214 KANSAS UNIVERSITY (^WARTERLV. 



general projective transformation leaving the points of two ravs 

 invariant. 



5. In this chapter we have to investigate whether there are 

 any projecctive transformations in space which leave the character 

 of the involution of the second kind unchanged. 



Writing again the equations of these involutions 



ax-f-by 

 X J = , 



z 



ex— ay ^^ 



a-+bc 



we deduce the ratio 



■by 



Yi ex— ay 



a ^-b 



Xi y 



}', c 



Designating the ratios ' and respectively by u^ and u, 



y 1 y 



this relation assumes the form 



au + b 



(7) 



cu — a 



which shows that the ratios u, and u are in the relation of an 

 involution in the straight line. From this we conclude //ui/ tlic 

 pencil of planes throiigli the axis z is Ira/isfon/ieil like the point-range 

 in the involution of a straight line. 



The double-elements of this involution are 



c ~" y 



a— ] a3 + bc X 



"= 1 =T' 



which with z= ±: 1 a^-j-bc confirms the previous result. 



In my article in the Quarterly* I have shown that the one- 

 termed group of transformations 

 *Loc. cit. 



bu^— bk ^_, 



Uo=, — -, (8) 



