326 THEORY OF HEAT. [CHAP. VIII. 



336. The expression for v is therefore formed of three similar 

 functions, one of x, the other of y, and the third of z, which is 

 easily verified directly. 



In fact, if in the equation 



dt~ 



we suppose v XYZ\ denoting by X a function of x and t, 

 by Y a function of y and t, and by Z a function of z and t, we have 



_ . , 



&quot; + + &quot; *W - F **-z&) 



i ax i dY i dz 



x&quot;^ + rW + ^^ 



which implies the three separate equations 



~dt ~ d^ di dy&quot; dt~ dz 

 We must also have as conditions relative to the surface, 



dV k V n 

 ^ + ^ F== 



whence we deduce 



=,=,. 



dx K dy K dz K 



It follows from this, that, to solve the problem completely, it is 



// ?/ ri ?/ 



enough to take the equation -^ = k -^ , and to add to it the 



equation of condition -p + ^u 0, which must hold when x = a. 



We must then put in the place of a?, either T/ or #, and we shall 

 have the three functions X } Y } Z, whose product is the general 

 value of v. 



Thus the problem proposed is solved as follows : 



, ; 



cos 



