PROFESSOR W. M. HICKS ON THE THEORY OP VORTEX RINGS. 
Also (21) 
747 
(C —e) 2 byfr _ 1 1 
001 
which in the square gives 
a 2 S bv 2^/2 'k'fP 
1 
16«tL 2 
A'o 2 /^ 2 (1 — cos 2r) 
For the first approximation we shall keep as far as cos v. In this case 
( C -cf hty _1 
iA' 0 - A' 0 (I*-*M 
4« 2 V 
a/2 
1 A ' §1 1 
* A °VJ 
a 2 S ' bk 2 
and in the square 
j r A' 0 =-A'„|A' 0 (L 3 -i)+ 4 ^-iA' 0 |}i 3 
k. 2 cos v 
2 a% 
or, to lowest orders 
cos v 
T— 1 A' 2 
8 a%* 0 
1 
1,1 2 a% | A ^ y/2 A 0 \ \ A 0 
So, to the first three orders of k, we get from (27) 
1 
\/2 
fe ^ = iA»+^ *»*+ { -iA 0 (L 2 -3)-^+^e a +^ (W+4/}&)} fc 
9 cos V 
Substituting for e x and A ] from (A.2) and (29) 
j_ J1A /1 _j_ a 
K) 1 V 
+ { W1 ■+3 *£)+— ( 9 -|i) y-i W ) !-h cos * 
v/2 
whence 
(C—c) 8 : , Hi _ 1 _ 
a 2 S 3 b/j a~kc,\/2 
i(Ao+^H 
+ {-|Ao(l-y)+^' 2 ( 1 -^ 4 )^ 
1 ^ ^ An — 4 ™ Xv 2 ) !• A cos V 
and the square is 
1 
2 a%* 
^An+^L 3 
4 -^0 
a/2 
4A« 4 
+( a «+ yI‘ y) {- 4 «( i -19+^( i -v) v-* f ( A «-^ k > » f * cos e 
