20 KANSAS UNIVERSITY QUARTERLY. 



having the direction to the centres and describes a certain curve 

 which passes through the centre. Any point A^, or | A-|-8A, as 



corresponding to A in regard to the conies K and j 8K, lies with 



A on a ray through the centre. The curve which A^ describes is 

 therefore a straight line through the centre and is invariant for all 

 perspective collineations of the system. 



Since any conic tangent to the axis may serve to determine a 

 perspective collineation with a given centre, axis, and character- 

 istic k, it is obvious that there is only one infinitesimal perspective 

 collineation belonging to a centre and an axis, or leaving these two 

 elements invariant. By integration all the ooi finite perspective 

 collineations may be obtained which leave the points of a straight 

 line and the rays through a point invariant. 



From the fact that the centre and axis determine an infinitesimal 

 perspective collineation, it follows that there are co'- infinitesimal 

 perspective collineations either leaving the points of a straight line, 

 or the rays through a point invariant. The integration gives in 

 both cases the co^ finite perspective collineations leaving the same 

 elements invariant. Moreover, it follows that the plane has co-* 

 infinitesimal perspective collineations which l)y integration give 

 the cd5 finite perspective collineations of the plane. 



It is not necessary to enter into a study of the infinitesimal 

 perspective collineation of the special cases, because the result is 

 essentially the same. It is sufficient to mention that the collineation 

 having the centre in its axis and the characteristic k = -j-i has 

 simply cxdI infinitesimal collineations leaving the points of the axis 

 invariant. By integration the oo^ finite collineations of this kintl 

 are obtained. 



In this last case as well as in the general case the whole of W- 

 curves"*' consists of the pencil of rays through the centre. This 

 pencil becomes a pencil of parallel rays if the centre is at infinit}-, 

 as it occurs in some of the special cases of perspective collineation. 

 We have here found the same result as Lie in his " Vorlesungen " 

 on pages 6g and 70. 



jJ5. Groups of Perspective Collineations. 



Suppose a system of perspective collineations which is restricted 

 by certain conditions, for instance, such as to leave given elements 

 or combinations of elements invariant, or to be characterized by a 



*•' W-Curveu." or " .selbstprojectiveCurveu " in Lie's Vurlesungen, page 08. 



