( 56 ) 
of order two, the locus of the lines y, resting on the conic of section 
ofse. Sand Bis a. VA 
4. If finally in Ss, two spaces S, are taken arbitrarily, these 
form with each other, as was mentioned above, in their point 
of intersection QO in general » angles differing from each other 
and the line at infinite distance of the plane of position of each of 
those angles in the space at infinite distance Soi of 92, intersects 
: ; ; iy ee ; 1 13 
again four spaces ni the spaces at infinite distance Ss! ) 2 S , 
oe a 
(1 (2) a (2 
of the two given spaces S° ) S and the spaces S° ef S ) normally 
n n i Nm ee 
conjugate to these. If now the special case presents itself that each 
E 1) (2 ld 4(2 
right line cutting three of the four spaces oe or) de Ss = es : 
n— I— — — 
also cuts the fourth, the 7 angles of position are mutually alike and the 
locus of the v-fold infinite system (y,), of the lines q,, at infinite distance 
. . wpm . . „1 ” . . . 
of the planes of position is a variety 7" of order 7 with n dimensions, 
n 
appearing likewise as the locus ofa simple infinite number of spaces S,—1, 
Le. in the form of (S,—1),, namely of the spaces Sj, determining with 
(1) (2) 2, = ) e xy (2) 2 5 : 
5 S Fey and now also w Si oe Ge 5 E85 OEFA 
Pe ee ya MR and n als ith & a jon the lines g, (Joo )n 
mutually projective series of points. In that case a plane connecting 
an arbitrary point at finite distance with a line g, of (yy)a is 
always a plane of position of each two of the spaces 5S, connecting 
O with spaces S,—1 of (Si), 
Bing dE : ° > . : . r2n : 
As is immediately evident, in this special case the section | of 
rn : ~y2 ; 2 5 > = : 
J” with the hypersphere S , with 27—2 dimensions common to all 
be | 
2 Y . 
hyperspheres 5 of Sa, breaks up into the two double spaces 
== 
aC (J oy : 2 ; = ¥ eat Ach EA 
S ) E Se of the involution of the spaces S,—1, S’n—1 Of (S,—1) 
n— nd 
é 3 -An—l) Er ik 
normally conjugate to each other and into a | ‚ „locus of the lines 
. i 
2 
Ja Of (J,)m lying entirely on B, . 
