126 KANSAS UNIVERSITY SCIENCE BULLETIN. 



same invariant point. These possess the same fundamental group 

 property and form a continuous group Gi(A) of type II. 



Theorem 10. The totality of transformations of type II, which 

 leave the same point invariant, form a continuous group ; the constant 

 of the resultant of any two transformations of the group is equal to 

 the sum of the constants of the components. 



Properties of the group G\{A). — The properties of the group 

 Gi(A) are not identical with those of the group Gi(AA') but very 

 similar. If T be the transformation 



_i •* _i_t 



x,— A x— A ~ L > 



its inverse, T" 1 , which transforms x back to xi, is 



t. 



x — A Xi — A 



Hence two transformations in Gi(A) whose constants are numerically 

 equal with opposite signs form an inverse pair. All transformations 

 in the group may be arranged in inverse pairs. 



The resultant of a pair of inverse transformations is the identical 

 transformation whose constant is t2 = t — t = o. The group Gi(A) 

 therefore contains the identical transformation. 



The only transformation in the group which is its own inverse is 

 the identical transformation, i. e., the group contains no involutoric 

 transformation. It contains one pseudo-transformation for which 

 t = oo. This transforms every point on the line to the invariant point. 



A transformation of the group whose constant t = $exp. iO is infini- 

 tesimally near to zero is an infinitesimal transformation. If an in- 

 finitesimal transformation is repeated n times, the resultant has the 

 constant nt. By a proper choice of n and this may be made any 

 number we please; hence every transformation in the group Gi(A) 

 can be generated from an infinitesimal transformation of the group. 



Theorem 11. The transformation of the group Gi(A) can be ar- 

 ranged in inverse pairs ; it contains the identical transformation, one 

 pseudo but no involuntoric transformation ; it contains an infinite 

 number of infinitesimal transformations, and every transformation of 

 the group can be generated from an infinitesimal transformation of 

 the group. 



Nuniber of one-parameter groups.— -We have thus found two types 

 of one-parameter groups of transformations of the points on a line, 

 viz., Gi(AA') and Gi(A). Evidently there areas many groups of the 

 first type as there are pairs of jDoints on a line, viz., oo' 2 . Also, there 

 is a group of type II for every point on a line; therefore, oo 1 in num- 

 ber. It is also evident that every transformation of the points on the 

 line belongs to one and only one of these one-parameter groups (ex- 

 cept the identical transformation which is common to all). 



