58 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
The equatorial axis therefore lies in the equatorial section in a similar position to 
that for a sphere. 
13. Dyad Spheroids. —Polj-ad spheroids clearly occur in the same way as for 
spheres ; they are, however, also unsteady. It will be sufficient merely to indicate 
the steps of the proof. 
Let n = u' and u = u" denote the two boundaries, ijj will involve terms in Z 2 and 
Zj. By applying tlie surface conditions in the same way as for the spheres to both 
sets of terms independently, the coefficients are determined, whilst the condition that 
the internal interface has the same translational velocity as the outer gives for Z 4 an 
equation which u' and u" must satisfy. This is 
where 
ri/Q/i 
^ ivr'' ^ iv/T' . ^ — Uo 
__ M - -g- M + g _ = 0, 
M X., 'y- - Y, 
i/v, du 
aud the dashed letters refer to values at the outer and inner boundaries 
The same applied to the 7a, terms give 
// / 
ll , tt . 
S" 
where 
N" - ~ N' + (C"S"' - = 0, 
N = X, - Y. 
dll du 
The existence of steaddy-moving spheroids depends on the possibility of finding 
values of u', u" to satisfy these two equations. 
It is easy to show that 
6M + N = iS(25S' + 14). 
Hei ice, adding 6 times the first equation to the second, there results an equation free 
of logarithmic terms and which can easily be reduced to 
1C’ 
C"S"' - C'S''‘ ~ 50'- - 1 ’ 
Butting C' = y, C" = X, the factor (x — yf divides out, and the equation may 
be put in the form 
(a:'l+ 2.x“y + Sxy~ — 2x) + (y- -- 1) (3y“ + 1) — 0. 
Now a; > y > 1. Hence Sxy~ — '2x = xy~ + 2x {y~ — 1) is positive. The expres¬ 
sion on the left is therefore always positive and no suitable Vcdues of x, y satisfy 
the equation. A prolate spheroidal dyad is therefore not steady. 
The condition for the oblate spheroid can be found by writing S^/ — 1 fur C. It 
can be shown that this also has no suitable root. 
