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



viz., the s-pole of L or z = ; the nodal tangents being 



or (92) - ......... (99) 



- by* = 0. 



The node, therefore, will be real or unreal, as 2 , 6 are of the same, or unlike signs. 



67. Since, now (94) 



a s & 3 2 2 , 



it appears that the nodal tangents (99) of v are the tangents to s at the points 

 * = 0, L, or 2 = 0, and will therefore only be real when L cuts its poloid in real 

 points, as shown before ( 50). 



The transversal L meets the Cotesian v, as well as the cubic u, at the points 



2 = 0, y = 0, z = 0, byt+Sa^^O, . . . (100) 



which last two will be real only when v is acnodal, or when L does not meet s in real 

 points. 



68. The tangents to 



u = &y3 + ex 3 + 3ckffy + 3a 3 o: 2 2 

 at the points u = 0, L, or z = 0, are (48) 



CT = v + 8PLs 

 if, as just above, 



viz., in this case, then, 

 tsr/4 = ?>a 2 s 



- 4a 3 2 6 3 2 3 /a 2 2 ), . . (101) 



since, for the above form of u, with = 0, 77 = 0, = n, 



P = 2&a 2 a 3 w 3 . .. . . . , .. : . . (102) 



Now, the terms within brackets in CT are equal to 



(by - & 3 2) (&V + 3&a 2 a; 2 -f 46632/2 + 4& 3 2 2 2 )/& 2 



or J. , (103) 



(by - 6 3 2) (6y + v ' - S&tt^ + 2& 3 2) (6y - v x 



J 



