PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
241 
so 
that 
Hence 
P = 
01 
0; 
C- ■ ' 
^ / 7\ r_^ 
^3+ ,2] ' (3 + 
V,)- 
3/2 • 
~ - cvc ( — 
4 
tr 
L . ±\ _^ 
- ^ / J„ («- + tCf (c- 
+ 
Z+2k-aM3+i)\’'^. 
(«- + u)~ (c^ + 
_ du 
^ ( r/2 4- )/V 
U 
k^- '•=" 
(,3(,_Z)-§(s-Z) 3) 
(z - Z), 
and P is a potential function, for it satisfies 
1 dV 0n' 
r 0r 0z- 
= 0 . 
It appears, then, that on attempting to obtain the values of the velocity com¬ 
ponents from the inolecnlar rotations by means of Helmholtz’s method, it is necessary 
to introduce the function P. This points to the existence of rotational motion outside 
the ellipsoid (as was previously remarked), P being’ the potential of the irrotational 
motion inside the ellipsoid due to the vortices outside the ellipsoid. 
If P be left out of account altogether, and an attempt be made to see whether the 
velocity components 0N/0y — 0M/02, 0L/02 — 0N/0a', 0M/0a; — 0L/0y, which give con¬ 
tinuous velocity at the surface of the ellipsoid, will not also give continuous pressure ; 
then inside the ellipsoid 
+ ' 
IV = ^’(Pc + ~ 
^ (1 
du. 
•2r^ _ P - Z)- \ 
- -f I'j c” + u j (fd -f 
1/3 > 
or putting 
then 
I — ko^c 
III = ka^c 
)i z=z ka^c 
dio 
(«- + ’ 
die _ 
(cP -1- (c- + ’ 
du. 
(ft- + ^e)~ {c~ + ’ 
r = nr [z — Z), 
IV = I — 2r^7n — (z — Zfn. 
2 I 
MDCCCXCIV.—A. 
