70 
tions as there are constant quantities. ‘The number of points there- 
fore is equal to that of the constants of the equation. See (600) 
Geometry. 
Prop. 
(3) To determine the greatest number of points in which a right 
line can meet an algebraic surface. 
Let 2 and y be eliminated by the equation of the surface and 
those of the right line, by substituting their values derived from the 
latter in the former. The result will be an equation of the form. 
Az" ++ Bs" oe" www ww Mz t+ N=), 
the surface being supposed of the n® order. The number of 
real roots of this equation cannot exceed n, and it therefore follows 
that the number of points where the right line meets the surface 
cannot exceed n. As the roots may some or all be impossible, the 
line may meet the surface in a less number of points, or in none. 
Hence every algebraic surface may be intersected by a right line 
mm as many points as there are units in the exponent of its order, 
but not in more. 
Prop. 
(4) To determine the curve of the highest order in which two al- 
gebratc surfaces, the exponents of whose orders are m and n 
can intersect. 
Assuming the general equations of the m” and n" orders between 
three variables, if the variables were successively eliminated the 
