Deflexions of Membranes, Bars, and Plates. 357 



the plate may move in the direction opposite to that of an 

 applied force. We may contemplate the arrangement of 

 fig. 2, where, however, the partition CD is now merely 

 supported and not clamped. Along the imperforated parts 

 CA, BD the plate must be supposed cut through so that no 

 couple is transmitted. And in the same way we infer that 

 internal nodes are possible when a supported plate vibrates 

 freely in its gravest mode. 



But although a movement opposite to that of the impressed 

 force may be possible in a plate whose boundary is clamped 

 or supported, it would seem that this occurs only in rather 

 extreme cases when the boundary is strongly re-entrant. 

 One may suspect that such a contrary movement is excluded 

 when the boundary forms an oval curve, i. e. a curve whose 

 curvature never changes sign. A rectangular plate comes 

 under this description; but according to M. Mesnager *, 

 " M. J. Resal a montre qu'en applicant une charge au centre 

 d'une plaque rectangulaire de proportions convenables, on 

 produit tres probablement le soulevement de certaines 

 regions de la plaque. " I understand that the boundary is 

 supposed to be "supported" and that suitable proportions 

 are attained when one side of the rectangle is relatively long. 

 It seems therefore desirable to inquire more closely into this 

 question. 



The general differential equation for the equilibrium of a 

 uniform elastic plate under an impressed transverse force 

 proportional to Z is f 



V i w = (d 2 /dx 2 + d 2 /df) 2 w = Z (3) 



We will apply this equation to the plate bounded by the 

 lines y = 0, y = 7r, and extending to infinity in both directions 

 along x, and we suppose that external transverse forces act 

 only along the line ^ = 0. Under the operation of these 

 forces the plate deflects symmetrically, so that w is the same 

 on both sides of x = and along this line div/dx = 0. Having 

 formulated this condition, we may now confine our attention 

 to the positive side, regarding the plate as bounded at # = 0. 



The conditions for a supported edge parallel to x are 



w = 0, d 2 w/df = 0; (4) 



and they are satisfied at y — and y = ir if we assume that w 

 as a function of y is proportional to sinn?/, n being an 



* C. R. t. 162, p. 826 (1916). 



t 'Theory of Sound/ §§215, 225; Love's 'Mathematical Theory of 

 Elasticity/ Chapter xxii. 



