CURVES Booh I 
on is written as here, 
express in the form 
singular points and 
nultiplicity r i +^i— 1 
3 multiplicity r i and 
nposed on the coeffi- 
Dmogeneous whether 
infinitely near sets, 
cult task of studying 
t the two curves be 
ie origin, where they 
e other intersections 
ies r^sp, r 2 ,s 2 .... Let 
2 ... a set through P 2 , 
we seek shall be in 
3 extremely uncouth 
a 2( pfl)... 
.*-)«•••«••• = 0 
>e large enough, the 
fulfilled. We may 
way that all of the 
led but the last one. 
It of the multiplicity 
onditions at O, or of 
•oints. Any identical 
ude the last tangency 
can manipulate the 
condition is fulfilled 
le others, and so on. 
d relation that may 
re the tangency con- 
i form 
~-r 0 +s 0 -2. 
,P 2 ... and all of the 
Chap. II DETERMINATION OF CURVES BY POINTS 27 
multiplicity conditions are fulfilled at 0 except an arbitrary one 
of the last set. Hence this condition cannot result from the 
others, or an identical relation could not include any one of 
the multiplicity conditions of highest order at 0. We go through 
the same reasoning when ocf-fî — r 0 -\-s 0 —3 and see that any 
identical relation among the conditions could not involve a 
multiplicity condition of next highest order, and so on. Finally 
we see an identical condition could not involve the origin at all. 
But this is any one of the intersections. These requirements are 
linear in the coefficients. Hence, they cannot introduce addi 
tional singularities in unspecified situations, for the conditions 
involved are not linear. Nor could they introduce undesired 
singularities at specified places, for we can avoid this by 
changing the Vs. 
Theorem 14] If the order of a curve be sufficiently high, the 
conditions which require it to have multiplicity at least r i f-s i —1 
at each point where one of the curves has multiplicity r i and the 
other multiplicity s { , neither number being 0, are independent, and 
additional singularities are not necessarily introduced. 
Let/be a curve of very high order which fulfils this condition 
with respect to two given curves f and f of orders n x and n 2 
respectively. Its coefficients have been subjected to 
yfo+^fo+Si— 1 ) 
i 
independent linear homogeneous conditions, so that the amount 
of freedom left is 
{n+l){n+2) ^ _, 
2 2, 2 
i 
Next consider a curve whose equation takes the form 
#'+#' = 0, 
Here cfV is a curve of order n—n 2 which has at each intersection 
of </» and f a multiplicity r i —1 at least, and we may imagine 
n so very large that the conditions imposed on are inde 
pendent, In the same way is a curve of order n—n x with 
multiplicities 1 all independent. The curve whose equation 
we have Just written fulfils all of the requirements imposed on/.