732 PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
Hence volume 
= Sn*a%\l + 4£ 1 3 +l/3 1 2 + 6/3^) 
or volume of core 
= 8 tt% 3 {Z; 2 2 — k ] *) + ±/3 l % z +6f3 l k 3 }=rn (say). ... (6) 
i.e., to lowest order 
m=8'7j z a 3 (Jcc?—Jc 1 2 ) .( 7 ) 
3. The circulation of the ring, taking the velocity through the ring as positive, is 
taken over the cross section. Hence by what has gone before, the circulation due to 
the actual vortex filaments of the core is 
^=-4irAa 8 {V-^ s +4(V“ifei 4 )+iW+6A^ 8 }- • • • (8) 
The outside stream function is 
ria= —-gVp 2 + A,/R, t cos nv 
Hence the circulation of the fluid outside the ring is [I., Eq. 4] 
M S =-’^ 2 (A„'+A 1 '+A 3 '+...) .(9) 
Again taking the circulation round the inner surface of the core, the circulation in 
the core additional to that due to its own rotation is 
/ x i = —■“(A 0 +Ai+A 2 4- • - .)—47rAa 3 (& 1 2 +4& 1 4 ') .... (10) 
For the sake of greater generality we shall suppose circulations, additional to that 
due to the core, as existing in the outer irrotationally moving fluid, and in the core 
itself. In the case where there is none added to the outside fluid 
V‘2 = H'1+P 
whence to the lowest order of small quantities 
4Aa 4 
A 0 —-A-oH - 
Z ;, 2 
