362 Lord Rayleigh on Vibrations and 



-and the summations to be considered are of the form 



Sw" 1 cos n/3e~ n * (30) 



This may be considered as the real part of 



Sn-V-fr-*), (31) 



that is, of 



- log (1 -*-<*-*«) (32) 



Thus, if we take 



2n- 1 e- n ( x - i *)=X + iY, .... (33) 



e -X-iY = l__ e -iz-i(S)^ an( J e -X+iY = l_ e -(x+if3)^ 



so that 



e- 2X = l + e- 2 *-2e-* cos J3 (34) 



Accordingly 



2 n' 1 cos w/3 e~ nx = — i log (1 + e- 2x —2e~ x cos /3) ; (35) 

 and 



dw _xZ n l + e- 2x -2e-> : cos(y- v ) 

 dx" 4tt g l + e- 2x -2e- x cos(y + ri)' ' { J 

 From the above it appears that 



W=«#log {l + e~ 2x — 2e -a: cos (z/ + t?)} = #log h 



must satisfy V 4 W = 0. This may readily be verified by 

 means of 



V 2 log7i = 0, and V 2 W = # V 2 \ogh + 2dlogli/dx. 



We have now to consider the sign of the logarithm in 

 (36), or, as it may be written, 



e x + e- x -2cos(y-y) ( „ , 



iU ° e x+ e -x-2cos{y + v ) y° ) 



Since the cosines are less than unity, both numerator and 

 denominator are positive. Also the numerator is less than 

 the denominator, for 



cos (y — ij) — cos (y + rj) = 2 sin y sin 77 = + , 



so that cos(y— rj) > cos (y + y). The logarithm is therefore 

 negative, and dw/dx has everywhere the opposite sign to 

 that of Zn. If this be supposed positive, w on every line 

 y = const, increases as we pass inwards from x = co where 

 w = to x = 0. Over the whole plate the displacement is 

 positive, and this whatever the point of application (77) of 

 the force. Obviously extension may be made to any 

 distributed one-signed force. 



It may be remarked that since the logarithm in (37) is 

 unaltered by a reversal of x, (36) is applicable on the 

 negative as well as on the positive side of x = 0. If y=v> 



