320 



Consequently the lower limit of a^k+i must be zero, and tliat 

 agrees with the fact that " tends to infinity, for Stifxtjes shewed 



Cn 



tiiat in that case no upper limit can be assigned to 



(Xn (Xr)X.\ 



The principal conclusion, however, is that the series ^«2/1+1 







diverges, that therefore the continued fraction 



11 11 11 11 11 

 \a^z I a, \a^z | a, \a^z 



converges except when z is real and negative, and that it is equal 

 to the integral J {z). Thus then, putting z^\, we have proved that 

 6' — \ can be expanded in the continued fraction 



1 I 1 I 1 I 1 I 1 I 

 I +1 +1 +1 +1 +•■•' 



l«l l«J l«I l«4 l«6 



the quantities at being rational and positive, whilst those of odd 

 index have the lower limit zero. More or less we are inclined to 

 believe that a fraction satisfying these conditions cannot represent 

 a rational number, and so the expansion of C' — \ again suggests 

 the conjecture that C' must be irrational. 



The result obtained is of no practical value; that after some 

 reductions we have 



1 6 79 2410 262445 



1— ' + 1-' + 1—' + 1 ' + I 



12 5 42 79 2651 



. I =: , — I + l-l + ,— I + l^^^l + l^^^-^l + . . . , 



is of small service in the evaluation of the constant, and though 

 numerator and denominator of any convergent can be expressed in 

 the BernouUian numbers, in approximating the constant C' other 

 methods. are to be preferred. 



1 



