WILLIAMS. — FINITE CONTINUOUS GROUPS. 101 



that is, unless m contains v. Therefore, if a^ -{- b^ =■ 2 m tt i :i^ and 



V 4: 1, we can always so choose m that c^ shall, in general, be infinite 

 and Tf^ T^^ singular. On the other hand, if 04 + J^ = 2 m tt i, and if 



V = 1 (i. 6. if a is an integer), one branch of c^ is always finite, and the 

 same is true for c^ and Co : so in this case 7], T^ can be generated by an 

 infinitesimal transformation of our group. 



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

 tions of the group. For let 



, 2 m TT ^ 2mvTri 

 «4 + 04 = 



— 1 



p. 



Then in general (i. e. provided the function of the a's and 5's found in 

 the second factor in the expression for c^ is not zero), Co, is infinite unless 



h A. ) = « . + ^4 + 2 K TT ^■ = ; 



IX — V J 



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



, , . . , . . . .r. 7 2m7rz'^ 



Therefore, whether a is rational or irrational, 11 a^ + 6^ = -^ 



a — 1 



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

 tain fx — v), e.j is in general* infinite, and consequently Tf, T^ cannot be 

 generated by an infinitesimal transformation of our group; i.e. 7], T^^ is 

 then singular. 



Among the singular transformations of our group obtained by putting 



a 4- (^4 = 4^ (where if « is rational and equal to — , the integer 



« — 1 V 



m does not contain /a — i/), let us consider those for which, further, 

 «g = ^3 = 0. These singular transformations are defined by the 

 equations 



Irmri 



(5) 



y' =y e ''-1 + 31 X + ^V {M^ 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""^*** + **) - e"* + «*4) + ^ (e^^* _ A), which in this case 



«4 °4 



becomes (-"e"* — ^) (e** - e"*«), is not 



\"4 ^i^ 



zero. 



