NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. ] 29 



invariant lines of the pencil, and x and xi any pair of correspond- 

 ing lines in the transformation. Then we have the cross ratio 

 0(AA'xxi) = k, and the theory requires no further development. 

 The second case, with one invariant element, may be deduced as the 

 limiting form of the first case in the following manner: 



Let O(AA'xxi) -k; whence O(AxA'xi) = 1 --k. Writing out 

 the last cross-ratio in full, we have : 



sin(A'OA) sin(x,OA) 



1 — k. 



sin(A'Ox) " siu(x,Ox) 



Whence 



sin(x,Ox) 1 — k 



sin( A'Ox) . sinu,OA) sin(A'OA) 



But (xOxi)==(AOx) — (AOxi); therefore, 



li m sin(AOx )cos(AOxi) — cos(AOx )sin(AOx 1 ) _ ij m 1 — k _ j. 



A' = A sin(A'Ox) . sin (x,OA) _ A' = A sin(A'OA) = ~~ l " 



Hence cot ( xxO A ) — cot ( xO A ) = t, 



or cot 0, = cot + t. (32) 



Theorem 14. In a transformation of a pencil of lines (or planes) of 

 type II, the difference of the cotangents of the angles made with the 

 invariant line (or plane) by a pair of corresponding lines (or planes) 

 is constant for all pairs of corresponding lines (or planes). 



PART II. 



§ 6. The General Projective Group EU and its Subgroups. 



The complex double parameter. — Hitherto we have considered the 

 natural parameters, A, A', k of T to be complex numbers, but have re- 

 garded them as single indivisible parameters. However, a complex 

 variable is of the nature of a double parameter and may assume oo 2 

 different values. From this point of view, the groups which we have 

 designated by Gi(AA') and Gi(A), whose parameters are respectively 

 k and t, are two-parameter groups. In like manner, G2(A) and G3 

 have, respectively, four and six parameters. The first point of view is 

 that of Lie and his followers, while the second point of view is more 

 that of Klein and other exponents of Riemann's function theory. 



According to Lie's theory of continuous groups, there are but four 

 varieties of continuous groups of transformations in one dimension, 

 viz., G3, G-2(A), Gi(AA'), Gi(A). But as soon as we regard a com- 

 plex parameter as a double parameter, we shall find several other va- 

 rieties of continuous groups within the general projective group of 

 points on the line. These will be determined in the next section. 



Hereafter we shall use H instead of G to designate a group whose 

 parameters are regarded as double parameters, and shall write He, 

 H 4 (A), H 2 (AA'), H 2 (A) instead of G,, G*(A), Gi(AA), Gi(A). 



»-Kan. Univ. Sci. Bull., No. 5, Vol. I. 



