point with the centre O' of the inversion; so the oblique cone with 
O' as vertex and circle C(M 
r,) as base must contain the centres 
of the spherical spaces of the series Sy,, and the locus of those 
centres is the conic of intersection of this cone with the plane «. 
Of this conie m is an axis of symmetry and the points M' and 1", 
0? 
becoming the centres of the inverted spherical spaces Sp", (J/', 1"), 
Sp"n(M',r') are vertices. This conic is an ellipse 4, a parabola 
P or an hyperbola H, according to none, one or two of the spherical 
spaces Sp", of Sy', passing through 0’, i.e. according to O' lying 
outside the two circles C(M!, 7!) and C(M"‚ 7), on one of those 
circles or inside one of them. Of these three cases fig. 2 represents 
the first and this will be furtheron exclusively under consideration. 
If we suppose that the conic obtained is an ellipse # the inverse 
spherical spaces Sp, (M', r') and Spa (JZ", r") of the spherical spaces 
Sp", CM’, 7) and Sp", (MY, 7’) will touch every spherical space Sp, (MZ, 7) 
of the system Sy,—2 in the same kind. From the triangle MZ M' M" 
