VARTATTON OF TERRESTRIAL MAGNETISM. 
471 
Table IV.—Force to Geographical North. 
Coefficients in the expansion 
Oj cos t + hi sin t -j- cos -j- 63 sin ‘It + V 3 cos 'M sin 2>t + «4 cos U -j- C, sin At. 
The unit of force is 1 C.G.S. X 10 “*^ ; t being astronomical time. 
Bombay. 
Lisbon. 
Greenwich. 
St. Petersburg. 
Summer. 
nter. 
Su 
iimer. 
VV 
nter. 
Summer. 
Winter. 
Summer. 
Winter. 
a. 
+ 
388-0 
+ 
332-4 
_ 
160-2 
_ 
116-1 
_ 
258-1 
162-9 
261-6 
115-2 
+ 
7-8 
+ 
38-8 
— 
7-6 
— 
60-4 
79-5 
— 
17-2 
+ 
142 2 
— 
9-3 
a.j 
+ 
1551 
+ 
125-1 
— 
70-5 
— 
70-1 
— 
149-8 
— 
93-4 
— 
152-8 
— 
68-5 
h 
— 
10-8 
-t 
7-7 
— 
12-8 
— 
39-6 
-9 
35-2 
— 
5-5 
-P 
41-6 
— 
15-3 
+ 
429 
+ 
40-3 
— 
4-5 
— 
20-0 
— 
24-6 
— 
28-5 
— 
17-2 
— 
19-2 
63 
— 
41-7 
— 
31-3 
— 
1-8 
0-0 
+ 
11-0 
15-1 
+ 
37-6 
+ 
21-9 
“4 
— 
40 
-f- 
15-9 
+ 
6-8 
— 
4-2 
+ 
8-2 
— 
5-3 
— 
9-1 
— 
4-0 
h 
— 
140 
— 
21 -3 
— 
9-5 
+ 
10-5 
+ 
7-0 
i- 
9-6 
-4- 
2-9 
5-9 
II. Some FormulcB useful in the Analysis hy Spherical Harmonics. 
In the expansion of mathematical expressions into spherical harmonics, it will often 
occur that we have to express powers of the cosine of an angle, or a cosine of some 
multiple of an angle, in terms of the differential coefficients of zonal harmonics with 
respect to the cosine of the argument. The necessary equations for the powers of 
cosines can be easily obtained by differentiation of the well-known formulae, giving 
the powers of a cosine in terms of zonal harmonics. But I think it will be useful here 
to give the general equations which I have had to use in expressing cos mu in terms 
of FVildpJ’, where p is any given number, p = cos u and the zonal harmonic of 
degree i. 
We may start from the expression 
cos m7i = -|- A,„_ 3 P ,«_3 + . . . A;Pi -f. (l), 
where 
_ _ ' {m - (t - 2)}(ffl. - {i - 4)} ■ . ■ (m - . . , {?»■ -f- (i - 2)} 
jw ~ (i -h — {i — 1)} . . . (w — 1) (m + 1) ... {ni + (i -f 1)} 
if w? and i be even, and 
_ _ , N - 2)1 (w- - (^ - 4)} . . . {m - 1) (m -f 1) . . ■ [m + (t - 2 )} 
' “ ' ' {m - f + l)}{m - (j - 1)} . . . (w - 2) (7/1 + 2) . . . {///. + (t -I 1)} 
if m and i be odd. 
