116 
(1) fails to satisfy the conditional equation (2), and still pos- 
sesses a single solution, this solution is gathered from the 
equation D^O. 
The solution here alluded to is not inaptly called the dis~ 
criminoidal solution by Sir James Cockle. 
5. The following view of the subject is intended to further 
elucidate the proposition in art. 4, and also to show a reason 
for the existence of these discriminoidcd solutions. 
By a reference to the general primitive and its derived 
differential equation, it will not be difficult to perceive the 
manner by which the harmony of the conditional equation 
(2) is disturbed by the legitimate process of elimination 
between the primitive and its differential equation. Take, 
for example, a simple case : 
Let [y + x^)z ~ y = ^ (3) 
be the primitive. Differentiate in the ordinary way, then 
(Z-l)g + 2^^+(2/ + *^)| = 0 (4) 
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 pro- 
duced on the discriminoid □ by eliminating ^xz from equa- 
tion (4) by means of its value as given by equation (3). 
From (3) 2xz 
2xy 
y + 
Substitute this value in equation (4) and it becomes 
(z-\) 
+ 
dx y -{-x 
( 5 ) 
Equation (5) has still the single solution (3) ; but it now fails 
entirely to satisfy the conditional equation (2). 
2 ^ 
The discriminoid Q = 
y + X 
{y-\-c^)z 
y 
( 6 ) 
The discriminoidal solution of equation (5) is, therefore, a 
factor of the discriminoid □, in accordance with the propo- 
sition of Sir James Cockle, as stated in art. 4. 
