( 21 ) 
Considered as equation in sc-variables , this relation determines the 
sphere S as the representative image of the complex of ^-spheres 
cutting it orthogonally, if considered as equation in §-■.variables , it 
determines the sphere X as the representative image of a complex 
of ^-spheres cutting it orthogonally. 
Thus , pentaspheric analysis disengaging itself from the impeding 
identy (1), the space 77 rises to the rank of a linear four-dimensional 
space in which , in virtue of the established, now completed, 
correspondence with S 4 , x-spheres and spheres appear as dual 
elements. 
4. Table of Correspondence. 
In pentasphenc space 77. In S 4 . 
x-Sphere-element Point 
Pencil o/ x-spheres , to be represented by the limiting points Line 
„ orthogonal circle Plane 
„ „ sphere S t 
S> 
„ common circle Plane 
„ „ couple of points Line 
„ orthogonal x-sphere Point 
In virtue of this choice of representative images, correspondence 
between S t and ^-sphere, plane and circle, straight line and couple 
of points, such as has been established at first, remains unaltered, 
it has only been rendered complete, now that the ^-sphere appears 
as the image of the arbitrary point in S 4 . 
The application of these principles reaching all over projective 
geometry of S 4 , pentaspheric geometry in its generalised form may 
be characterized as the projective geometry of our space 77, which 
does not ditfer essentially from that in S 4 . 
With this is stated as a matter of fact, that with special exploration 
of the space 77 it is all over now. 
We have still to show how construction in Z7 is connected with 
that in S 4 . 
Some few simple examples we* are going to deal with, may show 
that geometrical figures in 77 corresponding to such in S 4 , as admit 
effective construction with the aid of ruler only , may be determined 
by a set of constructions requiring only the ordinary use of ruler 
and compasses. 
5. To this end we call the attention to some properties concerning 
JSlet 
Complex „ „ „ „ 
§- Sphere-element 
Pencil of ^-spheres ,, „ 
xet ;, 
Complex „ „ „ v 
