POINTS OF THE INTEGRAL CALCULUS. [115] 



But y" is made infinite by the result : and the differential equation is not satisfied. Conse- 

 quently, the evolute has the general characteristic of a singular solution of the equation of the 

 involutes, but is really a curve passing through their cusps of infinite curvature. This was 

 noticed in my Differential Calculus, pp. 364—367. 



Let y = p, y\ = q be the differential equations of two families of curves, and let 

 /(#> y> y\ y'\) = be a relation which exists at every intersection. Then /(a?, y, p, q) = is an 

 identical equation, and 



fvP* +/«?*+/* = °. ftPy +/„?» +/, = 0. 



In all the cases in which p x = oo , p y = co make q x = oo , q y = -oo , the two families have the 

 same singular solution, or locus of infinite curvatures, or the singular solution of one is the 

 locus of infinite curvatures to the other. 



The manner in which the test just considered was first obtained, was, in its final step, as 



follows :-«...d'ou Ton tire/ (,) - ^^ , /-(fl, _gg^ . Or Equation 



primitive singuliere rend F'(<p) = ; done elle rendra infinie les deux fonctions f (x) etf'(y)...'" 



I quote Lagrange's words to recall the very decided manner in which he speaks of the effect 



of F(<p) = 0, without apparently casting a thought upon the possible accompaniment of 



(j>\y) =oo , or of F (tv) = and F'(y) = 0. But we are to remember that when Lagrange 



wrote, all general theorems were looked upon as liable to failure in particular cases : the tact 



of the mathematician detected the usual way in which an extreme case arose, and the unusual 



ways were left for consideration. The practice of demonstrating the universal truth of Taylor's 



theorem in one chapter, and its occasional falsehood in another, was not an isolated case, but a 



formal specimen of what was done whenever it was convenient. The reasonable alteration, in 



the case before us, would be to extend the result, and to state that jg, and y^ {which answer to 





 /'(*) ana\f'(y)\ are either infinite or else - or one of its congeneric forms. This is the test 



arrived at by M. Cauchy, and is subject to the following difficulty. If we could be secure of 

 arriving at ^ in the form P+(QR) we might certainly infer as above from R = 0. But 

 ■^y is subject to all the accidents of change of form which arise from disappearance of 



factors, &c. and it may well happen that P + (QR) shews the form -, while a transformed 



equivalent, being the one we actually obtain in a given example, shews the ordinary value to 

 which that form is reducible. The test deduced is, I venture to say, imperfect : for though 



admitting, of course, that - must be examined as well as other cases, I think I have shewn that 



unless - be, pro vice, infinite, there is no singular solution. The method followed is, first, a 



constructive solution of y = -^(x, y), by assuming that a? = a? gives y = y , and forming the 

 polygon in the limit of which the curve of solution is seen ; all done in algebraic terms, 

 without reference to geometry, and for values both real and imaginary. This first part 

 (Moigno, Vol. ii. pp. 385-434t) is a masterly specimen of a power which distinguishes all the 



39—2 



