + sign if the star S is west of the
meridian, and by the sign if east.
We next take the equations:
sin A cot C = cos 8 tan Si sin 5 cos A
sin r cot C = cos 5 tan <f> sin 8 cos r
. cos 5 sin C
sin A =
cos (f>
which are obtained from (5) and (3), Sph. Trig. From the
first of these we have
tan C =
sin A
cos 5 cot p sin 8 cos A
sin A
cos 8 cot p(l tan p tan 8 cos A) '
4S
= tan (l + tan p tan 5 cos A+),
cos 5
_ p sin (14-^ tan g cos a) (72 )
cos 5 v ' ;
neglecting p 3 . Again, from the second equation we have
cot C . tan
sin t . - + cos t = - ,
sin 5 tan 5
cot C .. t 2 tan ^
r T ^inT + 2 "tan 5 '
again neglecting the cube and higher powers of small quanti-
ties; /.
cot C f_ tan _ _ sin (0 8)
sin 5 2 tan 5 cos sin 5 '
Then assuming as a first approximation
cot C __ sin (0 5)
sin 5 cos sin 5 '
sin (0 5) _
or r = , tan C,
cos
we have by substitution for t 2 in the above equation
sin (0 5)
t = , tan v ,-
cos
sin (0 5)sin A /1 , s A \
s= : - (1 i p tan 5 cus A)
cos cos 5
by (72) ; or in seconds of arc
p sin(0 5)sin A /i , , . ir ,. f ... (73)
t=- - (l-*-/>sinl "tan 5 cos A)
cos cos 5
If the time star be observed below the pole, then 5 changes
its sign, and r becomes the hour angle reckoned from lower
culmination.
To find the azimuth we have from the third of the above
equations
COS
or by (72) , p sin A ,., . s . .,, . s (74.)
y v J A = -(l + sm 1'' tan 5 cos A)
COS
A and p being in seconds of arc.
Comparing equations (73) and (74) we see iu^t
, sin (0-5) (75)
t = A
cos 5
49
Example. The following observations were taken at
Toronto, Mar. 29, 1899:
PL obs'd. H.C.R. Watch
R.P. 45 18'
Polaris 73 33 .5 8 h 30 ra 51 s
r Hydrae 73 33 .5 8 34 43
The apparent places of the stars were:
a 8
Polaris l h 21 m 21 s +88 46' 23"
f Hydrae 8 50 06 + 6 19 35
We have then the following data:
A =111 13' (Eq. 71.)
4> = 43 39 36"
8 = 6 19 35
p =4417";
so that the computation proceeds as follows:
Eq. (74) log sin A =9.96952
logp =3.64513
Eq. (70)
Eq. (75)
log COS <j>
= 9.85941
3.61465
log 5692
log tan 5
log cos A
logp
log sin 1"
= 3.75524
= 9.04480
= 9.55858
= 3.64513
= 6.68557
log -5
= 0.68932
.'. .4=5687'^
As
R P
= 1 34' 47"
= 358 25' 13"
= 45 18
Rs
403 43 13
= 73 33 30
A P
= 330 09' 43"
log A
log sm(4> S)
= 3.75488
= 9.78280
log cos 5
= 9.99735
log 3470
50
3.53768
= 3.54033
r 3470" -231 s = 3 m 51 s
o(f Hydra) = 8 50 06
G r 8 46 15
L = 5 17 35
9(atGr.) =14 03 50
Equiv. M.T. int'l - 14 01 32
M.T. of sid. noon =23 33 23
37 34 55
13 34 55
Standard Time = 8 34 55
Watch = 8 34 43
Watch corr'n = +12
6th method By the observed angular distance of the sun
from a terrestrial point.
This method is useful when the sextant is the only instru-
ment available.
F/g.22
In Fig. 22 5 is the centre of the sun, and the terrestrial
point. The observation comprises:
Measuring the angular distance SO,
Noting the time of observation, and
Measuring the altitude of 0.
The latitude being known, the altitude and azimuth of
the sun's centre are computed by (4), (5) and (6). The
apparent altitude is then found by subtracting the parallax
and adding the refraction The measured angular distance
is corrected for semi-diameter. We have then
01 sin (s ZS) sin (s ZO)
tan^ 4 a = -
in which s =
51
sin 5 sin (s SO)
ZS+ZO+ SO
If then, h' =the apparent altitude of the sun
H = the altitude of
D = the angular distance SO
we find on substituting
,_ h'+H+D
S ~" 2
! sin(s'-#)sin(/- /Q (76)
cos s cos(s D)
If H is so small that it may be neglected, as is often the
case in hydrographic surveys, then (76) becomes
tan 2 |a = tan %(D+h') tan h(D-h') (77)
The azimuth of then is
A * a
If the correction of the watch is not known the observer
may proceed as follows:
Measure the altitude of the sun, then the angular distance
SO, then again the altitude of the sun, noting the watch
time of each of the three measurements. The altitude of
the sun at the instant of measuring SO may then be inter-
polated. The altitude of is measured as before. A may
then be computed from the data h 8 and <f> by either (8), (9)
or (10). The remainder of the reduction is as before.
82
6 Determination of Longitude by Observation.
The engineer is seldom called upon to determine longi-
tude, so that only some methods useful to the explorer will
be here :' escribed, and also in outline the most precise method
known, \ z., that by the electric telegraph.
The difference of longitude between two places may be
defined as the angle between the planes of their meridians.
It was seen p. 14 that the local times of two places
differ by an amount equal to their difference of longitude,
expressed in time. Any method, therefore, that serves to
compare the local times of the two places, at the same absolute
instant of time, will determine their difference of longitude.
1st method By portable chronometers.
If the correction of a chronometer on the local time of a
place A is found by observation, and also its rate, and the
chronometer is then transported to another place B, and its
correction on the local time of that place found, the local
times of the two places may be thus compared : Let
AT, 57" = the correction and rate found at A at the time T;
AT' = the correction found at B at the time T ( = 7"+/)
Then at the instant T the true time
itA = T+t+AT+t . 8T,
atB = T+t+AT;
the difference of which is
AL = AT+t . ST -AT,
or the difference of the corrections of the chronometer on
the times of the two places at an assumed instant of time.
2nd method By signals.
Any signal that may be seen at the two places may be used
to compare their local times. A chain of observing stations
may be established between the extreme stations, with inter-
* (VfV > x* f
F/o.23
mediate signal stations, so that the method may be used
between points at a considerable distance apart. The signal
used may be the disappearance of a light, a flash of gun-
powder, etc.
Let A ard B be the terminal stations, C and D intermediate
stations, and S\ S* and S3 signal stations (Fig. 23). Then if
a signal be made at Si which is perceived at A at the titr:? 7\
53
and at C at the time T%', and if then a signal be made at 5 2
which is perceived at C at the time T 3 and at D at the time
D i} etc.; then, A being the more easterly station, we have
AL = (T l -T i ) + (T 3 -T i ) + (T b -T 6 )
= T l -(T 2 -T 3 )-(T i -T 5 )-T 6 ;
which shews that it is not necessary to know the corrections
of the chronometers at the intermediate stations, but only
their rates. The times T\ and Tq, are the true local times at
A and B, respectively.
Eclipses of Jupiter's satellites are also used in longitude
determinations. As the satellite appears to fade out gradu-
ally the observed time of an eclipse will depend upon the
power of the telescope used. But for this objection this
method would be a useful one for finding longitude.
Reference may be made to the ephemeris.
3rd method By the electric telegraph.
The observer at each station must be provided with a
transit instrument, chronometer, and electro-chronograph,
for determining time with precision, and also a portable
switchboard by waich connections can be made with the
main telegraph line for sending signals to the other station.
F/e.24
The connections for observing the transits of stars in
determining time are shewn in diagram in Fig. 24, and for
sending arbitrary signals in Fig. 25.
The procedure at each station is to observe a set of stars
for determining time and the instrumental constants. Then
a series of signals is sent to the distant station, which are
also recorded on the local chronograph. A second set of
stars is then observed. By means then of the two time sets
the correction of the chronometer on local time at the epoch
of the signals can be interpolated.
These operations may be repeated on as many mutually
clear nights at the two stations as may be considered neces-
sary, say five nights.
54
In Figs. 24 and 25
C is the chronometer,
By the chronometer battery,
Rx the chronometer relay,
B t the chronograph battery,
M the chronograph magnet,
K the transit key.
F/g.25
Also in Fig. 25
LL is the main line,
R 2 the sounder relay,
S the sounder,
Rz the signal relay,
Rh a rheostat,
G a galvanometer,
K' the telegraph and signal key.
A signal is made by breaking the main line circuit by means
of the signal key, which may be a special break-circuit key.
If now at a time T\ at. station A a signal is made which
is recorded at B at the time T\ ; and if ATi A7Y are the chrono-
meter corrections on local time at the two stations, and x the
time of transmission of the signal; then the difference -of
longitude is:
AL=--(7\4-r-(7Y4-A7Y-*)
= AZ,i+x
in which AL 1 = (r 1 +A7 , 1 )-(2V'f A7Y)
55
If a signal now be made at B at the time 7Y, and recorded
at A at tne time T 2 ; then
AL = (r 2 +Ar 2 -.r) - (7Y+A7Y)
= AL 2 x
in which AL 2 = (r 2 +Ar 2 ) -(7Y+A7Y)
Taking the mean of these values of AL x is eliminated, and
we have
. T AL t +AL 2
- 2
4th method By moon culminations.
An examination of the moon's hourly ephemeris contained
in the N. A. will shew that the motion of that body in right
ascension is very rapid. If then a value of that co-ordinate
be found by observation, and the corresponding Gr. time be
interpolated from the ephemeris, the error in the time due
to the error in the observed quantity will not be excessive.
The Gr. time being thus found at the instant of the observa-
tion, which also serves to determine the local time, the longi-
tude follows by taking the difference of the two times.
To determine the moon's r.a. the meridian transit of the
moon's limb and that of some neighbouring star are observed.
Then let
and 6' = the sidereal times of transit of the moon's centre
and a star.
a and a' = their right ascensions
and we have
a-a' = 9-e'
or a = a'+e-e'
which gives the moon's right ascension.
To find the sidereal time of the semi-diameter passing the
meridian in order to correct the observed time of transit of
the limb, let
a = the sid. time of the S.D. passing the meridian
.S = the moon's angular S.D.
Aa = the increase of the moon's r.a. in l m of M.T.
then r =the increase of the moon's r.a. in 1 sid. second;
oU.lol
and
Act . . . , .
o" hn ifu ~ lts increas e in the interval a;
j . Act 5 sec S
and -' '-'60T64 = "IS"
as each side of the equation expresses the time of S.D. passing
the meridian if there were no change of r.a.; .".
56
5
a
15 cos5 O-ecok)
60.164 S
15 cos 5 (60. 164 -Aa)
This quantity is given in the N.A.
To interpolate the Gr. M.T. corresponding to an observed
value of the moon's r.a., let
a =the ephemeris value nearest to a,
To =the corresponding Gr. M.T.,
T = the Gr. M.T. corresponding to a,
x = T T (in seconds) ,
Aa =the increase of a in 1 minute of M.T. at the time T ,
ha = the increase of Aa in 1 hour.
