Mil. .1. .1. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 191 



which factors may be identified with the three tangents to u at the points u = 0, 

 2 =r ; viz., the points 



2=0, y=0; 2 = 0,i/=v Sa^/b x. 



These will be real points only when ba 3 are of unlike sign ; i.e., when L does not 

 meet s in real points, and v is acnodal. 



69. The Cotesian (98) discarding the factor 4&a i 8 < 



by* + Sa^y a^z b^z == 



and the primitive cubic u (93) are plainly intersected by L, or z = 0, in the same 

 three points, and the tangents to v at these points will be derivable from those to u 

 by changing therein 



3 3 into a,, 3& s into 6 8 ; 

 viz., their equations are 



3by + 6,2 = 0, 1 



)> ...... (104) 



3 V 36ojO; + 3by 26g2 = 0. J 



70. The equation to s (94) being 



- ba#? + 6y V 2 * = 0, 

 shows by its form that the tangent to u at the point 



(L or) z = 0, y = 0, 

 viz., the line (103) 



ty &3 2 = . 

 or 



<W + <h z = 0, 

 touches s at the point 



x : y : z = : 6 3 : 6. 



71. Again, throwing the equation to s into the form 



3s = - (&V + 36aj^ + 46%2 + 4 bjz*) + 46 2 y 2 - 4&&3y Z + 6 3 2 2 8 , 

 = { - 36 (a,a! + 36y 2 ) + 4 (26y + 6,2) (% - b s z) } + (26y 



the terms within brackets being (103) the tangents to u at the other two points at 

 which L meets it, it appears that these lines touch s on the line 



26y+ 632 = 0; ......... (105) 



