77 8 PROFESSOR W. M. HICKS OH THE THEORY OF VORTEX RINGS, 
keeping only the most important terms. Substituting the values already obtained 
^F(0-iU*+A.r/, 
2a 2 & 1 3 -^ : £—4 A a z k^{TT —v) 
+ U(U+2AaU-)^-2UA 
+ A {- 2a 8 HT£+ 4 v) + 4 Aa%*(L - 2) £} 
= F(0-iU 3 +Av//+4a^ 1 2 Lf+U(U-2Aa 2 Z;)^+4A 2 aUf(L —2)^ 
Hence along the inside 
F(0-iW+A^, + 4«^L,f+l^jA=(L 1 -2)a= + (^Y) ? -2^A«jvf=0 
Whilst along the outside boundary 
c;{F(-iU\+Afe+4^ 1 2 Lj+4a*|A 2 (L. 2 -2)a 8 +(^) 3 -^-Aa}vf 
=rf'{ ia^Uia^k^ } 
Remembering that the large terms have been already annulled, we get by sub¬ 
tracting the two equations and writing £ for k^, &c., as before 
(L 3 —{A% 2 (L 3 —Lj)+v 2 — id —A a(v — u )} £ 
=|{L s f+w»f} 
or 
{L 1 + (|-l)L 3 }f+[ ! »^-(i>-»)(f+«-A«)+AW(L 1 -L,)}f=0 
whence (if p=d'/d). The time of pulsation is 
_ L 2 (p —H+W _1 
pw 2 + A 2 a 2 (Lj — L 2 ) — (y — u) (y + n—Aa) J 
The numerator of this is always positive. Hence the condition for stability is that 
(79) 
(80) 
pw- +A 2 a 2 (L 1 — L 2 ) + {v — it) Act > F 2 —w 2 