Then the increase of Aa in the interval x is
x *
. da;
3600
.*. the value of Aa at the middle instant of the interval x is
Aa + ^oiT 5a
and .*. the increase of a in the interval x is
~6cr( Aa + w 5a )'
X / X "\
and.'. a = a + -^r-f Aa + ^qq 3a )'
Then x= 60 ( a ~ a ) _ 60(a-a o )
60 (a a ) /, X 6a \ ,
C 1 - -7200- Aa"/ nearly
Aa
, x' 2 8a .
= X ~ "720T -Aa~ nearly
... , 60 (a a )
in which x =
Aa
Then T=T + x
If then 6 is the Gr. sid. time corresponding to T we have
L = e-a
A more accurate method than the foregoing is to take
observations for determining a on the same night at the
57
station whose longitude is required and also at another
station whose longitude has been well determined. Thus
the increase in a while the moon is passing over the interval
between the two meridians is determined. This increase,
divided by the increase in 1 hour of longitude, gives the
difference of longitude in hours. Thus if
en and a.i =the values of a found at the two stations,
H = the increase of a in 1 hour of longitude while
passing over the interval between the two
meridians;
then ^^P"
H may be taken troin the ephemeris.
6S
7. The Theodolite and the Sextant.
The Theodolite.
For a knowledge of the construction and method of ad-
justment of the engineer's transit theodolite reference may
be made to any standard work on surveying.
A well constructed and adjusted transit should fulfil the
following conditions:
(1) The vertical and horizontal axes should pass through
the centres of the horizontal and vertical circles, respectively,
and should be perpendicular to their planes.
(2) The axis of the alidade of the horizontal circle should
coincide with the axis of the circle.
(3) The line joining the zeros of the verniers of either
circle (assuming that each is read by two verniers) should
pass through the centre of the circle.
(4) The extreme divisions of each vernier should coincide
at the same time with divisions of its circle.
(5) The horizontal axis should be perpendicular to and
intersect the vertical axis.
(6) The sight line of the telescope should be perpendicular
to and intersect the horizontal axis, and in all positions of
the focusing slide. It should also intersect the vertical axis.
(7) The two threads in the telescope, whose intersection
determines a point on the sight line, should be truly hori-
zontal and vertical, respectively, when the instrument is
adjusted for observation.
(8) The levels attached to the horizontal plate should read
zero when the vertical axis is plumb.
(9) When either vernier of the vertical circle reads zero,
and also the level attached to the alidade of that circle, the
sight line should be horizontal.
conditions 1, 2, 3, 4 and the second part of 6 are fulfilled
by the maker in the construction of the instrument; the
others, and sometimes 3, can be attended to by the observer.
With regard to 9, the alidade of the vertical circle of a transit
intended for astronomical observation should be provided
with a level capable of detecting a change of inclination
considerably smaller than the least count of the vernier.
The position of the alidade should be adjustable by means
of a slow-motion screw, so that the bubble of its level may
readily be brought to the centre, after plumbing the vertical
axis of the instrument.
It is proposed to examine the effects of these errors of
C(ns ruction and adjustment, shewing how in most cases
they may be eliminated.
59
(1) The effect of an inclination of the horizontal axis.
In Fig. 26, which is a projection of the celestial sphere on
the plane of the horizon, the horizontal rotation axis of the
transit is assumed to be inclined at a small angle to the
horizon, so that the collimation axis defined as a right line
through the optical centre of the objective perpendicular to
the horizontal axis traces on the celestial sphere the great
circle A'PZ'. P being any point and APZ a vertical circle,
F/g.26
the true altitude of P is the arc AP; and the apparent altitude,
affected by the inclination of the axis, the arc A' P. Z' is
the zenith of the instrument, and ZZ' is equal to the inclin-
ation b. It is clear that the effect of b on the H.C.R. is
shewn by the spherical angle AZA' . To find an expression
for this angle we have in the triangle PZZ'
+ p77 , tan PZ'
tan PZZ =
or
cot AA X
sin ZZ' . '
cot ti
sin b '
or tan Ayli = tan h' sin b;
or, as AAi and b are small, we may write this
AA^btanh', (79)
or the effect of an inclination of the horizontal axis on the
H.C.R. varies as the tangent of the altitude of the point
sighted.
In measuring the horizontal angle between two points it
is evident that the effect of b is nil if the altitudes of the two
60
points are equal, and that it increases with the difference of
the altitudes. A reversal of the instrument reverses the
algebraic sign of A^li, so that its effect on a horizontal angle
is eliminated by the reversal.
To find the effect of b on the measurement of a vertical
angle we again refer to the triangle PZZ', from which we
have
cos PZ = cos PZ' cos ZZ'
or sin h =sin h' cos b
Then denoting h' h by Ah and expanding cos b we have
sin(h'-Ah)=sm h' (l - -y)
or
sin h' Ah cos h' = s'm h' ^-sin h'
by expanding the sin and cos of Ah and neglecting its square
and higher powers; .*.
b 2
Ah = tan h'
It appears then that the effect of & on a vertical angle
varies as the square of b. Introducing the values of Ah and
b in seconds we have
., sin 1''
Ah = - b- tan h
(80)
This is a very small quantity; for, assuming 6 = 1' and h' =
45, we find A/z = 0".0087; it may therefore be safely neg-
lected. It is not eliminated by reversal.
fie. 27
61
(2) The effect of a collimation error; i.e., an error arising
from non-coincidence of the sight line and the collimation
axis as above defined.
Assuming that there is no inclination error the sight line
in this case will trace on the celestial sphere a small circle
parallel to the great circle traced out by the collination
axis. In Fig. 27 PZ' is the small circle, and A'BZ the great
circle traced out by the collimation axis. H and H' are the
poles of those circles, Z' being the zenith of the instrument;
ZZ' or PB is the collimation error, denoted by c.
To find the effect of c on a H.C.R., denoted by AAt, we
have in the triangle BZP
_ tan BP
tan BZP =~.^jr=-
sin BZ
or
tan AAz =
tan c
cos h'
or very nearly AA 2 = c sec h' (81)
or the effect of a collimation error on a H.C.R. varies as the
secant of the altitude of the point sighted.
F/&.88
The effect of this error on the measurement of a horizontal
angle evidently also increases with the difference of the
altitudes of the two points sighted, and is eliminated by a
reversal of the instrument.
To find the effect of c on the measurement of a vertical
angle we have in the triangle BZP
cos PZ. = cos BZ cos BP
or sin & = sin h' cos c
As this is the same equation as was derived in the discussion
62
of the last error, it follows that equation (80) also epressexs
the error in this case.
(3) To find the effect of a non-fulfilment of condition 1, 2
or 3, so that the line joining the zeros of the two verniers
does not pass through the centre of the circle.
The circle in Fig. 28 represents the graduated circle, of
which is the centre. 0' is the centre of the alidade. Also
the line joining the zeros of the two verniers does not pass
through the point 0'. It is clear from the figure that if in any
position of the alidade the reading of the vernier Vi is less
than what it would be if the line ViV 2 occupied a parallel
position passing through 0, then the reading of F 2 will be
in excess by the same amount. By taking the mean of the
two values of an angle, found by taking readings of both
verniers, the effect of eccentricity is therefore eliminated.
By a different process it may be shewn that the effect of
eccentricity may be eliminated by any number of equi-
distant verniers.
With regard to condition 8, it is convenient that the plate
levels should be in good adjustment, but in any case it is
advisable to use the more precise level attached to the alidade
of the vertical circle, or the telescope level, in plumbing the
vertical axis. The effect of the error arising from imperfect
leveling may be shewn as follows:
fiG.29
In Fig. 29 Z is the zenith, Z' the point to which the vertical
axis is directed. P is any point. The triangle PZZ' gives
the equation
sin 6' cot = sin d tan h' cos d cos d'
63
5
Then expanding sin d and cos d and neglecting all but the
first power of d we have
sin 6' cotd = d tan A'+cos 0'
or sin 0' cos cos 0' sin 6 = d tan h' sin
or sin (9' d)=d tan A' sin
or as 0' is small
d'e = d tan A' sin
Now if there are two points sighted in turn, and 0/ and 0/
are the values which 0' takes, respectively, we have
6i6i = d tan A/ sin di
e-i'-d 2 = d tan A 2 ' sin 2
so that, taking the difference
(0 2 / -0i / ) - (0 2 -0i) =d(tan A 2 ' sin 2 -tan /*/ sin 00 (82)
This expresses the error in the horizontal angle between
the two points. It appears to be a maximum when 2 = 27O
and 01 = 90, and for high altitudes its value may exceed d.
It is not eliminated by reversal.
To find the effect on a vertical angle, we have in the triangle
APA'
, tan h
cos /= rr,
J tan h'
, , tan k tan h i /, , f*\
or tan V = 7- = ^ = tan h I 1 + ~ I,
cos/ 1 _ /_. V 2 /
2
nearly. Then writing h' = h-\-Ah we have
tan /*' = tan(A+Aft)
= tan h-\-Ah sec 2 &
by Taylor's theorem.
f 2
AA sec 2 h = ^r- tan A
ft
or Ah= tan A cos 2 A
Zi
Again, in the triangle PZZ'
sin 0' sin J
cos A
d sin 0'
sin/ =
or / =
cos h
Substituting in the above expression for Ah we have
. , d 2 sin 2 0' 7,7
A/* = - T~r- tan h cos- #
2 cos 2 h
64
(83)
= sin 2 6' tan h
Zi
sin 1"
or in seconds Ah = = d 2 sin 2 8' tan h
Zi
This is never appreciable.
(5) It is convenient that adjustment 9 be nearly perfect,
the ugh not essential, as the effect of imperfect adjustment is
elin inated by reversal.
F/g30
In Fig. 30 the circle represents the vertical circle of the
transit; OP is the sight line, directed to some point P. The
error of VA , the reading of the vernier V, is evidently =
e+e'.
If the telescope now be transited, turned in azimuth, and
again directed to the point P, it amounts to the same thing
as transiting and directing to a second point P' which has the
same absolute zenith distance as P. The reverse reading is
then VA' whose error is =
~(e+e')
The mean of VA and VA', the two readings of vernier V,
is therefore the altitude of P freed from the effect of index
error.
To observe an altitude of a heavenly body with a transit.
It has been shewn that errors of adjustment have no
ippreciable effect upon a vertical angle, except the index
65
error, whose effect may be eliminated by reversal. In ob-
serving the altitude of a star, therefore, the method is to make
two pointings to the star, reversing the instrument between
the pointings. The telescope is first directed so that the
star is very near and approaching the horizontal thread at
a point a little to the right or left of the centre. The time of
crossing the thread is then noted, and also the V.C.R. The
instrument is then reversed and directed as before, with the
star at about the same distance on the opposite side of the
centre, thus eliminating the effect of any inclination of the
thread. The time of passage across the thread is again
noted, and the V.C.R. If azimuth is required as well as
time, the star must be observed on the intersection of the
horizontal and vertical threads. The mean of the two V.C.R's.
is then the observed altitude freed from the effect of index
error corresponding to the mean of the observed times.
