351] 
ON CUBIC CONES AND CURVES. 
407 
that is, S (64$ 3 — T' 1 ) = 0 ; where the equation 64$ 3 — T 2 = 0 would denote the existence 
of a nodal line, and consequent coincidence therewith of 6 of the 9 inflexions ; the 
equation S = 0 being left as the proper condition for the intersection by threes of the 
tangents at the inflexions in three lines. 
17. The investigation shows that the four systems correspond respectively to the 
four values of l which give S = 0 ; and that (reality being disregarded) there is no 
distinction between the four systems, or the corresponding values of l ; if however we 
assume that x, y, z, l are all of them real, then the cone has only the three real 
inflexions 1, 2, 3, lying in the real plane 123 ; and there is ah essential distinction between 
the real roots 1 = 1, 1 = 0 of the equation S = — l + l* = 0, viz. for 1 = 1, the tangents 
at the three real inflexions meet in a line ; whereas for 1 = 0 there is not any relation 
between the tangents at the real inflexions, and there is consequently no visible 
peculiarity in the form of the cone. 
18. I return to the analytical theory of the general case, as depending on the 
representation of the equation of the cone in the form 
(X+ Y + Zy + 6kXYZ= 0, 
where the coordinates are real, viz. X = 0, Y = 0, Z = 0 represent the tangent planes 
at the three (real) inflexions, or, as they have before been called, the tangents ; and 
X+Y+Z= 0 represents the plane through the three inflexions, or, as it has before 
been called, the equator. And we may assume the signs to be such that in one of 
the 2 trilateral regions the coordinates X, Y, Z shall be each of them positive : this 
being so the 14 regions will correspond to the following combinations of signs 
X 
Y 
X+ Y+Z 
+ 
+ 
+ 
+ 
i 
y 
J 
+ 
+ 
- 
- 
+ 
- 
+ 
- 
+ 
- 
- 
+ 
1 
! 
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+ 
+ 
- 
- 
+ 
- 
4" 
-- 
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+ 
+ 
/ 
- 
+ 
+ 
+ 
+ 
- 
+ 
+ 
+ 
+ 
- 
+ 
+ 
— 
— 
- 
+ 
- 
- 
- 
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+ 
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y 
. the 2 trilateral regions, 
the 6 trilateral regions, 
the 6 quadrilateral regions.