738 
PROFESSOR W. M. HICKS OH THE THEORY OF YORTEX RINGS. 
Hence (so far as it depends on B) = 
77V 2 
A 
iB 0 (l +W+ ***) +iW 2 (l +W) +&&&+ + 12 F + 42H) 
7T 72 
48 Aed\ 7 „ 
+ (i(B 0 +|B 1 + ^)+l(iB 0 +|B 1 +iiB ; +^|jFpco St , 
+ {-ftfB 0 +B 1 +fB 3 +i^')+*-(' 2 B »+VB 1 + i 8 i B s +^f)^}^cos2 l . [ (23) 
07 r 72/ \ 
ICOAa 4 
+Y 2 (5 B 0 +fTq -f ^B^-f-- 7 B 3 
7T-72 
d 3 cos 3 y 
■t'TY¥( 5 2 5 '-^u'i~15B 1 +- 2 -B ;J +35B 3 d--—^ 64 ,- 4-35 X 32. W * cos 4 y 
The next step is to substitute in this the value of h along the two surfaces, which 
will give the value of xjjJ y /2 along these surfaces. Suppose for the outer surface it 
becomes 
Po+Pi cos v-\-po cos 2y-by > 3 cos 3'y+yq cos 4-y 
The substitution Jc=Jc. 2 (l J r/3 1 cos v-\-/3. 2 cos 2 y+ . . . ) being made it will be found 
that 
+pAY(Bo+B 1+ |B 3 +^) + 7i ¥ (B 0+ lB 1+ ^f 
and the complete coefficient of cos v is found by adding the terms in A, <} viz., (15) 
— ^A 0 (Lo—2)do+-|-^+dA u /3 1 
This complete coefficient must vanish. Hence remembering that A 0 is of the second 
order with reference to At/d, it follows that 
Bo+fBp^e, 
(24) 
where e x is of the second order of small quantities, and can therefore be put equal to 
zero when multiplied by quantities of the third or higher orders. 
Again, it will be found that 
