372 THEORY OF HEAT. [CHAP. IX 



f 3 Fri 



is known, and is known to be equivalent to /- /^ (see the follow 

 ing article). Hence this last function of x and t agrees also with 

 the differential equation (b). It is besides very easy to verify 



_1 J 



P 4 



directly that the particular value -TF satisfies the equation in 

 question. 



The same result will occur if we replace the variable x by 

 x a, a being any constant. We may then employ as a particular 



Q-q) 2 



value the function - & -j= , in which we assign to a any value 



whatever. Consequently the sum I dzf (a) - p also satisfies 



J v t&amp;gt; 



the differential equation (6) ; for this sum is composed of an 

 infinity of particular values of the same form, multiplied by 

 arbitrary constants. Hence we can take as a value of v in the 



//7) CM 7J 



equation -j- = -3-- the following, 

 dt dx 



A being a constant coefficient. If in the last integral we suppose 

 ^ = j 2 , making also A ~r= , we shall have 



1 f* 00 



V/^-oo 



We see by this how the employment of the particular values 



or 



leads to the integral under a finite form. 



