232 Cambridge Philosophical Society. 



tion of a Differential Equation of the first order between two vari- 

 ables. By Professor De Morgan. 



By a singular solution of a differential equation is here meant any 

 solution which can be obtained by differentiation only, whether it be 

 a case of the primitive by integration or not. 



By a curve is meant all that is included under one equation, whether 

 resoluble into what are commonly called complete curves or not. 

 Thus, the equation 



(*-y)<>2+y 2 -l)=0 



belongs to a curve, having a rectilinear branch and a circular one. 

 By such a symbol as v x is meant the partial differential coefficient 



■ — , when obtained from an equation in which v is explicitly expressed 

 dx 



in terms of x and (it may be) other variables. 



Let <p(x, y, c) = be the complete primitive of the differential equa- 

 tion y'=%(a?,y). 



<p(x, y, c) belongs to two distinct classes of curves : — 



1. Continuous curves derived from such values of c, real or ima- 

 ginary, as will enable <p=0 to exist for points infinitely near to one 

 another. 



2. Systems of points, derived from 



A(j?,y,a,j8)=0, B(a?,y,a,/3)=0, 



where 



<p(x,y, a+/3V-l)=A(>,y, a, £) + B(a?,y, a,/3)v~l- 



When a curve is such that the points on one side of it are on 

 curves of the first kind, and those on the other side are part of 

 systems of the second kind, let that curve be called a separator ; and 

 the same when it separates points of both kinds from points which 

 belong to one kind only. 



No solution of the differential equation can be formed by combining 

 all those systems of the second kind in which a and /3 are connected 

 by a real relation. 



The curve which has at every point of it, either 



or 



or 



is a singular solution. And in the above are contained all the sin- 

 gular solutions. 



Every branch of a singular solution is either — 



A separator, only. 



A curve, every point of which has a contact of the first order at 

 least with some one real primitive, only. Or both. Or neither. 



If the first or last, it is a case of the complete primitive. And 



<P,V . 



<?c' 



= 00, £L = oo, 



21=00, 



■1-2- finite, x = const, 



&.= «,, 



i-l finite, y = const. 



<Pc 



<Pc 



