VARTATTOJ^ OF TERRESTRIAL MAGNETISM. 
479 
Y'^''sin 11 = cos 2X 
G4-59 + 9-42 
— 0-42 . „ — 1'47 — 
(Ifj? 
(Itj? 
tl-V r/-P ' 
+ »-'Gv+"'5‘;,A 
+ sin 2 X 
r72T> r72p rZ^P 
17-69 + 15-87 --7 + 1-27 4-^ + 0-68^ 
illjb~ a [ji- 
— 0-40 
cin\, 
— 0-80 
rzqy 
dfjr 
and from this, by multiplication with sin® u and integration with respect to X, 
- V<2Vfl = cos 2 X [2-95 T/ + 7-94 T 3 ® + 0-63 T,® + 0-34 T/ - 0-20 T,® - 0-41 T/] 
-sin 2X[10-76 T 2 ® + 4-72 T 3 ®- 0-21 T^®-0-73 15 ®+0-29 Tg® + 0-26 T.®J . [B], 
To find y® from the coefficients O 3 , 63 , a^, h^, the values of these coefficients were all 
divided by sin® u, for a reason which will appear. It was then found, however, that to 
represent the numbers so obtained satisfactorily, a greater number of terms was 
necessary in the expansion than the observations seemed to justify. I was thus led to 
give up calculating as far as tliese types are concerned the terms on which the differ¬ 
ence between summer and winter depends, so that, instead of using tlie coefficients from 
Table VI., the mean of corresponding values on both sides of the equator were taken, 
changing, of course, the sign of the coefficient given for the Southern hemisphere. In 
the next place, it was found that, owing to the smallness of sin u at high latitude, the 
three numbers corresponding to the colatitudes 0°, 7°-5,15° were veiy large, while their 
weight is very small, as they have only been obtained by graphiceJ extrapolation ; 
I have, therefore, discarded these numbers altogether, putting them into brackets in 
Table YI. Bessel has shown how the method of least squares may be applied to 
obtain a trigonometrical series for a succession of values some of which are missing. 
The equations thus calculated are as follows :— 
=74 cos u — 118 cos 3m — 101 cos 5m, 
63 /sin® u = 23 cos u — 111 cos 3m — 127 cos 5m, 
04 /sin® u = 39 sin 2 m — 6 sin Au — 9 sin Gu, 
7 > 4 /sin® M = 31 sin 2 m + 38 sin 4 m + 4 sin 6 m. 
The agreement with tlie numbers from which the series are obtained is not as good 
as would be desirable, especially for O 3 and 63 , but the types higher than the second 
will probably depend much on local circumstances, and the result would not, in 
my opinion, repay the trouble of taking account of further terms in the above series ; 
cos M, cos 3m, and cos bu have already been given in terms of c7^P;/cZp,® (Equations 
G, I, L, page 474), and, by differentiation with respect to ti, we can also get sin 2m, 
