1867.] Partial Differential Equations. 487 



where F and /'are definite, and is arbitrary. From this last we get 



F(2/) + F(^)2=f(/){/(y) ; 

 and ehminating 0' between these, we shall arrive at an equation between 

 ocyzpq which is the only partial differential equation of the first order free 

 from arbitrary functiojis which is obtainable from (2) without assigning to 

 (p a definite form. 



Hence (1) must be derivable from an equation of the form 



or of the form 



^■=q^f{xyzp\ . . : (3) 



where /is definite. Differentiating, we have 



0= )+/'(-')?+/'(;') '^'^ 



di/^ dx dif 



Multiplying the second equation by A (where A is any function of xyzpq) 

 and adding, we get 



If (1) admits of a solution of the form (2), (1) must be identical with 

 (4), or be capable of becoming so by virtue of (3). Hence 



= A+/'(/^). -a-=A/'(/^), 



^p + yz=f\y) + \f\^) + {Kp-f)f'{z). . . (5) 

 From the two first we get 



Hence f must satisfy both the equations 



=/'(y) W -(/±«i')/'(-')-(/3i'+y^)- 



The first gives us 



f-'¥{xyz)±cip, 



where F is arbitrary. Substituting this value in (6), observing that 



nx)=^Y(x)±p^£, /'(2/) = F(2/), f\z)=^Y'{z),^ 

 we get 



(6) 



dec 



dx 



VOL. XV. 2 s 



