TERNARY DIFFERENTIAL EQUATIONS. 103 



Let {ij-\-x'-)z-ij=o (3) 



be the primitive. Differentiate in the ordinary way; then 



I prefer writing differential coefficients to differentials. 

 Now equation (4) has evidently a single solution (3), and 

 satisfies the conditional equation (2). 



This is all right enough ; but now observe the effect 

 produced on the discriminoid D by eliminating 2xz from 

 equation (4) by means of its value as given by equation (3). 



From (3) 2xz=- — -^. 



Substitute this value in equation (4) and it becomes 

 , .dy ixy , ,.dz 



Equation (5) has still the single solution (3) ; but it now 

 fails entirely to satisfy the conditional equation (2) . 



The discriminoid □= /^ ^ -j (y + -^'^)g— y ?• • • (6) 



The discriminoidal solution of equation (5) is therefore 

 a factor of the discriminoid n ) in accordance with the 

 proposition of Sir James Cockle, as stated in art. 4. 

 Let us take another very general equation, viz. 

 fdw dv\dy V dw ^dv . dz _ 

 Xdy'' dy/dx w dx dx dx~ '' ' '' 

 where w, v are functions of x, y only. 



This equation does not satisfy the conditional equation 



(2) , as the discriminoid Q is readily found to be as follows : 



_ r I div dw d^w -\ „ 



~ ^ ' \iv dx dy dxdy J ' ' * ^ ' 



This equation vanishes only when one of the two factors 



of which it is composed vanishes. 



The first factor, viz. wz—v = o, satisfies equation (7). 

 therefore it is the discriminoidal solution of (7). 



