760 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then when the system is perturbed both by the oblateness of the earth and by the 
sun, we have from (107) and (109), 
£ . . dj , . 
y smp — = — ^rC sm 2 1 sm 2j sm (N—xp) 
£ . . dN 
sn\; 
k 
dt 
. di 
n sm i — = 
dt 
. . dxp 
n sm i -f = 
dt 
T ' p 
■-grf (cos 2 7 sin 2/+ sin 27 cos 2/ cos (N—xp) ' — g | sin 2y 
.{rC sin 27 sin 2 j sin (N—xp) 
— {sin 27 cos 2y‘+ cos 27 sin 2 j cos (N—\p)} — ^t£ sin 27 
>• (110) 
The second pah’ of equations is derivable from the first by writing 7 for j and j for 7; 
TV for xp and xp for X ; n for £/k ; n for 12 ; and for ^ in the term in t. 
The first pair of equations may be put into the form 
. d(2j) k .. 
cos 2 ) ——= — -tZ sin 27 cosp cos 2j sin (N — xp) 
■ 
8111 2 J Tt = 
fb . , . a . T . . « 
— - rf{cos 27 cosp sin 2y’+ sin 27 cos 7 cos 27 cos (N—xp)}—^ — sin 2y cos^ 
Now let 
Therefore 
or 
Again 
© 
y = t sin 2/ sin N, r] = ^ sin 27 sin xp 
2=1 sin 2j cos A, sin 27 cos xp 
n dz . d(2?) . , r . n .dN 
2 — = cos N cos 2/ — - -sm N sm 2 1 — 
dt 1 dt ■ dt 
k r 
=h rt[cos j cos 2/.2p-fi cos 27 cos/. 2y ]+■^ — cos j.2y 
dz /krt 
dt 
= ( cos 27 cosy+1^ cosj) y+-\ cos/ cos 2j. v 
krC 
. dy . -.j. . d(2j) n. T • ^ • dA 
2 --- = sm A cos 2; —r~+ cos A r sm 2; —— 
dt ■ dt dt 
kr£ r 
— — "--[cosj 1 cos 2/.2£+ cos 27 cosy.2zJ — cosj.2i 
(in) 
their common centre of inertia, then the three principal moments of inertia of the system are 
lfwic 2 /(Hf+m), Mmc 2 /(if + m), 0, and therefore the precessional constant of the system is A Thus the 
formula for dN/dt is precisely analogous to that for dip/dt, each of them being equal to t' X prec. const, 
x cos inclin. -5- rotation. 
