41 G THEORY OF HEAT. [CHAP. IX. 



we can regard f(x) as a function of the two variables x and y. 

 The function /(a) will then be a function of a and y. We shall 

 now regard this function f (a, y) as a function of the variable y, 

 and we then conclude from the same theorem (B), Article 404, 



1 f&quot; 1 &quot; 00 f 



that f(a, ;?/) = J^ / (a, ) jdq cos (qy - 



We have therefore, for the purpose of expressing any function 

 whatever of the two variables x and ?/, the following equation 



y) = **&f( $ cos (P*- 



/+oo 



J -00 



We form in the same manner the equation which belongs to 

 functions of three variables, namely, 



*, y, *) = ** A 7) 



jd/p cos (_p# |&amp;gt;a) /Jg cos (^ - 0/9) I?r cos (r ry) ..... (BBF), 



each of the integrals being taken between the limits oo 

 and 



It is evident that the same proposition extends to functions 

 which include any number whatever of variables. It remains to 

 show how this proportion is applicable to the discovery of the 

 integrals of equations which contain more than two variables. 



409. For example, the differential equation being 



we wish to ascertain the value of v as a function of (x, y, t), such 

 that ; 1st, on supposing t = 0, v or f(x, y, t) becomes an arbitrary 

 function &amp;lt; (a?, y) of x and y\ 2nd, on making t = in the value 



S/ IJ 



of -- y or f (x,y y t), we find a second entirely arbitrary function 



