96 
PEOFESSOR W. M. HICKS ON VORTEX MOTION. 
Therefore 
1 
a 
But 
Hence 
[X sin \ - 3J (X)} = — (-/(a- sin (a + Xp) - X')]. 
/(«, X» = (a, X'). 
X sin X — 3J (X) = 5 (Xy sin (a + Xj?) — 3y’(a, X^)} . . . (42). 
So, also, making dx^i^ldr — d^^/dr when r = l> v^e get 
T = §{/K) 
4 X' sin X' ] 
. . (43). 
Equation (41) determines a ; Equation (42) gives a relation between X, p, and q. 
We can therefore impress in general three further conditions. For instance, ratio of 
volumes, ratio of primary circulations, and ratio of secondary circulations. 
There is a natural connection of the various singlets which go to make up an 
aggregate of the kind first discussed. At any interface all the differential co-efEcients 
are continuous. In the polyad aggregates this is not so. Differential co-efEcients 
beyond the first are not continuous. Monads, &c., which go to form them, are arti¬ 
ficially combined. It is possible we may, on this basis, develop a theory of special 
aggregates which will unite with one another, or split up and be capable of uniting 
again in another manner. Some progress has been made with such a theory, but 
before an attempt is made to carry such a theory out it will be necessary to investi¬ 
gate the stability of the various systems. I hope soon to be able to take up this 
question. ' 
[May 6, 1898. —By the permission of Professor Hill, to whose careful reading of 
the MS. I owe a great debt, I ajipend an independent and very suggestive proof by 
him of the general theorem of gyrostatic vortices, based on the equations of motion.] 
Take as co-ordinates r, 6, z. 
X = r cos 9 
y = r sin 6 
Let p be the pressure, 
p the density, 
V the potential of the impressed forces. 
Let T be the velocity increasing r, 
cr be the velocity increasing 6, 
IV be the velocity increasing 2 :. 
Then the equations of motions are - . 
