400 THEORY OF HEAT. [CHAP. IX. 



here the constant represents any function of x. Putting for v&quot; its 

 value c&quot; + ldtv iv , and continuing always similar substitutions, we 

 find 



v = c+ jdt v&quot; 



\c&quot; +jdt (c iv +jdt vJ] , 



or v = c + tc&quot;+~d v + ^G + ^c + &c ............. (I 7 ). 



In this series, c denotes an arbitrary function of x. If we wish 

 to arrange the development of the value of v, according to ascend 

 ing powers of #, we employ 



d*v _ dv 

 dx*~dt 



and, denoting by &amp;lt; y , &amp;lt; /y , &amp;lt; //y , &c. the functions 

 d, d* d* 



a* a?* df^ &c -&amp;gt; 



we have first v = a + bx + \dx \dx v t ; a and b here represent any 

 two functions of t. We can then put for v its value 



a, + l&amp;gt;p + Idx Idx v /f ; 



and for v ti its value a tl + b^x + Idx Idx v 4llt and so on. By continued 

 substitutions 



v= a + bx + \dx Idx v t 

 = a + lx+ \dx\dx \a t + Ix 4- Idx Idx v J 

 = a + bx+ldx \dx a t + bx + Idx Idx (a u + b t x -f \dx \dx v\ | 



