228 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
Tvp [ c/R [ ddW' sin 6 (r^ + lO') 
J a Jo 
r® rn 
= TTO cm cWW sin e ~ (3 cos-d + 1 ) 
J a * 0 4 it” 
«6Z2 
Zed 
TrpcdZ- 
Art. 12. The Distribution of Matter ivhich would produce the Velocity Potentied 
of the Irrotational Motion. 
The velocity potential — a^Z [z — Z)/(2R^) at points ontside the sphere is due to a 
distribution of matter inside the sphere of density 
— 15Z ( 2 : “ Z)/(87ra~). 
and the potential of this distribution of matter inside the sphere is 
(LL), 
Z (2 — Z) [3R^ — 5a^]/(4a') 
For 
, 1 0 . 02\ /Z(2-Z)(3rd-5A2)\ , ^ ( lfZ{z-Zf 
P + 7^ + 0^ iid + -87^ 
. . (LIT.). 
::=0. (LIIL). 
Further, when R = a 
and 
Z(2 - Z) {3R' - 5ft"“}/{47) 
- cdZ{z - Z)/(2R3) 
- hZ{z - Z) 
-hZ (3-Z) 
(LTV.). 
Again, when R = a 
r- 
O 
4« 
, - Z) (3R3 - 5cr) 
3Z 
~ 2a- ~ 
and 
Also when R = a. 
a 
7 
■ z 
4a 
p; [- (. - Z)/(2R=)] 
, (z - Z) (3R" - 5a--) 
oZj / 
o.n{- 
z)"! 
I 
(LY.) 
-Z) 
and 
^,[-a^Z(.-Z)/(2R3)] = 
Z 3Z . 
■> + v; 
2 ^ 2a- ^ 
-Z) 
0,1 
>■ 
(LYL). 
