PROFESSOR M. J. M, HILL OH A SPHERICAL VORTEX. 
217 
These are, outside the ellipsoid, 
L 
M 
N 
where e is the parameter of the confocal ellipsoid through .r, y, z. 
Then 
cv^ 
{z - Zf 
div 
cd + u d (a® + (d + 
= (^ + 7) * f. (' - 
a- + u 
du 
{z - Z f _ 
c- + ?f. / ('d + (d + 
= 0 , 
0H 
Si/ 
dN 
dx 
-= ka^c 
«' c 
4 . 1 
du 
(rd + %)“ (d + 
du 
{od + iCf {(? + u'f‘1^ '■ 
0 M _ BL 7 .1 /4 , 2^2 {z-Zf\ du 
dx dy cV ), y cd + u d + ^6 y (fd + (d + u)y-^ 
To obtain the corresponding expressions inside the ellipsoid, it is necessary to 
replace e by zero. 
Outside the ellipsoid ^ ^ ^ ^ are the differential coefficients of 
oy oz cz ax ox ay 
the potential function 
{z — Z)'\ du 
d + ?t / (fd + ?0 (d + uf'^ ’ 
which, with a suitable value of gives the potential of the irrotational motion outside 
the ellipsoid moving parallel to the axis « wuth velocity Z. 
T, , . ., ... ., 0H 0M 0L 0H 0M 0L ^ . 
-Dut inside the ellipsoid respectively equal to 
the values of w, v, from which the investigation commenced. 
In fact 
dl 
0N 
0M 
u — 
dx 
dy 
■ 0^ ' 
0P 
+ 
0L 
0H 
V = 
Si/ 
dz 
dx 
0P 
+ 
0M 
0L 
w = 
05 : 
dx 
" S// ’ 
where P is the potential function 
MDCCCXCIV.—A. 
2 F 
