MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 1- "' 



ox 9 + 3eyz = 0, or by* + 3ezx = 0, 



and 



abxy 9e?z* = 0, 



there result 



a*bx* + 27e 8 2 s = 0, 



ab-if + 27c 8 2 3 = 0, 



showing that w is a tricuspidal quartic, two of the cusps being imaginary when the 

 line L meets the cubic u in one real point only, the real cusp being at the point 



x:y :z = 

 on the line 



a'x b k y = 0, 



which is the cuspidal tangent, and is harmoiiic-conjuyate to the connector of the s-pole 

 of L and its real intersection with u, relatively to the tangents to a from that pole. 



The curve w is shown in the figure, Plate 7, in the case when L is at infinity and 

 the cubic has a single real asymptote. 



61. Through any point x"y"z" on the polar line of a point x'y'z' in L, two chords of 

 u pass which have the first (x"y"z") as their Corns-point ; viz., the chord which forms 

 one of the pencil through x'y'z' and the other (53), 40 the line, 



(yV ~ a 



" 



x"y"z" being subject to the relation 



and the envelope of this line, or " alien " chord, may be found for any assigned 

 form of . 



In the case when L is 2 = 0, 



s = e*z* + abxy, 



u = ay? + by' A + cz 3 4- Sc^x + 3c.,z-y + Gexyz, 



i 8*u i &" " 



ate"* ~ C ' * 3^%" = Z ' 



MDCCULXXXVUI. A. 2 B 



