104 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Whence we derive 



^ = a a + *' • 



w -iL«t + «&/ J 



„ K + a.) r ck*) 5 / in 



+ t \, - w(a % +M ?+^ a )+^ aa ^(^)+ ea "v( & )T 



a ( o 8 + 6 3 ) 2 5 J 



Krom these ('([nations it follows thai <\ is Unite for finite values of the 



tt\ and 6's ; but if a„ + b :i = r}= 0, where « is an integer not /.< r<> 



a 



(u being cither rational or irrational), then Cj and c s are, in general, both 



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



that is, unless 



(5) -^- (e a "> _ 1) + A ( e «<«., + W _ e «<%) - 0, 



and 





+ ^(«) + a.e n "^(h) + e an 'i(,(b) = 0. 

 If (/)) is satisfied and (6) is not, we have c 9 finite and r, infinite to the 



"^ K' T ? ** /\" 71 J 



first order for o fi + />.,= - 4= 0. Therefore, if o. + 5. = - - :£ 0? 



where « is an integer, T/ t T„ is, in general, singular. 



Among the singular transformations of our group obtained by putting 



a f i 3 = - - 4 1 (* an integer), Jet us consider those for which, 

 « 



further, a., = b 2 = 0. Equation (">) is then satisfied ; and these singular 



transformations are defined by the equations 



I- If 71 £ 



a;' = a; H (k an integer ^ 0), 



(7) 



/ = y + J/e a * (J/i 0). 



