404 THEORY OF HEAT. [CHAP. IX. 



402. If the proposed differential equation is 



ifv dh efo / v 



dt*-dtf + d?/ z &quot; 



and if we wish to express v in a series arranged according to 

 powers of t, we may denote by D&amp;lt; the function 



S?* + ^* ; 



fPv 

 and the equation being -^ = Dv, we have 



v = cos (t &amp;lt;J D) $ (x, y). 

 From this we infer that 



-Ta -5-5 -7-5 



at dx* df 



We must develope the preceding value of v according to powers 

 of t, write (n + -rni , instead of D , and then regard i as the order 



n 



of differentiation. 



The following value \dt cos (t J- D) ^ (a?, #) satisfies the same 

 condition; thus the most general value of v is 



jdt cos (* 7^5) ^ ( x , y] ; 



and 



v is a function f(x y y&amp;gt; f) of three variables. If we make t = 0, we 

 have/= (a?, y, 0) = &amp;lt; (a?, y) ; and denoting ^/fe y, by/ (, y, &amp;lt;), 

 we have/ (a?, y, 0) = ^ (x, y}. 

 If the proposed equation is 



the value of v in a series arranged according to powers of t will 



