emch: projective groups. ig 



In the case of the relation of K and K^ being in a contact of the 

 second or third order, each point of the axis as a centre gives coi 

 perspective collineations (determined by the centre and the coi 

 corresponding points subtending different sects). Hence, the 

 points of a straight line are invariant for oo- such collineations. 



Dualistically, the rays through a point are invariant for co2 such 

 collineations. In other words there are ooS of those collineations 

 leaving a line-element invariant and again, there are coi such 

 collineations leaving the points of a straight and the rays through 

 a point on this line invariant. 



We have now investigated all the properties relating to number 

 and invariants of the general and special perspective collineations 

 which are essential from the standpoint of the theory of groups. 



In the next chapter we will consider some of the infinitesimal 

 properties of perspective collineations. 



§4. Identical and Infinitesimal Transformation and W-Curves 

 of Perspective Collineation. 



The axis and centre of perspective collineation being given, ooi 

 perspective collineations can be determined which leave these 

 elements invariant. Each of these collineations is determined by 

 the characteristic anharmonic ratio, or a pair of conies K and K^ 

 touching the axis at the same point and having for the other 

 two common tangents two rays through the centre. As is 

 known from §3 there are oo' such pairs or different collineations. 

 If especiall}' the two conies K and K^ coincide, the collineation is 

 an identical one. All the conies tangent to the axis at a certain 

 point and tangent to two fixed rays through the centre may be 

 taken for the representatives of the identical collineation. An 

 infinitesimal collineation differs from an identical collineation 

 b)' an infinitesimal amount, i. e. , a point and its corresponding one 

 have an infinitesimal distance and two corresponding lines include 

 an infinitesimal angle. Thus, in order to obtain the infinitesimal 

 from the identical collineation we have to choose for K' the conic 

 infinitely close to K in the same S3'stem. This conic shall be 

 designated by 8K. According to this proposition a point A is 

 transformed into A4-8A, and a line a into a-f-Sa. 



A and A-j-SA lie on a ray through the centre, and a f-Si intersect 

 each other in a point of the axis. Appljnng a whole S3'stem of 

 infinitesimal perspective collineations to each other and in a certain 

 succession, an integral, or finite, perspective collineation is obtained. 

 In this operation the corresponding point A^ to A starts from A 



