100 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where k and k are integers. From these equations it follows that a k 

 is an integer. Therefore, if « is irrational, k = 0. On the other hand, 



if a is rational and equal to - , where fx and v are integers relatively 



prime, k = A. j/, where X is an arbitrary integer. 

 We also derive from (3) and (4) 



o'i + ^i + 2 /< TT ^ C «3 . , , 6,^ K , T. . ) 



e"i + ^* — I C «4 04 ) 



«4 + ^4 + 2 K TT Z ( «2 . « ('^'^ + 64) O4 + a 64 Jo . a 64 J4. ) 



= (^ (a, &), 



«4 + ^4 + 2 K TT Z f 4> \p a{at + bi) 



Co — 



Ci = 



^a (04 + fti) 



■.TTl { <ji \p a 



"F^ («4 + ^4 + 2K^ir(a-l) ^' 



''l + ^4 -II \ 1 Cf2 ^3 / « (O4 + 64) (74 + a 64 a 6, 



— «e — 14-«;H — ^- -r (e — ae — e 



04- (« —.1) 



a 64. «l,a(a4 + 64) 064, ^2^3 /»64 A 



+ ae)H (e — e)-|- ^-r-r rr (e — ae 



04 O4" (« — 1) 



-1 + „) + *; (.-'-!)} 



If for finite values of the «'s and J'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 T;, T^, i. e. 

 T^ T^ is a singular transformation. 



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

 and S's, Cj^ is finite. But, if 04 + (^4 = 2 m it i for some integer m r}i 0, 

 Ca is infinite, provided 



«4 ^ O4 \«4 O4/ 



Similarly, if {n — 1) (^4 + ^4) = 2 m tt « 4^ 0, c^ is in general infinite; 

 and, if a (a^ + b^) = 2 m tt ^ =|= 0, Cj is in general infinite. 



Let now « = ~ ; then c^ is, as before, finite. In this case, as stated 



V 



above, k = X v, where X is an arbitrary integer ; and if a4 + J4 = 2 m tti 

 ^ 0, Cg, and therefore c^, are in general infinite unless 



2 TT ^ {in -\- \v) = «4 + J,j + 2 K TT ^' = 0) — 



