CURVES Booh I 
could be written in 
write 
dip. 
3e n—(%+w 2 ). The 
a curve compounded 
nt freedom, less the 
ounded curve will be 
+■!)(»—»i+2) 
n i+™ 2 ) + 2] 
1/ 2 
-WjWg—1. 
sections of 0 and ifj. 
-1) 
edom of/. We have 
ntly high, it can be 
way. 
ss high ? It is con- 
greater than n x -{-n 2 
ions of <p and ifj but 
urve of the highest 
s not obligatory. It 
ar expression, i.e. if 
bion of $ and ip 
+ty). 
Chap. II DETERMINATION OF CURVES BY POINTS 29 
Here d is an arbitrary curve of order or a con 
stant when that difference is 0. The curve 
ifj'—dip — 0 
contains the n 2 intersections of the line with ip. The total free 
dom of this curve is ———(%+^ 2 )+3] an( ^ - g 
greater than n— (%+w 2 )+l when n ^ n x -\-n z . We make use of 
this freedom to make our curve tp'—6ip = 0 go through other 
points of the line, or meet it altogether in n—Wj+2 points, 
i.e. include it as a part. We have, then, 
{ax-\-by+c)f==(P{ax+by+c)ip"+ip{(P'—6$). 
Clearly (p'—d<p must be divisible by ax-\-by-\-c. Dividing this 
factor out, j _ 
Let us suppose, lastly, that the order of / is 
n — n^n^—l l^.l^n 2 ^ : n 1 
Xf= #"+#"> 
where y is a polynomial of order Z. 
We may write this equally well 
Xf = 9H0"+/*/')+*/#"—P<f>)- 
Every curve (p"—pcp passes through all the intersections of x 
and (p, for (p and <p" both do so. If, then, we use p to make this 
curve include one more point of y it must include the latter 
completely. So will ip"-\-pip. As before, we may divide out x 
and § et f#'"- 
We may sum up all these results in a statement which is of 
absolutely vital importance in our whole theory.* 
Nother’s Fundamental Theorem 15] If two curves <p and ip 
have only ordinary points or ordinary singular points and cusps 
in common, then every curve which has at the least the multiplicity 
r i -\ r s i — 1 at every point, distinct or infinitely near, where <p has the 
* Cf. Nother 1 . There are many proofs extant of this famous theorem, we 
have followed Scott 1 . 
vvi* *