117 
Let us take another very general equation, viz. : 
( dw dv\dy v dw 
dy dyjdx w dx 
dv' dz _ 
— 0 
dx dx 
(7) 
where w,v are functions of x,y only. 
This equation does not satisfy the conditional equation 
(2), as the discriminoid □ is readily found to be as follows: 
^ , , { \ dw dw d^w ) 

This equation vanishes only when one of the two factors 
of which it is composed vanishes. 
The first factor, viz., wz—v~0, satisfies equation (7), 
therefore, it is the discriminoidal solution of (7). 
In consequence of ^oz—v—0 being a very general primi- 
tive, it is inferred that equation (7) is a very general diffe- 
rential equation which admits of discriminoidal solution. 
6. Boole, in page 285, has, by a reference to one of the 
dual equations, unnecessarily limited the values of the con- 
stants in the equation 
dy 
dz 
so as to admit of a single solution. 
This equation is satisfied by the single solution 
c^{z -c) + <£‘{pd‘ + - cd 
When a—h, the quantity C is introduced by integration. 
7. The elimination of an arbitrary constant between the 
primitive and its derived difierential equation does not dis- 
turb the harmony of the conditional equation □ 0. 
Let (C-R 1 XC-R 2 ) &c.=0. (9) 
be a primitive where (C) is an arbitrary constant and Bi,R 2 j 
& c., given functions of x,y,z. 
Differentiate (9), then 
c) + f(R.-c)}|=0 (10) 
- C) - 
dy 
+ 
dy 
This equation evidently satisfies the conditional equation 
(2). 
