NK.WSON: PROjKtIIVK IRANSKORMATIONS. 83 



whose characteristic constant is (/•/-!-(•/+//.... to oo)^k. Thus 

 any finite transformation of the group may be generated from this 

 infinitesimal transformation. 



Property f;. — T/ie it:;roi(/' Ci'^ co)itciins our and oiilv one infinitesimal 

 transformation; the luholr iiroi//> /nav /'/■ in^cucratcd froni t/iis infinitesi- 

 mal transformation. 



We have seen that there are two distinct types or kinds of 

 projective transformations of the points on a line; viz.: T_^r and 

 T^ . The first T^„ leaves two distinct real or imaginary points 

 invariant; the second T^ leaves a single point invariant. We can 

 see likewise that there are two distinct types or kinds of one- 

 termed groups of these transformations; viz.: GAB^ndG^^. The 

 first G^B leaves two distinct real or imaginary points invariant; the 

 second G( leaves a single point invariant. There are no other 

 kinds of one-termed groups. These two types of groups G^b and 

 G^ have many properties in common, and some points of differ- 

 ence. We shall now summarize briefly the results of this section. 



Theorem ~. — A one-termed i:;roup of projeetii'e transformations of the 

 points on a line eonsists of oo^ transformations ha?' ins;' the property that 

 the eombined effeet of anv tiuo or more of the transformations of the 

 ^i^ronp is eijuivalent to that of a sin^^le transformation of the same group. 

 Sueh a g^roup eontains one identieal, one infinitesimaf and t7V0 pseudo- 

 transformations ; all the other tran'iformiitions of the g;ronp may be 

 arranged in immerse pairs. The eombined effeet of the ttuo transforma- 

 tions composing any one of these pairs is equivalent to the identical 

 transformation of tJie group. Ativ fnite t/-ansformation of the group, 

 or the 7uhole group itself, max be generated from the infinitesimal trans- 

 formation. There are t7C'.> types of one-termed groups. 7'iz. : G^b) 

 and Ga • 



(/). The group G^b leaves t-ivo distinct points invariant. Tfie 

 variable parameter of the group is the characteristic anharmonic ratio. 

 The product of the characteristic anharmonic ratios of the component 

 transformations is equal to the characteristic anharmonic ratio of the 

 resultant transformation. This group contains one involutoric trans- 

 formation 7vhich is its oivn inverse. 



(2). The group (j^ leaves one point invariant. The variable para- 

 meter of the group is the characteristic constant. The sum of the 

 characteristic constants of the component transformations is equal to 

 the cfiaracteristic constant of the resultant transformation. This group 

 contains no involutoric transformation. 



A closer examination into the structure of these two groups G^j, 

 and G( raises some important (juestions concerning the sufficiency 



