618 Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



convenient to remove the origin to the point under examination. Suppose the resulting 

 equation to be 



x 12 + x u + x u y - aPy 2 + 2<r 7 y 3 - x*y* + y 6 - 3xy 9 + x u y la = 0, 



the character of which is to be determined both for infinitely small and infinitely great values 

 of x. Beginning with the former case, we write down the lower degrees as to ,v and y, and 

 form the first succession of fractions, as follows : — 



Three equal maxima, 12 : 6 the last. Six roots of the form y = x 2 (u + U), where u is to be 

 determined from 1 — u l — u* + u e = 0, of which the roots are +1, +1, - 1, — 1, ± v /_ *• P ro ~ 

 ceeding from and 6, we have -1:3 and -14:7; so that there are three solutions of the 

 form y = #~4 (u + U), where u is determined by 1 — 3u 3 = 0. 



Lastly, beginning with 1 and 9, we have - 13 : 4, giving four solutions of the form 

 y = x~~i (u + U), where — 3 + u*=0. 



To examine the origin as a sextuple point, write x 2 u for y, and reject the factor # 12 : we 

 have then 



1 + x* + xu - u 2 + 2xu 3 - u* +u e - 3# 7 m 9 + a? 28 M 13 . 



For u write u + U, preserve only the coefficients of £7°, IP, IP, and only the lower 



dimensions of these, which gives 



U* 

 (3x + ...) + (7* + ...) U+ (16 + ...) — + ... = 0, 



2 



when u = + 1. The fractions are : 1 and 1 : 2, and there are two solutions of the form 

 U = x$ (w, + £7i), where 3 + 8U* = 0. This pair of branches gives only an isolated point, 

 the vertex of a common parabola. But t*— — 1 writes (— 3x +...) for (3x + ...), deter- 

 mines m, from — 3 + 8w, 2 , and gives two real branches of the form U = xl (u t + £7,), or 

 y = x 2 \l + #i (u t + Ui)\. These two branches form a cusp with similar flexures, and the 

 axis of x for a tangent. Lastly, y = ± ■s/ — 1 • ** gives an isolated point. The sextuple point 

 consists of two isolated points and a cusp. The axis of y is an asymptote to three branches, 

 two on the positive side, and one on the negative. 



To find the ultimate character of the branches, we proceed with higher dimensions, and 

 examine 



14 1 4 7 8 11 14 



13 9 6 4 3 2 1 



which shews at once 13 solutions of the form y = x° (u + U) where 1 + u i3 = 0. There is then 

 one real case, approaching to the asymptote y = — 1, consisting of two infinite branches 

 approaching the asymptote on opposite sides. This will arise from the solution which begins 

 as y = x~$ (u + U) ; the other infinite branches will terminate at finite values of x, and the 

 branches which begin at a cusp will give a bounded figure. 



