1 CURVES Book I 
Chap. II ORDINARY AND SINGULAR POINTS 23 
a 
u 
= 0. 
), in which case there 
spidal tangent is not 
between a cusp and 
ing out at this time. 
sxional tangent, 
is double, 
ion-singular, 
count as three from 
has three-point con- 
of contact shall be 
ere all points of con- 
ry* multiple tangent, 
the plane 
ce: 
30 a curve, 
singular tangent. 
il tangent. 
le vitally important 
[■sections two given 
lete solution of this 
but we are able at 
The point in question shall be the origin; the two curves can 
be written 
/= {y—K x ){y~ A 2 x)...(i/— \.x)+cf> = 0 
f = {y—K' x )(y—K x )-{y—K' x )+ ( f>' = °* 
We make once for all the assumption that no tangent to one 
curve there is tangent there to the other also. The tangents to 
either curve may differ or coalesce as they choose. We write 
a third curve 
/" = {y—^){y—№)...{y—p s x)+{l-e)f = 0. 
How many intersections has this with / at the origin ? In 
general the pfs are distinct and not infinite, also different from 
the A’s. We have s developments for y in terms of x: 
y = b il x+b i 2 z 2 +.... 
Substituting in / since b' a each will give r intersections, 
hence in general the number of intersections is rs; in particular 
cases it might be more, but never less. If for a general value 
of e, and particular values of there were more, then 
there would be more when e= 1. But in this case there are 
always exactly rs provided 7^ A^. Hence there are always rs 
when t^Aj. 
Fundamental Intersection Theorem 10] If two curves have 
a common point, but no tangent to one is tangent to the other there, 
the number of intersections accounted for by this point is the pro 
duct of its multiplicities for the two curves. 
§ 2. Determination of a curve by points, Nbther’s funda 
mental theorem 
Suppose that we have a general curve of the nt\x order given by 
f n {x,y) = 0, 
/ is supposed to be the general polynomial of degree n. The 
number of coefficients, including the constant, is 
1+2+3+ ... + (H- 1 )= (tt + 1 f+ 2) =^ ) +i. 
Theorem 11] A curve of the nth order is completely and uniguely 
determined by independent linear homogeneous conditions 
imposed upon the coefficients.