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



But in virtue of its touching s also, if (xyz) is any other point on it, 



yz zy, zx xz', xy' yx' 

 must satisfy the reciprocal ( 30) or tangential of s, 



i.e., 



(Cy* + Bz a - 2Fyz) z' 2 + (Az 2 + Cz 2 - 2Gzx) ^ + (Ex 2 + Ay - 2H*7/)z' 2 



+ 2 ( Ayz Fz 8 + Gxy -f Hzz) y'z' + 2 ( - Ezx + Fxy - Gy* + HJ/Z) zV 

 + 2 ( - Cxy + Fzx + Gtyz - Hz 2 )y = 0. r : . f&u* ..... (43) 



The eliminant of these two quadrics (42, 43) with 



lx -\- my -f- nz' = 

 in the known form 



{(be' + b'c - 2f f)i 3 + . . . + 2 (gb/ 4- g'h - af - a'f) mn + . ..}* 

 - 4 {(be - P)P + . . . + 2 (gh - af) mn + . . . } {(b'c' - f 2 )/ 2 + ...} = 0, .(44) 



is, if 



L = lx + my + ns, 



(t> + 4PL*) 2 - 16FLV. 



34. For 



(be - f 2 ) Z 2 + . . . -f 2 (gh - af ) mn + . . . = 36s, 



4{(b'c'-f /2 )Z 2 + . . ,}^4{(BC-F 2 )x 2 + . . . + 2(GH- 



~ 4 (Discrt. of s) L 2 



Hence 



4 {(be -*)?+. ..}{(b'c'- f' 2 )/ 2 +...} = 36P 2 LV. . . (45) 



Observing now that 



4A= -.( V 4F-. 



~" ' 



_ 



3; 3j; Bx By d 



it appears at once from (30, 3 1 ) 24 that substituting s for <f> 



