KANSAS UNIVERSITY QUARTERLY. 



A coUineation resulting from two other collineations is related to 

 these in such a manner that the three centres lie in a straight line. 

 For, considering the triangles ABC and A"B"C,, it is seen that 



Fig.ii. 



their corresponding sides intersect each other in three points 

 S, B', A', of a', or that the triangles are homologous. Hence 

 AA", BB", CCj, intersect each other in a point, in the centre C" 

 of the third collineation. Hence C, C,, C", lie in a straight line. 



This fact leads us at once to a sub-group of perspective colline- 

 ation. By choosing the centres of perspective collineation constantly 

 on a straight line the centre of a collineation resulting from two of 

 these collineations is a point of the same line. To each perspective 

 collineation ma}' also be found its inverse belonging to the same 

 system. From §3 is known that there are » ~ such collineations 

 and we have therefore: 



Tlicori-in 12. The sxston of pcrspccti'i'r Lolli)ifaiio)is It'a^'i/ii;- the 

 points of a st)-aii^lit line in'oaiiant f(>iins a t^vo-ternted i:^roiip. 



Without needing a direct proof the following dualistic statements 

 of the two preceding theorems can be made: 



T/ieoreni fj. Tlie system of all perspeetii'e eoIli)ieations /ujTini^^ the 

 ravs tliroi/i^h a point i)ivariant foi'nis a tliree-tenneJ i:;i-oiip. 



And 



