24 



KANSAS UNIVERSITY QUARTERLY. 



i I, A\ A* 



-I I, X5, x-' 



{ r, A", X 



{ I, A. A« 



- I. AS A5 



I. A', A" 



I, AS A* 



. I, AS A^ 



( I, AS A 



I I, A, X^ 



S I, AS A8 



I I, AS A5 



Each of these triangles is the invariant triangle of two transfor- 

 mations of order 3. Thus we have twelve transformations of this 

 variety. The equations of these twelve transformations may be 

 written down by means of formulas (10). We give one example: 



P^ 



X y z o 



I X A8 I 



I A' A5 a 



I A-' A8 aS 



PYv 



I A* A» A*a2 



PZ] 



X y z o 



I A A8 A 



I A'" A5 A«a 



I A* A» A«a2 



These reduce topx, ^3A(a — a^)z, pyi=3A*(a — a^)x, pz^ =3A*(a — a')y 



or 



yj=x. The eleven others are obtained in like manner. 





The following sub-groups of G^^ may be noted: Gjg is an in- 

 variant sub-group; the nine transformations in the first row of the 

 table form a group G^. This group leaves invariant all three 

 sides of the triangle xyz=:o. The first and second rows of the 

 table constitute a group xGj g, which leaves invariant the side x=o. 

 In like manner the first and third rows and the first and fourth 

 rows form groups yG,g and zGjg, whose invariants are respectively 

 y=:::o and z==o. Rows i, 5 and 6 form a group G„.. 



• §6. The Groups G54(i) (i=2,3,4). 



Having determined the structure and properties of the group 

 G^^ (i) we can readily find from this the structure and properties 

 of its equivalent groups G.^(i) (i=2,3,4). We shall first confine 

 our attention to the group 05^(2). 



