436 THEOEY OF HEAT. [CHAP. IX. 



421. To give an example of the use of the last two formulae, 

 let us consider the equation -^ + -, ^ = 0, which relates to the 



uniform movement of heat in a rectangular plate. The general 

 integral of this equation evidently contains two arbitrary func 

 tions. Suppose then that we know in terms of x the value of v 

 when y = 0, and that we also know, as another function of x, the 



value of -7- when y = 0, we can deduce the required integral from 

 that of the equation 



which has long been known; but we find imaginary quantities 

 under the functional signs : the integral is 



v = &amp;lt;/&amp;gt; (x + y^l) + &amp;lt; (x - 2/7=3) + W. 



The second part W of the integral is derived from the first by 

 integrating with respect to y, and changing &amp;lt;f&amp;gt; into ^r. 



It remains then to transform the quantities $(x + y J 1) and 

 $ (# ~~ yj~ i)&amp;gt; m order to separate the real parts from the ima 

 ginary parts. Following the process of the preceding Article we 

 find for the first part u of the integral, 



1 /+ r+ 30 



u = ^- I da/(a) I dp cos (px -pa) (e 



00 ^ GO 



and consequently 



W= & F(a) cos (p - iw) (e- - e-)- 



The complete integral of the proposed equation expressed in 

 real terms is therefore v = u + W ; and we perceive in fact, 

 1st, that it satisfies the differential equation ; 2nd, that on making 

 y = in it, it gives v =f(x) ; 3rd, that on making y in the 



function -7- , the result is F(x). 



