26 
ON THE INTERSECTION OF CURVES. 
[5 
Consider generally two curves, U m = 0, V n = 0, of the orders m and n respectively, 
and a curve of the ? ,th order (r not less than m or n) passing through the mn points 
of intersection. The equation to such a curve will be of the form 
II u r _ m TJ m -f- v r _ n V n = 0, 
u r _ m , v r - n , denoting two polynomes of the orders r — m, r — n, with all their coefficients 
complete. It would at first sight appear that the curve TJ = 0 might be made to 
pass through as many as {1 + 2 ... + (r — m + 1)} + {1 + 2 ... (9— n + 1)} — 1, arbitrary 
points, i.e. 
| (r - w + 1) (r — m + 2) + | (r — n + 1) (r — n + 2) — 1 ; 
or, what is the same thing, 
\r (r + 3) — mn + | (r — m — n + 1) (r — m — n + 2) 
arbitrary points, such being apparently the number of disposable constants. This is in 
fact the case as long as r is not greater than m + n — 1 ; but when r exceeds this, 
there arise, between the polynomes which multiply the disposable coefficients, certain 
linear relations which cause them to group themselves into a smaller number of 
disposable quantities. Thus, if r be not less than m + n, forming different polynomes 
of the form x a y fi V n [a + /3 = or < m], and multiplying by the coefficients of x a y& in U m 
and adding, we obtain a sum U m V n , which might have been obtained by taking the 
different polynomes of the form x y y & U m [y + 8 = or < n\ multiplying by the coefficients 
of x y y s in V n , and adding: or we have a linear relation between the different 
polynomes of the form3 x a y p V n , and x y tfU m . In the case where r is not less than 
m+n + l, there are two more such relations, viz. those obtained in the same way 
from the different polynomes x a y 13 . xV n , x y y s .xU m , and x a y p .yV n , x y y s .yl T m , &c.; and 
in general, whatever be the excess of r above m + n—1, the number of these linear 
relations is 
1 + 2 ... (r — m — n + 1) = (r — m — n + 1) (r — m — n + 2). 
Hence, if r be not less than m + n, the number of points through which a curve of 
the r th order may be made to pass, in addition to the mn points which are the 
intersections of TJ m = 0, F n = 0, is simply %r(r + 3)—mn. In the case of r = m + n— 1, 
or r — m + n— 2, the two formulae coincide. Hence we may enunciate the theorem 
“ A curve of the r th order, passing through the mn points of intersection of 
two curves of the m th and n th orders respectively, may be made to pass through 
+r (r + 3) - mn + \ (m + 11 — r — 1) (m + n — r — 2) arbitrary points, if r be not greater than 
m + n — 3 : if r be greater than this value, it may be made to pass through \r (r + 3) — mn 
points only.” 
Suppose r not greater than m + n-3, and a curve of the r th order made to pass 
through 
\r (r + 3) — mn + ^(m + n — r — 1) (m + n — r — 2) 
arbitrary points, and 
mn - ^ (m + n — r — 1) {m + n — r — 2)