418 THEORY OF HEAT. [CHAP. IX. 



and we must take W to be the integral ludt, changing only &amp;lt;&amp;gt; 



into A/T. In fact, u becomes equal to (f&amp;gt; (a?, y), when t is made 

 = 0; and at the same time W becomes nothing, since the integra 

 tion, with respect to t, changes the cosine into a sine. 



Further, if we take the value of -7-, and make t = 0, the first 



part, which then contains a sine, becomes nothing, and the 

 second part becomes equal to ty (x, y). Thus the equation 

 v = u + Wis the complete integral of the proposed equation. 



We could form in the same manner the integral of the 

 equation 



&amp;lt;Fv c?v d?v cFv 



It would be sufficient to introduce a new factor 



2^ cos (rz - ry) , 

 and to integrate with respect to r and 7. 



410. Let the proposed equation be ;r^ + -7-2 + -7-* $ ; it is 



ctx cLy ctz 



required to express v as a function f(x,y,z), such that, 1st, 

 f(x,y,Q) may be an arbitrary function $(#,#); 2nd, that on 



making 2 = in the function -7- f(x,y,z) we may find a second 



ctz 



arbitrary function ^ (#, y). It evidently follows, from the form of 

 the differential equation, that the function thus determined will 

 be the complete integral of the proposed equation. 



To discover this equation we may remark first that the equa 

 tion is satisfied by writing v = cos^&amp;gt;#cos qij e mz , the exponents 

 p and q being any numbers whatever, and the value of m being 



We might then also write 



v = cos (px-p*} cos (qy - q(3} (e &amp;lt;v ^+ i -f 



