420 THEORY OF HEAT. [CHAP. IX. 



it is required to determine v as a function/ (#, y, t), which satisfies 

 the proposed equation (e) and the two following conditions : 

 namely, 1st, the substitution t in f(x,yji) must give an 

 arbitrary function &amp;lt;jf&amp;gt; (x, y) ; 2nd, the same substitution in 



f (x-, y, t) must give a second arbitrary function ty (x, y). 

 ctt 



It evidently follows from the form of equation (e), and from 

 the principles which we have explained above, that the function v, 

 when determined so as to satisfy the preceding conditions, will be 

 the complete integral of the proposed equation. To discover this 

 function we write first, 



v = cos px cos qy cos mt, 

 whence we derive 



d*v 2 d*v 4 d* 22 d v 



= -. m * v = tf v = p*g* v = ^ 



dt dx* dor dy* * dy* 



We have then the condition m=p* + q*. Thus we can write 



v = cospx cos qy cos t (p* + a ), 

 or v = cos (px px) cos (qy q/3) cos (p*t -1- q*t), 



or v = ldz \dpF(z, j3) Idp \dq cos (px pot) cos (qy q/5) 



cos (p z t + q*t). 



When we make t = 0, we must have v = &amp;lt;f&amp;gt;(x,y)\ which serves 

 to determine the function F (a., /9). If we compare this with the 

 general equation (BB), we find that, when the integrals are taken 



/ 1 \ 2 

 between infinite limits, the value of F(a, ft) is I \ (f&amp;gt; (a, /8). We 



have therefore, as the expression of the first part u of the 

 integral, 



J 



u = a cos ~ a cos ~ 



Integrating the value of w with respect to t, the second arbi 

 trary function being denoted by -\|r, we shall find the other part 

 W of the integral to be expressed thus : 