It is thus assumed that the change of altitude of a star,
during short intervals of time, is proportional to the time.
This assumption will seldom lead to an error exceeding 3 .1
for an interval of 3 m between the observations.
In observing the sun the same general method is followed
as in observing a star, but as there is no definite point at
the sun's centre that can be observed, the procedure is as
illustrated in Fig. 31. The sun's image is first brought to the
F/g3/
oosition shewn by the broken circle Si, so as to be in contact
.vith the horizontal thread and slightly overlapping the
vertical thread. It may then be kept in contact with the
horizontal thread by turning the altitude tangent screw;
its own motion will then bring it into contact with the vertical
thread, as shewn by the full circle Si. After noting and
66
recording the time and the readings of the circles the instru-
ment is reversed and the observation repeated, bringing the
sun into the position S 2 . The figure represents an afternoon
observation for time and azimuth, taken with an inverting
telescope. If time alone is required the contact of the sun's
image with the vertical thread is not important. The means
of the readings of the two circles may now be regarded as
corresponding to a pointing to the sun's centre at an instant
equal to the mean of the times.
A form of record is shewn on p. 43.
The Sextant.
The principle and construction of the instrument.
In Fig. 32 AB is the graduated arc, Mi the index mirror,
M2 the horizon mirror, M{V the index arm to which the
mirror M\ is attached, and carrying the vernier V at its
extremity. The instrument embodies the principle that if
a ray of light SMi be incident upon the mirror Mi, then
Fig. 32
reflected from it to the mirror M 2 , from which it is again
reflected, then the angle c between the first and last direc-
tions of the ray is equal to double the angle d between the
mirrors. This is readily proved, for in the triangles MiM 2 C
and MiM 2 D we have, respectively
2b = 2a+c
and b = a+d
or 2b = 2a+2d
c = 2d
The mirror M 2 is attached permanently to the frame of
the instrument, and half of its surface is unsilvered, while
67
Mi is attached to the index arm and turns with it. The
sighting telescope is directed along the line CM 2 . The mirrors
are so placed that when their planes are parallel the index
V is at the zero A of the graduated arc AB. The arc is divided
into twice the number of degrees that it subtends at its centre
Mi.
To measure the angle between two points the instrument
is held so that its plane passes through the two points, and
the left-hand point is seen in the field of the observing tele-
scope through the unsilvered half of the mirror M-i. The
index arm is then turned until the other point, seen by double
reflection from the two mirrors, appears to coincide with
the first. The reading of the arc is the angle subtended by
the two points at the point C. It is to be remarked that C is
not a fixed point for all angles.
Adjustment of the sextant.
To observe an altitude of the sun with a s?xtant and
artificial horizon.
The artificial horizon is a horizontal reflecting surface,
usually the surface of mercury contained in an iron trough.
In observing the altitude of a heavenly body the angle is
measured between its image, seen by reflection in the arti-
ficial horizon, and that seen by reflection from the mirrors of
the instrument. Fig. 32 shews that this angle is equal to
double the apparent altitude of the body. In observing the
sun, instead of superposing the two images seen in the field
of the telescope, it is best to bring them into external contact,
thus observing either the upper or the lower limb. As the
horizon image appears erect in the field of an inverting
telescope, and the other image inverted, the identification
of either image shews which limb has been observed.
To determine the index error of the instrument after
observing the sun, set the vernier nearly at zero and then
direct the sight line to the sun; the two images will now be
seen nearly in coincidence. Then turn the tangent screw
until the images are in external contact, and read the arc.
Then reverse the motion of the screw, causing the images
to pass one over the other until they are again in contact,
and again read the arc. One of the readings will be on the
extra arc. Half the difference of the two readings is the
index error, positive if the reading on the extra arc is the
greater. The sum of the readings is twice the sun's angular
diameter.
6S
8. Formula of Spherical Trigonometry.
cos a =cos b cos c+sin b sin c cos A
cos b =cos a cos c+sin a sin c cos B
cos c =cos a cos 6+sin a sin b cos C
cos .4 = cos B cos C+sin B sin C cos a
cos B = cos ^4 cos C+sin A sin C cos b
cos C = cos A cos 2?+sin ^4 sin B cos c
sin A sin 5 sin C
sin a sin 6 sin c
sin a cos 5 =sin c cos & cos c sin b cos ^4
sin a cos C =sin 6 cos c cos 6 sin c cos A
sin 6 cos A =sin c cos a cos c sin a cos 5
sin b cos C = sin a cos c cos a sin c cos B
sin c cos ^4 =sin b cos a cos Z> sin a cos C
sin c cos 5 =sin a cos b cos a sin b cos C
where
sin A cot 5 = sin c cot 6 cos c cos ^4
sin 5 cot A = sin c cot a cos c cos B
sin 5 cot C = sin a cot c cos a cos B
sin C cot jB =sin a cot 6 cos a cos C
sin .4 cot C =sin 6 cot c cos b cos ^4
sin C cot .4 =sin 6 cot a cos b cos C
\
sin
sin-
sin*
L . sin (s b) sin (? c)
sin b sin c
1R sin (s a) s'm(s-c)
sin a sin c
kr _ sin (s a) sin (s b)
sin a sin 6
5 =
cos 2 \A
cos- 2
cos'
iB =
^C
a+b-\-c
2
sin 5 sin(.y a)
sin 6 sin c
sin 5 sin(s b)
sin a sin c
sin s sin(s c)
sin a sin b
= sin (5 6) sin (5 c)
sin 5 sin(s a)
1 r sin (s a ) sin (5 c)
sin 5 sin(s b)
lr sin (s a) sin (5 6)
2^- ; -
sin s sin (5 c)
69
tan 2 \A =
tan 2
tan 2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
. 9 i COS
sin 2 a =
Scos(S-A) '
sin B sin C
sin 2 2
i, cos 6 1 cos (S
sin A sin C
(S-B)
sin 2 \c =
cos
where
5 cos (S
sin yl sin B
C)
cos 2 a =
cos 2 \b =
cos 2 \c =
^
c ,4+5 + C
= 2
cos (5-5) cos (S -C)
sin 5 sin C
cos (S- A) cos (S-C)
sin .4 sin C
cos (S- A) cos (S-B)
sin yl sin B
S cos(S-A)
(S-C)
i _ cos 5 cos(S-A)
' cos (S-B) cos (S-C,
tan 2 \h = cos 5 cos (<>-)
cos (5 -^) cos (5- C)
tan 2 A c = - _ cos ^ cos (^-Q
2 cos (S-A) cos (5-5)
s analogies
cos (S-A) cos (5-
Delambre's analogies
sin h(A+B) cos i( g _ft)
cos \C cos c
sin \(A -B) sin Ha~ft)
o ir 1 = -
cos \C
sin ^c
"-"j 2*-' 0111 2'
cos Q4+) cos | (a +6)
sin \C cos fc
cos \
HA-B) _
sin |C
Napier's analogies-
sin
i _
sin \c
HA+B)
tan |(.
tan h(A-B)
cos ^
1
(a-b)
cos ^(a+6)
sin_(a &)
cot \C
cot C
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
70
os a cos b
(C = 90)
(20)
sin a
. sin b
sin jD = :
sin c
(21)
sin c
tan &
tan c
tana
cos B =
tan c
(22)
tan a
sin b
D tan b
tan 5 = .
sin a
(23)
cot ^4 cot 5
(24)
cos B s
. cos A
sin 2> =
(25)
Formulae for right-angled triangles
cos c =
sin A =
cos A =
tan ^4 =
cos c =
sin ^4 =
cos b cos a
Solution of oblique-angled triangles.
Case 1. Given a, b and c, the three sides
Solution by means of equations (6), (7) or (8).
Case 2. Given A, B and C, the three angles.
Solution by means of equations (9), (10) or (11).
Case 3. Given two sides and the included angle, as a, b
and C.
1st solution By means of equations (16), (17) and (3).
2nd solution By means of equations (1) and (5), or
cos c = cos a cos 6+sin a sin b cos C
sin C cot A = sin b cot a cos b cos C
sin C cot .B = sin a cot b cos a cos C
These equations become, when adapted for logarithmic
computation
tan = tan a cos C tan Q x tan b cos C
cos a cos (b Q) tan C sin 0i
cos c = * tan B -
tan yl =
cos 6 s'm(a 6i)
tan C sin 6
sin (6-0)
Case 4. Given two angles and the included side, as A, B
and c.
1st solution By means of equations (18), (19) and (3).
2nd solution By means of equations (2) and (5), or
cos C cos A cos .B+sin A sin B cos c
sin A cot B = sin c cot b cos c cos ^4
sin B cot ^4 = sin c cot a cos c cos B
These equations become
tan 62 = tan A cos c tan 3 = tan B cos c
^ cos A cos 0B-|- 2 ) , tan c sin 3
cos C = tan b = . . . , *
ccs 2 sin(/l + 3 )
tan c sin 2
sin( + 2 )
71
tan a =
Case 5. Given two sides and an angle opposite one of
them, as a, b and A.
1st solution By means of equations (3), (16) and (18), or
. _ sin b sin A
sin B = - :
sin a
, cos| (a b) . , . , _ N
tan^C= 1) , , ; cot$(A+B)
2 cos (a +6) v '
, cos h(A+B) .. , ,.
tan|c = cosK^-^) tanKa+&) '
2nd solution By means of equations (3), (5) and (1), or
sin b sin A
sin B = :
sin a
sin C cot A =sin b cot a cos fr cos C
cos a = cos 6 cos c + sin b sin c cos A
These equations may be thus adapted for log's.
tan 4 = tan A cos b
sin(C+0 4 ) =tan b cot a sin 4
tan 5 = tan b cos A
. n . cos a cos 5
cos(c 5 )= 7
cos b
Case 6. Given two angles and a side opposite one of
them, as A B and a.
1st solution By means of equations (3), (16) and (18), as
in the last case, (3) being written
. . sin B sin a
sin b = -. -.
sin A
2nd solution By means of equations (3), (2) and (5), or
. , sin B sin a
sin b= : -.
sin A
cos A = cos B cos C+sin B sin C cos a
sin B cot A = sin c cot a cos c cos B
Adapting for log's, we have
tan 6 = tan B cos a tan 7 = tan a cos B
//- i \ cos yl cos 06 / n . .
cos (C+0 6 ) = 5 sin (c e-i) = tan B cot ,4 sin 0,
cos B
72
GEODESY.
1. Figure of the Earth.
In any survey the extent of which is such that the curva-
ture of the earth's surface must be taken into consideration,
the figure of the earth may be regarded as that of an oblate
spheroid, the elements of a meridian section of which are,
as determined by Col. A. R. Clarke, 1866:
Major semi-axis, a = 20926062 ft.
Minor semi-axis, 6 = 20855121 ft.
