186 MR. .7. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC, 



and 



since 



The equation of the " alien " chord is then simply 



ax"z"y'(zx" - xz") + (ax"z"x' + 2ez"Y) (zy" - yz") = 

 with (76) 



ax'V + fy'V + 2ezyz" = 0, 



in virtue of which the eqtiation of the " alien " chord reduces to 



ax'x"(xz" - zx") - by'y"(yz" - zy") = ; . . . ..'(87) 

 the envelope of which is 



ab (y'x + x'y}- + 8e (ax'-x -f by" 2 y + Zex'y'z) z = 0, . . . (88) 



a conic touching the line L at the point which is harmonic -conjugate to (x'y'O) with 

 respect to the intersections of L with the poloid s ; and touching the polar line of 

 (x'y'O) at the point in which it is met by 



y'x + x'y = 0, 

 viz., the point 



x : y : z = 2ex' 2 y' : 2ex'y' 2 : ax' 3 by' 3 , 



i.e., at the CoTES-point on that polar line (79), 48. 



62. The conic envelope just found may be called the "satellite-conic" to the polar 

 line of x'y'z'. Of course, this conic touches also the double chord of the pencil through 

 x'y'O, since it is at once an " alien " and a proper chord of the polar line of (x'y'Q). 



Arranging the equation (88) to the satellite conic as a quadratic in x'y' it is 



a (by 2 + 8ezx) x* 2 + 2 (abxy + 8e 2 2 2 ) x'y' + b (ax 2 + 8eyz) y = 0, 



giving at once, for the envelope of the system of conies satellite to the complex of 

 polar lines of the points on L, the quartic 



ab (ax 2 + 8eyz) (by 2 + 8ezx) (abxy + 8<rV) 3 = 0, 



which developed is 



8ez (atba? + a&V - 8e*z* + Gabexyz) = ; ..... (89) 



viz., the envelope of the satellite conies is the system of four lines L and the tangents 

 to u at the points L = 0, u = (67), 49. 



The explanation of this is 'that at the point of intersection of the polar line (76) 



