go KANSAS UNIVERSITY QUARTERLY. 



teristic anharmonic ratios of the transformation T, are now 

 A,,A-'SA"-i; those of To are A.,,A-^,A''"i; those of T„ can be expressed 



' ] 1 - - 2 3 



in terms of the others. The relations existing among these anhar- 

 monic ratios are given by the equations 



Aj A3=A3 ; A-»A;b=(A, A.)-'A;^-''^A:;'A''-'J; 



Xa-ixb-i = (A.A.,yi-Ui>=^=A''-iAi'-=^ (2) 



12 i -A i 



If now b^a these equations reduce to 



A,A„=A,; A-='A-='=--(A.A.O-''=A-''; \'^-iX^-'^ = {X,\.y-^^-\^-l (3) 



Hence we see that if the two transformations T, and T^ are so 

 related that their characteristic anharmonic ratios along one of the 

 invariant lines, for example along y, are each equal to the same 

 power of the corresponding characteristic anharmonic ratios along 

 another invariant line as x, then the resulting transformation T.^ 

 has the same property; i. e. its corresponding anharmonic ratios 

 have exactly the same relation. Thus we see that the two trans- 

 formations T, and T.^ of this particular kind are together equivalent 

 to a third T3 of the same kind; i. e. T3 is expressed in terms of A3 

 and a exactly as T, and T^ are expressed in terms of A, A3 and a. 

 This is the fundamental property of a group. Hence we conclude 

 that all the transformations of the group G„, which have the char- 

 acteristic anharmonic ratio along one of the lines as y equal to a 

 constant power of that along another of the lines as x, form a sub- 

 group. 



This is a one-termed sub-gro'up of G (ABC), the variable para- 

 meter of the group being the characteristic anharmonic ratio along 

 some one of the invariant lines. This one-termed group contains 

 00 1 transformations corresponding to the cc^ values of the variable 

 parameter. This constant power a is the same for all transforma- 

 tions of the group. If we give to a different values we obtain dif- 

 ferent one-termed groups, and as many as there are values of a, 

 viz: cci. Thus we see that our two-termed group G^,. falls apart 

 into co' one-termed groups G^. 



Theorem 13. The two-termed group of fransformations G^, which 

 leaves a triangle invaria7it, consists of 00' one-termed groups G ^. All 

 the transformations belonging to one of these onr-tcirmed groups have the 

 common property that the characteristic anharmonic ratio of each trans- 

 formation along one of the invariant lines is a consta7it poivcr of its char- 

 acteristic anharmonic ratio along another of the invariant lines; this 

 consta7it power is the same for all transformations of a one-termed 

 grot/p, but is different for different o?te-iermed groups. 



