300 KANSAS UNIVERSITY SCIENCE BULLETIN. 



E.— Mixed Groups. 



§ 1. Type XII. 



38. We now come to the consideration of what Doctor Wood terms * 

 "symmetry transformations," i. e., transformations which leave a given 

 quadric surface invariant, but interchange the two systems of gen- 

 erators. The discussion of such transformations brings to our notice 

 collineations of a type not yet considered. Choose as the invariant 

 figure of a transformation of type XII the vertex and axial plane as 

 pole and polar with regard to the given quadric surface, and next 

 choose one cross-ratio along each invariant line equal — 1. Then 

 each invariant line cuts the surface in two points which are har- 

 monically divided by the two invariant points on that line, since a 

 quadric surface divides every line through a pole and polar plane har- 

 monically. But this is merely stating, in a different way, that the 

 cross-ratio of the four points is -- 1 ; i. e., in the given transformation, 

 the two points on the quadric surface are interchanged. An effect of 

 the same kind is produced along each of the invariant lines ; i. e., every 

 point on the surface of the quadric is interchanged with another, but 

 the surface as a whole is transformed into itself. There are cc 3 such 

 transformations which leave the given quadric invariant, one for each 

 point in space taken with its polar with regard to a given surface. The 

 resultant of these transformations with those of the types heretofore 

 discussed give rise to the mixed groups — i. e., those groups which 

 leave a given figure as a whole invariant, but interchange its parts in 

 such a manner that a repetition of the transformation will bring it 

 back to the original invariant figure. 



A combination of this involutoric transformation of type XII with 

 any one of types XI or X cannot result in a transformation of either 

 of these types. Therefore, we can have no mixed groiqiis of these types. 



§ 2. Type IX. 



39. If we consider the group 9 G2(lmpF), we see that the invariant 

 figure consists of one generator of each system and the plane of 

 those ge aerators. We may choose an involutoric collineation of type 

 XII which will interchange these two generators. Their point of in- 

 tersection must be an invariant point of the involutoric transforma- 

 tion. For any plane through that point as axial plane and its pole as 

 vertex of an involutoric collineation, the point must remain in- 

 variant. Thus there are oc 2 involutoric collineations of type XII 

 which leave the figure of y Gr2(lmpF) invariant as a whole, though in- 

 terchanging parts of it. We see at once that our mixed group must 



* Annals of Mathematics, second series, vol. II, No. 4, p. 167. 



