290 KANSAS UNIVERSITY SCIENCE BULLETIN. 



This method of proof is a very general one, and is made use of in 

 showing the existence of higher groups of this class. There exist 

 four such groups, ^(ll'F), ^(lniF), ^(LF), and ^(F). The 

 proof of the group property in these last cases is left to the reader. 



21. Groups of second class. — If, to the conditions already placed 

 on the relations of cross-ratios in the preceding groups, we add that 

 of a constant r, we obtain several new groups. 



The first of these is a one-parameter group 'Gifll'mm'F),., whose 

 existence as a subgroup of 1 Gi(irmm'F) is evident.* The higher 

 groups are as easily shown, the existence or non-existence of the 

 group depending upon the corresponding group of the plane.f The 

 details of the work are omitted for the two remaining groups, 

 1 G 2 (lmF) r and 1 G 3 (lmF) r . 



Theorem. There are oc 6 collineations of space of type I which 

 leave invariant a given quadric surface, each of which, in addition, 

 leaves invariant a pair of generators from each system on the quadric. 

 These combine to form groups of two classes. Of the first class there 

 are two-, three-, two varieties of four-, five- and six-parameter groups. 

 Of the second class there are one-, two- and three-parameter groups, 

 but none of the higher order. 



B.— Combinations of Various Types. 



§ 1. Type XI. 



22. The resultant of two transformations of the same type we have 

 often found to be of a different type. We shall proceed next to ex- 

 amine not only the resultant of two transformations of one type, but 

 also the resultant of a collineation of one type taken with one from 

 each of the other types. Generally, this resultant is at once evident 

 and will be stated without proof. 



The resultant of two transformations of type XI with common in- 

 variant figure was shown to be of type XI. If, however, the invariant 

 figures differ in the generators of the second system, the resultant 

 one-dimensional transformation is not parabolic but loxodromic, and 

 the space collineation is of type X. 



Similarly, taken with different invariant systems of generators, the 

 resultant is of type IX. 



n T(^F) + 11 T 1 (^F)-= 11 T 2 (// A F) 

 11 T(Z /J tF) + 11 Ti(Z 1 /xF)=:= K, T(Z2^/ J tF) 

 U T(^F) -|- 11 T 1 (mXF) = !) T(lm^F). 



* Kansas University Quarterly, vol. X, p. 33. 



t American Journal of Mathematics, vol. XXIV, No. 2, p. 134. 



