240 
PROFESSOR M. J. M. HJLL OR A SPHERICAL VORTEX. 
Hence, 
0N 
0M 
/4/j 
Si/ 
I 
i 
II 
a 
0L 
0N 4 
/47- 
0s 
rs — Oj 0 
OX 
0M 
0L . 
_ — rt^n 1 
l\h 
du 
{a? + uf {g~ + 
du 
3 2 5 
{rr + uY (c" + '20 
3 2 ) 
d^c 
3,!/ 
/ J e \ « - + U C' 
- Z)^ \ 
du 
+ U j (cd + v)~ (c- + u)'^' 
The values inside the ellipsoid are obtained by replacing e by zero. 
Outside the ellipsoid the expressions 
where 
(f) = 
^ o]\I _ ^ 
dy dz dx 
0L 3N _ dcf) 
dz dx dy 
^ _ 0L _ ^ 
dx dy 0^ 
(d + u 
(z — Z)^\ du 
+ u J {cd + u) (cr + ?0*'^ 
as may be immediately verified by difierentiation. 
(f) is obviously a potential function, viz., it is what 
« I 
+ U 
{Z - Z)2\ du 
c^~ + ^0 ) (cd + {b^ + 0'^' (C' 4- 
becomes when a — h. 
Moreover, if k be suitably determined, it is the velocity potential for the fluid 
outside the ellipsoid moving with velocity Z parallel to the axis of 2 . (See Basset’s 
“Hydrodynamics,” vol. I., Art. 147.) 
Inside the ellipsoid the values of 0N/0y — dM/dz, &c., can be deduced by putting 
€ = 0, and it appears that they do not give the original expressions for u, v, w. 
Hence in this case tlie function P exists. 
It is such that 
2-^x{z — Z) — 
du, 
{cd + %df {d + uf'^ 
dy 
y 
(z - Z) f 
0 (cd + w)' (c® + 
0s 
a-^) - 2-^ (. - Z)^ 
— a'^c 
2d 
a" + u 
(z — Z)~ \_ du 
d + / {cd + u)~ {d + uf^"' 
