734 
PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
regard A' 0 , A 0 as determined by (eq. 9, 10) when the circulations are given, and B 0 
will be determined by the fact that the pressure along the free surface is zero, whilst 
it is IT at an infinite distance. There remain therefore 5n constants to be determined. 
The conditions to be satisfied are the following :— 
(1) A must be constant along the outer boundary of the core. This gives n equations, 
viz., by equating to zero the coefficients of cos v.... cos nv along the surface. 
(2) constant along both the outer and inner boundaries of the core. This gives 
2 n equations. 
(3) The pressure must be the same on both sides of the outer boundary. This 
gives n equations. 
(4) The pressure along the inner surface must be constant. This gives also n 
equations. Hence on the whole the surface conditions give 5n equations, 
sufficient therefore to determine the 5n constants. It is therefore possible 
to approximate to any order desired. 
It has already been noticed that A' 0 , A 0 are to be regarded as of the order Zd with 
respect to A a 4 . It will be seen later that when this is the case the B are of zero 
order with respect to the same quantity. Hence, if in the approximations account is 
taken of A 2 , the term in which B occurs must be carried as far as B t cos iv. It will 
be necessary therefore to carry the latter terms to the fourth order, except in the case 
where the added circulation is very large compared with that due to the actual rota¬ 
tional motion. This case is therefore a much easier one to discuss than the more 
general one. We proceed to apply the conditions given above to determine xp. 2 , xJ j v 
Our method of procedure will be first to express the functions in terms of h, and the 
cosines of multiples of v ; then to substitute the value of h along the surface, and 
reduce the expressions to a series of cosines of multiples of v, whose coefficients are 
functions of Zq or Zq as the case may be. The conditions above are then applicable 
at once. 
A The function xp 2 . It has been seen (1) that 
\/k — 
A! AfJb, cos nv 
Now, 
1 \y=ia°~v 
s 
=ia*V 
1 -k? 
C -cj " ' I 1+^ — 2 kc 
— ^a 2 V(l + 2Zr+4Z' cos v + Sln cos 2v) 
For later purposes the value o** (C—c) 1 must be carried to the fourth order, and 
then 
