220 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
Art. 4. The Surfaces ivhich contain the same 'particles of fluid. 
The next step is to find the surfaces which contain the same particles of fluid 
throughout the motion. 
If X = const, be one family of these surfaces, 
0A, 
dt 
1 
0X 0A. 
■S + '37,+ !" a; = 0 
Therefore 
^+2-r(2-/) 
The auxiliary equations for X are 
dr d 
(XII.). 
0 - Z)= + 27;! (2<-= - «“•) + 2/(0 
0\ 
= 0 (XIIL). 
2~ r (z-Z) - f (z - Zf - 2^ (2r- - a?) - 2f{t) 
. (XIV.), 
2-^ r {z — Zf + 2~ r {2m — cd) 
c" ' ' cr ' 
i/r + 2-^7’" (2 — Z) dz 
(y 
-frUz-Z) 
_f + 
d 
I. , 
—I z — 
O V ^ 
- 
24^" (2 - Z) 
■rfZ 
dt 
+ 2/(^) 
dZ 
Hence 'd‘f{t) ~ — T -^ = — |-Z, one solution of (XIIL) will be 
X = 
Hence the component velocities 
A + _ i' 
cr 
(XV.). 
r=2^r{z-Z) 
Lo = Z ~ 2-^ (27 " — af — 2 \ {z — Zy 
cv^ ct* 
(XVL), 
belong to a motion in which the surfaces X = const, given by (XV.) contain the same 
particles of fluid throughout the motion. 