Denoting the eccentricity by e we have
. _ a?-b> (1)
e-
a 2
The following log's are of frequent use:
log a =7.3206875
log e =2.9152513
loge 2 =3.8305026
log(l-e 2 ) =1.9970504
log ^ =3.8334522
le i
lo S l/i 2 =2.9167261
V 1=
Radii of curvature Any section of the spheroid by a
plane is an ellipse. If the plane contains the normal, or
plumb line, at a point, the resulting section is a normal
section. Any straight line so called traced on the earth's
surface is therefore a portion of an elliptic arc; for practical
purposes, however, if its length does not exceed 100 miles,
it may be regarded as a circular arc whose radius is the
radius of curvature of the normal section, of which it is a
portion, at its middle point. If the normal section coincides
with the meridian an expression for its radius of curvature is
0(1 -e 2 ) (2)
pm ~' (l-e 2 sin 2 </>)f
If the normal section is perpendicular to the meridian its
radius of curvature is
a (3)
Pn (l-e 2 sin 2 </>)
This is also the length of the normal AN or BN', Fig. 39
terminated in the minor axis of the spheroid. These are
termed the "principal radii of curvature" at a point whose
latitude is 4>. The radius of curvature of a normal section
whose azimuth is a may be expressed in terms of these; thus
73
1 cos 2 a sin 2 a (4)
Pa P P n
or = (1+ q ; cos 2 cos 2 a )
Pa Pn \ l-e- J
By substituting in (2) and (3)
sin 6 = e sin
they become p m =a{l-e 2 ) sec 3 (6)
p n =a sec (7)
Eq. (4) may also be placed in a convenient form for com-
putation. Thus writing it
Pm Pn
p n cos 2 a + pmsin 2 a
it may be thus transformed
Pn Pn
Pa =
sin 2 a + -^- cos 2 a sin 2 a (l + ^- cot 2 a^)
Pm \ Pm /
Then writing cot 2 a = cot 2 x
Pm*
it becomes
Pn Pn
Pa " sin 2 a(l+cot 2 x) sin 2 a cosec 2 *
sin 2 x
Pn ~ n
sin 2 a
p a is then given by the equations
/ p^~.._ sin 2 * (8), (9)
tan x = -i/ cHL tan a P a = Pn
sin 2 a
Pn
By expansion in series the log's of these radii of curvature
may be thus expressed:
log Pm =7.3199482 -[3.3448221] cos 20
+ [6.27371. .] cos 40- (10)
log p = 7.3214243 -[4.8677005] cos 20
+[7.79659..] cos 40- (11)
log p a =log p n [3.4712365] cos 2 cos 2 a
+ [5.00366. .] cos 4 cos 4 a- (12)
The numbers in brackets are the log's of constant numerical
coefficients.
For tables giving the values of p m p , etc., see the Supple-
ment to the Manual of Dom. Land Surveys, also Table IV.
74
A Trigonometric Survey.
Objects of such a survey.
Choice of stations. Well-conditioned triangles. The base
net.
Height of stations in order to overcome the earth's curva-
ture:
f/e.33
Let A and B be two stations whose heights above sea
level are H x and H 2 , and distance apart 5. is the centre of
curvature of the arc s. The curved line AB is the path of the
ray of light between the two stations, z is the zenith distance
of B observed at A. We have then in the triangle ABO:
B0_ _ sm BAO
AO '' = sin ABO
P+Hj sin(z+r) sin(z-fr)
p-\-Hi sin(z+r <r) sin(z-}-r)cos a cos(z+r) sin a'
Ml
P 1
or
or
or
1 +
1+^
P
1 -
o- cot(z-{-r)
Q + v) 0~ 7 1 ) =l+*cot( 2 + r) +
75
Hi-Hi. , . , , <r 2
or =o-(cot s r cosec- 2) +
P v ' ' 2 '
expanding by Taylor's theorem. Then as
per s, r ma,
m denoting the coefficient of refraction, and 2 is nearly 90,
we have
c-2
H 2 -H x = s (cot 2 j +
2 P
s 2
eq.
and
(13),
we
find
Hi-Hi
H'
-Hi
5
s'
Then so'
Iving
for
Hi we
have
Hi
H's
s
-H*s'
-s'
+ ks
= 5 cots + -jr- (l-2m) (13)
If i7' now be the height of the ray AB at a distance 5'
from A, we have
H'-Hi = s' cotz+4?- (1-2/n)
2p
1 2m
Writing & for and eliminating cot 2 between this
Zp
= k(s-s')
(14)
This gives the height necessary for a station at A in order
that a distant station B, of known height H 2 , may be visible
over an intervening elevation H'.
If we solve for H' we have
H' = H '~ Hx s'+Ht-ks'is-s'); (15)
which gives the height of the ray of light at a given distance
from A .
Clarke gives the following values for m:
For rays crossing the sea, m .0809
For rays not crossing the sea, m = .0750
Measurement of a base line Geodetic base lines are now
measured with tapes or wires of invar, an alloy composed
of iron and nickel in the proportion of 64 to 36. This material
has an extremely small coefficient of expansion, so that the
difficulty experienced in determining the temperature correc-
tion, when other materials are used, is thus obviated. Good
7B
results may also be obtained with a well standardized steel
tape by working in cloudy weather or at night so as to avoid
sudden changes of temperature.
In making a measurement the tape is stretched clear of
the ground by applying a considerable tension, and rests at
its zero points on supports in the form of tripods or stakes
driven firmly into the ground. The rear zero division of the
tape having been placed in coincidence with a fine mark on
the head of its support, the relative positions of the forward
zero division of the tape and the mark on its support may
then be measured with a scale. The distance between the
marks on the two supports may be found by applying certain
corrections to the tape length. These corrections are:
For temperature,
For tension,
For sag, and
For grade.
Correction for temperature:
cx = aL(t to) (16)
in which
L =the standard length of tape;
t = the temperature at which it is standard ;
/ = temperature at time of measurement;
a = coefficient of expansion.
Correction for tension:
c 2 = eTL (17)
in which
e = extension of unit length due to unit tension.
T= tension in lbs.
Correction for sag:
i _L_ / W\* (18)
24 T 2 24 V T )
cz =
in which
w =wt. of unit of length of tape
W = wt. of tape.
Correction for grade: Denoting the difference of elevation
of the end supports, determined by levelling, by h, we have
a = L-(L 2 -h 2 )i
~ L L y 2 U 8 U 16 u " )
1 W 1 h 4
2 L + 8 U +
t + t(t)'+
h h , h / h\*
2~
This first term in this expression is nearly always sufficient.
The following may be used as the coefficients of expansion
for steel and invar tapes:
Steel, 0.0000114
Invar, 0.00000041
In the absence of experimental data the extension of a
steel tape may be computed from its modulus of elasticity,
28000000 lbs. The extension of invar may be taken to be
0.00000004394 ft.
per lb., per foot, per sq. in. of cross section.
The distance between the supports, reduced to the horizon-
tal, then is
L =L+Ci-\-ci CzCi (20)
Reduction of a base measurement to sea level
F/g.34
o
Let, B = measured length of base, h being its height above
sea level ;
b =its length reduced to sea level.
Then we have
or b=B p
B p+h P +h
B-b =B (\ ^ =B
V 1 P+h) B P+h
=B -
p
p \ p p- p J
78
-tf-J-O-
(21)
The first term here is usually sufficient.
A broken base It is sometimes necessary to measure a
base line in two parts, deflecting through a small angle at
their point of junction.
Let a and b, Fig. 35, be the two parts, making the small
angle C with one another. It is required to find the length c
We have
C 2 =a 2 +b 2 +2 ab cos C,
= a?-+b 2 + 2ab(l \ nearly,
= (a+byabC\
-^ -()
= (a+b) i - | -^i), nearly,
= a + b-
or, if C is in seconds
x abC 2
2 a+6 '
sin 2 1" aC 2
(22)
* a+b
sin 2 1" _
log -s- - =11.0701198
7*0 interpolate a portion of a base Sometimes a portion
of a base cannot be directly measured. In Fig. 36, a and b
and the angles P Q and i? are measured ; it is required to find
the length x. We have
BE sin A CE sin A
sin Q
a
sin P a-\-x
BE
a sin Q
CE
(a+jc) sinP
79
6
F/g.36
Again BE - s[n ( A + R ) E sin (A + 22)
g ' 6+x sin (22-P) 6 " sin (22-0
BE = (b+x) sin (22 -Q)
CE bsin(R-P)
.*. equating, we have
ab sin Q sin(P-P)
sinPsin(P-(2) -^+*M&+*)
= aZ>-f-(a+&)#4-# 2
Then write
2 -, 4a&sin(gsin(P-P) (23)
(o-&) 2 sinPsin(P-<2)
and we have
x 2 +(a+b)x+ab-Ua-b) 2 tan 2 K =
x= ~%(a+b) y/\(a+b) 2 -ab+\{a-by tan 2 K
= -\{a+b) \/|(a-&) 2 +(a-&) 2 tan 2 K
= -\{a+b){a-b) secK
If a = b this solution fails. In that case write
2 , _ a& sin Q sin (22-P)
" sinPsin(i2-0
then we have
x 2 + (a +b)x+ab -tan 2 K' =
and x= -\ {a + b)V\(a+b) 2 -ab+tan 2 K'
= - \{a +b) Vi(a-6) 2 +tan 2 K'
= -|(a+6)tan K' (26)
Measurement of angles The angles of a triangulation
may be measured either with a direction theodolite, or one
of the repetition pattern. The circle of the former instrument
is usually read by three equidistant verniers or microscopes.
(24)
(25)
80
In measuring the angles at a station each of the distant
stations is sighted in order, from left to right, and the micro-
scopes read. The telescope is then transited, or reversed in
the standards, and each station is again sighted, in the order
from right to left, and the microscopes again read. A value
of each angle is thus obtained from each microscope, and in
each position of the instrument, direct and reversed. The
mean value of the angle thus obtained is free from the effect
of eccentricity and errors of adjustment of the instrument.
With three microscopes the effect of reversal is to give, for
each station sighted, six readings distributed at equal inter-
vals round the circle, thus minimizing the effect of division
errors of the circle. If the construction of the stand permits
the circle may now be turned to a new position and the angle
measurements repeated, etc., thus further diminishing the
effect of division errors.
A repetition theodolite is usually read by verniers, and
with this pattern of instrument the repetition principle may
be used to advantage. It may be thus described:
Let A (the left-hand station) and B be two stations, the
angle between which is to be measured.
Point to A and read verniers. Loosen upper clamp and
point to B and read verniers. Then loosen lower clamp and
again point to A. Then loosen upper clamp and again point
to B, thus obtaining a reading equal to double the angle.
This process may be repeated until a final reading is obtained
equal to, say, six times the angle between the two stations.
Next loosen the lower clamp, transit the telescope, and
point to B. Then loosen upper clamp, turn vernier plate
in a clockwise direction, and point to A, thus diminishing
the final reading of the first set of repetitions by the amount
of the angle between the two stations. Repeat this opera-
tion as often as in the first set, thus obtaining a final reading
approximating closely to the initial reading.
It is to be noted that in both sets of repetitions the vernier
plate is always turned in a clockwise direction; that in the
first set the instrument is turned from A to B with the upper
clamp loose and the lower clamp tight; and that in the second
set these conditions are reversed.
The required angle is now found by taking the mean of the
differences between the initial and final readings in the two
sets, and dividing by the number of repetitions. This result
is largely free from the effect of a drag of the circle by the
vernier plate.
Reduction of an observed angle to centre of station This
reduction is necessary when for some reason the centre of a
station cannot be occupied by the observer.
