EMCH: PROJECTIVE CtROUPS. 



29 



On account of the conspicuit}' in which the character of a group 

 appears in the groups of congruences, we give an iUustration in 

 Fig. 13- 



The pseudo-perspective collineations which have the character- 



istic k=o. 



00 , = — mas^ conveniently be represented in space. 



The cases arise when the centre of collineation is in either one of 

 the planes tt and tt', If the centre (C) is in tt then (CLAA')=: 

 (CLRoo)=(CLC 00 )=o. Every point of tt' is projected into C. 

 Taking C as a point of tt' (by revolving the plane tt about the axis 

 1 into tt') and performing the projection for another centre, say 

 Cj in TT, the point C is projected into Cj. But the points in tt' and 

 C may be taken as a new pseudo-collineation of the sj^stem. The 

 same can be said if the centre (C) is in tt', or if (C'LAA')^ 

 (C'Loo C')=±oo . The inverse of a collineation for both cases 

 k^o and k= 00 is indefinite, but as the other conditions for a group 

 are satisfied, we may say that the pseudo-perspective collineations 

 with either the characteristic o or 00 form a two-termed group. 



