106 PROCEEDINGS OP THE AMERICAN ACADEMY. 



(I) 



and the transformation T^ by 

 (2) 



Therefore, if T^ T, ^ 2\, 



C3 = «3 + *3 + 2 K 71 ^■. 



For finite values of the a's and 5's, every branch of Cg is finite, and at 

 least one branch both of Ci and of Cn is finite. For Ci and C2 can only be 

 infinite for a^-\- bz = 2 7nni {m an integer) ; but if «3 + Jg = 2 m tt /, 

 the branches of q and c.^ corresponding to k = — m are finite, being equal 

 respectively to 



L*3 (e«3 - 1) ^ + (e*» - 1)1 = ^'' (^"' - 1) (^' - 1 \ 



03 + 63 = 2 m TT I 

 and 



03 + 63 = 2nvi 



The groap 



is likewise continuous for values of r > 3 ; i. e. for values of r > 3, it 

 does not contain singular transformations. For, if r = p > 3 the trans- 

 formations Ta are defined by 







/ ai-\-a2^o(x)-ir as4>p,(^) + •  4- flp-ic^p-i (a;) V "p_i\ 



