156 



system <p(x,y) = const., the cusp of <£(#, j/) = must be an evanes- 

 cent loop, and the isolated point an evanescent oval, or bounded 

 portion. Some discussion of the meaning of y'=a + b V — 1 at an 

 isolated point is given. 



There have been two methods of treating the singular points. 

 The first has recourse to the theory of equations, using differentia- 

 tion, if at all, only to supply coefficients. The second attempts 

 canonical forms derived from differential coefficients, and examines, 

 in succession, the meaning and bearing of the successive orders of 

 differential coefficients. Mr. De Morgan affirms that this second 

 method cannot be what it pretends to be ; and, by treating it gene- 

 rally, shows that its questions are ultimately dependent upon the 

 theory of equations. An equation of the form t.Ay' x =0, when it 

 has no equal roots, decides the character of a singular point defini- 

 tively ; and reduces it to a number of intersecting branches without 

 contact, a number of coinciding isolated points without real tangents, 

 or some of one and some of the other. When the equation has some 

 real roots, each set furnishes either multiple branches with contact, 

 or cusps, or conjugate points with real tangents. All this is easily 

 illustrated by examining the curve in which q>(x, y) is an infinitely 

 small constant, near to the singular point of <p(x, y) = 0. 



A theorem given by Lagrange, and strongly indicated in the 

 writings of Newton, Taylor, Stirling, Cramer, and John Stewart, 

 but apparently nearly forgotten, solves the question of finding the 

 higher or lower degrees of all the roots of 'LA.y a =Q, where A is a 

 function of* of the degree a; that is, where A=* n (a+A'), and A f 

 vanishes when x= oo or when x=0. By this theorem (which is 

 also given in the first* Number of the Quarterly Journal of Mathe- 

 matics), y being x r (u-\-\J), all the values of r, and their corresponding 

 values of u, are very easily found ; and repetition of the process upon 

 a transformed equation gives U=x r '(u l + lJ l ), and so on. It obviously 

 follows, that when the origin is removed to any singular point of a 

 curve, the discussion of the branches which pass through that point, 

 and of their contacts with the tangent and each other, is made very 

 easy. In proof of this, the author takes the following instance, — 



x™ + x™ + x u y—x s y* + 2xiy 3 —x 4 y*+y 6 — 3xy 3 + x™y 13 = 0, 



and discusses its infinite branches, and the sextuple point at the 

 origin (which turns out to be a couple of isolated points, and a cusp 

 of similar flexures), with very much less space and trouble than ordi- 

 nary methods would demand from a much less complicated instance. 

 It is also shown that the lower form of Lagrange's theorem solves the 

 following question : — Given an equation with a certain number of 

 equal roots, what effect will be produced upon these roots by given 

 infinitesimal alterations in the coefficients, how many will remain 

 real, and how many will become imaginary ? 



Newton has given the foundation and the chief step of a geome- 



* There attributed to Mr. Minding, by a mistake caused by M. Serret, 

 who incorporates it with a theorem of Mr. Minding, without any notice of 

 its author. 



