PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
761 
whilst for the motion outside the core 
AR",= -a*XJ' ZK /(2k z )/3+^ (2& 3 )’S 
/y3 . 
CR"„= —a 3 U' 2V /(2& 3 )S -- (2 k^(3 
1 
^. 
I 
J 
(54) 
Denote the right hand members of these by £ r), rj, rj 1} rj\, then solving 
f-rp// riv 
A _ g l n — V 1 >» 
p/ rp// _p// m/ 
XL XL 72 J- n 
p_ — + h 
p/ rp// _p// rp/ 
with similar expressions for C, D, in r(. 
Suppose now 
2{(AJEt*+B*T„) cos wv+(C,R*+DX) sin nv} 
It has to be determined from the fact that e is uniform along the boundary, i.e., 
when k=k 1 {l-\-cc n cos nv-\-y n sin nv), &c., and that the whole circulation remains 
unchanged. 
If U denote the velocity along u, at a point of the boundary; V the velocity along 
v as determined by xp alone, and 9 the inclination of the boundary to u (and therefore 
a small angle); then the circulation is 
fx — ^■" J (A + -A„) +j( U cos #+V sin 9)ds 
Now 
U cos d-fV sin 9=( U 0 -f- ~bkj. 
dU n v 7 \/ <9 2 \ . dY 
2 /+ dk hU - 
<K 
dV 
- u »+ dk 6k+ di bke 
dY 
Now 9 is of order bk. Also —— is of higher order than we require ; for both 
a/c 
reasons therefore it may be neglected, and 
since 
f dY 
J(U cos 9-\-V sin. 9)ds=i± J r --j k(a cos nv-\-y sin nv) 
adv 
C—c 
dn a 
dv C — c 
