85]
NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER.
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the lines u = 0, v — 0, w— 0 (being lines which, as we have seen, pass through the
double points) touch the curve in three points lying in a line, viz. the given line
olu + /3v + 7w = 0. Hence the curve in question is a curve with three double points,
such that the points of contact of the tangents from the double points lie three and
three in two straight lines. Considering the double points as given, the functions
u, v, w contain two arbitrary ratios, and the ratios of the quantities a, /3, 7 being
arbitrary, the equation of the curve contains four arbitrary constants, or it represents
the general curve of the class to which it has been stated to belong.
As to the investigation of the above-mentioned theorem with respect to curves of
the fourth order with three double points, the general form of the equation of such
a curve is
a b c 2/ 2o 2 h
3 + 7; + -;+;— + — + —
x 2 y 2 z 2 yz zx xy
where the double points are the angles of the triangle (x = 0, y = 0, z = 0). It may be
remarked in passing, that the six tangents at the double point touch the conic
ax 2 + by 2 + cz 2 — 2fyz — 2gxz — 2hxy — 0.
To determine the tangents through (y = 0, z= 0), we have only to write the
equation to the curve under the form
the points of contact are given by the system
C B VF
—, +
V r yz
the latter equation (which evidently belongs to a pair of lines) determining the tangents.
The former equation is that of a conic passing through the angles of the triangle
x = 0, y — 0, z = 0 : since the tangents pass through the point (y = 0, z — 0) they
evidently each intersect the conic in one other point only. The equation of the
tangents shows that these lines are the tangents through the point y= 0, z — 0 to the
conic whose equation is
aA 2 x 2 + bB 2 y 2 + cC 2 z 2 + 2/BCyz + 2gCAzx + 2 liABxy = 0.
To complete the construction of the points of contact it may be remarked, that
the equations which determine the coordinates of these points may be presented under
the form
Ax = {A
C.
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