PROFESSOR W. M. HICKS ON THE THEORY OF YORTEX RINGS. 
737 
and along the surface 
r , . , /T . , , A', 2a 2 V"| 7 . 
= “i -i A o(Lo-2) + |—-j^sinv 
+ terms of the fourth order. 
Hence by (15) 
-7- sin v 4- terms of the fourth order 
^/2 ov 
■ (21) 
7. The function \jj ±.—The part depending on Ap 4 and B must be carried to the 
fourth order. The part depending on the A ;t is similar to the expression already 
deduced for xjj 2 , without the terms in V. 
=-g-Aa 4 {l +12& 2 +42Z4+ 8&(l + 6&' 2 ) cos r>-)-20/c 2 (l + H 2 ) cos 2v 
+ 40& 3 cos 3'r + 70& 4 cos 4vj . .(22) 
1 +^+***} 
= ~4~ B 0 {1 + -j- 3^-^ + Zj( 1 + fZ.’ 3 ) COS V 
+^F(3 + 2Z; 3 ) cos 2v+fZ; 3 cos 3y+ff^ 4 cos 4r} 
cos v =ffh4 l 008 * 
•Bi^{^+ 8^ 3 +(l + i^ 3 ) cos v-\-\h{l + f& 3 ) cos 2v 
+ f J& cos cos iv] 
1 ^)nrlf* 
B 2 To cos 2v=— :> , n -(1-|P) cos 2v 
vA c — c ) 
32v/(C~6')' 
1 ~ BA 3 {fZf+ 3 7,' cos v-f cos 2r+ p cos 3r+|Z; 2 cos 4r] 
—rB 3 T 3 cos 3r———-B 3 cos 3 y 
v/(C-c) 3 3 64^(0-c) J 
° 0 ^r^Bf 3 (lk cos 2-y+ cos cos 4-r) 
B/I\ cos 4r=7f v /2°^ -Z; 4 cos 4i» 
x/(C- C ) 
