104 PROCEEDINGS OF THE AMERICAN ACADEMY, 



Whence we derive 

 Cz — as + K 



= n {a, b), 



ea(a3 + 63)_l|_ „ I \ a(a, + b,)J 



« (flSg + 63) 2 5 J 



From these equations it follows that c^ is finite for finite values of tlie 



a's and 6's ; but if 03 + ^3 = =[= 0, where k is an integer not zero 



(« being either rational or irrational), then Ci and Co are, in general, both 

 infinite in all branches (and, indeed, Ci is infinite to the second order), 

 that is, unless 



(5) -^ (e««3 _ 1) 4_ ir_ (ea(<'3 + &3) _ e-%) = 0, 



a O3 « 63 



and 



— C 



(6) - 



L V «(«3 + 63)y «(«3 + ^3) 2 J 



+ ,/. (a) + a, e'^''3 ^ (J) + g^^ ^ (5) ^ Q. 

 If (5) is satisfied and (6) is not, we have Co finite and c^ infinite to the 



^ K ^ 2 ^ K 71 7/ 



first order for «. + ^3 = ^ — =1= 0. Therefore, if a, + b„ = =^ 0, 



a , " a 



where k is an integer, Tj T^ is, in general, singular. 



Among the singular transformations of our group obtained by putting 



^ K 7T t 



ciz + ^3 = 4^ (k an integer), let us consider those for which, 



a 



further, a.o, = bo = 0. Equation (5) is then satisfied ; and these singular 



transformations are defined by the equations 



, 2 K ni . . , ^- 



X ^^ X -] {k an integer df 0), 



a 



(7) 



/ = y + i¥e"^ (J/i 0). 



