358 Lord Rayleigh on Vibrations and 



integer. The same assumption introduced into (3) with 

 Z = gives 



(d 2 /dx 2 -n 2 ) 2 w = 0, (5) 



of which the general solution is 



w={(A + Bx)e- nx +(C + I)x)e ,u '} sinny, . . (6) 



where A, B, C, D, are constants. Since w = when oc— + co , 

 C and D must here vanish ; and by the condition to be 

 satisfied when #=0, B = nA. The solution applicable for 

 the present purpose is thus 



w = A sin ny . (l + nx)e~ nx (7) 



The force acting at the edge x = necessary to maintain 

 this displacement is proportional to 



d\/ 2 w . d 2 dw , . d 3 w . , /Q v 



^^ +(1 -^V 2 ^' s ^ simpy ' ■ () 



in virtue of the condition there imposed. Introducing the 

 value of w from (7), we find that 



d z wjdx z — 2n 3 A sin ny, (9) 



which represents the force in question. When n = l, 



w = Asiny.(l + x)e- x ; .... (10) 



and it is evident that w retains the same sign over the whole 

 plate from x=0 to ^ = co . On the negative side (10) is not 

 applicable as it stands, but we know that w has identical 

 values at +#. 



The solution expressed in (10) suggests strongly that 

 Resales expectation is not fulfilled, but two objections may 

 perhaps be taken. In the first place the force expressed in 

 (9) with n = l, though preponderant at the centre ?/=2 7r ; i s 

 not entirely concentrated there. And secondly, it may be 

 noticed that we have introduced no special boundary condition 

 at x = co . It might be argued that although w tends to 

 vanish when x is very great, the manner of its evanescence 

 may not exclude a reversal of sign. 



We proceed then to examine the solution for a plate 

 definitely terminated at distances Z, and there supported. For 

 this purpose we resume the general solution (6), 



w= smny{{A + Bx)e- nx +{C + Dx)e nx }, . (11) 



which already satisfies the conditions of a supported edge 

 at y — 0, y = w. At x = 0, the condition is as before dw/dx = 0. 



