226 
PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
T = 3Zr (z ~ Z)/(2«2) 
IV =■ Z 3 (z — Z)^ — 6r^}/(2a^) 
(XLY.) 
+ V = 
/3 ^ 8rP 
r*' — 
- ((z - Zf - 
n 
Also from (XXVT.) 
and 
l/; = SZ?’" [R2 — -f «~]/(4rt-) 
\ = 3Z?’‘2 [Pv 3 - rt2]/(4«2) 
Also from '(XXTV.) 
0 ) 
= 15Zr/{4rd) 
+ - . (XLVL). 
p 
. . (XLYIL). 
. . (XLYJTL). 
. . . (XLTX). 
It may b© noted that the value of pjp -j- Y given by (XL\I.) is least when 
(r® — is least, and {{z — Zf — is greatest, i.e., when r' — and 
s — Z = 0 ; and then py/p + Y = Tljp. 
Hence n/yo is the minimum value of py/p + Y throughout the whole mass of moving- 
fluid. 
Further, all points on the circle r — ajz —Z represent the surface 
X = - 3Za7(16); 
for this surface is 
rs (R3 „ ^ 2 ) = _ 0,4/4, i.e., (r^ - + ,.2 _ 2)3 = o. 
A neighbouring surface is 
(,.3_io3)2_p,.2(^_2)3:::,2e^ 
where e is small. 
Putting 
T — r + a. 2"^' 
z = z' + 2 
and retaining only the principal terms, it becomes 
(2eb'«)~ ^ 
proving that the section by a plane through the axis of z is an infinitely small 
elli])se, with its major axis double the minor axis, the minor axis being perpendicular 
to the direction in which the vortex sphere moves. 
