06 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
I 8. The moment of anoular momentum is 
o 
M 2 [ [ 'iTTpj'(Ir (Wvp s'm (j) 
/e“ sin 6 dr dO 
OJ 0 
2A, r<cr^ -^ 
a , 
['{J - a;'-’ J') dx ("siiE 6 dO 
S;A, — sni A. Jo^ -’o 
hi 
Si A — sill \ 
r- 
-V '- . 
I - - sm 
o\ A 
Si A - sill A 
\x — X' cos Vc; — ’ I dx 
sill A\ sill A1 
A 
where 
and 
where m denotes the volume of the aggregate. 
19. The internal energy of the aggregate, supposed without translation is 
E = I jl^TTp dp dz (v- COS" (^ + y' siid </») 
t/, = A{J-a;y')sin-d 
A = . 
Si A — sin A 
Hence, as in tlie usual way. 
Now along the boundary v// = 0. Also 
(I j 1 d^\r \ d / 1 dyjr x A" . J siir 0 X’yjr A’A 
,T:(br a;) + ati-v j. ) = - vt - vv: “ ~dP^ ’ 
dp \ p dp J d, 
therefore 
„ A= [{drxdrdd , A-AJ' 
Ft—+ w„. 
j'v/ip'r dr d6 
2 sill- Xc sill 2Ar , ^ , 
- ~ +2 COS" A. 
A-e- 
A'A'^ 
oird 
Ahl- 
37ra 
A^-A'^ 
OTTh \ 
1 T'2 
5 
/SsiiiA 3 cos A sin A\ . , sin 2 A 2 sin-A 
-^ - X + ' + Tx-x=" 
1 ^ \ T '2 I 2 sin A cos A 
^ ~ AV' ' A- ~ A 
fi _ 
^ A- 
„ + si.r A , 
