SOLUTIONS OF DIFFERENTIAL EQUATIONS. 171 



the custom hitherto, following the example of Lagrange, 

 to put in the place of the arbitrary constant a certain 

 function of x, y, such as to give the same differential 

 equation as that which is derived from the primitive before 

 the change of the arbitrary constant is made. 



In what follows a different procedure has been adopted, 

 viz. to impress upon the complete primitive such a form as 

 to produce two or more solutions of the derived differential 

 equation, neither of which shall be a particular case of the 

 complete primitive in the ordinary sense. 



By the former method the envelope species only of sin- 

 gular solutions is obtainable, whereas by the latter the 

 non-envelope species as well as the envelope species are 

 readily developed. 



At the Ordinary Meeting of this Society, March 2ist, 

 1882, Sir James Cockle, F.R.S. &c, read an interesting 

 paper " On Envelopes and Singular Solutions/'' in which 

 he refers to two theorems which I communicated to him 

 in 1867. 



The theorems are correctly stated by Sir James Cockle, 

 and are as follows : — ■ 



A. The condition of equal roots with respect to e, the 



arbitrary constant, in the complete primitive 

 <£(#, y, c) = o is a singular solution, if not con- 

 tained in the primitive by giving a constant value 

 to (c). 



B. The condition of equal roots with respect to p = -r- 



in the differential equation ty {x, y } p) = o is a 

 singular solution if it satisfies the differential 

 equation. 



These theorems, which facilitate the finding of singular 

 solutions both from the complete primitive and its derived 



n2 



