292 G. H. KNIBBS. 



point, O Fig. 20, throughout each kind of space is always iir 

 measured in steradians. 1 Consider a plane drawn through this 

 point and imagine a series of lines running out from it, to be 

 traced on the plane, so as to include angles of any given size, \tt 

 in figure. In elliptic space these must be curved toward one 

 another as AOA', B'OB', on the left side of figure, since otherwise 

 the three angles of every triangle will not total -k + e, hence the 

 line OA' must be identical with OB' and the voids A'OB', B'OO' 

 do not exist. Let OA and OA' be continued to O', QB' to O", 

 and 00' to 0'". These three if the curvature be constant will be 

 the one point, which, being impossible, proves that the curvature 

 cannot be regarded as existing in the initial plane. This however 

 is true of every other arbitrary plane in the homaloid of 3 dimen- 

 sions : hence the conception of lateral curvature is an impossible 

 one. 



Neither can it be assumed to act say perpendicularly to the 

 initial plane, for supposing the angles at Fig. 21 to be the same 

 as those at O Fig. 20, the curvature must, for the space to be 

 homaloidal, be indifferently in the direction 00' and 00" at the 

 same time: hence the curvature perpendicular to the plane angle 

 is equally impossible. 



Similarly if the angles from O, Fig. 20, are angles of triangles 

 the sums of whose interior angles are ir - e, the 'straight' lines 

 OA, OB etc., must be curved outwards: but AOA, cannot over- 

 lap BOB as shewn by double shading ; that is OB is the same 

 line as OA. As this must be equally true in all directions, 

 negative curvature cannot exist either laterally, or in any other 

 direction: i.e. the conception of curved 3-dimensional space is 

 absolutely incapable of geometrical representation in a homaloid 

 of three dimensions. We infer therefore, that in an isotropic and 

 homogeneous homaloid of n-dimensions the theory of curvature 

 for space of n-dimensions is invalid. 



1 A unit steradian is the solid angle subtended at the centre of a sphere 

 by a surface equal in area to the square on the radius. The surface 

 divided by this unit is 47rr 2 /r 2 - 4tt. 



