( 426 ) 
point three normals only on the parabola, as the line connecting 
P with the point P,, at infinity of the parabola must be consi- 
dered as an improper normal. Any point P of space is situated in 
seven normal planes of a skew ellipse or skew hyperbola, but only 
in six normal planes of a parabolic hyperbola and in ‘five normal 
planes of a skew parabola, as the plane through the connecting line 
PP, of P with the point of contact P, at infinity with the plane 
V at infinity, perpendicular to the tangent p, of the curve in P,, 
represents one improper normal plane for the last but one, and the 
coincidence of two improper normal planes for the Jast. 
Of course the particularity treated here can appear more than 
once. If »=0O contains the roots ¢,, t,..... tp respectively 
ky, ky,» + + hy times, where each of the p quantities exceeds unity, 
J, p 
the class of Zj, is represented by 3n-+ p— 2— En ne 
1 == 
3b If t=t, is a common Z-fold root of the s equations 2; = 0, then 
this value is at the same time a common 4—/-fold root of the s 
equations (';—=0 and the s forms of x; (5) are divisible by 
(t—tet, whilst +, contains the factor (t—t)?k—! ; then again (3) 
is divisible by (¢—t,)*—-! and the curve Zi is of class 3n —& — 1. 
1 
By the substitution of t—t; = 7 the case treated here presents 
itself in an apparently different form. It leads to the equations (6°, 
where now the s forms y; represent polynomia of degree n — k + 1 
in £ without constant term and xe is a general form of degree n 
in t. Regarded as equations of degree » in #’, the s equations yj = 
contain the common k—J-fold root t = and the common simple 
root t'=o. The « terms zj (@ = 1, 2,... s) become polynomia of 
degree 3n —k—1 in #', whilst 7, ascends only to degree 3n—2k—1 
in £. The corresponding equation (3) is then as above of degree 
3n—k— 1. 
Apparently besides the cases treated up till now where the 
equation (3) lowers its degree, another entirely new case can be 
pointed out, namely that where the s+ / equations /7/=0, v'=0 
have a common é-fold root t=. It is easy to see however 
that this apparent new case forms but a special case of what 
was treated above. If we start from the equations (1), because after 
all we shall directly have to transform the coordinates to parallel 
axes, then we have 
ef S(t tt Ok) im AL na 
