SOLUTIONS OF DIFFERENTIAL EQUATIONS. 185 



Diff. Eqs. p. 161, 2nd ed., and supp. vol. p. 17, by means 

 of the condition -r-= infinity, and thereby plunged him- 

 self into a complete mathematical labyrinth, from which 

 his great genius would not enable him to extricate him- 

 self. He states that "the value of (c) is not wholly 

 independent of (x)" and "regards y=o as a singular 

 solution;" whereas y=o is a particular solution, where 

 c=— infinity. For the value of (c) to be dependent, in 

 any degree, upon the value of (#) is in direct opposition 

 to the spirit of the definition of a singular solution. 



By paragraph (9) it appears that no differential equa- 

 tion, whose primitive is of the form c + v + zlogw=o } can 

 have a singular solution. 



10. There are three questions which demand special 

 consideration in the subject of singular solutions of differ- 

 ential equations, viz. — 



1 . The singular solution as derived from the complete 



primitive. 



2. The singular solution as derived from the differ- 



ential equation. 



3. Given a solution of a differential equation, is it a 



singular or a particular integral ? 



These questions have to some extent been replied to in 

 paragraphs 1 to 5. 



1 1. With respect to the first question, it may be observed 

 that every complete primitive can or cannot be put into 

 the form (22). When the former is possible then the sin- 

 gular solution is obvious from the primitive itself, but if 

 the latter prevails then the conclusion is that there is no 

 singular solution. 



It may be noticed here that other terms of the form 



SER. III. VOL. VIII. o 