81
In Fig. 37 A is the centre of the station, the point occu-
pied. The angles /3 and y are measured, and the distance
m. The angle A is required. We have
f/G.37
A = BDC-x = 0-x+y;
, . m sin /3 . m sin y
and sin x = sin y = -
c J b
Then x and y being small we may substitute their circular
measures for their sines, and write them in the form x sin 1"
and y sin 1", x and y being expressed in seconds, so that we
have
,4=0-
m sin /3 m sin y
csinl"
+
(27)
b sin 1"
Distant stations are rendered visible by means of acetylene
lamps for night work, and heliotropes for day work. De-
scription of some forms of heliotrope.
62
3. Computation of the Triangulation.
The portion of the surface of the spheroid contained within
a triangle is assumed to be a portion of a spherical surface
whose radius is the geometric mean of the principal radii of
curvature at the central point of the triangle.
Spherical excess of a triangleIt is shewn in spherical
geometry that the sum of the angles of a spherical triangle
exceeds two right angles by an amount termed the "spherical
excess" of the triangle.
To find the spherical excess of a given triangle :
f/G.38
Let ABC be a spherical triangle, and A'B' and C points
diametrically opposite A B and C. The surface of the hemi-
sphere is made up of the three lunes ABA'C, BCB'A, and
CAC'B this last being equal to the sum of the two triangles
CAB and CA'B' less twice the area of the triangle ABC.
Denoting these by Lune A, etc., and the area of the triangle
by A, we have
Lune A = 2wR 2 = 2AR 2
K
LuneJS =2BR*
Lune C = 2CR 2
2AR 2 +2BR 2 + 2CR 2 -2A = 2TrR t
A
or A+B + C tt= -=
R*
83
or, denoting the spherical excess by e we have in seconds
A (28)
=
R 2 sin 1"
For a triangle on the earth's surface this may be written
e = A (29)
p m Pn Sin 1"
The area of the triangle, in all but extreme cases, may be
computed as if the triangle were plane, so that we may write
ab sin C (30)
2p m p n sin 1"
a 2 sin B sin C (31)
Or = ;
2p m pn sin 1" sin (B-\-C)
The value of 1/2 p m p sin 1" which we may denote by m
may be computed by the expression
log 2 Pm Pn sin 1" = 10^372023+ [3.469754] cos 20 (32)
the number in brackets being the log. of a constant coefficient.
The following table was computed by (32) :
<f> log m
50 10.37151
51 141
52 131
53 121
54 111
55 101
56 092
57 082
58 073
59 064
60 055
Legendres theorem This theorem may be thus stated:
If the sides of a spherical triangle are small in comparson
with the radius of the sphere, it may be solved as a plane
triangle by first diminishing each angle by one-third of the
spherical excess of the triangle.
To prove this, let
A B and C be the angles of the triangle,
a b and c the sides, expressed in radians,
A'B' and C the angles of a plane triangle, whose sides
a /3 and y have the same lengths expressed in feet as
those of the spherical triangle.
S4
log m
40
10.37253
41
243
42
233
43
223
44
213
45
202
46
192
47
182
48
171
49
161
Then we have
cos a cos h cos c
cos A =
sin b sin c
2r 2 _ 24r* V 2r * 24r V \ 2 ' 2 24 >V
""* (A. _ ."\ /x _ t\
V r or % ) \ r Qr 3 J
i__ , *L_/i_ , Jl t_ , V , y^\
2r 2 ^24r 4 \ 2r 2 ~ r 24r 4 2r 2 ^ 4r 4 "^ 24^/
<3t 187 3 ~^T
r 2 " 6^ "" 6r*
/3 2 + 7 2 -a 2 a 4-^4_ 7 4_ 6/3 2 7 2
2r 2 + 24r*
py A /3 2 +7 2
r 2
(1 - ^r\
\ 6r 2 J
/ ^ + 7 2 -a 2 a 4-^4_ 7 4_ 6 ^2 7 2 X x ^+7 2 \
V 2/3 T 24/3 T r 2 / V 6r 2 )
^2 + 7 2_ a 2 B j8 4_ 7 _ 6/3 7 1
207 ' 24/3 7 r 2
l 8 4 4- / 3V-a 2 /3 2 + / 3 2 7 2 + 7 4 -a 2 7 2
12/3 7 r 2
= /3 2 + 7 2 -a 2 a 4 +/3 4 + 7 4 -2a 2 <3 2 -2a 2 7 2 -2 i 8 2 7 2 (c)
2/37 24/37Z- 2
Now in the triangle A'B'C we have
., )8 2 +7 2 -a 2 (b)
cosA = Wy
sin 2 ,4' = l-
/ j3 2 + 7 2 -a 2 Y
V 2 /57 /
a 4 +/3 4 + 7 4 ~ 2a 2 ff 2 ~ 2 "V ~ 2/3 2 7 2 (c)
4/3 2 7 2
.*. by (a) (6) and (c) we have
cos A = cos A sm~ A -^-r
Then assume ^4 = ^4'+0
and we have cos A =cos A' B sin yl'
by Taylor's theorem. Therefore comparing with (d) we have
sin ^' = sin 2 ,4' -J^-
or 1
85
By sin A' 1 . . , .
or d= P7 6r2 = 3^. iPy sin A'
3r 2 3
This proves the theorem.
If the three angles of a triangle are measured, the spherical
excess may be computed by (30) or (31) using the values of
the angles given by measurement. The closing error then is
180 + - (A +B + C)
which may be divided among the angles, giving to each a
correction which is inversely proportional to its weight. One
*hird of the spherical excess is then deducted from each angle,
o.nd the triangle solved as a plane triangle. If the three
jigles have equal weights the closing error may therefore be
iound as if the triangle were plane and divided equally among
them.
For triangles the lengths of whose sides do not greatly
exceed 6 miles the error due to the neglect of spherical excess
is not likely to amount to 0.01 ft.
In the case of a triangulation consisting of an intricate
chain or network of triangles, the angles must be subjected
to a rigid process of adjustment before the triangles are
solved. The adjustment of a triangulation constitutes a
subject in itself, which is beyond the scope of these notes.
T eading principles outlined).
88
4. Geodetic Positions.
The latitude and longitude of one of the stations, and the
azimuth of a triangle side extending from that station, having
been determined astronomically, the geographical co-ordinates
of all the stations of the triangulation may now be computed.
The problem thus presented for solution is:
Given the latitude and longitude of a point on the earth's
surface, and the length and initial azimuth of the line drawn
from it to a second point, to determine the latitude and
longitude of this point, and the azimuth of the first point as
seen from the second.
In Fig. 39 A is the first point and B the second ; C is the
oole. AC and BC are the meridians of A and B. is the
centre of the spheroid. AN and BN' are normals to the
87
spheroid at the points A and B. A'B'C is a spherical tri-
angle, the centre of the sphere being at N. We have given
then
0i ai and s
and are required to find
</> 2 AL and a 2
To find A0( = 2 0i)
In the triangle A'B'C we have given b c and A'( = ai), and
must find a( = 9O-0 2 '), C( = AL), and 5.
We have
cos a = cos 6 cos c-f sin b sin c cos A'
or sin 02' = sin 0i cos c+cos 0i sin c cos ai
= sin fa ( 1 s" j +c cos 0i cos ai
c 2
or sin 2 ' sin 0i = c cos fa cos ai ^-sin fa
m
But sin 2 ' sin 0i = sin(0i+A0') sin 0!
= sin 0i / 1 j+A0' cos 0i sin fa
, , A0' 2 .
= A0 cos 0i jr- sin 0i
A0' 2 . c 2 .
. . A0 cos 0i sin 0i = c cos 0i cos en -~- sin 0i
, A0 /2 c 2
or A0 ~- tan fa= c cos ai tan fa.
2
Assuming as a first approximation
A0' = c cos ai
and substituting in the term in A0' 2 , we have
c 2 c 2
A0' = c cos ai s~ tan 0i+ tan 0i cos 2 a\
c 2
= ccosai- -x- tan 0i sin 2 ai (33)
Then substituting c= N
we have (A0' being in seconds)
, ,, 5 cos ai 1 /^cosaiN 2 , . .. /<3/n
A * - AT sin-1" - 2 (iViETF') ta " * tan " "' Sm J (34)
This gives the difference of latitude on an imaginary sphere
whose radius is N( = p), whereas the radius should be
88
assumed equal to the value of p m for the mean of the latitudes
of A and B, or, with sufficient precision, for the latitude
0!-HA<'. We have then
. t N (35)
Pm
Also <j> 2 = (t> 1 +A<t> (36)
To find AL
Again, in the triangle A'B'C, we have
. _, sin c sin A'
sin 6 = ;
sin a
. T sin c sin <n
or sin AL = ,
COS fa
or, substituting arcs for sines
c sin a\
AL =
cos fa'
, AT 5 sin ai (37)
or in seconds AL = -=r=~. 777 r
N sin 1" cos 4> 2
To find Aa( = a' ai)
We have
tan \{A'+B') - ^f^TS cot ff
cos(a+o)
But 4'+3' = ai + 180-a',
= 180-(a'-a 1 ),
= 180-Aa;
a -& = 90-4>2-90 o +tf>i,
= -(<fe-fo) = -A0;
a+6 = 9O-0 2 +9O-0i
= l8O-(0i+fc);
cot|Aa= CO f ^ A0 cot \ AL; "
sin m
or tan ^Aa = sm 1 A w tan \ AL;
cos f A0
or, substituting arcs for tangents
Aa = AL*^-. < 38 >
cos %A<t>
This is termed the convergence of the meridians of A and B.
Then finally
a 2 = 180-fa'
= 180 o + a!+Aa (39)
89
An expression giving Aa directly in terms of the data is
sometimes useful. It may be derived as follows: Taking the
equation
sin A' cot J3' = sin c cot b cos c cos A',
it may be thus transformed
sin ai
tan B
sin c cot b
COS C COS di
sin
<n
COS a\ (
cos c-
sin c
cot &
cos aj
tan a\
c
2
1 --^--e
cot b
2 cos ai
c cot 6 c 2 c 2 cot 2 b '
(. , c cot b . c 2 c 2 cot 2 o \
cos o.\ 2 cos- 2 ai /
. , / c cot b , c 2 , c 2 cot 2 6 \
.*. tan 5' + tan ai= -tan ail h H ; -J
\ cos ai 2 cos^ ai /
But 5 = 180 -a', .*.
