24 
ELEMENTARY PROPERTIES OF CURVES 
Booh 1 
The most obvious way to impose a linear homogeneous con 
dition is to require the curve to contain a chosen point. There 
are cases where assigning a certain number of points to a curve 
will not impose independent conditions thereon. The simplest 
case is where n— 3. Two curves of the third order intersect, in 
general, in 9 points, so that although 
3x6 
= 9, there are cer- 
tainly cases where 9 points do not determine a single cubic 
curve. On the other hand, if we take 4 points on a line, and 
5 others on a non-degenerate conic, it is clear that any cubic 
through the 9 points must include the line, since it meets it four 
times. The remainder must be a conic through 5 given points, 
and this also is uniquely determined. There is thus but one 
cubic through these 9 points. 
If we can show that for a general value of n we can find such 
jiifi j 3) 
a set of — - points that there is but one curve of order n 
Z 
through them, then it follows that if an arbitrary set of this 
number of points be taken, the conditions which they impose 
upon a curve of order n are not necessarily dependent on one 
another, or through these points will pass, in general, a single 
curve of that order. We find the points by the following simple 
device.* Given an irreducible curve of order n, and n lines 
l v l 2 ,...,l n so situated that each meets the curve in n distinct 
points, no two lines being concurrent on the curve, or on a third 
line. Let P be a point on the curve, but not on any one of the 
lines, then choose two intersections of the curve with l x , three 
intersections of the curve with l 2 , and so on, so as to include 
finally n intersections with each of the last two lines. The 
number of points chosen is 
1 + 2+3+ ...+%+№ = 
n{n+\) n __n{n+ 3) 
If more than one curve of order n could pass through all of 
these points, there would be at least a one-parameter family 
of such curves. We might find one curve of the family to pass 
through an (w+l)th point of l n and so include the whole line. 
The remainder would be a curve of order n— 1 which meets 
* Berzolari 2 .