764 
PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
The values of <f> are needed along the inner and outer surfaces. Hence substituting 
the values of A, B, C, D already found, along the inner surface 
• v/(2A:,) f |(FAT / h + PA i T / ,)-2^r/„Th • , r(P , «T /, K + PAT / „)-2VR'„T' n 1 
<k= ~ j-^7777-W7777-Sin UV-\- -7W777-777777-COS HV 
p/ p// p// p/ 
JA n ± n AX n ±. n 
I )/ p// p// p/ 
1 n*- n Xt w 
2714-1 2 n —1 
Now the lowest terms in R,„ T„ are proportional respectively to k 2 , k~*~. 
Hence to the same order 
• V2f (k»* + k** y 2(k l k 2 )» > . 
^ 7 ) Wk-JT^*ryv / l,)suinv+{...)cosnv 
■K- 
We shall put for the future 
where it is to be noticed that 
With this notation 
s/ 
fc 3 2ra + 7y 2a 2(& 1 fc g )» 
_ k in —P> k Sn _ ^ 2 » ~ ? 
p° — q 2 =z 1 
<£=—-{ —(piy/h—qy\/h ) sin ^+{i^Vh-qvVh )cos • . (57) 
So along the outside boundary 
4>='~ {—{q.£\/ki —pv^/h) s i n nv +(<l£VK—PV cos nv] . . (58) 
whilst for the outside boundary in the outside fluid 
.(59) 
A vW, ■ . 
<p =—-— (—i7i sm wr4-7/1 cos nr) 
16. The pressure conditions. — The expression for the pressure at any point of the 
core has been already found (11), viz. :— 
P 
-= const 4-/(7) -<f>-±v z 4- At// 
At a point near a surface this is 
i> = const +/(<)-^_i(u+^6i + | 4 V 
d 
/ 6 m/ 
6U 
const +/W-^.-iU--U(^6l-+ A h)+A(</'„-2a=Uoi+ x ) 
2 a 2 /t 
U being the tangential velocity along the original surface. 
