TfTE GAMMA, ok FACTdKIAL, FUNCTION. 643 



therefore, putting a- = o we get the convergent expansion 

 (when X <^ _ ) 



log Ux = yx +-^'^S,- -'"'-.S3 . . . (-)-:'-" S„ . . . 

 23 11 



where S„ = 2_jJ' 



I 

 To expand ITa-, 



i- log Ha- = n^ - f "'■'■y--- n"o = (n'o)^ + .S, = y^ + S, 



n"'x n'An"A- , /r'aAs . s, , c 



so 3 — — ^ — • + 2 ( — I = ( - )3 2 ! S, 



■■■ n"'o = 3y(y^ + S,) -2y3 = y3 + 3y 5^ + 253-253 



and so on. We add a proof that if IIa- is expanded in positive 

 powers of a, the ultimate remainder is comparable with (— a)-^ 



and therefore the expansion is possible if, and only if, a- <^ _ . 



The proof has some points of interest, but of course the case is 

 covered by the theory of radius of convergence. 



PROPOSITION IX. 

 Proof that Ux can be expanded in powers of a:, if a<^ _ . 



We have Ha = 



log^j de, if A' > - I, 



.-. n^x) =, £(log ^)'(log log -^ dB, 



= [ /^(log /)" . e-'dl. 

 Divide the integral into two parts / = o to / = i and / ~ i 



to I = CO . 



The first part may be written ( — )" /^(?"M log j I <//. 

 Let / = X", where a is some positive quantity. 

 We get (-)''a'' + ' I \/\. /■' + '- ""e-Ylog^j J\. 



