SECT. IV.] PARTIAL DIFFERENTIAL EQUATIONS. 419 



or 



t = 



t, ft) jdpjdq cos (px -pi) cos (qy - qft) 



If 2 be made equal to 0, we have, to determine F(y, /3), the 

 following condition 



( y) = jdzldft F (a, /3) jdpjdq cos (^ -_pa) cos (^ - qft) ; 

 and, on comparing with the equation (BB) t we see that 



we have then, as the expression of the first part of the integral, 

 ^) 4P cos (P x -P*) d( l cos (^ ~ 2#) 



The value of w reduces to &amp;lt;/&amp;gt; (x, y) when = 0, and the same 



substitution makes the value of -j- nothing. 



dx 



We might also integrate the value of u with respect to z, and 

 give to the integral the following form in which i/r is a new 

 arbitrary function: 



IF= ^) | da. jd/3 ^r (a, ft) Jdp cos (^ - pa) jdq cos (jy - qft) 



The value of TF becomes nothing when = 0, and the same 



dW 

 substitution makes the function j~ equal to -^ (x, y). Hence 



the general integral of the proposed equation is v = u + W. 



411. Lastly, let the equation be 



f 

 dt 



* *-~*^dy*~ 



272 



