100 PROCE OF nil: AMERICAN ACADEMY. 



where k and it are integers. From Aese equations it follows that an 

 is an integer. Th< refore, if a is irratioual, k = 0. Ou the other hand, 



it a is rational and equal to , where /x and v are integers relatively 



V 



prime, K = A. r. where A is an arbitrary integer. 

 V\ e also derive from (8) and ( I) 



°i + ^4 + 2 K 7T I C a, V #- ) 



#/, + ft, + "J K 77 I j O, « ( ff4 + fc 4 ) n< + a 6 4 6- , a 6, 6 4 ) 



= </> (a, &), 



_ fl 4 + Q 4 + 2 K ff t / ^ a (fl. + b t ) 



1_ e «(«4 + *«)_ 1 l"(a 4 + 6 4 + 2kit0 9 («- !) 



"4+^4 1 , \ , "- " ■ / a ( a 4 + ^4) "4 + afc 4 o/» 4 



— ae —l + u)-\ Y7~ ~^T\ \ e — ue — e 



+ „ e '»«) + 2 (.-«* + *> - e a6 «) + - ^^ ( e a& « - „ e 6 < 



-1 + .) + £(."* -1)}. 



ft' for finite values of the o's and />'s. while some of the c's. may remain 

 finite, one (or more) becomes infinite in all branches, there is no 

 infinitesimal transformation of the group that will generate '/' /' . i.e. 

 T,, T . i- a singular transformation. 



Let a be irrational. Then k = ; and for all finite values of the a 'a 

 and 6's, c 4 is finite. But, if « 4 + 6 4 = 2 m it i for some integer m ={= 0, 

 c 3 is infinite, provided 



*> p + *4 _ ^ + £ ( /4 _ 1} = (+ _ bj\ ^ > Q 



Similarly, if (« — 1) ("., + 1\) = 2miri ^ 0, c 2 is in general infinite; 

 and, if « (n { + & 4 ) = 2/»7ri ^ 0, c { is in general infinite. 



Let now a = ; then e. is, as before, finite. In this case, as stated 

 v 



above, k = Ar, where X is an arbitrary integer; and if a, -f b t = 2 wi 7ri 



:£ 0, c 3 , and therefore c,, are in general infinite unless 



2 7r i (wj + A v) = a 4 + ft 4 + 2 k it i = o, — 



