PROFESSOR W. M. HICKS OR THE THEORY OF VORTEX RINGS. 
731 
To determine the similar quantities for the outer surface let it meet the plane of 
the ring in the points (//, k"), then 
k'=k 2 ( 1 —/3i) k"=k 2 ( 1 +A) (to the second order). 
Hence 
radius of cross-section=^{(R ,/ +r ,/ )—(R'— r')} 
radius of ring = |{(R/— r') + (R" + r")} 
From these it follows easily that 
radius of section — 2ak. 2 
radius of ring =a{l-\-2k. 2 -{-2k. 2 /3 1 ) 
and distance between centres of mean cross sections 
= 2a(k 2 2 —k l 2 +k 2 /3 l ) 
the centre of the outer lying outside that of the inner. 
The volume of the core will be 
dn dn' 
die dv 
dudv 
( 5 ) 
taken over the section. Let us first find the volume of the surface bounded by 
£=£ 3 ( 1 -)-/^ cos v . . .) 
Volume 
Now 
= 2 7 m 3 
' 2 tt 3 
= -7ra 3 ‘ 
r 
■ 1 “ 
Jo 
L(c c ) 2 J 
dv 
o r 2 - dv 
“""’Jo (0 -cf 
_ I _ = - (l 4.—4. 'l 
(C—c) 2 C ! \ ~C~C sT ■ ■ ■ / 
= 4F( 1 + 4& 2 +4& cos v-\-§k 2 cos 2v-\- . . .) 
Along the boundary k=k l (l-\-(3 1 cos v-\-/3. 2 cos 2v), 
Therefore to the second order in the bracket 
(C^ =4 ^^ 1 + 4 ^ 2+ 2/3i 2 +6^ 1 +i?cos'y~F(?cos2r+ . . .} 
