SECT. IV.] VIBRATION OF ELASTIC LAMINA. 409 



From this we conclude as the general integral of the proposed 

 equation (/), 



u = -L [ efe/( 



^7T J -oo 



405. If we propose the equation 



which expresses the transverse vibratory movement of an elastic 

 plate 1 , we must consider that, from the form of this equation, the 

 most general value of v can contain only two arbitrary functions 

 of x: for, denoting this value of v by f(x,t), and the function 



-rf(x, t) by / (a?, t), it is evident that if we knew f(x, 0) and 

 cit 



f (x, 0), that is to say, the values of v and - - at the first instant, 



at 



all the other values of v would be determined. 



This follows also from the very nature of the phenomenon. In 

 fact, consider a rectilinear elastic lamina in its state of rest: x is 

 the distance of any point of this plate from the origin of co 

 ordinates; the form of the lamina is very slightly changed, by 

 drawing it from its position of equilibrium, in which it coincided 

 with the axis of x on the horizontal plane; it is then abandoned to 

 its own forces excited by the change of form. The displacement is 

 supposed to be arbitrary, but very small, and such that the initial 

 form given to the lamina is that of a curve drawn on a vertical 

 plane which passes through the axis of x. The system will suc 

 cessively change its form, and will continue to move in the vertical 

 plane on one side or other of the line of equilibrium. The most 

 general condition of this motion is expressed by the equation 



d*v d 4 v ,, . , 



a?+- ........................ w - 



Any point m, situated in the position of equilibrium at a 

 distance x from the origin 0, and on the horizontal plane, has, at 



1 An investigation of the general equation for the lateral vibration of a thin 

 elastic rod, of which (d) is a particular case corresponding to no permanent 

 internal tension, the angular motions of a section of the rod being also neglected, 

 will be found in Donkiu s Acoustics, Chap. ix. 169177. [A.F.] 



