AND ON THE REMOTE HISTORY OP THE EARTH. 
457 
A number of multiplications have now to be performed, and only those terms which 
contain the argument n to be retained. 
The particular argument n can only arise in six ways, viz.: from products of terms 
with arguments 2 ( 77 -/ 2 ), n — 2/2; 2 n, n ; 2(77 + /2), 71+2/2; 71 — 2/2, 2/2; 77+2/2, 2/2 and 
from terms of argument n multiplied by constant terms. 
If < 1 ? and + and + and + be written underneath one another in the various com¬ 
binations in which they occur in the above expression, it will be obvious that the 
desired argument can only arise from terms which stand one vertically over the other; 
this renders the multiplication easier. The + X products are comparatively easy. 
Then we have 
(a) —= -^\_—E 1 p ! qsm(n — 2e 1 )-\-2Ep 3 q & (p l —q z )sm(n — 2e)-\-K 2 pq 7 sm(?i—2e. z )] 
(f3) +£¥'.$ = +i[~-E\p 7 qsin(n+e\) + 2E'p 3 q 3 (p 2 —qZ)sm(n+e)+E , z pqlsm(;a+e 2 )] 
(y) — ^T r +' = same as (/3) 
(8) +1++ =same as (a) 
(e) —|XpF' = — ^\_E"6p 5 q 3 sin ( n — 2e")—E"6p z q 5 sin (n+2e")] 
(£) = +l[E\Gp 5 q 3 sin {n — e\) —E' z 6p s q 5 sin (n— e' - 2 )] 
iv) + J iG+(l - Gp~tf) = -iE'2pq(p*-q z )(1 - 6p"q~) sin (n-e) 
% 
Now put -=F sin 72+G cos n. Then if the expressions (a), (/3) . . . (£) be added 
77* # 2 t 3 
up when 77=-, and the sum multiplied by —, we shall get F; and if we perform the 
same addition and multiplication when 77=0, we shall get C4. 
In performing the first addition the terms (a) (8) do not combine with any other, 
but the terms (/3), (y), (£), ( 77 ) combine. 
Now 
- \p\ +- 37/) 
_p¥(f* - r) - - f) ( 1 - ¥¥)= 
kpP - 1 pY = - YEW' - r/) 
- f p 5 ++ip V = — ip¥(¥ — ( f) • 
lp<l{p~ - T) (p 4 ’+<f - Gp~f) 
Hence 
\E v qPq cos 2 z Y — Ep\f(pr — cp) cos ^e—^E^pq 1 cos 2e. z 
- \E\phqY - 3 q~) cos e\ - \E’pq{jP - (f) (p 4 + q 4 - 6p~q~) cos e' - EE PP ( P (3p 3 — <f) cos e 3 
— %E"p 3c f(p~—T) cos 2e".(15) 
3 N 2 
