106 
ON THE CUBIC CURVES INSCRIBED IN A 
[400 
in fact, comparing the two sides of this equation, we have each of the seven coefficients 
of the sextic equal to a function of the seven quantities a \J (c), h (c), k \J (c), 
b V (c), j, l, /; so that conversely, these seven quantities are determinable (not however 
rationally) in terms of the coefficients of the given sextic. And when the sextic is 
expressed in the foregoing form, then it will presently be shown that we have 
(a, h, k, b\x, yf + 3z(j, l, fjac, yf + C z 3 = 0, 
or, what is the same thing, 
(a, b, c, f, 0, h, 0, j, kfjx, y, zf = 0, 
as the equation of a cubic curve touching the six given lines; and by what precedes, 
it appears that this may be taken to be the equation of the general cubic curve 
which touches the six given lines. On account of the arbitrary constant c, it is 
sufficient to replace z by ax + /3y + z, or, what is the same thing, to consider 2 = 0 as 
the equation of an arbitrary line, but without introducing therein an arbitrary multiplier. 
To sustain the foregoing result, consider the cubic 
(a, b, c, f g, h, i, j, k, Jx, y, zf = 0, 
then in general if A=(,Jx, y, zf, B = (,Jx, y, zf(a, /3, 7 ), G = (,Jx, y, z)(<x, /3, yf, 
D = (,,Ja, ¡3, 7) 3 , the equation of the pencil of tangents drawn from the point (a, /3, 7) 
to the curve is 
A 2 D 2 - 6 A BCD + 4 AC 3 + 4 B 3 B - 3B Z C* = 0, 
but writing for shortness 
(„$>> y, z f — (A\ B', C', D'\ 1, zf, 
so that 
A' = (a, h, k, b][x, yf, 
(j> l > /$>> y)\ 
(g, i\x, y), 
c 
then for the tangents from the point (x = 0, y— 0), writing (a, /3, 7) = (0, 0, 1), we have 
A=(A', B', C\ DJI, zf, 
B = (B', C’, DJI, zf, 
G= {G',DJl,z), 
D= D' 
and thence the equation of the pencil of tangents is 
A'*№ - ^A'BCD' + 4A'C 3 + 4>B' 3 D' - 3B' 3 C' 3 = 0. 
Hence for the curve 
B' = 
C = 
D' = 
(a, b, c, f, 0, h, 0, j, k, Jx, yf = 0,