406 THEOKY OF HEAT. [CHAP. IX. 



The functions &amp;lt;/&amp;gt; and i/r are determined as follows, denoting the 

 function v by /(a?, y, t), and ^/ (x, y, t) by/ (x, y, t), 



$ (*, y} =f (*, y&amp;gt; o), t fa y) =/x fa y. o). 



Lastly, let the proposed differential equation be 



dv 



_- = a 1-2 -y~4 c :r~6 

 dt dot? dx* dx 6 



the coefficients a, b, c&amp;gt; d are known numbers, and the order of the 

 equation is indefinite. 



The most general value of v can only contain one arbitrary 

 function of x ; for it is evident, from the very form of the equa 

 tion, that if we knew, as a function of x, the value of v which 

 corresponds to t 0, all the other values of v, which correspond to 

 successive values of t t would be determined. To express v, we 

 should have therefore the equation v e tj) ^ (x). 



We denote by D(f&amp;gt; the expression 



that is to say, in order to form the value of v, we must develop 

 according to powers of t, the quantity 



a.* + ca 6 + da. 8 + &C.) 



and then write -^- instead of a, considering the powers of a as orders 

 dx 



of differentiation. In fact, this value of v being differentiated 

 with respect to t only, we have 



dv de tD , N _. d*v , d*v d*v p 

 -T: = ^r 9 () = -Dv = a -j t + b -, 4 + c -j 6 + &c. 

 c?^ ai fic 2 dx* da? 



It would be useless to multiply applications of the same process. 

 For very simple equations we can dispense with abridged expres 

 sions ; but in general they supply the place of very complex in 

 vestigations. We have chosen, as examples, the preceding equa 

 tions, because they all relate to physical phenomena whose analytical 

 expression is analogous to that of the movement of heat. The two 

 first, (a) and (b), belong to the theory of heat ; and the three 



