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The substance of this knowledge may be given in two 
propositions, as follows : — 
First — That the differential equation 
Vdx + (^dy + ^dz = 0 (1) 
where P,Q,R are given functions of x,y,z, has a single solu- 
tion, or in other words, there is a single relation between the 
variables x,y,z which will satisfy it, providing the following 
conditional equation obtains 
/dQ dR\ dP\ dQ\ ™ 
□ is called, by Sir James Cockle, the discriminoid. 
Second . — If the conditional equation (2) does not obtain 
by virtue of the given functions P,Q,R, then the equation 
(1) has a dual solution, or in other words, there are two 
relations between the variables x,y,z which will satisfy it. 
2. The first of these propositions was known to Newton 
and his contemporaries, who, however, regarded all those 
differential equations which failed to satisfy (2) as absurd 
and meaningless; Monge supplied the dual solution, and 
thereby removed the reproach of absurdity and meaningless 
from those differential equations which failed to satisfy the 
conditional equation (2). 
3. Both of the above propositions are considerably circum- 
scribed by a third proposition discovered by Sir James 
Cockle, viz.: equation (1) may fail to satisfy (2) and still be 
satisfied by a single relation between the variables of which 
it is composed. 
The importance of this proposition can hardly be exagge- 
rated, especially in reference to the doctrine of curved sur- 
faces, &c., still, it might be hazardous, in the present active 
state of mathematical investigation both here and on the 
continent of Europe, to say it is new. It is not, however, 
referred to by Boole, Be Morgan, M. L’abbe Moigno, the 
ablest writers on differential equations. 
4. Sir James Cockle has shown, also, that when equation 
