422 THEORY OF HEAT. [CHAP. IX. 



the integrals of these equations, which have hitherto been 

 fruitlessly sought for, have a remarkable analogy with those which 

 express the movement of heat. 



413. We might also, in the investigation of the integrals, 

 consider first series developed according to powers of one variable, 

 and sum these series by means of the theorems expressed by the 

 equations (B), (BB). The following example of this analysis, 

 taken from the theory of heat itself, appeared to us to be 

 worthy of notice. 



We have seen, Art. 399, that the general value of u derived 

 from the equation 



dv d*v , N 



dt=dj ...................... ...... (a)&amp;gt; 



developed in series, according to increasing powers of the variable 

 t, contains one arbitrary function only of x ; and that when de 

 veloped in series according to increasing powers of x, it contains 

 two completely arbitrary functions of t. 



The first series is expressed thus : 



v = t(*) + tJ2tW + ft^4&amp;gt;W + to- --. ..... (T). 



The integral denoted by (), Art. 397, or 



v = ^- \ dy. &amp;lt;j&amp;gt; (a) I dp e~ pZ * cos (px ^?a), 



represents the sum of this series, and contains the single arbitrary 

 function &amp;lt; (as). 



The value of v, developed according to powers of x, contains 

 two arbitrary functions f(t) and F(t), and is thus expressed : 



There is therefore, independently of equation (/3), another 

 form of the integral which represents the sum of the last series, 

 and which contains two arbitrary functions, f(t) and F(f). 



