emch: projective groups. 23 



T/irore/ii i^. The system of all perspective colli/ieat/<>ns lea7'iiii:; the 

 rays through a point and a not lie r point in one of these ravs invariant 

 forms a two-termed gronp. 



P^or a proof of these theorems we refer either to Lie's theorem at 

 the beginning of this chapter, or to the first proof of theorem 

 II. In the dualistic case the reasoning is exactly the same. 



There is another three-termed group of perspective colHneations, 

 which is obtained by a special interpretation of the groups which 

 leave the points of a straight line or the rays through a point 

 invariant. The line-element, the point in which is taken as the 

 centre, as an invariant figure, is equivalent with the rays through 

 the centre. The system of collineations is therefore in both cases 

 the same. On the other hand the point of the line-element may 

 be a point of the axis. The invariant configuration is therefore 

 that of the points of a straight line and another straight line. But 

 the axis may be any of the rays passing through the point 

 of the line-element, so that the number of perspective collinea- 

 tions is co3 as before. Thus, the 



Theorem /j. All the perspective eollineations leaving a line-element 

 invariant form a three-terjned groi/p. 



The next and last sub-group of perspective collineation con- 

 cerns the points of a straight line and the rays through a 

 point as the invariant configuration. As usual let the point (centre) 

 be C and the line (axis) 1 and two collineations represented by 

 (CLAA') and (CLjAjAj'). Applying the second collineation to 

 A', the corresponding point A" of A' is obtained which lies upon 

 the same ray through C, as A and A'. The new collineation is 

 therefore represented by (CLAA"), i. e., by a collineation of the 

 same system. Since each of these collineations and its inverse 

 belong to the system the following statement may be made: 



Theorem 16. All the perspective collineations leaving the points of a 

 straight line and the rays through a point invariant form a one-termed 

 group. 



The groups of perspective collineation are also easily obtained 

 by the configurations in space. 



Assume the two planes tt and tt' intersecting each other in 1, and 

 a centre (C) without these planes as a perspective collineation in 

 space. Drawing the bisecting plane tt^ of tt and tt' and a perpen- 

 dicular from (C) to the bisecting plane, intersecting tt and tt' in C 

 and C, respectively, these, two points will coincide when one of 

 the planes tt, or tt' is revolved into the other about 1 as an axis. 



