776 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
where A and B are constants determined by the equation 
bcf> du 
bu dn 
— vel. inwards at the two surfaces 
dn dn du ■ dn, • , • . 
=!u = -toii i= ~^ 
Hence 
hep 
bu 
(C -c) 
;?= -4 
Now 
^= 27 (h)( A r«+BQo)+v/(C- 0 {Af + Bf»} 
bio 2v/(C 
-T7A-jsP„+2(C-e) 
dl\ 
B 
du j 2^/(C — c) 
Hence 
Therefore 
V(G-c)l 
= A x /‘2 to the lowest order. 
Ay2 = — — 4« 2 /c 2 2 ?? 
-{SQ 0 +2(C-o)^} 
^=-2 v '2a*i 1 V(C-c).fP 0 
(75) 
Again the circulation is unaltered. A term must therefore be added to xjj; let it be 
denoted by y. Then y is of the form 
1 
X s/(C-c) 
The circulation of this is 
2{A 0 R 0 +£> 0 T 0 + • • • } 
-w / i 2A „ 
a 
But the circulation of ip round the new boundary is 
^ U + ~dk) 2vr. 
Along the inner boundary this is 
=W' 1 {u 1 +u 1 f+'^ l f} 
= 47ra& 1 U 1 — Sv A aPJc-fg by (61) 
Hence since the circulation is unaltered 
-8^Aa 3 Vf- T 7 ? A„=0 
A 0 = — 4 v /2Aa%| 2 £= — 4 x /2Aa i JcJ i r) 
(76) 
