AND ON THE REMOTE HISTORY OF THE EARTH. 
4G7 
g a 
t a 
— a£ a &c., - -=-a P, &c. 
T, ft ' 
Also if x, y, z and x,, y„ z t be the moons and sun’s direction cosines, we have as 
in (7), 
— &c., yf- s / 3 =cf> / +X / 4 -l(l-GpV), &c. 
Then using the same arguments as in Section 3, the couples about the three axes in 
the earth may be found, and we have 
H 
c 
{ T (y. 
d_ d_ 
dz dy 
ft ft. 
4-rly --s y- 
dz, 'dy, 
where in the first term x, y, z are written for £ 77 , £ in cr-fcy, and in the second term 
x t , y p z, are similarly written for <:, rj, £. 
Now let %nm / indicate the parts of the couple % which depend on the 
moon’s action on the lunar tides, the sun’s action on the solar tides, and the moon’s 
and sun’s action on the solar and lunar tides respectively, then 
% 
C 
, ny 
c ; c_ 
d W , / d d\a 
~ Z dy)a +T \ y 'dz Z/ dy)a 
Then obviously 
y-y=(«-bte+(«,-b ,)y*+ &c. 
As before, we only want terms with argument n in %„ m/ , and non-periodic 
terms in 0 mm/ . 
The quantities a, b, &c., x, y, z with suffixes differ from those without in having 
fl, in place of fl, and it is clear that no combination of terms which involve fl, and 
fl can give the desired terms in the couples. Hence, as far as % mm ,, mm „ are 
concerned, the auxiliary functions may be abridged by the omission of all terms 
involving fl or fl,. 
Therefore, from (4), we now simply have 
= = cos 2 n, T / = x F / = —2 pq(p 2 — q~) cos n, X = X =0. 
But c—b only differs from c / — 1) in that the latter involves fl, instead of fl, and the 
same applies to yz and yz,. 
Hence, as far as we are now concerned, 
(c—b)y t z=(c,—b,)yz 
and similarly each pair of terms in % mm , are equal inter se. 
