oe 
( 597 ) 
Although it would not be difficult to trace by means of the 
equations (10) the complex of the connecting lines PM, we shall 
avoid this for brevity’s sake. 
4. Cyclographic representation of the spheres of JOACHIMSTHAL. 
If we wish to extend F1epLeER’s cyclographic representation of circles 
lying in a plane to spheres in space, we must suppose that the 
three-dimensional space containing the spheres forms part of a space 
S, with four dimensions. We represent in Sy a sphere lying in Ss 
and having M as centre and g as radius by the two points M,, M, of 
the normal in M on 8S, at a distance MM, = MM,=¢ from M. 
We shall now first investigate what is the representation of the 
net of the spheres of JOACHIMSTHAL passing through the points fg, t3 
of the skew parabola. If the ordinate in the direction perpendicular 
on S; is indicated by w, the equations 
(vw — ts)? + (y — (2)? + (2 — 13)? = w? 
(12) 
(et) + GD Heze? 
will indicate the two quadratic hypercones forming successively the 
representation of all the spheres through ¢, and all the spheres 
through #3. By subtracting these equations from each other we find 
that the section of the two hypercones lies in a three-dimensional 
space perpendicular to Sz along the plane (ll). So the locus 
of pairs of points M,, Mz corresponding to this net of spheres 
of JOACHIMSTHAL is, as the section of a hypercone of revolution 
with a three-dimensional space parallel to the axis of the hypercone, 
a hyperboloid of revolution with two sheets, and the orthogonal one. 
Passing on to the investigation of the curved space containing 
the pairs of images of all the spheres of JOACHIMSTHAL, we have to 
deal with a simple infinite number of orthogonal hyperboloids of revo- 
lution with two sheets. To find the degree of that curved space we 
have but to observe that the point common to all the normals on S, does 
not belong to the locus and that the number of points of that curved 
space situated on a definite one of those normals is double the 
number of planes (11) passing through the point (M) where that 
normal meets S;. This number being five, the curved space must 
be of order ten. It is not difficult to deduce the equation of the 
locus; we have but to eliminate t, and ts between the two equations 
1 
(12) and t,t; = 3° To this end we give the equations (12) the form 
