MB. J. J. WALKER OX THE DIAMETERS OF A PLANE CUBIC. 183 



eliminating z between this equation and v, the result is 



= 0, 



viz., the points in which v is met by L are points of inflexion on v, as was manifest 

 from the form of its equation. When the node of v is real, only one of these points of 

 inflexion is real ; but if v is acnodal, all three will be points of real inflexion. Thus 

 much is known of v generally. In Plates 7, 8, it is figured for two cases when L is 

 the line at infinity, as will be further described. 



58. The coordinates of the CoTES-point on the polar line of (x'y'O) 



by'*y + Zex'y'z = 

 may be found as those of its point of intersection with 



y'z + z'y = 0, 

 the s-polar of (x'y'O) ; viz., they are 



x : y : z = 2ea/y : 2ex'y'* : *' 8 by" A ..... (79) 

 or, what is the same thing (36), p. 165, 



i a*' a' &' du 



x :y :z = 



a/ ay' ay 



The double chord (56) 42, through (x'y'O) is determined as the polar with respect 

 to * of that CoTES-point (79) on the polar line of (x'y'O). Its equation is, therefore, 



abx?y'( y'x + x'y} + e(ax^ - by' 3 )z = 0. 



Arranging this as a binary cubic in x'y 1 , 



aezx* + abyx'*y' abxx'y" bezy* =0, ..... (80) 



the discriminant will be the envelope of the double chord as the point (x'y'Q) varies on 

 the transversal L, viz., 



3w = 4ab (a.c- + 3eyz) (by* + 3exz) (abjcy - 9e*z*)~ = 0, . . . (81) 

 or, developed, 



wSa'Mzy + 4d-bex*z + 4a6 2 ey 3 z + I8abe-xyz 2 27V = 0. 



59. If the first derived of the cubic (80), i.e., 



3ex'*z + Zbx'y'x- by'-x = 0, 

 -f 2ux'y'x-ax'-y = 0, 



