Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

328 INFLEXION POINTS OF A CUBIC CURVE [CH. XVIII

where A, is the cofactor of at in A, B, that of bi, and C{ that of
ci. Hence
io x C x iXoB1t
Thus any equation f(x, y) =0 can be expressed as a homogeneous equation 0(x1, x2, X3)=0 of the same total degree, and
conversely. In particular, any straight line is represented by
an equation of the first degree in xi, X2, X3, and conversely.
For example, Xi =0 represents a side of the triangle of reference.
Let yi, y2, ya be the homogeneous coordinates of the same
point (x, y) referred to a new triangle of reference having the
sides L'. As before,
(1') py = a'x +b'{y+c' (i=1, 2, 3),
where the right member equated to zero represents L'\. Solving equations (1') as we did (1), we obtain x and y as linear fractional functions of yi, y2, Y3. Inserting these values into (1),
we get formulas like
(3) Xi = c1yI +c2y2+ca3y3, IcUl C 0 (i=1, 2, 3).
Thus a change of triangle of reference gives rise to a linear
transformation of the homogeneous coordinates.
Let f(x, X2, X3) be a homogeneous rational integral function
of the nth degree. Under the transformation (3), let it become
~(yl, y2, y3). Then ~ =0 represents the same curve as f=0,
but referred to the new triangle of reference. Let
t=kxxla2bx3c (a+b+c=n)
be any term off. Then
at at at
Xi- = at, X2 - =bt, X3 a=ct.
axi aX2 aX3
Their sum is nt. Hence we have Euler's theorem:
(4) X1 +X2 +X3 =nf.
axi aX2 xs3