408 THEORY OF HEAT. [CHAP. IX. 



We might then also take 



v _ e -mt cos 



giving to the constant a any value. We have similarly 

 v**fd*&amp;lt;j&amp;gt; 0) e-*(^+ 6 * 4 +^ 6+&c ) cos (px -pz). 



It is evident that this value of v satisfies the differential equation 

 (/) ; it is merely the sum of particular values. 



Further, supposing t = 0, we ought to find for v an arbitrary 

 function of x. Denoting this function by/(#), we have 



/ (x) = I dz (f&amp;gt; (a) I dp cos (px p%). 



Now it follows from the form of the equation (/), that the most 

 general value of v can contain only one arbitrary function of x. 

 In fact, this equation shews clearly that if we know as a function 

 of x the value of v for a given value of the time t, all the other 

 values of v which correspond to other values of the time, are 

 necessarily determined. It follows rigorously that if we know, 

 as a function of t and x, a value of v which satisfies the differential 

 equation; and if further, on making t = 0, this function of x and t 

 becomes an entirely arbitrary function of x, the function of x and 

 t in question is the general integral of equation (/). The whole 

 problem is therefore reduced to determining, in the equation 

 above, the function &amp;lt; (a), so that the result of two integrations 

 may be a given function /(#). It is only necessary, in order that 

 the solution may be general, that we should be able to take for 

 f(x) an entirely arbitrary and even discontinuous function. It is 

 merely required therefore to know the relation which must always 

 exist between the given function f(x) and the unknown function 

 &amp;lt;j&amp;gt; (a). Now this very simple relation is expressed by the theorem 

 of which we are speaking. It consists in the fact that when the 

 integrals are taken between infinite limits, the function &amp;lt; (a) is 



~ / (a) ; that is to say, that we have the equation 



I r+oo /+ 



~fc.-l & /( a )| 



^?r j - oo j - 



