30 KANSAS UNIVERSITY QUARTERLY. 



In the pseudo-perspective collineation with the centre in the axis 

 and k indefinite there is no group at aU. 



With this we close the study of groups of perspective collinea- 

 tions, perspective collineations taken in its common meaning, and 

 proceed to that particular kind of perspective collineations in which 

 centre and axis coincide and k^|^ i. This kind of collineations 

 might be considered as the supplement of involution. For, the 

 involution being obtained by revolving one plane, say tt, into the 

 other ir' through an angle of a", the collineation in question is 

 obtained by revolving w into tt' through i8o — a. According to Lie 

 (page 202 of his " Vorlesungen ") we may call this collineation an 

 elation. 



But here the word does not mean entirely the same as in Lie's 

 definition. We use it here only as a convenient word to express 

 those collineations. 



In Fig. 12 C and L, C, and C„; C" and L" coincide, so that the 

 characteristic of an elation is -f i> 



The fundamental relation is 



( + i)X(+i)= + i 

 and since there are 00^ elations restricted to a line we have 



Theorem 27. The system of elalioiis leaving the joints of a straight 

 line invariant fu-ms a tjuo-termed group. 

 and dualistically: 



Theorem 28. The system of elations leaving the rays through a point 

 invariant forms a tioo-termed group. 



Taking A, A', A" on the same ray through the centre, there is 

 CA CA' _ CA 

 CA^ ■ CA""" CA"' 

 and the same is true for LA, LA ', and so on, since L coincides with 

 C. 



Therefore the relation as above: 



(+i)X(+i)=+i- 



Theorem 2g. The system of elations leaving the points of a straight 

 line and the rays through a point invariant forms a one-termed group. 



The centre of elation can also be considered as the point of a 

 line-element. From this point of view we have another theorem. 



Theorem jo. The system of elations leaving a line-element invariant 

 forms a t7vo-termed group. 



Summing up, the following groups of perspective collineations 

 are possible. (The Roman numerals denote the types of trans- 

 formations as given in Lie's " Vorlesungen iiber Continuierliche 



