410 THEORY OF HEAT. [CHAP. IX. 



the end of the time , been removed from its place through the 

 perpendicular height v. This variable flight v is a function of 

 x and t. The initial value of v is arbitrary; it is expressed by any 

 function (/&amp;gt; (x). Now, the equation (d) deduced from the funda 

 mental principles of dynamics shews that the second fluxion 



of v, taken with respect to ,-or ~ z , and the fluxion of the fourth 



(Jut 



d*v 

 order taken with respect to x, or ^ 4 are two functions of x and t, 



which differ only in sign. We do not enter here into the special 

 question relative to the discontinuity of these functions; we have 

 in view only the analytical expression of the integral. 



We may suppose also, that after having arbitrarily displaced 

 the different points of the lamina, we impress upon them very 

 small initial velocities, in the vertical plane in which the vibrations 

 ought to be accomplished. The initial velocity given to any 

 point m has an arbitrary value. It is expressed by any function 

 ty (x} of the distance x. 



It is evident that if we have given the initial form of the 

 system or &amp;lt;/&amp;gt; (x) and the initial impulses or ty (x), all the subse 

 quent states of the system are determinate. Thus the function 

 v oif(x,t), which represents, after anytime t, the corresponding 

 form of the lamina, contains two arbitrary functions &amp;lt; (x) 

 and ijr (x). 



To determine the function sought f(x t t), consider that in the 

 equation 



we can give to v the very simple value 

 u cos (ft cos qXj 

 or else u cos ft cos (qx &amp;lt;?a) ; 



denoting by q and a any quantities which contain neither x nor t. 

 We therefore also have 



u = I doL F(OL) Idq cos ft cos (qx q 1 *), 



