282 KANSAS UNIVERSITY SCIENCE BULLETIN. 



variant figure a quadric surface and one generator of each system on 

 that surface, together with a constant relation between the cross- 

 ratios of the one-dimensional transformations along these generators. 

 We have indicated the various parts of the invariant figure as follows : 

 T = transformation. 

 ex Gr SU b = group of type (ex.); parameters (sub.) 



F = "Flache," a non-degenerate quadric surface. 

 1,1'....; m, m'. . . ; = generators of first and second systems of gen- 

 erators on the surface, respectively. 

 X ; ft = all generators of first and second systems on the 



surface, respectively. 

 n, n' = reciprocal polar lines with regard to the quadric. 

 t, t'. . . . = lines tangent to surface. 



I, m. . . . . = lines of invariant points in their respective po- 

 sitions as generators, etc. 

 A, B, C, D = points on the surface. 



P = point without the surface. 

 p = tangent plane at A to surface, 

 s = polar plane to P with regard to surface. 

 K, K'. . . . = conies on surface. 



^ = variation of the portion indicated as x. 

 ~ = contains. 



A. — Transformations of Six Types. Number and Groups. 



§1. Type XL 



2. Fundamental group. — If we consider the invariant figure of a 

 transformation of space of type XI (Newson's classification), we find 

 that this consists of a straight line, together with all points on, and 

 all planes through, that line ; and a pencil of lines in each plane, the 

 vertices of which are distinct, but all lying on the axis of the pencil of 

 the invariant planes. These lines, then, form the generators of a sys- 

 tem of quadric surfaces, on each of which are left invariant all the 

 generators of one system, and the generator, common to all, of the 

 other system ; therefore, also, the common tangent plane. In this 

 work we shall choose one of the quadric surfaces and neglect any 

 changes that may occur on the other quadric surfaces of the system. 

 This group we shall call n Gri(Z/u.F) — the only parameter being the 

 constant of the parabolic, one-dimensional transformation along each 

 of the generators of system /*. The one-dimensional transformation 

 along I is identical. 



3. Number of transformations. — Of these groups there are double 

 infinity leaving invariant the chosen quadric surface, each group 



