234 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
Also 
8 
d(z-Z) 
x-u+f(L + f|z^)(?( 
- 5X 
^ (^ - Z) f 
i-lr 1 
I(L + - r^f~j 
= — 5\ 
4a- 0:' (L + 
[_ ^]Z _t[i,_ 
J?’^(L + r2«2 _ J2,.i 
dr 
(L + 7'hd — 7’*)®''^ 
15Z 
4cd 
■0L2 
i [__ I 1 3 _ 
s ] rML + r-a2 - dz J ?- 
fZ?’ 
(L + 7 
(L 4- 
15Z 
4a- dz 
L= 
fZ?’ 
_ 15 Z 3 r r 
— 4a- 0 (z - Z) L J ?’■ 
5.4 + r^a^ - 
dr 
(L + rhi^ - 
(LXXXYIIL). 
Now by (LXXXVII.) and (LXXXVIII.) 
X - U + f(^ + If z^) d, = % + 
Therefore 
X — ^^ ~ [( tf d" "gy f I (J _|_ ,. 4^13 d“ const. (LXXXIX.), 
where, after the integration has been performed, L must be replaced by 
4a“X/(3Z). 
Art. 14. 2 I 1 C Figure. 
The figure has been constructed from the two following tables. 
Table I. gives the form of the surfaces 
^r2(R2 ft2) = _ d\ 
which aie inside the sphere, and which always contain the same particles of fluid 
throuo’hout the motion. 
O 
* For the time taken by the particles on one of these surfaces to go once completely round, see the 
Note at the end of the paper. 
