452 
MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
of thirteen terms which have the desired form, and a fourteenth which is independent 
of the time. 
It will now be convenient to introduce some auxiliary functions, which may be 
defined thus, 
cp( 2 n) = Up 4 cos 2(n—n)-\-p~q 2 cos 2 n + cos 2 (w+/ 2 ) 
R(«.)= 2 p 3 <7 cos (n — 2 / 2 ) — 2pq(jo? — <f) cos n — 2p<f cos (n + 2 / 2 ) 
X(2o) = cos 2/2 
• ( 4 ) 
<t>(2 «—|tt), R(m— \tt), X(2q—^ 7 r) are functions of the same form with sines replacing 
cosines. When the arguments of the functions are simply 2 n, n, 2/2 respectively, they 
will be omitted and the functions written simply <b, R, X; and when the arguments 
are simply 2 n — \tt. n — \tt, 2/2 — r, they will be omitted and the functions written < b ', 
R', X'. These functions may of course be expanded like sines and cosines, e.g., 
x l r (n —a) = R cos a-f-R' sin a and R'(n — a) = R' cos a. — R sin a. 
If now these functions are introduced into the expression for Y, and if we replace 
the direction cosines sin 6 cos <f>, sin 6 sin (f>, cos 6 of the point P by y, £, we have 
LVTi 
— R, 2 ^ 77 , ££, rjl, + — 2 £ 2 ) are surface harmonics of the second order, and 
the auxiliary functions involve only simple harmonic functions of the time. Hence we 
have obtained Y in the desired form. 
We shall require later certain functions of the direction cosines of the moon referred 
to A B C expressed in terms of the auxiliary functions. The formation of these 
functions may be most conveniently done before proceeding further. 
Let x, y, z be these direction cosines, then 
cos PM=x£+ yrj-\~zl 
whence 
cos' I’M - = (.rf+ i/ri + z £)-—Jr If '+r + £ 2 ) 
= + + • ( 6 ) 
But from (5) we have on rearranging the terms, 
COS' 
PM- 
— <b+ 3 X+rr( 1 — 6p 2 <f )} -fi p - {<b -f- -^X +£(1 • 
-2t ? ^R'-2^.1R-2^cP / . 
.( 5 ') 
