( 422 ) 
points taken on the curve coincide in a same point P, the limit of 
the first point is the centre C, of the circle of curvature, that of 
the second point the centre S, of the sphere of curvature, i.e. the 
centre of spherical curvature of R in P. If P describes the given 
curve PR, then C, and S, describe twisted curves related to R, of 
which the latter is also the cuspidal edge of the developable enve- 
loped by the normal planes of R; this locus of centres S, of spherical 
curvature may be indicated by the symbol R; . 
From the wellknown theorem according to which the line of inter- 
section ¢ of two planes @, /?, perpendicular to the intersecting lines 
a and b, is a normal to the plane y of these lines a and b, ensues 
that reversely the osculating planes of R are also perpendicular to 
the corresponding tangents of &,. These osculating planes of & 
however, not passing at the same time through the points of con- 
tact of the corresponding tangents of R,, are not normal planes 
of R, and so the relation between the curves R and R, is generally 
not reciprocal. A wellknown striking example derived from trans- 
cendent twisted curves, where this reciprocity really exists, is the 
helix or curve formed by the thread of a screw; moreover for this 
curve the two loci of the points C, and S, coincide. 
Let us go a step farther and suppose that P,,P,,P3, Py, Ps... 
are successive points of a given curve ZR, which is contained in a 
four-dimensional space, but not in a three-dimensional one, which 
curve we therefore call a “wrung curve’; then besides the centres 
of the circle and sphere of curvature the centre H, of the hyper- 
sphere of curvature appears, which is the limit of the hypersphere 
P, P, Ps Py Ps, when the five determining points coincide in point P 
of the given curve. A third locus has then to be dealt with, and 
so we can extend these considerations to a space with any given 
number of dimensions. 
In the following pages we wish to deduce the characteristics of 
the locus Zj, of the centre of hyperspherical curvature of the highest 
rank in relation with the general rational wrung curve R; of 
degree », which 7s contained in a space with s dimensions but not 
in a space with s—l dimensions. 
2. “The row of characteristic numbers from class to degree of 
“the locus Zj, of the centres of hyperspherical curve of the highest 
“rank belonging to the general rational wrung curve in Ris 
“3In—2, 2(8n—3), 3(8n—4),... s(Bn—s—1).” 
