110 
DR. G. R. GOLDSBROUGH OX THE INFLUENCE OF 
where A and X are, as has been said, purely periodic functions of period 2 ? r. On 
substituting in equations ( 6 ) we find 
c 2 A + 2 cA' + A"— 2 /c (cX.+ X') + AZ 0 ,. r cos + X 20 2 , r sin r<p = 0 
c 3 X + 2 cX / + X // + 2 k (cA +A') +A 20 4 , r sin /’0 + XZ 0 -, r sin r<p = 0 
Let us now assume that A and X can be represented in the most general way by a 
series of terms in 0 with suitable coefficients, the coefficients being periodic functions 
of (p with period 27r. That is, let 
A = A 0 sin (n<f> 
X = X,, cos 
-r) + tt A r . s 0 r s + ttttB,, , p . ? 0, S 0 P , q+ 
-t) + 2£X r . ,0 r .. + SSSSY* ? 0 r . s 0 p> q +... 
5 
In these expressions A 0 and X 0 will be arbitrary constants, n is an arbitrary 
integer* and t a parameter which will be defined presently. 
We shall assume that the index c is of the form 
ttc 0 
- r, s ^r, s 1 - ,u " 
r, s, p. q 
0 ,. 
S®p, q + 
Then, if we substitute these values in equations (7) and equate to zero those terms 
which do not involve any 0 except 0 4 , o and 0 5iO , which are large compared with the 
others, we find 
{ (0i, (j — n 2 ) A 0 + 2 /cnX 0 } sin (n<p — r) — 0 , 
{ 2 /cuAy + ( 0 5 iO — n") X 0 } cos (ncp — t) = 0 . 
On eliminating A„ and X 0 we find 
(0i iO —n 2 ) (0 6iO —n 2 ) —4/cV = 0 
(9) 
In general, the given values of 0 1O and 0 5 O will not satisfy the identity (9) for any 
integral value of n. Let us replace 0 1O by cq 0 , where cq , 0 is a quantity which 
satisfies the relation 
(rq.u-rA) (0 5iO -» 2 )-4<cV = 0. 
( 10 ) 
For some suitable value of n, it will usually be found that « 10 approximates closely 
to 0 ], o. 
Following the method of Whittaker previously referred to, let us now assume 
that 
(Qi. 0 —n 2 )(Q bi0 —n 2 ) — (a li0 — n a ) ( 0 5 l o— n J ) — -tu riS 6 r _ s -\-S^Sv riSiP<s 0 rtS 0 p<g + c. , 
or 
4 k n 
0i,o - n 2 + 2 +sta r , s e AS +ttttb r _ s ^ q e r , s e p<q +, 
0; n — n 
( 11 ) 
7 5 0 
* The use of n is to be distinguished from 
a former use where it referred to the number of particles in 
the ring. 
