PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 735 
_1_ -y/ (2k) 
-v/(C — c) ^/(l +/fc 2 — 27ic) 
= \/(2^)| 1- + + gV c ' + (^ + P' J ) cos v+(—+ iVd^ cos 2v 
+|/c 3 cos 3-y+Hcos 4v| (12) 
Substituting the values of II given in [I., Eq. 11], 
■A-'qPo+A 1-^1 cos -v+A'jjEg cos 2v 
— — hr- {^L — 1 +-|-(L+ 1 )I 3 } A 0 +^ j { 1 — f(L— ^)I 3 } A^ cos v 
-j-7r*(l—-p 3 )A' 2 cos 2v 
Putting in these values and reducing, it will be found that 
Xfc=-iA' o fL-2+i(2L-l)^3+iA' 1 -0 2 (l + 2,M 
+ {-iA' 0 (L-2)+i^-^U 
COS V 
+ 
-|A' 0 (L-2) + |f+ §-S^}Fcos 2i 
J 
Now, along the outer surface 
whence 
Jc=k 2 (l J r(3 l cos v+fii cos 2v) 
L — Lo-f-^^i'—/3^ cos /3. 2 ) cos 2v 
k 2 =lcp {1 + lySf 1 + 2/3 l cos v J r (2/3. 2 -\-^/3 2 ) cos 2v] 
(18) 
(14) 
Suppose that when these values are substituted xf/ 2 becomes p 0 -\-J)i cos v-\-p z cos 2v. 
This gives the value of i fj. 2 along the outer surface, and since this is constant, p v p. 2 
both vanish. 
Make these substitutions and reduce, then it will be found that 
V~2^=' 
El 
a/2" 
p 2 _ 
A2 : 
iA' 0 -] L s -2+i(2L s -l)y j- +iA,-nj|(l + 2V) 
-lA(iA' 0 (L,-3)J»+i^+iA' (J 3i+^} 
■iA' 0 (L 2 -2)4+i^-^ s +|-AA' 0 =0 
■fA' 0 (L 2 -2)V + iA' 1+ ^-^V-i(iA 2 -A)A'o 
) • (15) 
r A' 2 a 3 V 1 
+ 2& j ——3)^ 3 A-- l ——^h 2 \ = 
5 B 2 
