101 



May 13, 1850. 



Results* connected with the theory of the singular solution of a 

 Differential Equation of the first order between two variables. 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 



(a:-y)(a«+y 9 - 1) = 



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

 By such a sjrmbol 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, e) = be the complete primitive of the differential equa- 

 tion y'=x( x, 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 = to exist for points infinitely near to one 

 another. 



2. Systems of points, derived from 



A(x,y,a, |S)=0, B(x, y, a, |8)=0, 

 where 



<p(x, y, a-\-fiV— l)=A(a;, y, a, |3)+B(a?, y, a, /3). V— 1. 



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 



£f.= oo, ^ = oo, 

 or 



^_£.= oo, ^-finite, x = const., 



* This communication is the abstract of a part of a paper not yet com- 

 pleted, and tf as forwarded to the Society for the purpose of ascertaining 

 whether any examples could be produced destructive of the perfect gene- 

 rality of the results. 