, / c cot 6 . c 2 . , c 2 cot 2 6 \
tan a tan a x = tan ai f -\ H 5 1
V^ cos ai 2 cos 2 ai /
Also a' = ai+Aa, .'. by Taylor's theorem
tan a' = tan(ai+Aa)
= tan ai+Aa sec 2 ai+Aa 2 tan ai sec 2 ai
.'. , substituting, we have
/ ' c cot b c 2 c 2 cot 2 b*\
Aa sec 2 ai+Aa 2 tan ai sec 2 ai = tan ai( h~sr + - * I
\ COS ai 2 COS' 1 ai /
c 2
or Aa+Aa 2 tan ai = c cot b sin aiH =- sin ai cos ax
-\-c 2 cot 2 6 tan oi
Assuming as a first approximation
Aa = c cot 6 sin oi
and substituting in the term containing Aa 2 we find after
reduction
c 2
Aa = c cot b sin a x -\ ~-sin ai cos ai(l+2 cot 2 b)
or in seconds
^ ta
~N~ sin 1" ' 2\N J sinl"
^ tan 0i sin ai .1/5 \ 2 sinai cos ai /t . , , J N
Aa=- h 2(lvy sinl" ( 1 + 2tan "^
(40)
90
By writing
x = s sin oi y = s cos a\
equations (34), (35), (37) and (40) become
y x 2 tan </>i (41)
A0 =
AL =
p w sinl" 2p m psinl"
x (42)
p cos fa' sin 1"
A " = OT+wlrr (1+2tan! *' )(43
These equations should not be used for distances exceeding
20 miles. (38) should be used in preference to (40) or (43)
when all the unknown quantities are required.
For longer distances approaching 100 miles the following
equations may be used :
^ sin on 5 cos ai
x = y =
Pn Pn
A , _ y y s tan 2 en x 2 tan <j>' (44)
* = sin 1" + 3 sin 1" 2 sin 1"
</>' = 0i + lst two terms
Atf> = A0' -^
Pm
...2 1//
AL =
(45)
_____ sin 2 1" / _ cos 2 2 ' \
cos <h' sin 1" + 6 lAjL J V 1 sin 2 ai /
AZ/ = lstterm 2 ' = </>i+A0'
Afl -- AL sin *" _ sin2 *" rA ai /i _ cosHA0\ (46)
cosiA0 12 K - a) \ l sin 2 <j> m J
</>m=0i + |A0 Aa' = lstterm.
The following log's are here useful :
log 1/ sin 1" = 5.31442513 log sin 2 l"/6 = 1^.59300
log 1/3 sin 1" =4.83730 log sin 2 1"/12 = 12.29197
log 1/2 sin 1" = 5.0133951
Example. Let s = 20 miles, 0i=44 30', ai = 48 20'.
To find A0', eq. (34)
log 5 (in ft.) = 5.0236639
log cos ai = 9.8226883
logp = 7.3214108
log sin 1" = 6.6855749
91
4.8463522
2.0069857
log 690.8225 = 2.8393665
5.67873
log 0.5 = 1.69897
log tan 0! = 9.99242
log tan 2 ai =10.10129
log sin 1" = 6.68557
4>x
log 1.4355 = 0.15698
Ad/ = 689".387
= 11' 29".387
To find Ac/), eq. (35)
log Ad/ - 2.8384631
log Pn ~ 7.3214108
\ ogPm = 7.3199151
10.1598739
logAd, = 2.8399588
Ad, = 691".765
= 11'31".765
= 44 30'
to -44 41' 31".765
To find AL, eq. (37)
log 5 =5.0236639
log sin ai =9.8733352
log Pn =7.3214108
log sin 1" =6.6855749
log cos to 1 =9.8518109
4.8969991
1.8587966
log 1091.952 =3.0382035
AL = 1091".952
= 18' 11".952
The second term in eq. (45) in this example =0".0005.
92
To find Aa, eq. (38)
log AL
log sin <t> m
= 3.0382075
= 9.8464016
log cos |A<
= 9.9999993
log 766.672
Aa
2.8846091
= 2.8846098
= 12'46".672
The second term in eq. (46) here amounts to 0".001.
To find a 2 , eq. (39)
ai = 48 20' 00"
Aa = 12' 46".672
180 00' 00"
o 2 =228 32' 46". 672
The above equations (41), (42), (43) and (38) may readily
be adapted for the solution of a variety of problems. Thus
given </>i </>2 and AL
to find ai a 2 and s.
We have x = AL . p n cos </> 2 sin 1" (47)
A , 1 // I 1 X * tai1 <^1 1 //
;y = A0.p m sin 1 + - -77 p w sin 1
p m pn sin 1
AJL . " /y x 2 tan^, (48)
= A4> . Pm sin l"-i
2p
x (49)
Then tan a x =
Aa= AL
sin 4> m
cos | A0
a 2 = 180 + ai+Aa
x 3> (50)
5 =
sin ai cos ai
Again, given 0i < 2 and on,
to find .s AL and a 2 .
We have from (48) and (49)
y 2 tan 2 a\ tan 0!
y= A<f> . p m sin 1" +
2p
* , i// i /-a j i//\9 tan- oi tan <j5>i
= A$ . p m sin 1" + (A< . p w sin l") 2 -
*Pn
x y tan ai (51)
93
X
s =
AL =
sin ai cos ai
x
p n cos 02 sin 1"
Any other problem in which three of these six quantities
are given may be solved in a similar manner.
The foregoing equations may be used in reducing to differ-
ences of latitude and longitude the courses of a traverse line.
Only the first terms are here necessary, so that we may write
x = s sin a y = s cos a
A0 =
AL =
Pm sin 1
x
p n cos 4> sin 1"
. x tan 4>
Aa= : 777 = AL sin <j> (52)
p n sin 1
In latitude 45 the maximum values of the second terms of
the above expressions, for a length of 1 mile, are, respectively
0".0066
.0093
.0098
The use to be made of Aa is to correct the azimuth of a
course referred to the meridian of the initial station of the
traverse, to refer it to the meridian of the initial point of the
course. As a correction it is additive. The algebraic s gns of
x and y must be carefully observed.
94
5. Certain Problems which occur in the Dominion
Lands System of Survey.
A general description of that system of survey. .
(1) To find the amplitude of a meridian arc having a given
length; and conversely.
We have
A0 =
(53)
Pm sin 1"
A4> being in seconds; and conversely
s = A(f) . p, sin 1" (54)
If the arc is at a height H above sea level, then
A0= (p M +tf)sinl"
Pm (l+ V inl "
V p>/
VO-S (55)
Pm sin
nearly. Conversely , #. (56)
s = A<t> . p m sin 1
Example. Find the amplitude of an arc whose length is
24 miles, middle latitude 52, and height above sea level
1200 feet.
Eq. (55)
log 24
log 5280
= 1.3802112
= 3.7226339
log 5 (in ft.)
= 5.1028451
log Pm
log sin 1"
= 7.3204817
= 6.6855749
2.0060566
log 1249.650
logH
= 3.0967885
= 3.07918
log p m
= 7.32048
log 0.0717
6.17597
= 2.85549
&4>
= 1249.578
= 20' 49".578
95
7
For rinding the length of a meridian arc exceeding about a
degree the following expression may be used:
s = [5.56182842]A</> (in" degrees)
-[5.0269884] cos 2< sin A<
+[2.0527848] cos 4<t> sin 2A0
-[1.17356. .] cos 6</> sin 3A0 +
in which
A0 = the difference of latitude of its extremities,
4> = the mean of the extreme latitudes.
The numbers in brackets are logarithms.
This expression is sufficient for finding the length of a whole
quadrant.
(2) Given two points on the same parallel of latitude, at
a given distance apart, to find their difference of longitude,
and the convergence of their meridians.
F/g.40
A and B are the two points; ADB a normal section, and
AEB a parallel of latitude. PD is drawn at right angles to
ADB. The triangle PDB gives
sin BD
sin BPD =
sin PB
s
or
sin
AL
sin
2N
2 cos 4>
or, as AL is assumed to be small, this rray be written
s *
AL =
lV cos
A AT S {L ' b}
\>r in seconds aL = -r= - : 777
N cos 4> sin 1
If the higher powers of AL and s/27V are retained in the
expansions, this becomes
AL = Tt s . + ^^ (AL') 3 sin 2 <f> (59)
N cos sin 1 24
in which AL' is the first term. As
TV cos cf> = P,
the radius of the parallel of latitude, this may be written
^ = ^7, + ^f-c^-h^)' * (eo)
P sin 1 24 \P sin 1 /
For a chord 6 miles in length, in latitude 52, the second
term of (60) amounts to only 0". 00008, a quantity quite
inappreciable, so that the first term may be considered exact.
Again, in the triangle PDB we have
__ tan BD (61)
LUS J. J
tan PB
or
Aa
sin T
tan 27V
cot
or
A,
5 tan 4>
x= - N
Aa
being
small-
; or in
seconds
5 tan <t>
(62)
"" iVsinl"
The higher terms are here also inappreciable. From (58)
c n 1 (62) we have
Aa = AL sin
(See eq. 52).
The deflection angle between two consecutive chords of
the same length is clearly
. s tan 4>
Aa = , T . -j,
TV sin 1"
and the azimuth of a chord at either extremity
90-^-
To find the difference in length of s and the arc of the
parallel p we have
AL = -^ + ~ (?- Y sin 2 <f> t
24 \N cos <j>J
and AL = P
N cos <f>
_P .
~N cos <f> '
97
Equating these we have
p-s = ( ) sin 2 0iVcos0
24 y N cos </> /
24 v^vy
tan* <{> ^
To find the length of an offset from the chord to the parallel
of latitude.
Applying eq. (33) to the arc DE, Fig. 40, we have, denoting
AD and DE by x and y, respectively,
x
N N' 2 V N_
y x l / x v
-^ cos a- -,- 1 tan 4> sin- a
and by*(61) cos a = tan <f>
.'. writing sin 2 a = l we have
i s
'N'T ^N tan0 ~ 2^ tan< *
x(5 x)
= "2i^" tan<A
6. Trigonometric Levelling.
A and B are two stations whose difference of elevation is
to be determined; A' and B' are the apparent positions of
A and B, affected by refraction. The altitude h of B, observed
at A , and the distance s, are assumed to be known.
F/G.^-I
Denoting the height BC of B above A by H, we have
sin BA C
H = AC
sin ABC
a
BAC' = h-r + CAC' = h-r+- ,
= h ma + -tt ,
ABC = 90 -h + r- a
= 90-h+tn<r-o
= 90-{/; + (l-m)<x}.
99
sin {h + {\-m)c } (65)
cos! A+(l m)a\
See Supp. to Manual of Dominion Land Surveys.
For the numerical value of m see p. 76.
In eq. (65) it is assumed that the distance 5 is equal to the
chord AC. If A and B are stations of a trigonometric survey ,
and 5 is obtained by the solution of a triangle, then it is the
distance AB reduced to sea level. The correction to 5 for
elevation is
H,
s ,
P
Hi being the height of A above sea level. Also the correction
to reduce from th ; arc to the chord is
24V, p)
so that the length of the chord A C is
<<+f)i>-i(i)T
the second correction only becoming appreciable for con-
siderable distances.
Reciprocal zenith distances
If the zenith distances z and z' be observed simultaneously
at the two stations the effect of refraction is eliminated, if
it can be assumed to affect the two zenith distances equally.
Thus, returning to the above equation for H, we have
BAG' = 90 -z-r+-^-
ABC = 180 -z'-r
But we have also
A'AB = z + r = l?0-(z' + r)+a
so that r =
which therefore becomes known. Substituting this we have
BA C = Z ~^
ABC = 90- -'
2
.". substituting in the first above expression for H gives
H = s sin \{z'-z) (66)
cos Kz'-s + o-)
* having been corrected for elevation, and if necessary for
curvature.
