DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
175 
The general problem will be treated in the Appendix, where it is shown that for 
practical purposes y and y may be treated as small quantities, the squares of which 
may be neglected. The equations may then be written 
p cot 9 ^ = Pu + (1 — y r cos 9— y sin 0 cos X) —- 
ak d9 sm 0 dk 
a dip c?S i f n 'a \ \ dRi 
-po cos 0 = P 0 - (1 —y cos 0-y sin 0 cos X) 
' dv sm 0 dk <10 
(15). 
Neglecting y and y , our previous results give R in terms of i p. Let Q/ be one 
part of It thus obtained. The next approximation is found by substituting Q/ for It 
in the terms of (15), which contain y and y'. 
The complete value of It, as far as it depends on Qy, will then be Q/ + R’, where 
R’ is determined by 
/ / a . - a w dQ," c/S’ 
(y cos 9 + y sm 0 cos X) -. " = p 0 —r^ 
sin 0 dk 
(y' cos 0 + y sin 0 cos X) —= p 0 
(It 7 
+ 
dR’ 
d0 sin 0 dk 
ai 
sm 
_ dR! 
9 dk <10 
(16). 
In the two cases which specially interest us we must substitute for Q/ the values 
respectively of Ha 1 and pR s 2 as determined by (5). The solution of (16) involves the 
elimination of S'. 
Treating the terms containing y and y separately, we find for It' as far as it 
depends on y', 
y 
cot 0 ^3? + ~ sin 0 cos 9 C -^3?- ) — —tn . n + 1 . sin 0 It’ 
dk 2 d0 
10 
if It’ is expressed as a series of harmonics, n being the degree of one of the terms of 
the series. 
The left-hand side may be transformed as shown in the Appendix, the result being 
given by (25) ; we obtain in this way 
< [{n + 2)n(n — or+ l) n<r 
7 1—2^n— y 
(pi — I ) (pi + 1) (jl + cr) 
2)1+ 1 
%n (pi El) lt’„. 
R' is therefore expressible by two terms, ItL+i and R\_ l5 so that 
(2n+1 ) (pi+1 ) RA+i = y'n(n-a-+ l)Q' n+1 , 
(2 n+ l) n . RVi = y' (pi +1 ) (n + o - ) QA_i. 
As regards the terms in y, the elimination of S' leads to 
y ( — cos X -j— cos X ~ sin 2 0 '! ) Q/ = — %n (n+ 1 ) sin 0R’ n . 
\dk dk d0 d0) 
