emch: projective groups. 25 



according to the theorem of Menelaos we have the relations: 



(i). LA • LjA' • L"A"= LA' • L^A" • L"A, or 



. , . LA L.A' L"A , . ^, 



(2). • — 1 ^ , and m the same way: 



^ ^ LA' L,A" L"A" ^ 



(3). CA • C,A' • C"A"=CA' • CjA" • C"A, or 

 . CA^ . C,A' _ C"A 



^^^' CA' ' C^'~~C^' 

 Dividing (4) by (2) we have 

 CA . CjA' 



CA' C^A" 



LA 



Xa' 



L^A' 

 LjA" 



C"A 



(TA^' 



L"A 



L"A" 



(CLAA') (CjLjA'A")=(C"L"AA") 



Designating the characteristics respectively by k, k^, k'', we find 

 the required fundamental relation in the form: 



k"=k. k, 

 If these three characteristics are equal, say equal to k, the 

 relation becomes 



k^k", or 



k^ — k=o ; 

 Whence either k^o, or k=-f i. Excluding the singular case 

 ki^o we can therefore say, that only those perspective col- 

 lineations with a constant characteristic are liable to form a group, 

 for which kr=-]-i. If a perspective collineation (CLAA')=k is 



given we find its inverse (CLA'i\)=— — , which is a number of 



k 



the same kind. Thus, to each perspective collineation we cam. 



