MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 198 



The former value of w shows that the polar of y 0, 2 = 0, one of the three points 

 common to L (2 = 0) and n, viz., 



*=0, 

 is the tangent at a cusp at the point 



x=0, by + 3b s z = 0, . . ''; ' ..... (110) 

 the tangent meeting w for the fourth time at the point 



x = 0, a.,y + a 3 2 = 0, or by 6 3 z = ..... 



73. Now, since any real intersection of L and M might be taken as the point y = 0, 

 2 = 0, it follows that when these three intersections are real there will be three real 

 cusps to w, the cuspidal tangents being the polars, with respect to the poloid s, of the 

 points common to L and u. 



This is shown, independently, for the two intersections of L and u other than 

 y = 0, 2 = 0, by the second form of w, which gives simultaneously (109) 



\ ^F 3 / \ / 3 / 



of which the former has been shown to be the (square of the) s-polars of the two 

 intersections in question (106). 



74. It is plain from the forms of w that the triad of coordinates (110), (112) satisfy 

 the first differential coefficients of w ; in fact these are 



- (2by - 3b s z) 2 } 



- 6632) -y(by + 36 3 z) 2 }, 

 (by - Gb^) - y (2by - 



fk 



~ = 4a 3 (Sajftx 8 + (by + 3& 8 2) 3 } (2by - 



- 40, {36(a^ 8 + 3&y)(2fy - 3b 3 z) - (iby 

 Also 



, i = 4a, 8 { - 36 (a^* + by'-) + (2by - 86,2)*}, 



MDCCCLXXXVIII. A. 



