Camh'idge Philosophical Society. 565 



-Kl'b lisnoibn CAMBRIDGE philosophical SOCIETY. '"^^ ^ " 

 ^''':?V?! !^'''\^';^;-; (Continued from p. 339.) 



' *e1)!'24, 1851.— On several points of the Integral Calculus. By 

 Professor De Morgan. 



Some time ago, Mr. De Morgan communicated to the Society an 

 abstract of some unfinished views on the connexion between the or- 

 dinary and singular solution of a diflFerential equation. The present 

 paper completes those views, and also contains sections on the solu- 

 tion of differential equations by elimination, on the proof of the 

 number of constants which a solution may contain, and on the cri- 

 terion of integrability of a function of x, y, and differential coefficients 

 of j^^ 



J, On singular solutions. — As to equations of the first order, the 

 tests obtained in this paper may be described as follows : — 



Mr. De Morgan means by a singular solution any one which is 

 obtained by other process than integration, whether it be contained 

 in the integrated primitive, or not. When the singular solution is 

 not contained in the primitive, he calls it an extraneous solution. 



Let <}>(x, y, c) = be the primitive equation, giving c= <!>(«, y). 

 The differential equation then is 



" . . i'O HI noiJoor 



and«>,,= -^ *y=-^ (-.::nr.Tnrrr-c 



Every relation between x and y which satisfies either of the fol- 

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

 definition, a singular solution. 



"^ 1. 4>j and <l>y both infinite. ;<• 



i~, ., . 2. <I>j^ o«/v infinite, and .r=const. 



rt;-rq ICTiL'^p A - ,^ . £ -^ J 



3. 4>j, only mfinite, and y=const. 

 And all possible finite solutions of the differential equation are 

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



Y 1 V Y lit 



Let X= ' " ■ ■ Then all relations between x and y which 



satisfy either of the following collective conditions are solutions of the 

 differential equation, and are singular solutions. 



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

 1"" " ',, 2. Y only infinite, x=const., and X=0. 

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



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

 the cun'c 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~^' Kvery 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 .t=oo , or 

 v= 00, is a singular solution. 



