( 593) 
Mathematics. — “The theorem of JoacuimstHaL for the normal 
curves”, by Prof. P. H. ScHOUTE. 
The circle through the feet of the three normals, which we can 
let fall from any point of the plane of a parabola on this curve, 
passes through the vertex of the curve. In other words: 
“The circles of JOACHIMSTHAL presenting themselves for a para- 
“bola form a net with one basepoint, the vertex of the parabola’. 
And the relation between the point P through which the three 
normals pass, and the centre M of the corresponding circle of 
JOACHIMSTHAL can be expressed as follows: 
“If P describes the point-field of which the plane of the parabola 
“is the bearer, M generates in the same plane a point-field affinely 
“related to this.” 
We shall now investigate how far these theorems can be extended 
n . . . 
to the normal curve NV, of the space S, with » dimensions, and we 
shall commence this investigation with the simple case n= of the 
skew parabola. | 
1. The spheres of JoacuinsrmaL for the skew parabola. If the 
skew parabola is represented by the equations 
n=; Y= i, B=; 
then 
AEL NONE ye en 
is the equation of the normal plane in point ¢ ‘This equation 
in ¢ being of degree five, through any point P five normal planes 
pass; the feet of these normal planes we shall call “conormal points” 
of the curve. These conormal points form on the curve an involution 
of degree five with three dimensions, for, if three points of such 
a quintuple are taken arbitrarily, the point P in space through 
which the five normal planes must pass, and in this way likewise 
the supplementary pair of feet, is unequivocally determined. So 
there must exist two relations between the parametervalues ¢ of five 
conormal points. If in general am represents the sum of the products 
» 
1 by J of & quantities m, we deduce immediately from (1) 
eh =O, OE a AP mie f°) 
5,1 5,2 
On the other hand the six points of intersection of the given 
curve with the sphere 
43* 
