xewson: projfxtive iraxskormatioxs. 87 



we have 00' transformations which do not form a grou)-). These 

 statements may readily be verified from the construction. 



Every transformation of the kind T ' depends on two parameters, 

 in and k, coordinate of invariant point and characteristic constant. 

 If m be fixed and k variable, we have a one-termed group; but if k 

 be fixed and m variable, we do not have a group. If m and k botli 

 vary, the resulting 00^ transformations of the kind T' do not form a 

 group. The verification is easy. 



We thus recognize two distinct varieties of parameters; m and n 

 constitute one kind and k the other. If either one or both of the 

 parameters of the first kind are fixed quantities, we have one- and 

 two-termed groups; but if the parameter of the second kind is 

 fixed, we do not have a group. Thus G^ and G[ are the only 

 possible variety of one-termed groups; and G^ is the only possible 

 kind of two termed group. 



Til ear cm 11. — Two distinct kinds of parainctcrs enter into the determ- 

 ination of a projective transformation, viz. : co-ordinates of invariant 



. ■ . , . ■,■ \ anharmonic ratio. I ,,- , , ^ 



points, cliaracteristic - , ^ II e liave one- and tivo- 



\ constant. \ 



termed groups lo/ien and onlv wlien the parameters of t/ie frst kind are 



fixed quantities. 



^A. The General Projective Group and Some of Its Si)ecial 



Sub-Groups. 



We shall now proceed to show that all the projective transforma- 

 tions of a line form a group, using the word in the same sense as 

 heretofore. Any two or more of the projective transformations of 

 a line when carried out in succession produce a result equivalent 

 to that produced b}' some single transformation. This follows 

 immediately from the theory of projective ranges, a knowledge of 

 which is here presupposed. For if we project any range R^ into 

 another range R.,, then project R., into R3, and so on until we 

 obtain n ranges; then R,) is projective with each of the preceding 

 ranges, including R^. Let us carry out in succession 'i\ — i) pro- 

 jective transformations by means of conies touching the lines 1 and 

 1.' We shall finally obtain a range R„ on 1 ' : this being projective 

 with the first range R, on 1, the lines joining corresponding points 

 on these two ranges touch a conic which also touches 1 and 1'. 

 Thus we see that this last conic determines a transformation equiv- 

 alent to the combination of the (n — i) preceding ones. This proves 

 that the 00'' projective transformations on a line form a group. We 

 shall call this the general projective group of the line, and desig- 

 nate it by G3. This general projective group is three-termed since 

 it contains oo-^ projective transformations. 



