24 KANSAS UNIVERSITY QUARTERLY. 



From this we see that the line 1 is invariant for every perspective 

 collineation having any point of space as its centre. The 

 number of such collineations is coS. A point C and a line 1 are 

 invariant for ooi such collineations, for there are ooi centres, (C), 

 on the perpendicular to the plane ttj in C, and ooi planes through 

 the perpendicular. A plane through such a perpendicular inter- 

 sects TT and tt' in two lines, which after revolving one plane into the 

 other become an invariant line. As there are ooS points (centres) 

 in the plane perpendicular to ttj, the points of 1 and another 

 straight line will be left invariant by c»2 perspective collineations. 



In this manner we can successively deduce all the groups from 

 intuition in space. We shall not carry on the enumeration of the 

 other groups by this method; it is sufficient to have shown the 

 possibility of this method, which in fact is identical with the other. 



In the general case we had considered the co3 perspective col- 

 lineations leaving the points of a straight line, and, dualistically, 

 the rays through a point invariant. Each collineation of the 

 system is characterized by a certain anharmonic ratio. Two 

 perspective collineations with the characteristics k and kj applied 

 one to the other determine a new perspective collineation of the 

 same group, whose characteristic may be designated by k. Now 

 it is known that k'' is an algebraic relation between k and kj, say: 



k" = f(k, k^. 



Suppose now that two perspective collineations of the same 

 characteristic applied one to the other produc<^ a perspective col- 

 lineation with the same characteristic, such that 



k"=f(k", k"). 



If this relation shall hold for all values of k", it must be an 

 identical one; i. e., k"=k", as it occurs in the identical perspective 

 collineation. From this we conclude that in general a system of 

 perspective collineations with a constant characteristic does not 

 form a group. From the above relation we can find the special 

 values for which 



k"=f(k", k") 

 by resolving the equation 



k" — f(k", k")=o. 



It is therefore necessary to know the form of f (k, kj). 



For this purpose we consider the three perspective collineations 

 (CLAA'), (CjLjA'A"), (C"L"AA"), of which the last results 

 from the two others as described before. 



The sides of the triangle AA'A" are intersected by two trans- 

 versals 1 and the line joining the centres C, Cj, C". Thus, 



