102 MR. ROBERT RAWSON ON 



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 repf oach of absurdity and meaningless 

 from those differential equations which failed to satisfy the 

 conditional equation (2) , 



3. Both of the above propositions are considerably cir- 

 cumscribed by a third proposition discovered by Sir James 

 Cockle, viz.: — Equation (i) may fail to satify (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 exag- 

 gerated, especially in reference to the doctrine of curved 

 surfaces, &c.; stiU 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, De Morgan, or M. FAbbe 

 Moigno, the ablest writers on differential equations. 



4. Sir James Cockle has shown also that, when equation 

 (i) fails to satisfy the conditional equation (2) and still 

 possesses a single solution, this solution is gathered from 

 the equation 0=0- 



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 discriminoidal solutions. 



By a reference to the general primitive and its derived 

 differential equation, it will not be difficult to perceive the 

 manner in which the harmony of the conditional equation 

 (2) is disturbed by the legitimate process of elimination 

 between the pi'imitive and its differential equation. Take, 

 for example, a simple case : — 



