Deflexions of Membranes, Bars, and Plates. 301 



over the whole range from x — O to x = l is the opposite or! 

 the sign of w when # = 0* ; and since w = when # = Z, it 

 follows that it cannot vanish anywhere between. When 

 ??=1, w retains the same sign at ,x = whatever be the 

 value of y, and therefore also at every point of the whole 

 plate. No more in this case than when the edges at x— +1 

 are merely supported, can there be anywhere a deflexion in 

 the reverse direction. 



In both the cases just discussed the force operative at 

 x = to which the deflexion is due is, as in (8), proportional 

 simply to d 3 w/dx z , and therefore to sin ny, and is of course 

 in the same direction as the displacement along the same 

 line. When n = l, both forces and displacements are in a 

 fixed direction. It will be of interest to examine what 

 happens when the force is concentrated at a single point on 

 the line x = 0, instead of being distributed over the whole of 

 it between y = and y = 7r. But for this purpose it may be 

 well to simplify the problem by supposing I infinite. 



On the analogy of (7) we take 



w = %A n (l + nx)e- nx smny, . . . (23) 

 making, when # = 0, 



d*w/dx* = 2X'n?A n smny. .... (24) 



If, then, Z represent the force operative upon dy, analysable 

 by Fourier's theorem into 



Z = Z 1 siii3/ + Z 2 sin2?/+Z3sin3y+..., . . (25) 



we have 



2 C n 2 . 



T L n = - \ Zsmny dy= —Z n sin, nr), 



if the force is concentrated at y = r]. Hence by (21) 



(26) 



7T 



so that 



Z7T ft 3 V . X ■ 



where n = l, 2, 3, &c. It will be understood that a constant 

 factor, depending upon the elastic constants and the thickness 

 of the plate, but not upon n, has been omitted. 



The series in (28) becomes more tractable when differ- 

 entiated. We have 



* Z u g cos w(y - q)- cos n(y + 9) .,^. ^ 



dw 



dx 2ir 



* This follows at once if we start from x—l where w — 0, 



A»=%^, (27) 



