Deflexions of Membranes, Bars, and Plates. 359 



At x = l the conditions for a supported edge give first w = 0, 

 and therefore dhv/dy 2 = 0. The second condition then 

 reduces to dho/dx 2 = 0. Applying these conditions to (1.1) 

 we find 



D = Be- 2nl , C=-e~ 2nl {A + 2lB). . . (12) 



It remains to introduce the condition to be satisfied at 

 «i' = 0. In general 



+ e«*{n(C + Da?)+D}]; . . (13) 



and since this is to vanish when x = 0, 



-ttA + B + nC + D = (14) 



By means of (12), (14) A, C, D may be expressed in 

 terms of B, and we find 



+ e-*<- 2l - x '){-(2l-a:)+we- 2nl }]. (15) 



In (15) the square bracket is negative for any value of x 

 between and Z, for it may be written in the form 



- xe - nx {l-e- 2n V l - x 1}-{2l-x)e- 2nl {e nx -e- nx }. (16) 



When x =0 it vanishes, and when x — l\i becomes 



— 2le- 2nl (e nl —e~ nl ). 



It appears then that for any fixed value of y there is no 

 change in the sign of dw/dx over the whole range from 

 # = to x = l. And when n = l, this sign does not alter 

 with y. As to the sign of w when x = 0, we have then 

 from (11) 



/a , n\ -d • e 2nl — e~ 2nl —±nl 

 w=smny(A + C) = Bsmny w ^ni + g -ni) » 



so that dw/dx in (15) has throughout the opposite sign to 

 that of the initial value of w. And since w = when x = l, 

 it follows that for every value of y the sign of w remains 

 unchanged from x = to x = l. Further, if n=l, this sign 

 is the same whatever be the value of y. Every point in the 

 plate is deflected in the same direction. 



Let us now suppose that the plate is clamped at &=+l, 



