349] OX A CASE OF THE INVOLUTION OF CUBIC CURVES, 
341 
so that the terms in (Y, Zf are 
= - * ft w y@ s (r* -1 ™) - ypr, (I № + *)' 
and the equation, omitting the factor — ^ li 3 l 2 s , is 
U0M, 
Y-Z I — 
k 
'1*0a®a Ido 
FF 2 + ^ 
55. But the term in [ ] is 
(F_ ^ Gss + mß) + pä (* ~ ¿) + 5*^0, (~ 1 _ w) YZ2j 
= 0. 
which is 
TimjwsrwJkm 7 -m z] ’ 
= (Y~Z)(t£*+ Z ‘ 
and the equation of the nodal cubic is finally 
+ft < 7 ■-*» (wm + im) ~ £ (<£ - m) - °- 
The lines F = 0, Z = 0, = 0 each pass through the node and meet the cubic 
4 3 <s> 2 4 3 ®3 6 
in a third point; the three points of intersection lie in the line l x X + 0 X (Y — Z) = 0. 
Article Nos. 56 to 66. The Cubic Locus, Hctrmoconics and Harmonic Conic. 
56. Suppose that the line \x + fiy+vz= 0 passes through a given point (a, b, c), 
then we have 
\a -4- ¡jib -l - vc — 0 \ 
and observing that 0 + 6 + fi, 0 + v, 0 are proportional to 
we find 
111 
X * y ’ z ’ 
2 
X + y + Z 
respectively, 
a + - + c _ 2 {a + b + c) _ () 
x y z x-\- y + z 
the equation of a cubic curve, the locus of the critic centres corresponding to the 
several lines \x + fiy + vz = 0 which pass through the point (a, b, c). The cubic curve 
passes, it is clear, through the six points which are the angles of the quadrilateral 
x = 0, y = 0, z = 0, x + y + z — 0.