( 425 ) 
We shall now consider the influence of each of those suppesitions 
on the class of the locus J,. 
3%, If t=t, is a k-fold root of y=0, this value is at the same 
time a k—1-fold root of y'=0 and each of the forms r of (5), and 
so (3) too, is divisible by (¢—é,)*—. The curve Ly is then of class 
3Sn—k— 1. 
y 
By the substitution of t — tj = ee the case of the k-fold root t‚ of 
y =0 assumes an apparently different form, It transforms the 
equations (2) into 
LE ea) eee rn oan Se HO 
where the s forms y; represent polynomia of degree n in t without 
constant term, whilst « contains t to the degree n—k only; so 
it leads to the case that #=0, considered as an equation of 
degree , possesses a k-fold root =o. Then the s forms 
Ti, (i = 1,2, .. . 8) of (5) become polynomia of degree 3n — 2k — 1 
in ¢', whilst 7) ascends to degree 3n—k—1 in t. Then the cor- 
responding equation (3) is also of degree 3 —k— 1 in t and so 
Ry remains of class Sn — k— 1 as it should do. 
In passing we draw attention to the fact that the degree of « being 
lower than 2 it will be impossible to lower at the same time the 
degree of all the s polynomia 7; by a transformation of coordinates 
to parallel axes, as this would inelude at the same time the possi- 
bility to lower the order of Bs. 
The particular case treated here refers to the position of the 
points of Rs at infinity. If v is divisible by (¢—4)* the point at 
infinity of the curve belonging to t will count & times among the 
n points of intersection of the curve with the space at infinity with 
s—1 dimensions containing all points at infinity of the space with 
s dimensions. 
So we find for s=n=8: 
“The class of the locus As of a skew ellipse or a skew 
hyperbola is seven, whilst this number passes into six with the 
parabolic hyperbola and into five with the skew parabola.” 
What we find here agrees with the wellknown results for 
s=n=2. Although through any point P of the plane of an ellipse 
“or hyperbola four normals of this curve pass, we can fall from this 
OL 
Proceedings Royal Acad, Amsterdam. Vol. LL. 
