PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
220 
The equations (LIV.) show that the potential function in (LII.) is continuous with 
the velocity potential of (XXXI.) at the surface of the sphere. The equations (LV.) 
and (LVl.) show that the differential coefficients are also continuous. Finally (LIII.) 
shows that the density of the distribution of matter is that given in (LI.) 
Art. 18. Expression of the Velocity Components of the Rotational Motion in 
Clebsch’s Form. 
Clebsch has proved that the velocity components can be expressed as follows :— 
T - ^ _L \ ^ 
- dr + ^ dr ■ ■ 
+ X . 
cz oz 
where 
/h 0 
■dt + ’■ dr + ® 
0 0 0 \ 
',¥ + ’■ 87 +“" 37 ]'" = ®. 
V + ^ V + “’ s) X = - (]] + v) + i (- + -) . 
The value of X may be taken as 
SZ)- (IF - F)/(4F). 
(See equation XLVIII.) 
To find ju., there are the equations 
dt dr dz d fjd 
1 T VJ 0 
Therefore 
dt dr dz dfj, 
^ “ SZr{z - Z)j{2a~) ~ Z{5F _ 3(3 _ Zf - 6V}l(2a~) ~ 0 
(LVII.) 
(LVIII.) 
(LIX.) 
(LX.) 
(LXI.) 
(LXII.) 
(LXIII.) 
One integral of (LXIII.) is 
X = constant, 
LC., 
3ZF {R2 - a3}/(4F) = 3ZL/(4F) 
(LXIV.), 
where L is some constant. 
