SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. H 
when its axis of rotation had come to depart from its axis of symmetry.* * I propose 
then to discuss the subject shortly, and to establish the law which was there assumed. 
Suppose that the earth is rotating with an angular velocity w about the axis of z, 
but that at the instant at which we commence our consideration the axis of symmetry 
is inclined to the axis of z at an angle a in the plane of xy. and that at that instant 
the equation to the free surface is 
r=a j" l+^f^-[cos a cos d+sin a sin 9 cos </>] 3 ^ j- 
where m is the ratio of centrifugal force at the equator to pure gravity, and therefore 
. , co"a 
6Q U cti to -. 
9 
5ma /I 
Then putting i= 2 in (12), and dropping the suffixes of S, s, cr, s— ——[ ]- 
We may conceive the earth to be at rest, if we apply a potential 
so that 
By (12) we have 
wr~ S=~ (rihvr~ ^ — cos 3 6 
s=;W^—cos 3 e 
5crS 
cr- 
2 g 
l—exp 
2wgat\ 
19v ) 
exp 
iwga 
Iwgat 
Uh7 
Then, substituting for S and 5, and putting k= |() 
er=' 111 ^ j^; —cos 3 djj 1 — exp(— [cos a cos d-fsin a sin 6 cos (ff^jexp(—nt) j>- 
Now 
[1 — exp>{ — Kt)~] cos 3 9 + exp ( — xt) (cos a cos d-f-sin a sin 9 cos </>) 3 
= cos 3 9 [l — sin 3 a exp>( — «:/,)]+sin 3 a sin 3 9 cos 3 <f> exp(~Kt ,) 
+ 2 sin a cos a sin 9 cos 9 cos exp(-Kt). 
Therefore the Cartesian equation to the spheroid at the time t is, 
+z - =c p —^ [2 2 ( 1 — sin 3 a exp (~Kt))-\- x z sin 3 a exp (— Kt ) + 2 xz si n a cos a exp ( — Kt)} 
0772 / 
1 + T 
or 
* “ On the Influence of Geological Changes on the Earth’s Axis of Rotation,” Phil. Trans., Vol. 167, 
Part I., sec. 5. 