100
TABLE I. Mean Refractions. (Bar. 29.6 ins., Ext. therm. 48.)
App.
Refr.
App.
Refr.
App.
Refr.
h
ro
h
ro
h
ro
/
/ it
o /
/ //
/
i ir
00
34 10.5
14 10
,3 44.2
27 20
1 50.9
30
28 26.3
20
3 41.6
40
1 49.4
1 00
24 05.2
30
3 39.0
28 00
1 47.9
30
20 43.3
40
3 36.5
20
1 46.4
2 00
18 04.4
50
3 34.0
40
1 44.9
30
15 56.9
15 00
3 31.6
29 00
1 43.5
3 00
14 13.4
10
3 29.2
20
1 42.1
30
12 48.0
20
3 26.9
40
1 40.7
4 00
11 36.7
30
3 24.6
30 00
1 39.4
30
10 36.4
40
3 22.4
31 00
1 35. 5
5 00
9 45.0
50
3 20.2
32 00
1 31.9
30
9 00.7
16 00
3 18.1
33 00
1 28.4
6 00
8 22.2
10
3 16.0
34 00
1 25.2
30
7 48.6
20
3 13.9
35 00
1 22.0
7 00
7 18.8
30
3 11.9
36 00
1 19.1
10
7 09.7
40
3 09.9
37 00
1 16.3
20
7 00.9
50
3 08.0
38 00
1 13.6
30
6 52.4
17 00
3 06. 1
39 00
1 11.0
40
6 44. 3
10
3 04.2
40 00
1 08.5
50
6 36.4
20
3 02.4
41 00
1 06.2
8 00
6 28.9
30
3 00.6
42 00
1 03.9
10
6 21.6
40
2 58.8
43 00
1 01.7
20
6 14.5
50
2 57.0
44 00
59.6
30
6 07.7
18 00
2 55.3
45 00
57.5
40
6 01. 1
10
2 53. G
46 00
55.6
50
5 54.8
20
2 52.0
47 00
53.7
9 00
5 48.6
30
2 50.4
48 00
51.8
10
5 42.6
40
2 48.8
49 00
50.0
20
5 36.9
50
2 47.2
50 00
48.3
30
5 31.3
19 00
2 45.6
51 00
46.6
40
5 25.9
10
2 44. 1
52 00
45.0
50
5 20.6
20
2 42.6
53 00
43.4
10 00
5 15. 5
30
2 41. 1
54 00
41.8
10
5 10.6
40
2 39.7
55 00
40.3
20
5 05.8
50
2 38.3
56 00
38.8
30
5 01.1
20 00
2 36.9
57 00
37.4
40
4 56.6
20
2 34.2
58 00
36.0
50
4 52.1
40
2 31.5
59 00
34.6
11 00
4 47.9
21 00
2 28.9
60 00
33.2
10
4 43.7
20
2 26.4
61 00
31.9
20
4 39.6
40
2 23.9
62 00
30.6
30
4 35.7
22 00
2 21.5
63 00
29.3
40
4 31.8
20
2 19.2
64 00
28.0
50
4 28.1
40
2 17.0
65 00
26.8
12 00
4 24.4
23 00
2 14.8
66 00
25.6
10
4 20.9
20
2 12.7
67 00
24.4
20
4 17.4
40
2 10.6
68 00
23.2
30
4 14.0
24 00
2 08.6
69 00
22.1
40
4 10.7
20
2 06.6
70 00
21.0
50
4 07.5
40
2 04.7
72 00
18.8
13 CO
4 04.3
25 00
2 02.8
74 00
16.6
10
4 01.2
20
2 01.0
76 00
14.4
20
3 58.2
40
1 59.2
78 00
12.3
30
3 55.3
26 00
1 57.5
80 00
10.2
40
3 52.4
20
1 55.8
85 00
05.0
50
3 49.6
40
1 54. 1
90 00
0*00.0
14 00
3 46.9
27 00
1 52. 5
TABLE II. Corrections to Mean Refraction.
F*
ctor B depending on
the
Factor t depending on the External
Barometer.
Thermometer.
Ins.
B
B
Fahr.
t
Fahr.
t
0. 845
30.9
1.044
C
O
. 1
0. 84*
31.0
17
- 22
1.163
36
1.026
.2
0. Sol
- 21
1.160
37
1.024
1.022
.3
0. S55
20
i. :
.4
0. 85S
- 19
1. 1
39
1.020
. 5
0.81 -
- 18
1.152
40
1.018
.6
0. -
- 17
1.150
41
1.016
7
0. -
- 16
1.147
42
1.014
-
0.872
- 15
1 . 145
43
1.012
.9
-
0. 87a
0. 878
- 14
- 13
1. 142
1.139
44
1 010
Fact r T
depending
45
1 '. 007
. 1
0.882
on the
Attached
- 12
1.137
46
1 . 005
2
0.88a
rmometer
- 11
1.134
47
1 . 003
4
0. 8*
0.- _
- 10
- 9
l l!
Jv
i roi
Fahr.
T
.
0.999
. o
895
- 8
1.1.7
50
0.
.6
- 20
1.'
- 7
1.124
51
0.996
7
0.1' -
- 1C
1.004
- 6
1.122
0.994
-
0.91
1.'
5
1. 119
53
0.992
.9
0.909
- 10
1.002
- 4
1.117
.54
0.990
27
0.9 -
-
-il
- 3
1.114
0:98*
. 1
0.916
30
l.l 1
_ 2
1.112
:
0. 9
2
0. 919
40
0.999
- 1
1. 1
57
984
3
0.922
50
0.9
1. 107
58
0. 9 - -
.4
0.92
60
0.'
- 1
l. :
0.9*
. 5
0.929
7
0.997
2
l. li .
60
0.97*
0.933
0.9
3
1.100
61
0.97
.
0."
0.9 "
4
1.0 "
62
0. 974
8
0. 939
-100
0.994
5
1.095
63
0.973
.9
0.943
0. 9
g
1 093
64
0. 971
7
1.090
:
0.
1
0.949
-
1.088
66
0. '.' 7
0. 9 '
9
1.086
"
0. 965
_
0.956
10
1.'
68
0. '
.4
0.960
11
1.081
0.961
5
0.963
12
1 . 079
70
0. 960
.6
0.966
13
1.070
71
0. 9 3
7
0. 97
14
1.074
72
C.
-
973
15
1.072
73
0. 954
.9
0. 97
16
1.069
74
0. 952
2
0.980
17
1.(07
75
0. 951
. 1
0.9*
1.065
76
0.94
0.9^7
~
19
1.063
77
0.
_
3
0.990
20
1.060
7--
0. 945
.4
0.993
21
1. 05*
79
0. 9
5
-
22
1.056
0.942
.6
1.000
~
-
1 . 0.54
81
0. 940
7
1.003
1.052
S2
0.9
-
1 007
_-
1.049
0. 936
1.010
-
1.047
0.9 "
30.0
1 . 014
27
1.045
ss
0. 933
1
1.017
2*
1.043
86
0. 931
1.020
29
1.041
-7
0. 930
_
1 . 024
30
1.039
38
0.92*
i
1.027
31
1.'
0.926
.5
1.031
_
1. ,34
90
0.924
1 . 034
33
1.032
91
0. 92
7
l.< "
34
1 . 030
92
0.921
-
1.041
"
1.02^
TABLE III. m =
2 sin
2 I
sin 1
T
m
l m
om
gm
4 m
5 ra
gm
7 m
8 m
s
//
"
"
"
it
/
//
1'
//
0.00
1.96
7 85
17.67
31.42
49.09
70.68
96.20
125.65
i
0.00
2.03
7.98 '
17. s7
31.68
49.41
71.07
96. 66
126.17
2
0.00
2.10
S. 12
18.07
31.94
49.74
71.47
97. 12
126.70
3
0.00
2.16
18.27
32. 20
50.07
71. 86
97.58
127 22
4
0.01
2.23
S. 39
18.47
32.47
50.40
72.20
98.04
127! 75
5
0.01
2.31
8.52
18! 67
32. 74
50.73
72. 66
98. 50
128. 2S
6
0.02
2.38
S. 66
18.87
33.01
51.07
73.06
98.97
128.81
7
0.(12
2.45
8.80
19.07
33.2 7
51.40
73.46
99.43
129.34
8
0.03
2.52
8.94
10 2s
33. 54
51.74
73.86
99.90
129. S7
9
0.04
2.60
9.08
19.48
33.81
52.07
74.20
100. 37
130.40
10
0.05
2.67
9.22
19.69
34.09
52.41
74.66
100.84
130. 94
1
0.06
2. 75
9.36
19.90
34.36
52. 75
75.06
101.31
131.47
2
(Mis
2.83
9.50
20.11
34. 64
53.09
75.47
101.78
132. CI
3
0.09
2.91
9.64
20.32
34.01
53.43
75.88
102. 25
132.55
4
0.11
2.99
9.79
2C.53
35. 19
53 77
70 29
102.72
133.09
5
0.12
3.07
9.94
20. 74
35.46
54. 11
76. 69
103. 20
133. 63
6
0. 14
3.15
10. 09
20.95
35. 74
54.46
77. 10
103 67
134.17
7
0. 16
3.23
10.24
21.16
36.02
54. SO
77.51
104. 15
134.71
8
0. IS
3. 32
10.39
21.38
36. 30
55. 15
77.93
104.63
135.25
9
0.20
3.40
10. 54
21.60
36.58
55. 50
78.34
105. 10
135. SO
20
0.22
3.49
10.69
21. S2
36. S7
55. S4
7S. 75
105. 5S
136.34
1
0.24
3.58
10. S4
22.03
37. 15
50. 10
79.16
106.06
136.88
2
0.26
3.67
11.00
22.25
37.44
56. 55
79.58
106. 55
137.43
3
0.28
3. 76
11. 15
22.47
37. 72
56. 90
SO. 00
107.03
137.98
4
0.31
3.85
11.31
22 70
38.01
57.25
so. 42
107.51
13S.53
5
0.34
3.94
11.47
22.92
38.30
57.60
SO. 84
107.99
139.08
6
0. 37
4.03
11.63
23.14
38. 50
57. 96
81.26
108.48
139. 63
7
0.40
4.12
11.79
23. 37
38. ss
58. 32
81.68
108.97
140.18
8
0.43
4.22
11.95
23.60
39. 17
5S.68
s2. 10
109.46
140.74
9
0.46
4.32
12. 11
23.82
39.46
59.03
S2.52
109.95
141.29
30
0.49
4.42
12.27
24.05
39.76
59.39
S2.95
110.44
141.85
1
0. 52
4.52
12.43
24.28
40.05
59. 75
S3. 38
110.93
142.40
2
0.56
4.62
12.60
24.51
40.35
60.11
S3. SI
111.43
14 2.00
3
0.59
4.72
12.70
24.74
40.65
60.47
S4.23
111.92
143.52
4
0.63
4.82
12.93
24.98
40.95
60.84
s4.66
112.41
144. OS
5
0. ('.7
4.92
13. 10
25. 21
41.25
61.20
S5.09
112.90
144.64
6
0. 71
5. 03
13.27
25. 45
41. 55
61.57
85. 52
113.40
145. 20
7
0. 75
5.13
13.44
25.68
41. S5
61.94
85
113.90
145.70
8
0.79
5. 24
13.62
25 02
42.15
62.31
86.39
114.40
140. 33
9
0.83
5.34
13.79
26.16
42. 45
62.68
86. 82
114.90
146. S9
40
0.87
5. 45
13. 96
26. 40
42.76
63.05
S7. 26
115.40
147.46
1
0.91
5, 56
14.13
26. 64
43.06
63.42
87. 70
115.00
14S. 03
2
(t. 96
5. 67
14.31
26. ^
43. 37
63. 79
88. 14
116.40
14s. 00
3
1.01
5. 78
14.49
27. 12
43. 68
64. 16
ss.,-,7
116.90
149. 17
4
1 . 06
5.90
14.67
27. 37
43.99
64.54
89.01
117.41
149.74
5
1.10
6.01
14. So
27.61
44. 30
64.91
89.45
117.92
150.31
6
1.15
6.13
15.03
27.86
44.61
65. 29
89. so
lis. 43
150.88
7
1.20
6 24
15.21
28. 10
44.92
65. 67
90. 33
IIS. 94
151.45
S
1.26
ti. 36
15.39
28.35
45. 24
66. 05
90. 7s
119.45
152.03
9
1.31
6. 18
15. 57
2S. 60
45. 55
66.43
91.23
119.96
152.61
50
1 . 36
6 60
15. 76
28.85
45. 87
66. 81
91. 6S
120.47
153.19
1
1.42
6 7.'
