MH. J. J. VVALKKR ON THK DIAMKTKRS OP A PLANE CUBIC. 177 



giving for the point of contact of the polar line of (x'rfo) 



x" : y" : z" = betf* : neat* : - ob-sij : ..... (61) 

 and conversely 



x : y' = - ez" :ox" = by": - ez". = '; WM? 3 ( 62 ) 



IV. SPECIAL FORMS OF CUBIC AND POLOID. 



48. T now treat some of the questions, hitherto considered without reference to any 

 particular form of , in more detail, by means of special forms in which the line L is 

 used as one of the lines of reference say 



L = z = 0; ........... (63) 



and for the two other lines of reference I take 1st, the tangents to the poloid * from 

 the pole of L, as 



x = 0, y = 0. 



The conic s is now (10) 19 s being dropped reduced to 



(a 3 6 3 e'-)z 2 + (ab a^xy = 0, 

 with the conditions 



a^-a^O, (i.) 



&a 2 &J 2 = 0, . ........ (ii.) 



ba 3 + a 2 6 3 26 1 e = 0, . . ; ; j ...... (iii.) 



a6 3 + ctg&j 2aje = 0- ... - . ; .-., (v.) 



From the first and second of these, 



a 2 (ab Uj&j) = 0, 

 whence 



a 3 =0, 



since ab a s b l = is excluded by the form of s. 

 Hence, from the second 



*>i=0, 

 and from the third and fourth 



fl s =0, 6 3 =0, 



MDCCCLXXXVIII. A. 2 A 



