454 Lord Rayleigh on the Finite Vibrations of a 



When the process is pursued to the next stage, the ratio 

 H^Ki may become determinate. In illustration of this let 

 us suppose that V is an even function o£ both X and <f> 2 . 

 Thus V 3 =0, and 



Y^h^ + h^W + ^W- • . • (17) 

 Using this as before, we obtain 

 H = 0, K = 0, H 2 = 0, K 2 = 0, H 4 &c. = 0, K 4 &c. = 0. 

 To determine H 3 , K 3 we have 



(ci-^XJHs+^iHi' + i^HiKx^O, . . (18) 



(^-^VKs + ^K^ + ^ImHj^O. . . (19) 

 Also from the terms in cos nt 



H 1 [ Cl -n 2 a 1 + 3S 1 H 1 2 + p 3 K 1 2 ]=0, . . (20) 



K 1 [c 2 -n 2 a 2 + 3S 5 K 1 2 + 3S 3 H 1 2 ]=0. . . (21) 



Equations (20) > (21) can be satisfied by supposing either 

 Hj or Kj to vanish while the other remains finite. Thus if 

 H! = 0, (20) is satisfied and (21) gives 



c t -n f o f +f« 1 H 1 f =0, .... (22) 



determining n. From (19) we see that in this case K 3 = 0, 

 while Ho is given by (18) with K x =0. 



There is also another solution in which both H x and K x 

 are finite. Since by supposition 



c 2 /a 2 = ci/ai, 

 2S 1 H 1 2 + S 3 K 1 2 = d-rfci! = cj 

 S 3 H 1 2 + 28 5 K 1 2 Cl -n 2 a 2 c 2 > 



(23) 



which determines K^/H^. and then either (20) or (21) gives 

 n 2 . Equations (18), (19) determine H 3 , K 3 with two alter- 

 natives according to the sign of K^/Hi. 



In certain cases the ratio Kx/Hi may remain arbitrary ; 

 for example, if 



c 2 = c x and 2S X = 2h- — 8 3 , 



making V 4 a complete square. 



The other class of cases demanding further examination 

 arises when 



c 2 /a 2 = 4tcja l9 (24) 



and it requires that K 2 should be treated as a quantity of the 



