288 KANSAS UNIVERSITY SCIENCE BULLETIN. 



with common invariant point is parabolic. The resultant is of type 

 III, and we have a three-parameter group 3 G3(lmF) established. If, 

 however, we allow m to vary, along 1 and 1', the resultant of two 

 parabolic transformations without common invariant point is in gen- 

 eral loxodromic. Hence, we can have no group 3 G 3 (ll'mF). 



Next consider 3 T(ll'mF) and 3 Ti(hl'imF). Along m the resultant 

 of two loxodromic, one-dimensional transformations without common 

 invariant point is loxodromic, while along the new 1 and 1' the two 

 parabolic transformations give a parabolic transformation. The en- 

 tire resultant is of type III, and we have established the four-parameter 

 group 3 G4(mF). 



The resultant of any two collineations of type III, taken in more 

 general position, is of type I. Therefore, there are no groups of 

 higher orders. 



Theorem. There are of space oo 4 collineations of type III which 

 leave invariant a given quadric, and each of which leaves invariant, 

 in addition, two generators of one and one of the other system on the 

 surface. These collineations combine to form two-, three- and four- 

 parameter groups of the first class. 



17. Second class. — In the collineations of our two-parameter group 

 3 G-2(ll'mF), let us assume the relation between the t along 1 and the 

 k along m to be k= a 1 , where a is a constant ; and take the resultant 

 of two transformations having different t's. Along 1 and 1' these com- 

 bine by the law. t2 = t + ti, and along m according to the law 

 k2 = kki = a t a ti = a t + t *. Therefore, the a of the resultant transforma- 

 tion is the same as in each of its components. There is a one-param- 

 eter subgroup of 3 G2(H'mF) for each value of a, the symbol being 

 3 Gi(ll'mF) a . 



18. Higher groups. — Take two collineations, 3 T(H'mF) a and 

 3 Ti(ll'imF) a , and form their resultant. In the plane lm the plane 

 transformations are of type II, and have in common two lines and their 

 point of intersection. But this is the case in which the plane col- 

 lineations with constant a combine according to the lawt-2 = t+ti. 

 Hence, the resultant of the space transformations is of type III with 

 constant a, and there is determined a group 3 G 2 (lmF) ;1 . It may be 

 easily seen that there are no higher groups of type III of this class. 



Theorem. There are cc 5 collineations of type III which trans- 

 form a given quadric surface into itself. These combine to form 

 one- and two-parameter groups of the second class ; but there are no 

 groups of this class of higher orders. 



