383] 
PROBLEMS AND SOLUTIONS. 
579 
Conversely any given cubic curve may be taken to be a cubic of the series; and 
the points 1, 2, 3, 4 will then be determined as follows, viz., 1, 2, 3, 4 are the points 
of contact of the tangents to ’ the cubic from an arbitrary point 0 on the cubic; and 
then taking as before A, B, C for the intersections of 14, 23, of 24, 31 and of 
34, 12, respectively, the points A, B, 0 will lie on the cubic, and the tangents at 
A, B, C, 0 will meet the cubic in a point O'. I call to mind that a cubic curve 
without singularities is either complex or simplex; in the simplex kind there can be 
drawn from any point of the curve two, and only two, real tangents to the curve 
in the complex kind, there can be drawn four real tangents or else no real tangent, 
viz. from any point on a certain branch of the curve there can be drawn four real 
tangents, from a point on the remaining portion of the curve no real tangent. 
Hence, in the foregoing construction, in order that the points 1, 2, 3, 4 may be real, 
the given cubic must be of the complex kind, and the point 0 must be taken on 
the branch which has through each of its points four real tangents. 
The foregoing results may be established geometrically or analytically; but for 
brevity I merely indicate the analytical demonstration. Suppose first, that the points 
1, 2, 3, 4 are given as the intersections of the conics U=0, V=0; let a, /3, y be 
the coordinates of the point 0, and write D = a8 x + ¡38 y + yS Z) so that JDU= 0 and 
D V = 0 are the equations of the polars of 0 in regard to the conics U = 0, V = 0 
respectively. The equation of any conic through the four points is U+kV=0; and 
the equation of the polar of 0 in regard thereto is D U + kD V = 0; eliminating k 
from these equations, we have UD V— VDU = 0, which is the equation of the given 
locus. We see at once that it is a cubic curve passing through the points 
(U = 0, 1^=0), that is, the points 1, 2, 3, 4; and through the point DU = 0, DV= 0, 
that is, the point O'; it also follows without difficulty that the curve passes through 
the point 0. But for the remaining results it is better to particularize the conics 
U=0, V=0. Let the equations of 12, 23, 34, 41 be x = 0, y — 0, z= 0, w = 0 
respectively, (where x-\-y + z-\-w = 0); and in the same system, let a, /3, y, 8 be the 
coordinates of 0 (a + /3 + 7 + 8 = 0), then xz = 0, yw = 0 are each of them a conic 
(pair of lines) passing through the four points; and we may therefore write U = yw, 
V = xz; the equation UD V — VD U=0 thus becomes yw (az + yx) — xz (/3w + 8y) = 0, or, 
as this equation may also be written, 
X y z w 
73—2