360 Lord Rayleigh on Vibrations and 



instead of merely supported. The conditions are of course 

 w = 0, d w/dx = . They give 



D = e- 2nl {2nA + B{2nl-l)\ .... (17) 



C=-e- 2nl {A(l + 2nl) + 2nl 2 B}. . . . (18) 



The condition at x — is that already expressed in (14). 

 As before, A, C, D may be expressed in terms of B. For 

 shortness we may set B = l, and write 



K = l + e- 2nl (2nl-l) (19) 



We find 



-nA + l = 2n 2 l 2 e- 2nl /T{, 



D = (2nl + l-e- 2nl )e- 2nl /R> 



nG + D=-e- 2nl .2n 2 l 2 /R. 

 Thus 



~ = sin ny [e~ nx (— nA + 1—nx) + e nx {nC + D + nDx) ] 

 = H" 1 sin ny . e-™[2n 2 l 2 e~ 2nl -nx{l + e~ 2n \2nl- 1) }] 

 + H' 1 sin ny . e<*-^ [ - 2nH 2 + nx{2nl + 1 - e~ 2nl }] , 



vanishing when # = 0, and when x = l. 

 This may be put into the form 



dw _ _ H _ 1 gin ny ^ 2n n(l-x)e- 2n \e nx -e-^) 



+ nxe- nl (l-e- 2nl ){e n «-^-e- n{l - x ^)], . (20) 



in which the square bracket is positive from x = to x = l. 



It is easy to see that H also is positive. When nl is 

 small, (19) is positive, and it cannot vanish, since 



e 2nl >l>l-2nl. 



It remains to show that the sign of w follows that of 

 sin ny when # = 0. In this case 



w = (A + C)sinm/; (21) 



and 



W ( A ^°) =l-e- 2nl (2-2nl + 4;n 2 l 2 ) + e- Anl 



= e -2ni( e 2ni + e -2ni_ 2 + 2nl-ln 2 l 2 ). . (22) 



The bracket on the right of (22) is positive, since 



We see then that for any value of y, the sign of dw/dx 



e 2nl 



