128 Lord Rayleigh on the Passage of 



The functions P, Q, R being now known, we may express 

 a, b, c. From (4) 



da . . r, dR m 2 + PdR 



so that 



m 2 + k 2 dR m 2 +P dR c = Q (u) 



a ~~ ipP dy' ~ ipP dx* " \ ) 



In vibrations of the second class R=0 throughout, so 

 that (2) and (7) are satisfied, while k 2 is still at disposal. In 

 this case 



T = df/dy y Q=-<tyAte, . . . (17) 

 and 



(V lJ -f& 2 )^=0. (18) 



By the third of equations (4) 



do . dR dQ 2 . 72 . 



so that ^= —ipc/P, and 



ipa\ Q _*pdc_ R _ fl91 



Also by (4) 



imdc , imde /aM 



a =Wdx> h= WTy- • • • • < 20 > 



Thus all the functions are expressed by means of c, which 

 itself satisfies 



(V 2 + F)c = (21) 



We have still to consider the second boundary condition (8). 

 This takes the form 



dc d% de dy _ 



dyds dxds ; 



requiring that dc/dn, the variation of c along the normal to 

 the boundary at any point, shall vanish. By (21) and the 

 boundary condition 



dc/dn=0, (22) 



the form of c is determined, as well as the admissible values 

 of P. The problem as regards c is thus the same as for the 

 two-dimensional vibrations of gas within a cylinder which is 

 bounded by rigid walls coincident with the conductor, or for 

 the vibrations of a liquid under gravity in a vessel of the 



same form *. 



* Phil. Mag. vol. i. p. '272 (1876). 



