PROFESSOR VV. M. HICKS OK VORTEX MOTION. 
55 
It is thus the same as a mass of one-third its own mass moving with its velocity of 
translation. Now 
Eo = 3-5 fc ’ 
therefore 
E ... Y 5c 
E„ ~ ’ V„ “ / ’ 
therefore 
7 /vy 
E„ ^^\yJ' 
This only holds, however, when V/Vq is small. It is a small part of a parabola in 
the figure touching the axis of E/Eq. 
12. Spheroidal Aggregates. — As is known from Hill’s investigations, the spheroid, 
although an instantaneously possible form, is not steady. It proceeds at once to 
change its shape into a non-spheroidal one. It seems, however, advisable to give the 
general outline of the method as adopted in this paper and as applied to the 
spheroids, in order to investigate whether by superposing a second or third layer it 
may be possible to obtain a steady form. 
The functions involved and the differential equation for rp are given in Eqs. 8, 9. 
Writing C for cosh w and S for sinh u, the differential equation in xp is 
since 
1 
cos V 
du \ S du j 8 
dLl Jl 
dv \cosv 
Ank-X* S cos V (C" — siir v), 
CD = hp = Z:XS cos V. 
As in the former case, a particular integral is 
i// = — ^ hp*^ =z — T Trews'* cos^ V. 
It remains to integrate 
1 d fl_ d^\ , 1 ^ 0 
cosr du, \ kS du) S dv \cosr dv ) 
This can be satisfied by writing \p = where X and Z are functions respec¬ 
tively of u and v only, and 
dv Vcos V dv 
m 
Z 1 
cos V 
pL /J_ _ 
dn \ S du 
viX 
S 
J 
m being any constant. These equations are 
