( 566 ) 
must likewise touch the spherical space Sp,(Q) or Sp,(Q’) passing 
through this point in the same point, ete. 
We have now arrived at the first part of our investigation proper 
concerning the system of the spherical spaces Sp, touching 7 spherical 
spaces Sp, (Mz, rx), (= 1, 2,...m) given arbitrarily in S, and we 
reduce the general case — following the way indicated by Rye for 
our space — to a simpler one, in which the centres J/; of the n 
spherical spaces which must be touched lie in a space Sn 
The centres J/; of the n given spherical spaces Sp, (Mx, rj) deter- 
mine a space S,—;, intersecting these spherical spaces according to 
“central spherical spaces” Sp,—1 (Mr, 7), thus intersecting them at 
right angles. Let O be the radical centre of these 2 spherical spaces 
Spr: and 7? the power of this point with respect to the spherical 
spaces Sp, provisionally supposed to be positive. Then the spherical 
space Spr1(O0,7) lying in S,—1 intersects at right angles the n 
spherical spaces Spy (My, 77), thus also the 7 spherical spaces 
Sp (Mi, rj). So an inversion with an arbitrary point OQ’ of the 
surface of the spherical space Sp,—1(0,7) as centre makes the 7 
given spherical spaces Spn (Mr, 17) and the spherical space Sp (O, 7) 
cutting them at right angles to pass into n new spherical spaces Sp’, 
and a space SS,» cutting them at right angles. This special case 
where the centres WM, of the nm spherical spaces which must be 
touched lie in a space S,—2 shall be treated first. 
5. If Sp", is a spherical space touching the new spherical 
spaces Sp’, then this spherical space Sp", rotating round the space 
Ss through the centres WV will touch in any position the 2 
spherical spaces Sp’, and will thus form a singular infinite series of 
tangent spherical spaces. In an arbitrary space .S,;—; through the 
axial space S,-2 we find according to the results obtained above 
2-1 pairs of spherical spaces Sp",—1, touching the central spherical 
spaces Sp',-1— lying in S,—1— of the # spherical spaces Sp’, and as 
a matter of course each of these pairs consists of two spherical spaces 
Sp"n—1 lying symmetrically with respect to S,-2. As each of those 
pairs by rotation leads up to a singular infinite series there are 2”—! 
of such series. The spherical spaces of each of those series are 
enveloped — compare my preceding communication on page 492 — 
by an n-dimensional torus T,,,; their centres lie on a circle. And if 
we confine ourselves to one of the 2%! series, we can extend the 
system of the touched spherical spaces Sp', to a n—2-fold infinite 
series by representing to ourselves all the spherical spaces described 
