722 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
But if oj v (o. 2 , to 3 be the component angular velocities of the planet about A, B, C 
respectively, and if we may neglect (C —A)/A compared with unity, the equations of 
motion may be written 
d &) | H f/a> 2 fit dco.. 
dt C ’ dt C ’ dt C 
as was shown in section (6) of my previous paper on “ Precession.” 
Then since x : — nt , we have by integration, 
1 / . dm dW\ 1 d W . 
—. COS i —- — cos v- — sin x 
n sm i\ d x dy ] n di 
1 / . dW cW\ . 1 dW 
—— cos ^ -- - ) sm x -cos x 
n sm % \ »x / n ch 
Then substituting these values in the geometrical equations, 
di 
It 
= — to, 
cos x+<yj sm x 
We have finally, 
/yjr 
Sill l— = 
dt 
q sm x — cos X 
. . di .cm 
n sin l v = COS l —r~ 
dt d x 
. . dcir dm 
n sm i — =—T 
dn dm 
dt d x 
dm - 
d.yjr 
y 
(18) 
These are the equations which will be used for determining the perturbations of the 
planet’s rotation. 
We now see that the same disturbing function W will serve for finding both sets of 
perturbations. 
It is clear that it is not necessary in the above investigation that a should actually 
be a tide wave ; it may just as well refer to the permanent oblateness of the planet. 
Thus the ordinary precession and nutations may be determined from these formulas. 
§ 3. To find spherical harmonic functions of Diana’s coordinates with reference to axes 
fixed in the earth. 
Let A, B, C be rectangular axes fixed in the earth, C being the pole and AB the 
equator. 
