( 20 ) 
Omitting dashes we establish analytical identity between the geo¬ 
metry of the hypersphere H and that of pentaspheric space 77. 
2. Corollaries. 
I. One linear equation in pentaspheric point-coordinates: 
= 0.(2) 
represents a sphere E, this being the stereographic projection of the 
section with H of the S t that has the same equation. 
The coefficients & are the coordinates of E. 
II. Two linear equations: 
, 2ru*i= 0.(3) 
represent a circle E/H. 
III. Three equations: 
2&*i = 0 , 2m*i = 0 , 2$ixi = 0 . (4) 
represent the cowple of points E/H/Z. 
IV. Four equations: 
= 0 , 2^*= 0 , = 0 , 2 8i*i = 0, . (5) 
have in general no geometrical equivalent. 
V. E is a zero-sphere if the corresponding S , is tangent to H, 
which is the case if 
2&' = 0 i . . ..( 6 ) 
VI. Two spheres E and H intersect orthogonally if the corre¬ 
sponding spaces S t are conjugate with respect to H, i.e. when 
= 0.(7) 
3. Now considering in their mutual connection the relations: 
2 Xi = 0 : identity between pentaspheric point-coordinates, (1) 
2 Xi = 0 : equation of the sphere E, (^) 
2 & = 0 : condition that E zero-sphere, .(6) 
2 §, tj, = 0 : „ „ two spheres cut orthogonally, (7) 
the following we may remark. 
We may occasionally consider a set of independent x { to represent 
a sphere X having the same coordinates as a point X in £ 4 ; now 
this sphere is the stereographic projection of the section of the 
polar space of the point X with the fundamental hypersphere. 
Viewing matters thus, the relation a* = 0 appears as the equation 
of a three-dimensional manyfold of zero-spheres ; the space n appearing 
as a four-dimensional space of spheres. 
At the same time JS & = 0 appeals as the condition that the 
spheres E and X intersect orthogonally. 
