868] 
ON THE INTERSECTION OF CURVES. 
501 
2°. If however the 8 points B are on a curve of the order <y — 3, then the 
mn — 8 points A are a system such that every curve of the order r passing 
through mn — 8 — 1 of these points passes through the remaining point; and in this 
case the general curve of the order r, which passes through the mn — 8 points A, has 
for its form of equation 
0 = H r -f- L r _ m P m + M r _ n Q n , 
il ; . = 0 is a particular curve through the mn — 8 points A, which does not go 
through any of the points B; and consequently the curve of the order r does not 
pass through any of the points B. 
For the proof of the theorem I premise as follows: 
A curve of the order r depends upon \r (r + 3) constants, or to use a convenient 
expression, its Postulandum is = \r (r + 3): if the curve has to pass through a given 
point, this imposes a single relation upon the constants, or say the Postulation is = 1; 
similarly, if the curve has to pass through k given points, this imposes k relations, 
or say the Postulation is = k. The points may be however a special system, for 
instance, they may be such that every curve of the order r which passes through 
k — 1 of the points, will pass through the remaining point; the Postulation is in this 
case —k — 1; and so in other cases. Assuming the Postulation of the k points to 
be = k, then the Postulandum of a curve of the order r through the k points is 
= ^r(r + 3) — k. I stop to remark that the Postulation has reference to the particular 
curve or other entity in question; thus in the case of a curve passing through 
k points, the Postulation for a curve of a certain order may be = k, and for a 
curve of a different order it may be less than k. 
Considering now, as above, two given curves of the orders m and n intersecting 
in the mn — 8 points A and the 8 points B, then assuming that the mn — 8 points 
are not a special system, viz. that their Postulation in regard to a curve of the 
order r is = mn — 8, the Postulandum of a curve of the order r through the mn — 8 
points is 
= \r (r + 3) — mn + 8, 
which is 
= i ( r — ni + 1) (i— m + 2) + ^ (r — n + 1) (r — n + 2) — 1, 
viz. this is identically true when for 8 we write its value 
= \ (m + n — r — l)(m + n—i' — 2). 
But we have through the mn — 8 points A, a curve 
L r —m Pin A M r - n Qn — 0 
with the proper Postulandum: viz. L r _ m contains ^ (r — m+ l)(r— m + 2) constants, 
M r - n contains £ (r — n + 1) {r — n + 2) constants, and there is a diminution —1 for the 
constant which divides out; hence this is the general equation of the curve of the 
order r through the mn — 8 points A; and the curve passes through the remaining 
8 points B.