Partial Differential Equation of the Second Order, 223 

 equation becomes 



and if we put for fax, fay the above values, we shall have in the 

 case now being considered, instead of equations (7), 



0=fJ(x)-mJJ(i,)+f a .fJ{z) + *l + yz + 8 



+ { m - m 2-fJ( g ) + m i*+P+-^~ m *-jfr) -f, , 



(8) 



° = »»i/,-/.+/„+ 



«i) 



<fy 



•/• 



Two cases occur. 



I. Suppose that f af which does not contain ft m) (i. e. U) or 

 higher derivatives of <£, is also destitute of (^ m_1) . 



In this case f a , fj (x), fj (y), fj (z) are alike free from U ; but 

 /contains U; hence the coefficient of/ in the first of equations 

 (8) must vanish, i. e. we must have 



0=fX.r)-m 2 f«'( V )+f rl .fJ(z) + a f a + J z + S, 



From the latter we have 



where 



If o d m i dm^"] 



?n 2 — m i \ ' 1 dx 2 dy J 



Hence 



fa=P*-t-f b (xy); • • • 

 and substituting this in (9), observing that 



(9) 



(10) 



fJ(y)= d £ J z+fJ(y), 



we get the result, 



in which, since / 5 contains xy only, the coefficient of z must va- 



