284 KANSAS UNIVERSITY SCIENCE BULLETIN. 



surface, each of which leaves also invariant all the generators of one 

 system and two generators of the other. 



7. Higher groups. — Let us take the resultant of two transforma- 

 tions, 10 T(//'/xF) and 10 Ti(^'i/aF), in which the invariant figures are the 

 same with the exception that l\ is a distinct generator from l', though 

 still of the same system. Along each of the generators I, V, l\ the re- 

 sultant of two identical transformation is identical, while along each of 

 the generators of tt, the resultant of two loxodromic transformations 

 with one common invariant point is loxodromic. The remainder of 

 the invariant figure is common and the resultant space collineation is 

 of type X. The generator I' has been given one degree of freedom 

 without destroying the group property. Therefore, there is a two- 

 parameter group 10 Gt2(Z/aF) in which 1! varies over the invariant surface 

 in the system A. 



Further, let us consider two transformations 10 T(ZZVF) and 

 10 T(ZiZ'i/aF), where I and I' are generators of the system A but distinct 

 from l\, l\. Each of these transformations leaves invariant every 

 generator of /a, and must therefore transform each generator of A into 

 another of the same system, i. e., changes a straight line cutting all the 

 generators of /a into another straight line cutting all the generators of 

 fi. The resultant of two loxodromic one-dimensional transformations 

 is generally loxodromic. Moreover, since by each component the 

 generators of A as a system remain unchanged, the resultant also 

 leaves the system A unchanged, and the two new invariant lines are 

 generators of the system A. Hence, the resultant space collineation 

 is of type X, and we have established the group property for a three- 

 parameter group, 10 G3(ju.F). 



Furthermore, it is readily seen that there can exist no other groups 

 of type X which leave invariant a non-degenerate quadric surface. 



Theorem. There are 2 co 3 transformations of space of type X which 

 leave invariant a given quadric surface, each of which leaves also in- 

 variant all generators of one and two generators of the other system 

 on the surface. These combine to form one-, two- and three-para- 

 meter groups but no groups of higher order. 



§3. Type IX. 



8. Fundamental group. — The fundamental invariant figure of type 

 IX, third class, is a pencil of planes, a pencil of conies in each plane, 

 together with a line of invariant points, and a pencil of invariant lines 

 in one of the invariant planes. This can mean only a system of 

 quadric surfaces to each of which the plane determined by the line 

 of invariant points is a tangent. There are oc 1 transformations which 



