42 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
(4) Toroidal Functions .—Here [‘Phil. Trans.,’ 188 L, Part III., p. 614j 
p + e + zi sinh cho sinh u 
u vl = log 
p + a + ;u 
= a 
cosh u — cos V 
(hi 
whence 
(I /C—COSV(IF\ . 1 P" 
*\-^ * j + y * ”) ih) = - y"X. 
4:Tr(C‘ 
(C — cosH 
2 CO sin X • (10). 
5. Equations 1, 3, 4, 5 or 6 give the conditions for a possible motion. It is ojDen 
to us to choose xfj arbitrarily. In this case the equations give v, co, y, (p. The motion 
is instantaneously possible, but in general it will at once proceed to change the 
configuration—the motion will not be steady. The application of this theory to 
values of xjj which are already known (Hill’s vortex for example) leads to interesting 
results, but the absence of steadiness robs the theory of importance. If we impose 
the condition of steady motion, it is no longer open to us to choose xp at will. Let 
us then impose this condition. The condition that the motion shall be steady 
involves:— 
(1) xjj must be a surface containing both vortex-lines and stream-lines. 
This is already the case. 
(2) V(x) sin {(p x) dn must be constant over the surface, 
It must therefore be of the form Fdxp, where F is a function of \h. Hence 
VO) sin 
(^ + X) = rg' 
(11). 
Expanding this, and substituting from 1, 3, 4, 7, 
or 
1 J_ ^ 
+ “V 
cot 6 d){r ] 
d0 I 
87r"p"F, 
d'-ik , _^ cot 0 ^ _ Q 2 p df 
tV 1” der- ' 1- de ~ p n / 
( 12 ). 
where f and F are arbitrary functions of if/. Choosing these, equation 12 will give 
the type of if//'' 
We proceed to apply these general theorems to certain special cases of spherical 
aggregates. In order to exenqolify the method employed we will take first the case 
in which there is no secondary spin, the type in which Hill’s spherical vortex is the 
simplest case. 
* For aiiotlier proof of tEi.s equation, due to one of the referees, see end of present paper. 
