x { m ( 1+ 1) − m′ ( 1+ 1) } − y { 1+ 1− ( 1+ 1) } + 2m′ − 2m = 0,
x3 x4x1 x2x3 x4x1 x2

or, what is the same thing,

( 1+ 1) (y − m′x) − ( 1+ 1) (y − mx) + 2m′ − 2m = 0,
x1 x2x3 x4

which by what precedes is the equation of a line through the point Q. Substituting herein for 1/x1 + 1/x2, 1/x3 + 1/x4 their foregoing values, the equation becomes

−(A + Bm) (y − m′x) + (A + Bm′) (y − mx) + C (m′ − m) = 0;

that is,

(m − m′) (Ax + By + C) = 0;

or finally it is Ax + By + C = 0, showing that the point Q lies in a line the position of which is independent of the particular lines OAA′, OBB′ used in the construction. It is proper to notice that there is no correspondence to each other of the points A, A′ and B, B′; the grouping might as well have been A, A′ and B′, B; and it thence appears that the line Ax + By + C = 0 just obtained is in fact the line joining the point Q with the point R which is the intersection of AB and A′B′.

15. In § 8 it has been seen that two conditions determine the equation of a straight line, because in Ax + By + C = 0 one of the coefficients may be divided out, leaving only two parameters to be determined. Similarly five conditions instead of six determine an equation of the second degree (a, b, c, f, g, h)(x, y, 1)² = 0, and nine instead of ten determine a cubic (*)(x, y, 1)³ = 0. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely determined. There is, however, a remarkable exception. Considering two given cubic curves S = 0, S′ = 0, these intersect in 9 points, and through these 9 points we have the whole series of cubics S − kS′ = 0, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (k = S0 ÷ S′0, if the coordinates are (x0, y0), and S0, S′0 denote the corresponding values of S, S′). The resulting curve SS′0 − S′S0 = 0 may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (x0, y0); and from the equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection.