1 8 KANSAS UNIVERSITY QUARTERLY. 



Every cubic of the pencil will be transformed into itself when 

 k^ = i, k'3:=;.i, and kk'=i. These relations are satisfied by k— a 

 and k':=a^ or by kr=a- and k'=-a. Thus we have two transfor- 

 mations 



Xj=X Xj=X 



yj=ay and yi^a^y (12) 



Zj=a^Z Z^=aZ 



of this kind which transform every cubic of the pencil into itself. 

 These are a pair of inverse transformations and each of period 3. 



In like manner it ma)' be shown that each of the other inflec- 

 tional triangles is the invariant triangle of a pair of transformations 

 of period 3, such that they transform every cubic of the pencil into 

 itself. In this way we find eight transformations of this variety. 

 The equations of these eight transformations may be written down 

 by means of formulas (10), making use of the values given in (g). 

 These equations are as follows, numbered according to the tri- 

 angles: 



jxj=x, X t y, z iy, z (y, z 



I. ^ yi=:ay, a2y; 2. -<^ Z, x; 3. -^ aZ, a^x; 4. -^ a^z, aX . (13) 

 (z,=tt^Z, az ( X, y ( a^X, ay ( aX, a^y 



These eight transformations, together wuth the nine perspective 

 transformations given above, and the identical transformation con- 

 stitute a group Gjg, every transformation in which transforms 

 every cubic of the pencil into itself. The fact that these eighteen 

 transformations form a group may be verified by applying the test 

 of forming all possible resultants. There are no other transforma- 

 tions possessing this property. 



It is evident from the character of the transformations contained 

 in Gjg that the group contains four cyclic sub-groups of order 3 

 and nine cyclic sub-groups of order 2. G^^ also contains a sub- 

 group Gy of order 6 and one Gg of order 9. These are given as 

 follows: 



Xj=x, X, z. y, y, z, 



G6^^^yi=y, z, y, X, z, x,=:^G3+3G2. (14) 



Z|=z, y, X, z, X, y, 



All transformations of this group are real; one is of order i, three 

 of order 2 and two of order 3. The group Gg is as follows: 



Xj=x, X, X, y, z, y, z, y, z, 

 GgZ:zyi=y, "^y> "'3'' ^' ^> °-^' u'^x, a"z, ax.^^4G3. 



Zj=Z, u-Z, aZ, X, y, a-X, ay, aX, a"y, 



