ON THE THEORY OF ALGEBRAIC CURVES. 
50 
[10 
being coincident in the two curves, the number of points of intersection is mn — r — s ; 
but the converse of this theorem is not true. 
In a former paper, On the Intersection of Curves, [5], I investigated the number 
of arbitrary constants in the equation of a curve of a given order p subjected to pass 
through the mn points of intersection of two curves of the orders m and n respectively. 
The reasoning there employed is not applicable to the case where the two curves 
intersect in a number of points less than mn. Iu fact, it was assumed that, W = 0 
being the equation of the required curve, W was of the form uU+vV; u, v being 
polynomials of the degrees p - m, p — n respectively. This is, in point of fact, true in 
the case there considered, viz. that in which the two curves intersect in mn points; 
but where the number of points of intersection is less than this, u, v may be assumed 
polynomials of an order higher than p — m, p — n, and yet uU+vV reduce itself to 
the order p. The preceding investigations enable us to resolve the question for every 
possible case. 
Considering then the functions U, V determined as before by the equations (3), (4), 
suppose, in the first place, we have a system of equations 
a = A, /3 = B 0 — H (t equations) (8). 
Assume P = (y — ax — ...) (y — /3# (y — Ox — ...), 
Q =(y-Ax- ...) (y -Bx (y -Hx- ...); 
T = (y-i.x-...) ... (y-KX - ...), 
y =(y-\x-...) ... (y-Kx-...) ; 
whence U = PT, V = QV. 
Suppose T = PT + AT, T = PT + AT, 
PT. U - PT . V= JAY . PT - PT. QV, 
= P'T . P (PT + AT) - PT. Q (E'V + A'P), 
= PT.P¥.(P- Q)+ E'V .P. AT-PT.Q. A'T, 
= P {PT . E'V . (P - Q) + E'V. PAT - PT . Q. A'P}, 
= II suppose. 
In this expression PT, E'V are of the degrees m — t, n — t, AT, AT' of the degree 
- 1, and P, Q, P — Q of the degrees t, t, t- 1 respectively. The terms of II are 
therefore of the degrees m + n — t — 1, m — 1, n — 1 respectively, and the largest of 
these is in general m + n — t — 1. Suppose, however, that m+n — t — 1 is equal to 
rn — 1 (it cannot be inferior to it), then t = n; V becomes equal to unity, or AT 
vanishes. The remaining two terms of II are PT (P — Q), PAT, which are of the 
degrees m- 1, n - 1 respectively. II is still of the degree m — 1, supposing m> n. 
If m = n, the term PAT vanishes. II is still of the degree m — 1. Hence in every 
case the degree of n is m + n — t— 1: assuming always that P — Q does not reduce 
itself to a degree lower than t- 1, (which is always the' case as long as the equations