II. On the question. What is the Solution of a Dijirential Equation f A Sup^de- 

 ment to the third section of a paper, On some points of the Integral Calculus, 

 printed in Vol. IX. Part II. By Augustus De Morgan, of Trinity College, 

 Vice-President of the Royal Astronomical Society, and Professor of Ma- 

 thematics in University College, London. 



[Read April 28, 1856.] 



Trusting that it will be sufficient excuse for a very elementary paper, that writers of the 

 highest character are not agreed with each other on a very elementary point, I beg to offer 

 some remarks upon the usual solution of such an equation as dy'^ — a^dar' = 0, to which Euler 

 assigns the integral form (y - aa> + b) {y + ax + c) = 0, where b and c are independent 

 constants. Most other writers insist on the condition b = c. 



Lacroix refers only to Euler and to a paper by D'Alembert (Berl. Mem. 1748) which I 

 have not seen. All the reasons which have been given on the subject are reducible, so far as 

 I have met with them, to those which I shall cite from Lacroix himself and from Cauchy. 



Lacroix (ii. 280) in his explanation of this case, and in defence of the substitution of 

 (y - a.v + b) (y +ax + b) for (y - ax + b) (y + ax + c), makes two remarks. The first, — 

 chacun de ses facteurs doit etre considere isolement; the second, alluding to the form with two 

 constants, is — on n'en tire pas d'autres lignes que celles qui resulteraient de Tintegrale renferm- 

 ant une seule constante. M. Cauchy (Moigno, ii. 456) says — On ne restreindra pas la generalite 

 de cette integrale en designant toutes les constantes arbitraires par la meme lettre... : and 

 grounds the right to do this on the possibility of thus obtaining all the curves which can satisfy 

 the equation. 



In searching out this matter, I found it by no means clearly laid down what is meant by 

 the solution of a differential equation : and, on looking further, I found some degree of ambi- 

 guity attaching to the word equation itself. The following remarks will sufficiently explain 

 what I mean. 



A connexion between the values of letters, by which one is inevitably determined when the 

 rest are given, may be called a relation. But an equation is the assertion of the equality of 

 two expressions. Every simple explicit relation leads to an equation, to one equation : but every 

 equation does not imply only one relation. The object of the pi'oblem being relation between 

 y and x, the equation {y — x) (y — x^) = implies power of choice between the relations y = x^, 

 y '^ w. The equation {y - x') (a; - 1) = implies the relation y = od' with a dispensation from 

 all relation in the case of a; = 1. 



Now I assert that in mathematical writings confusion between the equation and the simple 

 relation is by no means infrequent : without dwelling on instances, I think we shall find, by 



