MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 187 



with one of the tangents (66), that tangent is itself the " alien" chord for that point : 



for (76) (66) 



aax" + by' s y" + Zex'y'z" = 0, 



a6.T" + aWy" - Zez" = 0, 



give 



x" : y" : z" = -2&ey' : 2a*cx : 



and the substitution of these values in (87) gives for the " alien " chord, after dividing 

 out the extraneous factors 2ftbex'y'(a*x f fcty), simply 



aWx + cWy ( 2ez = ; 



viz., the tangent to u at the point x : y : z = 6* : a* : 0. 



Thus the system of satellite conies is inscribed in the quadrilateral formed by the 

 line L and the three tangents to u at the points in which L meets it. A satellite para- 

 bola is shown in Plate 8, when L is the line at infinity. 



63. The tangents to * from the pole of the transversal L being real only when L 

 meets the cubic u in a single real point, it is desirable to use as lines of reference 

 another pair in connexion with L, which shall be real in all cases for purposes in which 

 their reality is essential. 



Consider the connector of the pole (C, fig. 1) of L (z = 0) with a real point (A) in 

 which L meets u, taking it, say, as y = ; and let the s-polar of the point z = 0, 

 y = 0, be taken as the third line (BC) of reference, x = 0. The equation of s is then 

 reduced to the three terms, remembering that still / = 0, m = 0, 



(a&,-a 8 )^+(&a 2 - V)2T + W>3- 2 ) 2 =0 ..... (90) 



since the triangle x = 0, y = 0, z Ois self-conjugate with respect to that conic ; 

 and as conditions for the terms yz, zx, xy disappearing from its equation 



&a 3 + A = 2 V> ] 



& 3 + s*>i = 2a^, I ......... (91) 



ab aA = 0. 



But since the equation of the three tangents to u at the points 



u = 0, z = 

 is now of the form 



ku Jfc'z 2 = 0, 



and one of them passes through the point 



y = 0, 2=0, 

 2 B 2 



