n40 
Thus 
s = 1 _ 
.AIT. E. AV. BAEXES OX THE THEORY OF THE 
a j oj^, CO.,) + 
s — 1 
— ~ ^'' 2(^0 + 2(m + m')77tnS/'-(a). 
Again differentiating, we have 
t r _c 
27r J L (1 — 
Also, if e be small, we have 
e z)(h 
e “!') (1— c '^■^) 
^.(2 — e, a j 0 )^, CO.,) 
_ t r(e — 1) _2 
2 MeTri 
'.IT 
zf-^dz 
JL (1 — “ c~'^-^) 
1 T./ 1 \ (1 — 76 . . . )(1+ 6 + . . . ) 1 f 1 , /I N 1 
and I (e- 1) z= =--- ~ e ^ ^ • ' ’J' 
Thus 
y, I \ 1 I 7 — 1 + SM-TTi 
4.-.(2 — e, a ! oj,, co.,) = — . - -|-- 
ewj CO., cojco.. 
A r 6 «--(_ g)Iog(-,^) 
27rjL (1— 
ch, 
so that, since 
(a I 
we have 
O)^, CO 
^)=2bf 
Lt 
S = 2 
^ 0 ( 5 , nl&ij. Wo) — 
c z) log (— z)(b 
L (1 — 6“1®) (1 — 
1 
(s — 2 ) 
+ [2(M + w + + y]oS/=^'(a) , 
— 2(m + rn')77-toS/-'^^(a). 
0)1 Wo 
Tabulating our results, we see that 
C 2 (■% a I w^, Wo) = ^p.2^%a), 
when s>2 ; 
^ _g _ J . _ 2{m + 7H')7noSi<^^(a), .s— 2 ; 
6 — S Wj^Wo 
3^1^) _ _g 2(m + 7jf)7rnSi<-^'(«)> « = ^ ; 
S ““ X 
= ,S'i(a) 
= oS_j(<7) A 3Bi-«(wn Wo) 
,5=0 
, s < 0. 
These formulae hold for all values of the varialde a, though we have onff" established 
them for the case when a is positive with resj^ect to the w’s. They evidently agree 
with the results established for the case w^ = Wo in “The Theory of the Cf Func¬ 
tion,” § 34. 
§ 56. In a note appended to the “Theory of the Gamma Function,” it was stated 
that a theory of the simple Biemann ^ function had been developed by Mellix."^ It 
■*■ AIellin, ‘Acta Socictatis Feunicae,’ vol. 24, No. 10, 1899. 
