106 PROCEEDINGS OP THE AMERICAN ACADEMY. 



(1) 



and the transformation T b by 

 (2) 



Therefore, if T h T a — T e , 



«, + ^ + 2k^« C e „ fl3 _ a, 



e «3 + 6. _ 1 ( 



->»■ 



c z = a 8 -f b B + 2 k n i. 



For finite values of the o's and //>, every branch >>t c is finite, and at 

 least one branch both of c t and of c, is finite. For r, and c, can only be 

 infinite for a a + b a = 2 m n i {m an integer) ; but if <i, + b t = 2m n /. 

 the branches of c, and c 2 corresponding to k- = — m are finite, being equal 

 respectively to 



p, (c », _,) a + ( a _,)]=^ ( ^ -Dg;-i). 



<j 3 -f 6 3 = 2 m ir i 



and 



^ ( „_ 1) | +(e »._ 1) |] = ^ (e ._ 1) (l_g = ^ ( ^_ 1) g4 : ). 



« g + b. t — 2 m 7r (' 



7. &(*)?> • • • ^r-i(- r )7» yy> 



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

 dors not contain singular transformations. For, if >• = p > 3 the trans- 

 • irmations 7], are defined by 



= x 





