MR. J. J. WALKER ON THK DIAMETERS OP A PLANE CUBIC. 155 



sions as much as possible, a preliminary investigation of a form into which the result 

 of substituting for the variables in a ternary quadratic form the expressions 



ty ty ,0d S^> 3A ,ty 



n ~ T m v I -f n ^-, m ~ r I ft 



oy as vz ox ox < i 



< being any ternary form and /, m, n any three constants, is introduced ( 24-27) ; 

 and the special cases of its application in the sequel are considered. 

 15. The notation employed throughout is as follows : 



(i.) The right line, considered generally, is written 



* + W + fc = : 



(ii.) The particular transversal considered in connexion with a cubic u is always 

 written 



L =E lx + my + nz : 



(Hi.) The cubic u being, in point coordinates, 



= ax* + fry 3 + ez 3 + Sajarty + 3a^z + . . . + Gexyz, 

 is written in line coordinates or its reciprocal is 



u = 



(iv.) The poloid of L with respect to the cubic in point coordinates is represented 

 by s as in 1, or ( 19) 



and in line coordinates or its reciprocal by or, where 

 4<r = 4u 23 M - -* + . . . + 2 uu - 



(v.) Any special point on L regarded as the " pole " of a pencil of chords of the 

 cubic u is marked (x'y'z), and any special point on the polar line of a/v/V is marked 

 x"y"z" ; a " chord " being a line, or transversal, considered particularly in connexion 

 with its intersections with the cubic u. 



II. PRELIMINARY. 



16. The intersections of a line r + r)y -f z = 0, with a curve u = of order p, 

 may be studied through the equation 



UV-'M + P 'f'~ 1 . p"-' . D"- 2 u + +/Y>Du + u = 0, . . (3) 



x2 



