412 THEORY OF HEAT. [CHAP. IX. 



In fact, this value of v satisfies the differential equation (d) ; 

 also when we make t 0, it becomes equal to the entirely arbitrary 



function fy (x) ; and when we make t = in the expression -7- , 



cLii 



it reduces to a second arbitrary function ^r (as). Hence the value 

 of v is the complete integral of the proposed equation, and there 

 cannot be a more general integral. 



406. The value of v may be reduced to a simpler form by 

 effecting the integration with respect to q. This reduction, and 

 that of other expressions of the same kind, depends on the two 

 results expressed by equations (1) and (2), which will be proved 

 in the following Article. 



dq cos ^ cos qz = p-sin I-T + T) (1). 



* * * v 



/: 



Ciq sin q*t cos qz .-= sin f-r -T- ) (2). 

 -** -^ /. \ ZL AiT 1 / \ / 



(k/ \ **/ 



From this we conclude 



Denoting j- by another unknown p, we have 



a = x + 2/,6 Jt, da. = 

 Putting in place of sin (^ + A fc2 J i ts value 



1 



v 

 we have 



u = -TT= f ^ (sin ^ 2 + cos fS) &amp;lt;j&amp;gt; (OL + 2/4 V/) ........ ( ). 



V ^7T J -oo 



We have proved in a special memoir that (5) or (8 ), the 

 integrals of equation (d), represent clearly and completely the 

 motion of the different parts of an infinite elastic lamina. They 

 contain the distinct expression of the phenomenon, and readily 

 explain all its laws. It is from this point of view chiefly that we 



