868] 
ON THE INTERSECTION OF CURVES. 
503 
which is 
— L ( r — n + 1) (r — n + 2). 
In fact, substituting for 7, 8 their values = m + n — r, and |(m + n — r — 1)(to 4- n — r — 2) 
respectively, this equation is satisfied identically. The system of the to (r — n) 4- 8 points 
D thus depends upon |-(r — n + l){r — n + 2) constants, or say the Postulandum of 
the points D is = ^ (r — n+ 1) (9— n+ 2). It follows that the curve of the order r 
through the (mn — 8) points A and the m(r — n) + 8 points D cannot have an equation 
of the form 
Lf—niPm 4* Mr—nQn = 0 , 
for the intersections of this curve with the basis-curve P m = 0 are given by the 
equation M r - n Q n = 0, which contains only the 
^ (r — n + 1) (r — n + 2) — 1 
constants of M r _ n (one constant of course divides out, giving the diminution — 1). 
The equation must have the more general form 
L r — m l m 4- Q n = 0 , 
and it thus appears that the Postulation of the mn—8 points A, instead of being 
= mn — 8, must be = mn — 8 — 1. This completes the proof. 
I notice that, combining the last-mentioned identity 
^ m (m - 3) — m (7 — 3) 4- 8 = (r — w4-l)(r— n 4-2) 
with the like identity 
\n (n — 3) — n (7 — 3) 4- 8 = |(r — m 4-1) (r — m 4- 2), 
we obtain 
\m (to — 3) 4- \n (n — 3) — (to 4- n) (7 — 3) 4- 28 
= -|(r — n 4-1) (r — n 4- 2) 4- 2 ( r ~ m 4-1) (r — to 4- 2), 
and consequently, referring to a former result, the left-hand side should be 
= \r (r 4- 3) — mn 4- 8 4-1; 
substituting for 7, 8 their values, this is at once verified. 
As appears by what precedes, Bacharach’s special case is that in which the 
= x (7 _ 1) (7 _ 2) points B satisfy the single condition of lying on a curve of the 
order 7 — 3. We may have between the points B more than a single relation; in 
particular, the points B may be such as to include among themselves the complete 
intersection of two curves of the orders a, b respectively (ab 5 8): this will be the 
case, if the given curves are of the form 
P m — \ a ¿(n—« 
Qn = ~^a Pn—a 
fXjj R m - b , 
— U n —i) , 
it being understood, here and in what follows, that the values of a, b are such that 
the suffixes are none of them negative.