740 
PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
These give the following’ values 
\ r A 
p _ _2_p 2 60 8 ■ a - ct ' 
- d 4— 6 3-°0 3 1.5 V^/2 
A /yl 
p _ 2 _p 2.3.2_ 
^3- 3-5^0- 3 5 Vx/2 
P — —--R — -fi-4 
^2— 15 D 0 15 • 
A« 4 
77^/2 
Ar/ 4 
p 2 p 8. 
131 ~ 3 130 3 Vn/2 
>• 
(26) 
These values of the B greatly simplify the coefficients of cos v and cos 2v. 
In that of cos v occurs the expression 
JL 
24A« 
IBo+IB. + pfBo- 
rv/2 
By means of the above equations it follows at once that its value is 18A a‘ i /(Tr s /2). 
The coefficient of (cos r , )/ v /2 now becomes 
- 2)1 2 +||i+|a a+ p A+^ V+^*A*V 
So also the expression 
4B 0 +^B 1 +¥B 3 
2^2 
320AP 
7Ta/2 
occurring in the coefficient of cos 2v is found to be 248 A<x 4 /(7t v / 2), and the coefficient 
becomes 
~P, l (iA 3 -A)-fA 0 (n-2)y+iA 1 +X_ i/ 3 1 jA 0 (L 3 -3)fc + Al} 
+~ V+ 8 AVi+¥^|‘ AV + (A+ JAW pf- 
Now these coefficients vanish both at the outer and inner surfaces—their values 
for the inner will be found by writing k { for k. 2 , /3, = 0 and a 2 for fi 2 . They are 
written down later in equations B. 
The pressure conditions require a knowledge of k^xjj/Sk. So far as it depends on 
the B it is given by (differentiating 23) 
1 _ ]Hi 
7T \/ 2 Sk 
=i(i s, +ia)B 0 +-A-(2^ + ^)B 1 +||B i a+A^( 3 i*+2U<) 
7T V Z 
+ 1+4U=) + iB 1 «l +p 3 )+MBA s +^~ (1 + 181 s )} cos v 
+ {jB 0 F(3 + 4t 3 ) + fB 1 l 3 (l + P) + i|B il l i +JtB 3 l*+ ^~(A+8l*)}cos 2« 
the coefficient of cos 3?’, cos 4v vanishing by what has gone before. 
