AND ON THE REMOTE HISTORY OF THE EARTH. 
459 
F-*--=pP0(l-f0 3 ) cos 2e-E'PQ{l-%Q 2 ) cos e'-f E"F<? cos 2 e" 
5 
G-f-= -\EPQ{ 1 -f<?) sin 2e — fE'PQ s sin e'+P"# 3 sin 2e" 
9 
(19) 
§ 5. Development of the couple Jfl. 
In the couple fl about the axis of rotation of the earth we only wish to retain non¬ 
periodic terms, and these can only arise from tire products of terms with the same 
argument. 
By substitution from (7) and (10) in the last of (13) 
. ( 20 ) 
' h 5 
Then as far as we are now interested, 
2<E> f ffi'= —2<E> , £ <1 >=E 1 sin 2e 1 d -Ep\f sin 2e-f E. 2 sin 2e 3 
— f^h^'—E'i \p ()< f s i n e i+ E'hp~<p(p ! — <pf sin e'-f E\ fp~cf sin e' 3 
Hence 
-r-—=E 1 p 8 sin 2e t J r Elp [ (f sin 2e +Epf sin 2 e 3 
-\-E\2p Q cf sin e'Y^YpY(Y —cpf sin e E'fphef sin e' 3 . . . (21) 
If as in the last section we group the semi-diurnal and diurnal terms together and 
put Ey—E^—E, &c., and observe that 
then 
^ + 4 i 9Y+g 8 =(p 4 +g 4 ) 3 +2i ? Y=(l-^T+i^=^+t^ 
2p Y+ 2 \p % ff p : — (ff +2y> Y = iplfl 'p lJ r f —P V]= Q 2 ( 1 — f Q~), 
f-P=-E(P 2 +f<?) sin 2H-£'<2 3 (1-|0 3 ) sine' .... (22) 
V_ 
§ 6. The equations of motion of the earth abouts its centre of inertia. 
In forming the equations of motion we are met by a difficulty, because the axes 
A, B, 0 are neither principal axes, nor can they rigorously be said to be fixed in the 
earth. But M. Liouville has given the equations of motion of a body which is 
changing its shape, using any set of rectangular axes which move in any way with 
reference to the body, except that the origin always remains at the centre of inertia. 
