PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
227 
Art. 10. The Cyclic Constant of the Spherical Vortex. 
Tlie cyclic constant of the vortex is 
a r \Hv?- — -Xi 
- a JO 
dz d r 
2c(r 
=i:i 
dz d r 
dz 
15Z 
~ 4^ 
= 5aZ 
0 
a-’Z — .7 
o 
+ a. 
(L.) 
Hence the cyclic constant of the vortex sphere is equal to live times the radius ^of 
the sphere multiplied by the uniform velocity with which the vortex sphere moves 
parallel to its axis. 
Art. 11. The Kinetic Eneryy of the Vortex. 
The kinetic energy of the vortex 
,'Z + « rv{a2_(, _Z)2} 
= np dz dr r 
• Z-a Jo 
7^- ^ 
8a* ^ ^ ^ ^ m 7 “ i 
_ 'l'dTrpZ-(d 
~ 21 ■ 
The kinetic energy of the irrotational motion outside the voi’tex is 
2 G 2 
"25a^ — 30a^ {z — Zf + 0 (z — Zf " 
+ 45r^^ (z - Zf — 60«b- + 3G/'‘^_ 
_ -Kpi? p + “ r[25a" - 30a2 (z _ Z)^ + 9 (z - Zf] {a^ - {z - Zf] 
8a* 1 t45(z_ Z)3 - 60a~}[G3 - (z- Zf j2+12 {a2_ (2_ 
W ir" (z - ZY + 3 (z - ZVi 
= TTp 
Z + ft 
dz [ 
Vp,,2 - (3 - Zf] 
dr 
■ 4a* 
