SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 417 



From the form of the differential equation (c) we can infer 

 that the value of v which satisfies this equation and the two pre 

 ceding conditions is necessarily the general integral. To discover 

 this integral, we first give to v the particular value 



v = cos mt cos px cos qy. 



The substitution of v gives the condition m = Jp* + q*. 

 It is no less evident that we may write 



v = cosp (x a) cos q (y ft) cos t J$ -f (f, 



or 



v = I dx I d/3 F (a, /3) I dp cos (px - pot) Idq cos (qy - q@) cos t Jp* + q* t 



whatever be the quantities p, q, a, ft and F (a, @), which contain 

 neither x, y, nor t. In fact this value of t is merely the sum of 

 particular values. 



If we suppose t = 0, v necessarily becomes $ (x } y). &quot;We have 

 therefore 



( x &amp;gt; y) = jdzldP F ( a , /3) J dp cos (px - POL) jdq cos (qy - q/3). 



Thus the problem is reduced to determining F (a, /3), so that 

 the result of the indicated integrations may be &amp;lt; (x, y). Now, on 

 comparing the last equation with equation (BB), we find 



*&amp;gt; y} = ( *-} f k f + ^ &amp;lt;/&amp;gt; ( a&amp;gt; /S) f + 



\^7r/ J-ao J-x&amp;gt; J - 



cos - 





 



Hence the integral may be expressed thus : 



We thus obtain a first part i of the integral; and, denoting 

 by W the second part, which ought to contain the other arbitrary 

 function i/r (x, y), we have 



v = u+ W, 

 F. H. 27 



