PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
739 
a= *(b»+b 1 +*b,+4 3 +a(2B 0+ ^b 1+ ¥B 3 + 
+i(B 0 +fB I +^)ft* i +AW(B 0 +B 1 +|B s +^) 
+AA V (fB 0 +f b 1+ j-/b 2+ vb s + 6 ^|) 
72Art 4 \ 
12 Art 4 ' 
+|/3i^ 3 ^B 0 +fB 1 +f|-B :2 +^-^-J+i(2^ ;2 +i/5 1 2 )^ 3 ^B 0 +fB 1 -|- /o 
+iAi 2 (B 0 +|B 1 + 4 ^) 
and the complete coefficient of cos 2v is found as before by adding 
Az) — fA 0 (Lo—2)&v + ^Ai + ~ — 5/8 i | A 0 (L 3 — 3)/c 2 -f q J j 
Bemembering the remark as to the order of A 0 , &c., it follows that 
Bo+Er+fB,, 
40A« 4 
37r^/2 
= e 0 
where e z is of the second order of small quantities. 
The coefficients of cos Sv, cos 4v are easily found, viz.:— 
i%=*{lB„+fB 1 H-J-/B 2 +VB 3 + 8 ^|‘}A 3 =0 
jh— Hs j % B 0 + 3B,+fB. -f 7B 3 +- 4 -B 4 + j k% + f—0 
These equations give, to find the values of B to the lovjesi order. 
9 94 A r/ i 
P 0 +3B 1 +|B 3 +7B 3 +^B 4 +^- = 0 
,90 An 4 
IBo+^+^+W + ^ =0 
B 0 +Bi+|B 2 
40 Art 4 „ 
+ o 7o :=0 
07 T\/ l 
+ =° 
7Tv/ 2 
(25) 
