Introduction 
v 
Example 
1863. March 13. 
Chronograph Readings 
Declina¬ 
tion 
Chronograph Readings 
R. A. from Sun’s Center 
Declination 
Scale 
Set I 
Set II 
Divisions 
Set III 
Set IV 
I 
II 
III 
IV 
Mean 
Divisions 
// 
©Limb 
oh inm 11 S4 
4I m 33 ?C 
+ 46.2 
Tj 
to- 
CO 
C 4 
e 
CO 
X 
0 
oh eo m C 75 1 
4 - 4 . 2 .0 
A a 
24 -3 
45-8 
— 20.7 
40.9 
51 9.6 
— 5 I? 7 
-52 *3 
-52*2 
— 52 ? 1 
-52*05 
-23.0 
-521 
a 1 
24.9 
46.6 
— 20.8 
41.8 
10.4 
5 ii 
5 i -5 
5 i -4 
5 1 -3 
51-3 
-23.1 
-523 
Bb 
58-3 
42 20.2 
— 6.0 
49 15-2 
44.0 
17.7 
17.9 
17.9 
17.7 
17.8 
- 8.3 
—190 
Cc 
40 5.2 
27.6 
- 4.0 
22.2 
S °-9 
10.8 
10.5 
11.0 
10.8 
10.8 
~ 6.3 
-144 
d 
7.6 
29 -5 
+ 0.4 
24.6 
53-3 
' 8.4 
8.6 
8-5 
8.4 
8-45 
— 2.0 
- 45 
d l 
9.8 
31-9 
+ 0.7 
26.8 
55-5 
6.2 
6.2 
6-3 
6.2 
6.2 
— 1.6 
~ 37 
Dd * 
10.2 
32-3 
+ 1.9 
2 7-3 
56.0 
5-8 
5-8 
5-8 
5-7 
5-75 
- 0.4 
- 9 
d 3 
II.9 
34 -o 
+ 1.4 
28.8 
57-5 
4.1 
4.1 
4-3 
4.2 
4 -i 5 
- 0.9 
— 21 
J 4 
14.0 
36.2 
4 - 0.6 
311 
59-6 
2.0 
1.9 
2.0 
2.2 
2.0 
- i -7 
“ 39 
e l 
28.3 
5°-3 
+ 3 - 6 
45-2 
52 13-8 
+ 12.3 
+ 12.3 
+ 12.1 
+ 12.1 
+ 12.2 
+ 1-3 
4 - 30 
E e 2 
29.0 
5 I- 2 
+ 3 -° 
46.0 
15.0 
13.0 
i 3 -i 
12.9 
I 3-3 
131 
4 - 0.7 
4 - 16 
e 
3 2 ■ 7 
54-6 
+ 1.0 
49.8 
18.5 
16.7 
16.6 
16.7 
16.8 
16.7 
- i -3 
- 3 ° 
Penumbra 
4 i 3-5 
43 25.0 
+ 3°-3 
.... 
49.2 
47-7 
47.0 
.... 
47-5 
47-3 
4-28.0 
4-631 
F Center 
4.2 
2 S -7 
+ 29.8 
50 20.9 
49-9 
48.2 
47-7 
47-7 
48.2 
47-95 
+ 2 7-5 
4-620 
Penumbra 
5 -o 
26.8 
+ 29.3 
.... 
5 °- 7 
49.0 
48.7 
.... 
49.0 
48.85 
4-27.0 
4-609 
©Limb 
20.6 
42.7 
— 41.6 
37-9 
53 6.3 
.... 
.... 
.... 
.... 
“ 43-9 
.... 
© Center 
o h 40 111 15597 
oh 42 m 38505 
+ 2.3 
° h 49 m 33 ?I 3 
oh 52 m I ?7 
64*58 
64*60 
64*78 
64.60 
64564 
“The mean of the times is o h 46 111 7?2, and the correction of the clock + 8 S 5, so that the observations as taken 
are valid for o h 46 111 15^7 mean time. 
“The probable error of an observation was deduced to be ±oTi from a number of determinations of the time 
of transit of the sun’s radius, by means of the deviations from the average of each day. For a spot the error in 
general will be somewhat greater; and since on the one hand this has to be multi plied by a number larger than 
V 2, while on account of being the average of four determinations, it must on the other hand be divided by 2, 
we may estimate the accuracy of the determination as if5. The probable error in declination will be about the 
same, since here the limit of reading is from to ^ of a scale-division.” 
The formulae now involved are simply derived by applying the three fundamental formulae of spherical trigo¬ 
nometry to the triangle having as its vertices the spot, the sun’s pole and the center of the disk. 
Let A a and AS represent the differences in right ascension and declination of the spot and the center of the disk. 
Let p be the “zenith distance of the earth as seen from the spot,” and n be the geocentric distance of the spot from 
the center of the disk. Then n=p sin p , where p is the sun’s radius. P is the symbol for the position angle of the 
spot, reckoned from the east through the north, and N is the position angle of the sun’s pole. B is the heliographic 
latitude of the spot, B 0 that of the center of the disk; L and L 0 are the heliographic longitudes of the spot and of the 
center of the disk from the node. 
Then we have ^ ,,, 
-=sin p cos P; 
P 
AS" . . „ 
-= sin p sm P . 
P 
sin B =cos ( p—n ) sin B 0 +cosB 0 sin {p—n) sin ( P—N ); 
cos B cos {L — L 0 ) = cos {p—n) cos B 0 — sin B 0 sin {p—n) sin {P — N ); 
cos B sin {L—L 0 ) = sin {p—n) cos(P— N ). 
For adaptation to logarithmic computation, let 
g sin £=sin {p—n) sin {P—N), 
g cos£=cos {p—n ). 
Then we get sin B=g sin {B 0 + £); 
cos (L— L 0 ) cos .B=g cos {B 0 + £); 
sin {L—L 0 ) cos B—sin {p—n) cos {P—N) .* 
1 It seems that it would have been more expeditious to use addition and subtraction logarithms and employ the first and third of the funda¬ 
mental formulae, solving (L—L 0 ) by the tangent only when necessary. 
