•26 KANSAS UNIVERSITY (QUARTERLY. 



find its inverse. In the construction this fact is self-evident. If 

 the characteristics of two perspective coUineations are of the same 

 sign the sign of the resulting third perspective collineation is 

 always positive; if they are of a different sign, the sign of the 

 third is always negative. From this we conclude, as we know 

 already, that the system of involutions does not form a group: 



— IX— i = + i- 

 Among the general cases of perspective coUineations we have to 

 study the system of dilations in the first place. The centres are 

 all at infinity. Hence we have only the relation 

 LA L,A' L"A 



LA' LjA" L"A" 



CA CjA' _ C"A 



while 



CA' C,A" C"A" 

 The characteristic k"of a dilation resulting from two other 

 dilations of the same system is therefore expressed as before 



k"=k. k,. 

 To each dilation can also be found its inverse , so that the 



respective characteristics are k and -— . To sum up we can sa}': 



k 



Thccyrcni 17. The system of dilations leaving t lie points of a straight 

 line invariant forms a two-termed group. 



If the centre at infinity and the axis are kept fixed the dilations 

 differ according to their characteristics. A, A'; A', A" being two 

 pairs of corresponding points on a ray through the centre at infin- 

 ity, A, A", will be the corresponding pair of the resulting dilation, 

 and there is evidently 



LA LA' LA 



LA' LA" LA" 



k • k,=k"; i. e., 

 the third dilation belongs to the same system. As there are coi 

 such dilations we have 



Theorem 18. The system of dilations leaving the points of a straight 

 line and a point at infinity {centre of dilation) invariant jorms a one- 

 termed group. 



The same is true for the system of dilations leaving the points of 

 a straight line and another straight line invariant. 



The dualistic interpretation, however, does not lead to the same 

 result. From the general case it is known that all the coUineations 

 leaving the rays through a point invariant form a three-termed 

 group. If this point is at infinity the coUineations are dilations. 

 Hence the 



