152 

 where 



Lord Rayleigh on the Maintenance of 



©to 



..[-2,0], [-1, -1], [0, -2], [1, -3], [2, -4],.. 



..[-2,1], [-1,0], [0,-1], [1,-2], [2,-3],.. 



..[-2,2], [-1,1], [0,0], [1,-1], [2,-2],.. 



..[-2,3], [-1,2], [0,1], [1,0], [2,-1],.. 



..[-2,4], [-1,3], [0,2], [1,1], [2,0], .. 



and 



[n,r] = (c + 2n)' 2 ® r -i(c + 2n)W r -® r . . . (37) 



By similar reasoning to that employed by Mr. Hill we may 

 show that 



S(c) = A (COS 7TC— COS TTCq) 



+ B (sin 7rc — sin 7rc ) . . . , 



where A and B are constants independent of c ; and, further, 

 that 



2)(0) = A(l-cos7rc)-Bsin7rc. . . . (38) 



If all the quantities <I> r , ^ r , © r vanish except <3> , ty Q , ® , 

 2)(0) reduces to the diagonal row simply, say 2/(0) . Let 

 c 1; c 2 be the roots of 



, dhv , T , dw ^ ,. 



(39) 



then 



= 0. 



(40) 



SV (0) = A ( 1 — cos TTCi) — B sin irc^ 

 = A (1 — cos 7rc 2 ) — B sin irc 2 

 so that the equation for c may be written 



® (0), 1 — COS 7TC, Sill 7TC, 



®'(0), 1— cos7tc 1? sin7rc!, 

 2)'(0) ? 1— COS7TC 2 , sin7rc 2 , 



In this equation < S(0)-^'S / (0) is the determinant derived 

 from 2(0) by dividing each row so as to make the diagonal 

 constituent unity. 



If . . . ^_i, ^ , ^ . . . vanish (even though . . . <£_!, <l> , c^ . . . 

 remain finite), 2)(c) is an even function of c, and the co- 

 efficient B vanishes in (38). In this case we have simply 



1 — cos i re _ S)(0) 

 l-coswV0 o ""^(O)' 



exactly as when <I>i, <E>_], <E> 2 , ^-2 • . • vanish, 



