The collected mathematical papers of James Joseph Sylvester...

39] On Certain Ternary Cubic-Form Equations 357
independent of X and 1a. Consequently we may make 8z = 0. The two connectives then become
x, y, z
X +&r ), y + y, Z;
and the co-ordinates of the tangential will therefore be proportional to
yZ (x + 8X)2 - Z (y + 8y) 2: zx (y ~ 8y)2 - Z (x + 8x) y2: Z2 {xy - (x + 8x) (y + 3y)}
that is, to x (2ysx - Ay): y (2x~y - y~x): z (xy + yC~x)
where 8X: 8y:: y2 + ~cxz: x2 kyz.
Hence the co-ordinates required are as
x {2y' ~ x3 + 3klxyz}: y {- 2x3 - y3 - 3kXyz}: z (X3 - y3),
that is, as X (y3 -Z): y (Z3 - X3):Z (X3 _ y3),
a result which appears to have been first found by Cauchy for the general
form, but previously by Euler, and before him by Fermat, for the case i = 0.
If we write a, b, c, instead of x, y, z, and call the co-ordinates of the
tangential x, y, z, we might find their values by virtue of the condition that
the connective of a, b, c and x, y, z is a, b, c over again. This furnishes the
equations
bCX2 - a 2yz = am
cay2 - b2zX = bm
abz2 - C2Xy = cm,
which may be satisfied by writing
x=a(b3-C3)p; y=b(C3-a3)p; z=c(a3-b3)p;
(a' + b6 + c6 - a3b3 - bGC3 _ a3c3) p2 = m;
but whether or not the above is necessarily the only possible solution is not
quite clear a priori, and a posteriori it looks as if the solutions might be.
manifold.
The co-ordinates of the point whose index is 4, that is, of the second
tangential, will be those of the first tangential to the point
X (y3 _- Z3): y (23 - X3): Z (XI _ y3),
namely,
X (y3 - ZI) {y3 (X3 -- Z3)3 + (3 - y3)3}: y (ZI _ X3) JZ3 (y3 - X3)3 + i3 (y3 -: 2 (Xi3 _ y3) {i3 3 - Y3)3 + Y3(Z3 _ X3)31)
and are of the order 16.
To find the co-ordinates of the point whose index is 5, we may take the
connective of the one last found, and of i, y, z, that is, of 4 and 1. Let us