500 
[868 
868. 
ON THE INTERSECTION OF CURVES. 
[From the Mathematische Annalen, t. xxx. (1887), pp. 85—90.] 
It is only recently that I have studied Bacharach’s paper “ Ueber den Cayley’schen 
Schnittpunktsatz,” Math. Ann., t. xxvi. (1885), pp. 275—299: his theorem in regard 
to the case where the 8 points lie on a curve of the order y — 3 is a very 
interesting and valuable one, but I consider it rather as an addition than as a 
correction to my original theorem; and I cannot by any means agree that the 
method by counting of constants is to be rejected as not trustworthy; on the 
contrary, it seems to me to be the proper foundation of the whole theory; it must 
of course be employed with due consideration of special cases. I reproduce the 
theorem in what appears to me the complete form. 
Writing with Bacharach 
r^m, r ^ n, rzsm + n — 3, y = m + n — r, 8 = ^ (y — 1) (y — 2), 
and assuming n ^ m, (these equations and inequalities are to be attended to throughout 
the present paper), I consider two curves of the orders m, n respectively meeting in 
8 points B, and in (mn — 8) points A ; and I state the theorem as follows: 
1°. The mn — 8 points A are in general such that a curve of the order r, 
which passes through mn — 8 — 1 of these points, does not of necessity pass 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 — L r _ m P m + M r _ n Q r 
where P m = 0, Q n = 0 are the equations of the given curves and L r _ m , M r _ n denote 
functions of the orders r — m, r — n respectively; and it thus appears that the curve 
of the order r through the mn — 8 points A passes also through the 8 points B.