218 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
h . , . /4 . 1' 
2 ( „2 + ,,2 ) (',,2 ^ „, V 3 
clu 
+ 
Ti 2h — kci^c ( 0 + ~ 
cV '0 («■ + uf {c^ + 
clio 
cV 'o + uf (c“ + 
{rM^-Z)-f(.-Z) 
(. - Z). 
S') 
7. The expressions^^ — ^ ^ ^ cannot be taken by themselves to 
^ oy oz oz ox ox oy 
represent the velocities inside and outside the ellipsoid, for, though they would 
furnish continuous values of the velocities at the surface of the ellipsoid, they would 
not make the pressure continuous. 
Art. 1. The Equations of Motion. 
If the velocity components of a mass of incompressible fluid at the point x, y, z be u, 
V, IV at time t ; if the pressure be jy, the density p, and the potential of the impressed 
forces V, then the equations of motion are 
du 
+ 
dio 
+ 
du 
du 
0 / 
dt 
dy 
dx \ 
dv 
+ 
dv 
01’ 
0y 
0 / 
dt 
dy 
+ ”& = 
dy \ 
dir 
+ 
dw 
+ ^ 
dio 
dvj 
9 / 
dt 
u-^ 
ox 
% 
+ w~ — 
oz 
07 ( 
P 
P- 
P 
JL 
P 
• • 
CIO 
. dv dw 
a,. + % + a; = 0 
If the motion be symmetrical with regard to the axis of 2 , let r 
let the velocity perpendicular to the axis and away from it be r. 
Then 
u = Txjr 
V = ry/r 
and the equations of motion become 
0T 0r 0T 
■a^ +’"S + - 
dw dvj dvj 
dt + ’■ av + = 
0T T , w.. 
“a;' + 7 + & = ® 
—ff-^ + v 
0^ \ P / J 
dw 
> 
. . . . (IL). 
+ yff and 
. . . (Ill), 
(IV-)> 
(V.), 
