PKOFESSOR W. M. HICKS ON VORTEX MOTION, 
whence it is easy to prove that 
-1C, 
C + 1 
C - 1 
n = - A-C(:i5C*- 13). 
With these values the particular integral is 
~iX,)(2Z,- IZ,). 
The terms in X 2 Z. 2 , X 4 Z 4 may be supposed mei’ged in tire general solution. We 
may then write 
i//i = (A2X2 TrkX'^^i) Z2 + (A4X4 — i.\Trl'X*^ 2 ) 24, 
\po = B.2Y2Z2 + B4Y4Z4. 
From these it is easy to deduce the values of A 2 , etc., for a single free aggregate, 
by applying the conditions xf/i = i //2 and clxf/j/du = dxfj-ijcht at the surface. It is 
unnecessary to do this, as from Hill’s work we know that it is not steady. 
The case of motion inside a rigid spheroidal boundary is also given by Hill.* 
Tlie solution follows immediately by impressing the condition = 0 when u — u. 
Hence 
Ao = - 7-5 , 
A 2 
A, = , 
where thick type denotes values at the surface, and 
xjj, = ^5- rrkX^ Z2 - A Z4, 
which easily reduces to 
xfjl = 
'Itt Ji'X 
4 + 5S^ 
^ (S^ ~ 8'“^) S'‘^(S^ + cos^i’) cos"r, 
The total circulation is f 
The equatorial axis is given by 
du 
— 0, when v = 0. 
That is by the equation 
2S"S - 48® = 0, or 8 = ^ S. 
* ‘ PEil. Trans.’ Part 11., 1884, p. 403. 
VOL. CXCII.-A. 
I 
