WILLIAMS. — FINITE CONTINUOUS GROUPS. 101 



that is, unless m contains v. Therefore, if a 4 + b t = 2 m tt i ^ and 

 v 4= 1. we can always so choose m that c 3 shall, in general, be infinite 

 and T b T a singular. On the other hand, if a A + 6 4 = 2 m tt i, and if 

 v = 1 (i. e. if « is an integer), one branch of c 3 is always finite, and the 

 same is true for c x and c 2 : so in this case T b T a can be generated by an 

 infiuitesimal transformation of our group. 



When a is rational there are, however, always singular transforma- 

 tions of the group. For let 



. 2nnri 2 mviri 

 « 4 + h - 



a— 1 



fi — v 



Then in general (i. e. provided the function of the a's and b's found in 

 the second factor in the expression for c. 2 is not zero), c 2 is infinite unless 



2 v 7T i ( — h A. j = a 4 + \ + 2 k ir i = ; 



which is impossible if m is so chosen that it shall not contain jx — v. 



Therefore, whether a is rational or irrational, if a. + b, = - — ± 



« — 1 



(where m is an integer which if « is rational and equal to - does not con- 



V 



tain /a — v), c 2 is in general * infinite, and consequently T h T a cannot be 

 generated by an infinitesimal transformation of our group; i.e. T h T a is 

 then singular. 



Among the singular transformations of our group obtained by putting 



, 2 miri .. . . . , , m, 



a , + o 4 = - — — 4 1 (where it « is rational and equal to -, the integer 

 a — 1 v 



m does not contain /x — v), let us consider those for which, further, 

 a 3 = b,, = 0. These singular transformations are defined by the 

 equations 



(5) 



2amiri 



(Jf*0)- 



The singular transformations T defined by equations (5) leave invariant, 

 as a whole, the system of lines x = const., but change each line into 



* I. e. provided "? {e a ( a * + b J - e a * + ab *) + f (e ab * - e 6 *), which in this case 



«4 "4 



becomes ( — e a * — ■— ) (e 6 * — e abi ), is not 



zero. 



