(565 ) 
Ee had SF crt sE, 
where the two solutions of a selfsame pair correspond in all signs 
or differ in all signs. 
The construction of the tangent spherical spaces proves the above 
assumed concerning the number of the solutions and their connection 
with the 2” spaces S,—. We give it here — to avoid prolixity — 
for the case » =4 in a form, in which it is immediately transferable 
to the case of an arbitrary ». It is: 
5 4d 5 veal y (2 y (5 5 
a. “If in S, the spherical spaces Sp4 ) Sp§ Bee Spy are given 
: " sp Je . 
arbitrarily, if d, is the space through the points 
Te T he p 
U» U» U 23) Us Li Ls Lis Ls Ls Is 
and if P, is an entirely arbitrary point of Spy”, then the antihomo- 
logous points P,, P,, P,, P, of P, in the inversions U (1, 2), U (4, 3), 
11,4), 71,5) are to be determined and the spherical space Sp, (/) 
through the five points P,, P,, P,, P,, P;.” 
b. If e, is the plane of intersection of din with the radical space 
of Spa” and Sp,(P), let us bring through ¢, two spaces touching 
Spy and let Q, and Q,’ indicate the points of contact.” 
c. Finally must be determined the pairs of points (Q,, Q,’), 
(Qs Qs) (QQ), (Q@;, Q;") which are antihomologous to (Q,, Q,’) 
in the inversions U (1, 2), U (41,3), 71,4), 7,5) and the spherical 
spaces Sp,(Q) and Sp, (Q’) passing through the quintuples of points 
Qi, Q;,..-,Q, and Q,’, Q,’,...Q,’. These spherical spaces Sp, (Q) 
and Sp, (Q/) form one of the 2” pairs of solutions of the problem.” 
The proof of this construction is plain. When P, moves over 
y (1 . q EN 1 . . : 45 
Sp” the power of each of the ten centres of similitude lying in d,,, 
with respect to the spherical space Sp, (/) remains unchanged; con- 
sequently the spherical spaces Sp,(2?) which are possible form a 
5 5 15 ° . 
pencil with d, as common radical space and ¢, is a common radical 
plane of Sp? with each of the spherical spaces Sp, (P) of that 
pencil. If now we choose for P one of the two points of contact 
° (1 : 5 
Q or Q of Spy’ with a space through ¢,, then this tangent space 
