107 



lowing collective conditions, is a solution of the equation ; and, by 

 definition, a singular solution. 



1. ® x and <frj, both infinite. 



2. $> x only infinite, and x= const, 



3. <fr y only infinite, and y = const. 



And all possible finite solutions of the differential equation are 

 given either by the original primitive, or by these relations. 



Let x= X * + XjjX . Then all relations between x and y which 

 Xv 

 satisfy either of the following collective conditions are solutions of the 

 differential equation, and are singular solutions. 



1. v and v both infinite, and X = 0. 



2. x oru y infinite, a? = const., and X=0. 



3. y only infinite, y = const., and X=0. 



But when one of these sets fails only in that X does not vanish, 

 the curve so obtained, instead of having contact with a primitive 

 curve at every one of its points, passes through the points of infinite 

 curvature of the primitives ; and the differential equation which is 

 satisfied is y'=^—X. Every evolute is related in this manner' to 

 its involutes, passing through all their cusps. 



The above tests do not give the possible case in which x= go , or 

 y= oo, is a singular solution. 



Mr. De Morgan proposes the following geometrical illustration of 

 the connexion between the primaries and the singular. Let c be the 

 third ordinate of a surface (usually denoted by z) having the equation 

 <p(x, y, c)=0. The projections upon the plane of xy of sections 

 parallel to that plane are the primaries : the singular solution is the 

 base, upon the plane of xy, of a cylinder perpendicular to that 

 plane, and which always touches the surface. By means of this 

 illustration, it may be made manifest that certain cases of singular 

 solution which have always been discarded as unmeaning, are 

 limiting cases of the kind which are admitted in analysis so soon as 

 the way up to the limit is clearly seen. 



Taking a general . equation with two arbitrary constants, so that 

 a relation between those constants selects and designates a family 

 of curves, it is shown generally (without examination of exceptional 

 cases) how to find the families which have with their singular curves 

 contact of the second order. The equation of these singular curves 

 is a differential equation of the first order : but its singular solution 

 is the singular curve of a family of curves which have with it a con- 

 tact of the third order. 



2. Solution of differential equations by elimination. — This is an idea 

 derived from the method which Mr. De Morgan communicated 

 (vol. viii. part 5) relative to partial differential equations, and which 

 he found, after his paper was finished, had been given by M. Chasles, 

 as he supposed, from knowledge of the results of Monge. But it 

 afterwards appeared that the authority for Monge having obtained 



