newson: continuous groups. gi 



We shall now proceed to study in detail the properties of one of 

 these one-termed groups. Since the variable parameter of the one- 

 termed group in the plane is the characteristic anharmonic ratio of 

 a one-termed group of one-dimensional transformations along one 

 of the invariant lines, we may expect that the properties of the 

 group of the kind Gj on a line, (Sec Part I.) 



The characteristic anharmonic ratios of a transformation T along 

 the invariant lines are A, A"=^.A='"^; hereafter we shall speak of A as the 

 characteristic anharmonic ratio of the transformation T. We have 

 already shown in equation (r) that Aj, t'.\2 characteristic aaharmon- 

 ic ratio of the reiultaat transformation Tg, is equal to the product 

 of A, and A,, the characteristic anharmonic ratios of the component 

 transformations T, and To- By combining T^ with T^ we obtain 

 Tj; so that P. is equivalent to the coaibination of Tj, T.,, T^; tlitis 

 TjTgT^^Tj. Also A jA^-=A- i. e. AjAjA^ — Ag. The same reason- 

 ing may be extended to any number of transformations. 



Property I. Any liuo or more transfoniiations of the grotip G^ arc 

 equivalent to som: sin':;;le transformation of tli3 sam: group; the eltarac- 

 t eristic anharmonic ratio of the resultant transformation is equal to the 

 continued product of the characteristic anharmonic ratios of the compon- 

 ent transformations. 



If A=i, then A'-'-^i, and A"'"!^!; but for A^=i the transformation 

 along an invariant line is an identical transformation. Hence every 

 point on the invariant lines x, y, and z are invariant points; also all 

 lines through A, B, and C are invariant lines; therefore every line 

 of the plane and every point of the plane is invariant, The trans- 

 formation of the group given by A = i is therefore an identical trans- 

 formation. 



Prop. 2. Tiie group G contains one identical transformation whose 

 eiiaracteristic ratio is unity. 



Two transformations of the group whose characteristic anharmon- 

 ic ratios are reciprocals of one another are said to be inverse trans- 

 transformations. It is evident that all transformations of the group 

 may be arranged in inverse pairs, and that the two transformations 

 of a pair are together equivalent to the identical transformation of 

 the group. Hence we see that if any transformation T moves P to 

 P', the inverse transformation T' moves P' back to P. 



Prop. J. Pile transformations of a group G may be arranged in in- 

 7'erse pairs; the characteristic anharmonic ratios of t/ie transformations 

 forming an inverse pair are the reciprocals of ofie anotJicr. Any trans- 

 formation of tlie group and its inverse are together equivalent to the iden- 

 tical transformation of the group. 



