PROFESSOR M. J. M. HILL OH A SPHERICAL VORTEX. 
239 
Now, Helmholtz’s method gives the following values for u, v, w as deduced from 
t V’ C, 
_ 0P 0N 0.H 
“ 0^; ■ di/ 'dz ’ 
0p . 0L ex 
V - -^ ’ 
cz d:c 
0P , 0M 0L 
IV — + V W" ’ 
dz dx dy 
0T a=p_ 
+ df 3:= 
where 
andL, M, N, are the potentials of rj/'In, il^rr respectively, taken throughout the 
rotationally moving fluid. 
Hence, if the rotationally mo\dng fluid be limited to the ellipsoid of revolution 
above, the values of L, M, N may be w^orked out completely. 
For it is known that a solid ellipsoid of density, jxx, gives for potential outside the 
ellipsoid, 
a;" _ ^_\ _ du _ 
a? + w 5- + u c- uj (ft- + u)^ ' (//- + «)'" (c~ + ^ ’ 
where e is the positive value of \ satisfying 
or 
ft- A 
+ tI 
y. 
+ A c2 + A 
Inside the ellipsoid the potential has the same value if the lower limit of the 
integral, e, be replaced by zero. 
(See a paper, by Mr. Dyson, “ On the Potentials of Ellipsoids,” in the ‘ Quarterly 
Journal of Mathematics,’ vol. 25, 1891.) 
Hence, outside the ellipsoid, 
{z — Z)-\ chc 
~ / (fP- + (c“ -F ’ 
{z - Z)2\ du 
^u) {o? + uf (c^' + m)"'" 
N = 0. 
