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



giving for the cuspidal tangent 



2 



when in -., are introduced the values x = 0, by = 3b s z, 



as in (110) ; when the values a/x* = 3by~, 2by = 3b s z, 



= 



as in (112) above. The same values substituted in the second differential coefficients 

 will be found to make 



_ 

 "~- 



75. Considering the intersections of w and s, the elimination of x 2 between their 

 equations leads to 



(Wy* - 3bb s *yz 2 - b^z*) z = 0, 

 or 



(by - b,z) (2by + 6 3 z) 2 z = 0. ^ ^,..(113) 



Keferring to the equation of w (108), 72, it appears at once that 



a2/ + 3 2 = 0> 

 or (92) 



ty & s z = 0, 

 gives 



x 2 = 0, 



viz., the tangent to u at the point y = 0, z = 0, (103), 68, touches w at the point 



x :y:z= : b s : b, 



in which w meets s ; and the same line has been shown ( 70) to touch s at that point. 

 From this it would follow at once that the other two tangents to u, at the points 

 L or z = 0, u = 0, touch both w and s at the same points ; but this is shown by (113) 

 above, since w and s appear at once from it to have double contact at the points in 

 which s is met by the line 



2by + b.z = 0, 



