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out of the points of S,-2, in such a way that they touch one of 
the spherical spaces of the singular infinite series, thus all the 
spherical spaces of that series too. 
6. If we now confine ourselves to a single series of the 27—! 
singular infinite series we have found two systems of spherical spaces 
possessing the remarkable property that each spherical space of 
one touches all the spherical spaces of the other. Of these two 
systems one is a singular infinite series of equally large spherical 
spaces with a circle C(M,, 7,) having M, and 7, as centre and as 
radius and lying in the plane «, as locus of centres, whilst the other 
is an n—2-fold series with the space Sa perpendicular in WM, to 
& as locus of centres. How do these two systems transform them- 
selves if we apply to both — in order to return to our ” given 
spherical. spaces Sp, — the inversion with O’ as centre and the 
formerly used power? 
To answer this question is made easy by the observation that the 
n-dimensional figure consisting of the two systems S/,, Sy',~2 and 
their inverse systems Sy, Sij,—2 have a plane of symmetry, the 
plane o through 1/,, O' and the projection O” of O' on Ss. This 
plane o forming the. plane of fig. 2 has with ¢, in common the 
diameter m' parallel to O' O" of the cirele C'(M/,,7,) and is according 
to that line m’ perpendicular to €,; so it is a plane of symmetry for 
Sy. It has moreover with S,—2 the line J/,0" in common and is 
according to that line « perpendicular to Sy,—2; so it is also a plane 
of symmetry for Sys. And if it is a plane of symmetry for Sy’, 
and Sy',—2, then it is so too for Sy, and Sy,—2, because it contains 
the centre ©’ of the inversion. 
We prove to begin with that the centres of the spherical spaces 
of Sy, lie in a conic. To this end we regard in the plane of sym- 
metry o (fig. 2) the points of intersection J/’, J/" with the cirele 
C(M,,7,), the circle of section C(J/,,7,) with the spherical space 
Sp'n(M,,7,) of Sy’, and the point O of the line J/'O', for which 
MOM Or. Then point A of a, which is at an equal distance 
-from O' and O, is the centre of a sphere Sp,(4,A0') with A) as 
radius, intersecting Sp", (J/',7") and so all spherical spaces Sp", of 
the singular infinite series at right angles. This sphere is transformed 
by the inversion with O° as centre into a plane ¢ perpendicular to 
O'A, intersecting 6 according to a line m normal to O'A; this plane 
e must contain the centres of the spherical spaces of Sy, as it cuts 
all those spherical spaces at right angles. And farther, when inverting, 
the centre of a spherical space remains on the line connecting this 
