Mathematics. — -'On Eulkr's Constant". Bj Prof. J. C. Kluyvkk. 



(Communicated at the meeting of May 26, 1923). 



In calculating the value of Euler's constant 6' the summation 

 formula or any other asymptotic series is used, and one term at 

 least in the expansion is always a transcendental quantity. It would 

 be preferable to represent 6' as a convergent expression containing 

 rational terms only, because such a representation of the number 6' 

 perhaps eventually will furnish the means to establish its irrationality. 

 As yet Vacca's series ') 



f I 1 1 \ /I 1 



1 



17 ^ ■■ ~ 31 



is the only result in the desired direction, and as a second I will 

 add the proof that C — ^ can be expanded in a convergent continued 

 fiaction 



11 11 11 1 



,-' + ,-' + ,-' + |- 

 a, \a. a. \a. 



the quantities a,^ being throughout positive and rational. 



Following Stieltjes' method ') for converting an integral into a 

 continued fraction, I consider the integral 



J{z)^ 



supposing z ]> 0. Expanding the integrand in powers of-, terra-by- 



z 



term integration gives the divergent series 



'^~\^\ + '-"-Sr. + 



z z^ z' «"i-i 



the coefficients of which are determined by the equation 



1) Q. J. Math , London, vol. XLI, p. 363. 



-) Recherches sur les fractions continues. Oeuvres completes, II, p. 402. 



