( 563 ) 
‘ er 2) (I 
U. and the n —1 pairs of lines (/j,, /3;) through /,, — where 
12 I b + ce 12 
successively p assumes the 7 — 1 values 3,4, ...,”n-+1 —, we 
see immediately that each space S,—-; through »n —1 lines / through 
U. (or 7,,) —with all indices p differing mutually — contains one of 
12 13 / o « 
the two centres of similitude of each of the +1), pairs (U, Ly). 
Thus a space S,—: through 7 —1 lines / through U, will contain 
I 2 - 1 
the point U, or the point /,,, according to the two lines / with 
| Pq Pq S 
p and g as third index being of the same kind or not; just the 
reverse is found for a space S,—; through »—1 lines / through 
I,,. As the choice of the lines / corresponds in both cases to n— 1 
bifureations 2”—! of those spaces S,—; pass through each of the two 
! I gs 
points U,,, J,,. So the theorem holds good: 
“We can indicate 2" spaces S,-;, each of which contains (n-+-1), 
centres of similitude of a system of » +1 spherical spaces Sp, given 
arbitrarily in S,, and namely one of each of the (n +1), pairs 
(Tyr Lpg)” 
We need not enter into further details about the situation of the 
centres of similitude for the purpose we have here in view. 
3. From the well known properties of the figure consisting of two 
circles and their centres of similitude we read (fig. 1): 
u 
UP," : UP, rn: se r, | } Ou zae 7 WIE == Jit ‘ vol 
EN eaeey tp IP, 
1 
5 
IP; — IC, é zen 
With the aid of these relations we can easily find the following 
A 
u 
Fig. 1. 
theorems, where for Sp, (M,,r,) and Sp,(M,,7,) we shall write the 
B (1) > (2 
abridged form Sp,’ and Spe. 
