( 55 ) 
case, when in S, two three-dimensional spaces SD, 8,2) are given 
arbitrarily. The line g, at infinite distance of the plane of position 
of one of the angles formed by those spaces in their point of inter- 
section © is then a common transversal of the planes at infinite 
distance el, «2 of S,%,.S,@) and the planes &), &(2) normally con- 
jugate to these. As the common transversals y, of the three planes 
el), eV), €( form a “variety” W,? of order three of three dimensions 
and this curved space is intersected in three points by &(), the spaces 
5,0, 5, make in O three angles with each other. This variety V, 
is not only (compare i. a. the first part of my “Mehrdimensionale 
Geometrie”, vol. XX XV of the Sammlung Scuusert’, N°. 102, 103) the 
locus of a twofold infinite series of transversals g,, but at the same 
time the locus of a simple infinite series of planes ¢, , each of which 
determines with ed, ed, el on the lines y, quadruples of points 
with a definite anharmonic ratio, i. e. V,* is the bearer of two 
nm? 
incident systems which we represent according to the nature and 
the multiplicity of the elements by (y,,), and (¢,),. Beside the general 
case, where &!?) has in common with J’,? three points not situated 
on a right line, the particularity may arise that e€@®) has a line in 
common with one of the planes ¢, of (e‚), or coincides with one of 
those planes; to this answer the particularities that two of three generally 
different angles of position or all three of them become equal to each 
other. In the last case of the three equal angles of position, to which 
we restrict ourselves here, there is between the two systems of incidence 
this connection, that the plane passing through an arbitrarily chosen 
point © at finite distance and an arbitrary line g, of (g,,), 1s a 
plane of position for the pair of spaces connecting © with each of 
two arbitrarily chosen planes ¢, of (¢,),. If namely we make on 
each of the lines yg, of (y,), those points to correspond to each 
other which are conjugate to each other in the involution determined 
by the pairs of points of intersection of that line with the planes 
ai), ef) and es, e@, then between the planes «, of (&,), is 
formed a quadratic involution, from which ensues that the plane ¢, 
normally conjugate to the plane ¢, of the variety J’,* lies likewise 
im Vas etc: 
The two double planes ¢;, ¢; of the involution (¢,, €,,) form part of 
the imaginary hypersphere £,° with four dimensions at infinite 
distance common to all hyperspheres £,? with five dimensions of 
S,- So the section J," of J’,* and 4,° consists of these two planes 
and a J,‘ which must now necessarily be the locus of the trans- 
versals g, of (g,), lying entirely on /,*. Whilst as is known the 
lines g, vesting on any arbitrary right line of ¢, form a surface 
