260 
MISS C. A. SCOTT OX PLAXE CUBICS. 
and the “ numerical characteristic ” h 
(= 64SVT2) = - X (4 - X)3 / (6 - 6X + X^)^. 
The cubic is therefore bipartite or unipartite according as 2 X — 9 is positive or 
negative. 
The Hessian is 
— &ixx'y'z' — 0 , 
where 
(6 —\)x—2v — Xx, &c., 
v = x-^y + z = x -^y' + z, 
P = (6 — Xf /3 (4 — Xf, 
therefore 
2p - 9 = - X^ (2X - 9) / 3 (4 - X)l 
The inflexional tangents to the Hessian are 2v — Xx = 0, &c. ; these are concurrent 
if X — 6 ; they coincide with ToTg, &c., i.e., with v — 2x = 0, &c., if X = 4. 
The Cayleyan is 
iv^ — — dj 
where 
2 (3 - X) f = - it; - (2X - 9) &c., 
r+ v + r, 
P=2(3-X)3/3(4~X), 
therefore 
2 p - 9 = - X (2X - 9)2 / 3 (4 - X). 
The cusps are ( 2 X — S)^d- 7 ;-l-^ = 0 , &c. ; i.e., they are at ( 2 X — 8 , 1 , 1 ), &c. ; 
they are therefore collinear if X = 3 ; and they are on the inflexional tangents to the 
Hessian if 
i.e., if 
2 ( 2 X - 6 ) - X ( 2 X - 8 ) = 0, 
X2 - 6 X + G = 0 ; 
thus for the harmonic cubics X = 3 rh a/S. 
When X assumes the values 6 (fig. 7), 3 + s/S (fig. 8 ), 9/2, 4, 3 (fig. 11), 3 — v/3 
(fig. 12), the numerical characteristic has the values 4/3, oo, 1, 0, — 1/3, — co . 
19. The diagrams here given have been made by means of § 12 ; but from the 
analytical expressions just quoted a graph can be constructed, by means of which 
these may be readily drawn, and the variation possibly more easily grasped. 
