502 
ON THE INTERSECTION OF CURVES. 
[808 
In the case where the 8 points B are on a curve of the order 7 — 3, (observe 
that this is a single condition imposed on the 8, = 2 (7 — 1) (7 — 2) points, for a curve 
of the order (7 — 3) can be drawn through 7 (7 — 3) points), it is to be shown that 
the Postulation of the mn — 8 points A is = mn — 8 — 1; for, this being so, the 
Postulandum of the curve of the order r through the (mn — 8) points A will be 
= ^r (r + 3) — mn + 8 + 1, 
and the equation of the curve will no longer be of the foregoing form, but it will 
be of the form 
ilj. + L r — m P m + M r _ n Q n = 0, 
fi r =0 being a particular curve through the mn—8 points A, which does not pass 
through any of the points B. The proof depends on the theory of Residuation: which 
for the present purpose may be presented under the following form. 
Let A, B, ... denote systems of points upon a given Basis-curve, for instance 
the foregoing curve P m = 0, of the order m. And let A = ci denote that the points 
A are the complete intersection of the basis-curve by some other curve; (this implies 
that the number of the points is = km, a multiple of to, and the intersecting curve is 
then of course a curve of the order k). It is clear that, if A — ci, and B = ci, then 
also A + B = ci. But conversely we have the theorem that, if A + B — ci and A = ci, 
then also B = ci. And we at once deduce the further theorem : if A + B = ci, B + G = ci, 
C+P = ci, then also A + D = ci. For the first and third relations give A +B AC+D — ci, 
and the second relation then gives A + D = ci. 
mn—8 n S 
A 
B 
D 
C 
m(r-n) + S m- 3 m(y-3)-d 
Starting now (see the diagram) with the 8 points B on a curve of the order 
7 — 3, suppose that we have through these points the basis-curve P m = 0 of the order 
to, and another given curve Q n = 0, of the order n; and let these besides meet in 
the mn — 8 points A. Let the curve of the order 7-3 besides meet the basis-curve 
in the to (7 — 3) — 8 points G; and through these let there be drawn a curve of the 
order to — 3, which besides meets the basis-curve in the m(r—n) + 8 points D. We 
have here A + B = ci, B + G = ci, G + D —ci; consequently A+D = ci, that is, the 
mn — 8 points A and the m(r-n)-\-8 points D lie on a curve of the order r. The 
curve of the order to- 3 passes through the to( 7 -3)-8 points G; its Postulandum is 
thus 
= (to 3) m (7 — 3) + 8,