128 On Differential Equations of the First Order* 



the latter part of this memoir the author gives, as a particular 

 case of more general results, the following theorem : — 



If (f>(xj y,p)=0 be a differential equation, and M be a func- 

 tion of x, y } and p such that 



then 



M.(dy—pdx) 



is a complete differential; and the integral of (£ = is obtained 

 by eliminating p between 



c/>=0, 



§M(dy—pdx)+c = 0. 



The value of this theorem is materially affected by a circum- 

 stance pointed out by the author, viz. that in many cases the 

 difficulty of effecting the integration of the complete differential 

 is so great, that nothing but the assurance of its being actually 

 integrable would induce one to continue the search for the inte- 

 gral. Numerous examples are given which at once illustrate 

 the method and warrant the remark which I have quoted. On 

 a careful inspection of these examples, I observed that io nearly 

 every instance the equation to be solved was of such a form as to 

 give for its primary solution 



v=fu, 



v and u being correlated functions, one of which entered into the 

 original equation, and the other of which was obtained by finding 



I r-sr- -r-~\dx when practicable. With the knowledge that u 



and v are correlated functions, I found it easy to determine the 

 form of/, and thus to escape the difficulty pointed out by M. 

 Malmsten. I prepared a paper having this object in view; but 

 in the course of the investigation the more general results which 

 I have put forward in this paper presented themselves to my 

 mind, and appeared practically to supersede the limited object 

 which I had previously had in view. 



12 Fitzwilliam Square, Dublin, 

 January 5, 1865. 



