Linear Geometry of the Cubic and Ouartic. 



BY H. B. NEWSON. 



INVARIANTS OF THE SYSTEM OF A POINT AND A CUBIC. 



Let there be given two binary forms, one of the third degree and 

 the other of the first. Let the former be written C=ax^-(-3bx2y4- 

 3Cxy2-|-dy3:i^o; and the latter, P=ajX-[-bjy=;0. Interpreted accord- 

 ing to the conventions of Linear Analytic Geometry, the equation 

 P=:o, represents a point on a fixed line; and C=o represents a group 

 of three points on the same line. Two fixed points, A and B, on the 

 line are the Ground Points of the system of binary co-ordinates. I 

 shall call the group of three points represented by the equation, C^o, 

 a cubic; in like manner a group of two points given by a quadratic 

 equation will be called simply a quadratic ; etc. 



These two groups of points P=:0 and C=o, give us a group of four 

 points on a line. I propose to investigate the Invariants of this 

 system, P and C, by means of the anharmonic ratio of the four 

 points. 



Let x-LR^y, x-j-Roy, x-j-Rgy be the three factors of the cubic; 



where 



1 1 1 



^ b p^ H b wp^ w'-^H t w^pS 



'a a 12a a^-^a a 



ap'^ ap'* 



.^llL, andH=^ac— b2;G=a3d— 3abc+2b3;p=^(— G + -,/G2+4H3); 

 ap^ 



-^ + l/-3 l _ 



2 J 



The anharmonic ratio k of the four points given by x-(-Rjy^=o, 

 x-j-R^y:=o, x-j-Rgy^o and x -] ^-y=o is 



p^ H w^p^ wH b, b p^ H 



a i a ia^aa ^k 



ap^ ap-* I ap-* 



w is one of the imaginary cube roots of unity, viz: 



1 1 • 1 



w^p-* wH wp^ w^H b b^ wp^ w^H 



ap* ap3 1 ap^ 



(85) KAN. UNIV. QUAR., VOL. IF., NO. 2, OCT., 1893 " 