15.95
29.10
46. is
07. 19
92. 12
L20 us
153. 77
Q
1.48
6 84
16. 14
29.36
46. .".<
07 5s
92.57
121.49
154.35
3
1 . 53
6.96
16. 32
29.61
46. s-
67.96
93. 02
122.01
154.93
4
1.59
7.09
16.51
29.86
47.14
68.35
93.47
122 53
155.51
5
1.65
7.21
16. 70
30. 12
47. 46
68. 73
93.92
123.05
156.09
6
1.71
7 34
16.89
30. 38
47. 70
00. 12
94 :.s
123 57
156.67
7
1.77
7 47
17 08
30. 64
is. 11
69.51
94. 83
124.09
157.25
s
1 . 83
7.60
17 28
30.00
4S.4:-!
09 00
95.20
124 61
157 84
9
1.S9
7.72
17 17
31. 16
4s. 76
70. L!)
95. 74
125.13
158. 43
TABLE III. Cont.
m :
2 sin 2 |t
sin 1
T
9 m
10 m
ll m
r_ m
lS m
14 m
15 m
(
16 m
s
//
It
//
ft
ft
ft
//
tf
159.02
196.32
237. 54
282.68
331.74
384. 74
441.63
502. 46
i
159.61
196.97
238.26
283.47
332.59
385. 65
442.62
503. 50
2
160.20
197.63
23S. 9S
284.26
333. 44
386. 56
443.60
504. 55
3
160.80
198. 28
239.70
285. 04
334. 29
387. 48
444. 58
505.60
4
161.39
198. 94
240.42
285.83
335. 15
388. 40
445. 56
506.65
r>
161.98
199. 60
241.14
286. 62
336. 00
389. 32
446. 55
507. 70
6
162.58
200.26
241.87
287.41
336.86
390. 24
447.54
508. 76
7
163. 17
200.92
242.60
288.20
337. 72
391.16
448. 53
509. 81
8
163.77
201.59
243. 33
2S9.00
338. 58
392.09
449.51
510.86
9
164.37
202.25
244.06
289. 79
339. 44
393.01
450. 50
511.92
10
164.97
202.92
244. 79
290. 58
340. 30
393:94
451.50
512.98
1
165. 57
203. 58
245.52
291.38
341.16
394.86
452.49
514.03
2
166. 17
204. 25
246.25
292. 18
342.02
395. 79
453.48
515.09
3
166.77
204.92
246. 98
292.98
342.88
396. 72
454. 48
516. 15
4
167.37
205. 59
247.72
293.78
343. 75
397. 65
455. 47
517.21
5
167. 97
206.26
248. 45
294.58
344. 62
3E8.58
456.47
518.27
6
168. 58
206.93
249. 19
295. 38
345. 49
399. 52
457.47
519.34
7
169. 19
207.60
249.93
296. 18
346. 36
400.45
458. 47
520.40
8
169.80
208.27
250. 67
296.99
347. 23
401.38
459.47
521.47
9
170.41
208. 94
251.41
297. 79
34S. 10
402.32
460. 47
522. 53
20
171.02
209. 62
252. 15
298. 60
348. 97
403. 26
461.47
523. 60
1
171.63
210.30
252. 89
299. 40
349.84
404. 20
462. 48
524. 67
2
172.24
210.98
253. 63
300.21
350.71
405. 14
463. 48
525 74
3
172.85
211.66
254.37
301.02
351.58
406.08
464. 48
526.81
4
173.47
212.34
255. 12
301.83
352.46
407. 02
465. 49
527. S9
5
174.08
213.02
25.5. 87
302. 64
353. 34
407.96
466. 50
528.96
6
174.70
213.70
256. 62
303.46
354.22
408. 90
467. 51
530.03
7
175.32
214.38
257. 37
304. 27
355. 10
409.84
468. 52
531. 11
8
175.94
215.07
258. 12
305. 09
355. 98
410.79
469. 53
532. IS
9
176. 56
215.75
258. 87
305. 90
356.86
411.73
470. 54
533. 26
30
177. 18
216.44
259. 62
306. 72
357.74
412.68
471.55
534.33
1
177. 80
217. 12
260. 37
307.54
358. 62
413.63
472. 57
535 41
2
178.43
217.81
261. 12
308. 36
359. 51
414.59
473. 58
536. 50
3
179.05
218.50'
261.88
309. 18
360. 39
415. 54
474. 60
537.58
4
179. 68
219. 19
262. 64
310. 00
361.28
416.49
475.62
538.67
5
180. 30
219.88
263. 39
310.82
362. 17
417.44
476. 64
539.75
6
180. 93
220. 58
264. 15
311.65
363. 07
41S. 40
477. 65
540. 83
7
181.56
221.27
264.91
312.47
363. 96
419.35
478. 67
541.91
8
182. 19
221.97
265. 68
313. 30
364. So
420.31
479. 70
543.00
9
182.82
222.66
266. 44
314. 12
365. 75
421.27
480. 72
544.09
40
183.46
223.36
267. 20
314. 95
366. 64
422.23
481.74
545. 18
1
184.09
224.06
267. 96
315.78
367. 53
423.19
482.77
546. 27
2
184. 72
224.76
268.73
316.61
368.42
424. 15
483. 79
547. 36
3
185. 35
225.46
269.49
317.44
369.31
425. 11
484.82
548.45
4
185. 99
226. 16
270.26
318.27
370.21
426.07
485.85
549. 55
5
1S6. 63
226.86
271. C2
319.10
371. 11
427.04
486. 88
550.64
6
1S7.27
227.57
271.79
319.94
372.01
428.01
487.91
551.73
7
187.91
228.27
272.56
320.78
372.91
428.97
488.94
552.83
8
188. 55
228.98
273. 34
321.62
373.82
429.93
489.97
553. 93
9
189. 19
229.68
274. 11
322.45
374.72
430.90
491.01
555.03
.30
189.83
230. 39
274.88
323. 29
375. 62
431.87
492.05
556. 13
1
190.47
231. 10
275. 65
324. 13
376. 52
432.84
493. 08
557. 24
2
191. 12
231.81
276. 43
324.97
377. 43
433. 82
494. 12
558. 34
3
191.76
232.52
277.20
325. SI
378. 34
434.79
495. 15
559.44
4
192.41
233. 24
277. 98
326. 06
379. 26
435. 76
496. 19
560. 55
")
193. 06
233.95
278. 76
327. 50
3S0. 17
436. 73
497.23
561.65
fi
193. 71
234.67
279.55
328. 35
381.08
437. 71
498.28
562.76
7
194. 36
235.38
280.33
329.19
381.99
438. 69
499.32
563. 87
8
105. 01
236. 10
281.12
330. 04
382.90
439. 67
500. 37
564.98
9
195. 66
236.82
281.90
330. 89
383.82
440. 65
501.41
566.08
TABLE IV. Values of Log p n & Log p m (in feet).
o /
Log p
Log p m
4>
/
Log p n
Log p m
40 00
7.3212921
7. 3195588
50 00
7.3215482
7.3203271
10
2963
5715
10
5524
3398
20
3006
5842
20
5567
3524
30
3048
5969
30
5609
3651
40
3090
6096
40
5651
3777
50
3133
6223
50
5693
3903
41 00
3175
6350
51 00
5735
4029
10
3218
6478
10
5777
4155
20
3260
6605
20
5819
4281
30
3303
6733
30
5861
4407
40
3345
6861
40
5903
4532
50
3388
6988
50
5944
4657
42 00
3431
7116
52 00
5986
4782
io-
3473
7244
10
6028
4907
20
3516
7372
20
6069
5032
30
3559
7501
30
6111
5156
40
3601
7629
40
6152
5281
50
3644
7757
50
6193
5405
43 00
3687
7885
53 00
6235
5529
10
3730
SOU
10
6276
5652
20
3773
8142
20
6317
5776
30
381.5
8271
30
6358
5899
40
3858
8399
40
6399
6022
50
3901
8528
50
6440
6145
44 00
3944
8656
54 00
6481
6268
10
3987
8785
10
6522
6390
20
40.30
8914
20
6563
6513
30
4073
9042
30
6603
6634
40
4115
9171
40
6644
6756
50
4158
9300
50
6684
6878
45 00
4201
9428
55 00
6725
6999
10
4244
9557
10
6765
7120
20
4287
9686
20
6805
7240
30
4330
9814
30
6845
7361
40
4373
7.3199943
40
6886
7481
50
4416
7. 3200072
50
6925
7601
46 00
4459
0200
56 00
6965
7721
10
4502
0329
10
7005
7840
20
4544
0458
20
7045
7959
30
4587
0586
30
7084
S07S
40
4630
0715
40
7124
8196
50
4673
0843
50
7163
8314
47 00
4716
0972
57 00
7203
S432
10
4759
1100
10
7242
S550
20
4801
1228
20
7281
8667
30
4844
1357
30
7320
8784
40
4887
1485
40
7359
8901
50
4930
1613
50
7398
9017
48 00
4972
1741
58 00
7436
9133
10
5015
1869
10
7475
9249
20
5058
1997
20
7513
9364
30
5100
2125
30
7552
9479
40
5143
2253
40
7590
9594
50
5185
2380
50
7628
9708
49 00
5228
2508
59 00
7666
9S22
10
5270
2635
10
7704
7. 3209936
20
5313
2763
20
7742
7.3210049
30
5355
2890
30
7779
0162
40
5398
3017
40
7817
0275
50
7.3215440
7.3203144
50
7854
0387
60 00
7.3217891
7.3210499
f 1
J II
REFERENCE COP>
l
-^^i
=j
! J
29
~ J
I
=1
1
2
I
1
1
I
