( 564 ) 
: , (1) , (2 ‘ : 
“The spherical spaces Sp, and Spy’ are homothetic and directly 
similar with U, homothetie and inversely similar with / as centre 
1 . . . er rn . 
of similitude and + — as quotient of similitude. The points corre- 
lis 
; “ : : i 
sponding to each other, P," and P, in the first case and P, and P, 
in the second, are called homologous.” 
“The spherical spaces Spi” and Sp correspond to each other 
in an inversion with U as centre and UC“. UC, as positive power 
and in an inversion with J as centre and / C: . IC, as negative 
power. The points P,“ and P," corresponding to each other in the 
first case and P, and P, in the second are called antihomodlogous. 
And the two inversions appearing in these theorems shall furtheron 
be indicated for shortness’ sake by the symbols U (4, 2) and J (4, 2).” 
“Each spherical space Sp, through a pair of antihomologous points 
Ie rant > (2 5 
Pand ot Sp? and Spo cuts these spherical spaces at equal 
angles. If the: spherical space Sp, through P, and P, touches the 
° SHAD) . : B (2 
spherical space Spo? in P,, it will touch the spherical space Spo 
in P,. And these contacts will be of the same kind or not, according 
to U or J being the centre of the antihomologous correspondence.” 
In connection with the general theorem concerning the situation 
of the centres of similitude the second and the third of these three 
simple theorems form the foundation of a method of solving the 
problem to construe a spherical space Sp, touching n-+-1 spherical 
spaces Spo, Spe, oe, Spor) given arbitrarily in ,. As will 
immediately be evident, to each of the 2” spaces 5"! through m1), 
centres of similitude answers a pair of tangent spherical spaces Sp, 
and the contact of one of these spherical spaces with Spt and Sp? 
is of the same kind or not, according to the chosen space S,—1 con- 
as, ‘Ave em ‘ Cl 
taining of the centres of similitude U, J), of Sp? and Spx either 
the first or the second. So 2! is the number of the theoretic solutions. 
And if we indicate external contact by —+ and internal contact by 
—, then the 2” pairs of solutions are indicated by the pairs of 
completely opposite combinations of signs of the series consisting of 
m+ 1 terms 
