emch: projective groups. 15 



spective collineation, we obtain 006 ■ ccI^tcb'' such pairs, or per- 

 spective collineations; but we have seen above, that among these 

 are oo'* that represent the same perspective collineation. Hence, 

 there are only ooi : coi^co^ different collineations left having a 

 certain straight line of the plane for their axis. From this follows 

 that to each pair of tangent-conies of a S3'stem confined to a certain 

 point of the axis corresponds a pair of tangent-conies of a system 

 confined to any other point of the axis. As to the other perspective 

 collineation of the plane it is sufficient to say that there are 

 oo3x 008=035 different perspective collineations in a plane; each 

 straight line of the plane giving oo3 such collineations. But it is 

 obvious that each of those cd2 systems has the same properties as 

 any of the rest; each and any configuration of the one system can 

 be made coincident with a certain configuration of any of the 

 other systems. It is of no importance for our purpose to stud}^ 

 up relations between two different S3^stems in a general position. 

 However, we shall have to consider two different systems with a 

 common centre. 



This relation occurs in the study concerning the invariant 

 properties of perspective collineations. Here we have to consider 

 the collineations in regard to those elements which by all collinea- 

 tions do not change their position, or even remain invariable in 

 their intermediate parts and as a whole. Such elements are said 

 to be invariant in the collineation. We found that there are oo3 

 perspective collineations belonging to a straight line as their axis. 

 Hence the 



Theorem 5. There are 00 3 perspcetive coUiueatioiis leaving the points 

 of a straight tine invariaiit. 



Dualistically a point and the invariant ra3^s through it, or the 

 centre of collineation, can be taken as the invariant element. The 

 co2 straight lines of the plane and the characteristic anharmonic 

 ratio combined w'ith the centre give also oo3 different perspective 

 collineations with the same centre, and we have therefore, 



Theorem 6. There are 00 3 perspective eollineations leaving the rays 

 through a point invariant. 



This is the relation between two collineations with a common 

 centre to which we drew attention while considering all the 

 perspective collineations of the plane. 



The next invariant element to be considered is the line-element,* 

 or a straight line and a point on it. 



*See Lie's deflnitiouof it in hiis " Vorlesuugen." page 203. 



