338 
mi E. W. BARNES ON THE THEORY OF THE 
Utilising the results of §§ 49-52, we find, when a is positive with respect to the cu’s, 
(- r)-i{loo-(- z) + 7} 
-logr.O) = LOUf 
877 JL (1 — 
dz — 2M7ri.iS/ («) 
+ [oS/-^(c< + w) — (w) l0gp)&j] + Fn[oSQ(« fi- 0})\0gp(x)] 
+ F>g ( 0 ) — oS/ («) j — log (ojj^, 6p) 
-]-«[— oS/"’ (o) 2 (»i + m) TTi + ( 0 ) log n — Fn{hSf'-^ (oj) logpoj} 
— ?tFo{oS'®^ (aj)^taj logp/oj] ] 
+ ~ [ — { 0 ) 2 (m fi- in') TTL -}- ( 0 ) log n — FoUS^ '^^ (w) logjtw} ] | , 
the logarithms having their principal values with respect to — oj^). 
Now oS/ (a) = oS/ ( 0 ) + ( 0 ) + f ,8/^^ ( 0 ), 
for the higher terms in Taylor’s expansion all vanish. Hence 
i _Z 2 h0_ _ i. f u"'~(--^UM]og(- + 7 } 7 
p .2 ("1 > " 3 ) 2 '^ J h (1 “ (1 ~ 
-j- .iSq (rt) 2 (jVI -j- VI -|- VI) TTL -f- .i8| (^ 0 ) 2iM7rt, 
which is the expression which has been jtreviously obtained in § 45. 
§ 54. We next proceed to consider the values of the double liiemaim ^ fimctiou, 
^o(s, a 1 w^), when s is a negative integer. 
By the definition of § 39, 
(«,«!<»„ a,,) = AiLV;)<,=M». 
277 
. f _ ez 
L ri - fi- 
c~“~ (— 
L (1 — c-“0 (1 — 
dz. 
a being ])ositive witli respect to the &j’s. 
Now in § 15 it has been proved that 
(1 - 6;-“r)(l 
Therefore 
—i'iU_ _ (0, / X _J_ 2^/ (^0 . 1 
// ; 
4 (,s «1 0 ,, ,[ ( _ 3 ).-^ ,h ] U; - (“) 
Ztt j I ct)]^a).T 
n ; 
the latter integral being taken along a small circle surrounding the origin. [Since 6‘ 
is an integer, the two line integrals in the standard reduction of ^ 4G destroy one 
anotlier.] 
