736 
PROFESSOR W. M. HICKS OK THE THEORY OF YORTEX RINGS. 
Hence 
A \= 4 ^V+ (P-2 
Ah_ l A' j_i A' /T ' 2a 2 V 1 /3 t f . , ,, __1A' 
“* A o*a+s a o(4--) + ^2 +5 4 { a ° 1V-)+ 72 * A °tJ ’ 
Now by (9) 
Hence to the second order 
A' 0 +A' 1+ A' 2 =-^ 
A' J 1 _i _I T _9_ 3 1 —_ thh_ 4&~'V j 3 
A oi 1+ ( L3 - 77 x/ 2 \Z2^ 2 
or 
Ah 
3* I f /t o A\ 4«“"\ ] J 0 
7H+^ r75^- 2 - 
r \/2 
h) a/2 
Again, differentiating (13) with respect to k 
_A_7.^3—1 a ' (i _ 
^2 St' “^oi 1 
+ -iA' 0 (L-3). 
I-fA' 0 (2L-5)-^ 
07 
r—1 
1 
2 a*V 
^2 
A' 
i A i 
2« 2 V] 
2 ¥ 
a/2 J‘ 
2A' 2 
6« 2 V1 
1 ¥ 
\/2 J 
1 
Jc z cos 2v 
Along the surface this becomes 
i^=|A' 0 fl-(L 2 - W } 
2 a*V 
V2 
r A' 9 «2\r 
+ I -iA' 0 (L 2 -3)-iY_^|j i3Cosl , 
+ 
2Ah 6a 2 V 
|A'o(2L 3 -5)- 
+i|{ -iA' 0 (L 3 -4) + i f)-^J JO «* 2* 
( 16 ) 
(17) 
( 18 ) 
(16) 
>-• (20) 
Further, to the third order of small quantities 
