238 
PROFESSOR M. J. M. HILL OR" A SPHERICAL VORTEX. 
and as 
is equal to the same expression, it is obvious that the normal velocity is continuous at 
the surface of the ellipsoid. 
But 2 ')Ip + V is not continuous. 
For 
7" 0^ ' 2 yv%y V0^ 
+ ( 57 ) + f^) ) = an arbitrary function of A 
and since (taking Z constant), 
and since, in this case, h = a 
dt~ . ^ dz' 
dll 
^ + Y - ixZi 
P ^ Jo («' + «) (C“ + H) 
3i + 
+ 
Jo (<*" + il') 
= an arbitrary function of A 
Therefore 
f- 
Jo («" 
dll 
2vVZ (. - Z)^ 
o?& 
W (2 - Z)~ 1 ^ 
d ii 
(rc^ J 0 (<^“ + td) (c" + ii) 
V.+ 
Aifp?{z-Zf 
ik J- V -J- \ y _ 
p cdif [ ^ Jo {(d + u) (c- + uy' ' arc 
= an arbitrary function of A 
But 
therefore 
Z = fji C — 
Jo («•“ 
dll 2p, 
0 («“ + 'll) (c- + %i)^'^ arc * 
V 
+ v=HA£(i^ + 
«%6 
an arbitrary function of A 
This value of 'plp + V is not continuous with the value of pjp + V inside the 
ellipsoid. 
Further, on returning to rectangular axes in three dimensions, 
u 
-2-.r(.-Z), 
= 2~y(.-Z), 
io=zZ-^l~ (2x^ + 27 - a-) --Z^iz- Zf. 
a~ ' ' & ^ 
Hence, if iq, { be the components of the molecular rotation. 
