766 
PROFESSOR W. M. HICKS OX THE THEORY OF YORTEX RIXGS. 
with a corresponding expression from the second equation. In these equations alter 
the meaning of the a, so as to write a for lep a, y for \ 2 y, ft for h z z (3, 8 for 
Further, write u, v, w for Uy^adq), U 2 /(2a& 2 ), U / 2 /(2a£. i ) respectively. Then these 
equations become 
— Axia.-\-py + { n(np -f 1 )u—nAa}uy-\-7iq(y—u)/3—q8—nhivqb = 0 ~| 
pot-\- {n(np-\- l)u—nAa}ua.-\-Aay—ql3—n z quv/3—nq(v — n)S = 0 j 
(62) 
Along the outside surface, we find the corresponding expressions by interchanging 
1\, h 2 ; a, /3 ; y, 8; u, v, i.e., put —p for p and —q for q. 
Then 
P 4a 2 
- = —[{ —y>/3 (— np-\- l|r— Aa)nvfi-\- AaS+ga+n^r^a—n^(r—n)yj cos nr 
+ { — A a/3—pS + (— np +1 |r—Aa)nrS-}-ng{ v — n)a+qy+ n~uvqy } sin nr] 
Ao’ain 
P v"(2 h),-, • • s TT' i 1 V 
d'= ~ a h 1 C ° S m ~^ SU1 "YU « + 
\/ (-A) / * / ■ • \ 
= —--(77 j cos nv—rj x sin nr) 
+const 
7l> 
— Uhl —U' 2 (/3 cos nr + S sin nr) + o, cos n ^ J r r } 1 sin nr) 
Substituting for the 77 from (54), and altering the meaning of a, /3, y, 8 as before 
P 4« 2 •• 
— [_{/3-f--j- l)w 2 /3] cos nr+{S-bn(n+l)iFS} sin nr] 
Ct 01/ 
Hence, since the pressure is the same on both sides the boundary 
qa + rPquvoL—nq(v—u)y —pi3 + n {(— np +1 )r — Aoj v(3-\- A aS 
d’ 
= -j{(3+n(n+l)iv~/3 
nq( v — u)a -f qy + n 2 qu v y —A«/3 —y>8 + n {(— np -j-1) r — A a } rS 
d' 
= {S-|-n(n+ l)nrS} 
When the waves are travelling in the positive direction round the ring, bk will be of 
the form 
SZq = L sin (nr+Ai) 
8& 2 =M sin (nr+Xt) 
or 
a=L sin Xt , y —L cos Kt 
/3=M sin \t , S = Mcos\^ 
