400] 
105 
400. 
ON THE CUBIC CURVES INSCRIBED IN A GIVEN PENCIL OF 
SIX LINES. 
[From the Quarterly Journal of Pure and Applied Mathematics, voi. ix. (1868), 
pp. 210—221.] 
We have to consider a pencil of six lines, that is, six lines meeting in a point, 
and a cubic curve touching each of the six lines. As a cubic curve may be made 
to satisfy nine conditions, the cubic curve will involve three arbitrary parameters ; but 
if we have any particular curve touching the six lines, then transforming the whole 
figure homologously, the centre of the pencil being the pole and any line whatever 
the axis of homology, the pencil of lines remains unaltered, and the new curve touches 
the six lines of the pencil ; the transformation introduces three arbitrary constants, 
and the general solution is thus given as such homologous transformation of a 
particular solution. To show the same thing analytically, take (x = 0, y — 0, 2 = 0) for 
the axes of coordinates, the lines x = 0, y = 0 being any two lines through the centre 
of the pencil, so that the equation of the pencil is {*\x, y) 6 = 0, then if 0 (x, y, z) = 0 
is the equation of a cubic curve touching the six lines, the equation of the general 
curve touching the six lines will be 0 (x, y, ax + ¡3y + <yz) = 0 ] or what is the same 
thing, considering the coordinate 2 as implicitly containing three arbitrary constants, 
viz. an arbitrary multiplier and the two arbitrary parameters of the line 2 = 0, then 
the equation 0 (x, y, z) = 0 may be taken to be that of the cubic touching the six 
lines. 
Now the given binary sextic y) 6 may be expressed in the form P 2 + Q 3 , 
where P is a cubic function, Q a quadric function, of the coordinates (x, y) ; or, what 
is the same thing, but introducing for homogeneity a constant c, we may write 
(*$#, y) 6 = c [(a, h, k, b\x, ,y) 3 ] 2 -f 4 [(j, l, /$>, y)f ; 
C. VI. 
14