if 
b 
n 
n 
sS 
5] 
of the mn points of intersection above. Such a curve passes through \r (r + 3) given 
points, and though the mn — £ (m + n - r - 1) (m + n — r — 2) latter points are not perfectly 
arbitrary, there appears to be no reason why the relation between the positions of 
these points should be such as to prevent the curve from being completely determined 
by these conditions. But if it be so, then the curve must pass through the remaining 
i ( m + n — r — 1) (in + n — r — 2) points of intersection, or we have the theorem 
“If a curve of the r th order (r not less than m or n, not greater than m + n- 3) 
pass through 
mn — | (m + n — r — 1) (m + n — r — 2) 
of the points of intersection of two curves of the m tYl and n th orders respectively, it 
passes through the remaining 
\ (m + n — r — 1) (m + n — r — 2) 
points of intersection.”