[116] 



PROFESSOR DE MORGAN ON SOME 



later writings of M. Cauchy on fundamental points, that of recurring, in the treatment of 



elevated subjects, to the very consideration of successive finitely differing values of a function, 



by which a beginner learns common differentiation and integration. The conclusion is, that 



y = v has a solution in which x = x , y = y whenever y y is finite and continuous from w = x , 



through any finite extent. This limitation is not meant to be an exclusion of what is without : 



and it is followed (pp. 435, &c.) by proof that when two or more solutions agree in giving 



„ 



.v = x , y = y , we must have either ^ = - , or ^ y = - or y^ ■ oo , for the case in question. 



Hence the singular solution is detected, provided it be granted that whenever only one solution 

 gives x = x a y = y , that one is an ordinary primitive. This point is asserted in p. 435, and 

 I cannot find any proof of it in what goes before. Until the contrary be disproved (the 

 geometrical illustration I have given does disprove it in real cases) we have no right to assume 

 that every extraneous solution has at every point, concourse with an ordinary primitive. I may 

 also remark that M. Cauchy does not arrive at the cases in which only y, is infinite. Thus 



y = x (1 - x v )"^ has x = ± 1 for an extraneous solution : but ^ is 0, neither oo nor - . And 



Lagrange also omitted these cases. 



The whole of this examination was suggested to me by what appeared a strong prima 

 facie case against the tests derived from ^ and ■% : one of the strongest, I think, which was 

 ever finally refuted. The primitive and singular have contact, generally of the first order ; 

 and y has the same form in both. But this contact is not generally of the second order ; 

 y" is therefore not the same in both*. Now y = ^ must give y" = X* + XyV > w' 1 ' ' 1 can 



only have twice as many values as y by an escape into - or some other singular form : and 



=•= co =f co seems of necessity to be the most common way in which this can happen. But 

 sometimes the contact will be of the second order: may not then x* + Xy^' De ordinary, 

 and y" the first which exhibits singularity of form ? And so on : for the contact may be 

 of any order. The following investigation shewed that in cases giving contact of the second 



order, the general forms of v f and y y exhibit a tendency to the form -, though I do not find 



the same in reduced examples. 



Let d> (#, y, a, b) = be the equation of an infinite number of families of curves, each 

 family being defined by a relation between a and b, to which relation b' is the differential coeffi- 

 cient of b. This relation existing, the singular solution for the separated family is determined by 



*££&£«% and &±fel,Q, 



omitting (in this sketch of main points) the cases of x = const, and y = const. And the 



* How does it happen that, though y' = x( x >y) mav > f° r 

 ;v given point, have two solutions, with different values of y" , 

 yet x« +v X ■ X> reduced, shews no symptom whatever of the 

 second value? An examination of instances will shew that 

 y" = \j/x is obtained as y" = \j/x+ P-P, and that P= ■ in the 



ambiguous case. So that it seems the following reasoning is 

 inaccurate ; — " Since P — P = 0, however great P may be, then 

 P-P = 0, when P is infinite." All we ought to say is that. 

 one value is 0, and that the form is an indication of the pos- 

 sibility of more values than one. 



