356 
.ME. E. AV. BARNES ON THE THEORY OF THE 
And tlierefore 
1)1 =■ 1 /<t= 1 
l( )g’ II n 
;■ = 0 s =0 
/ rwy + so)n' 
m 
r.T (a + 
pAu)i, Wj) 
— {m m')277(,(l — 
+ oS/(?7ia) log m + log 
lA {run) 
which is the result required. Tliis result has of course only been proved by means 
of the contour integral, under the assumption that a is positive with respect to the 
w’s. To establish it in general \ve should appeal to the principle of continuity. 
§ 66. Before concluding the multiplication theory, we deduce expressions for the 
values of 
, = 1,2,3 
. = n ' m / 
^1 = 0 5 = 0 
We recall from § 29, that within a circle of sufficiently small radius surrounding 
the origin, we have 
log lh(:) = — log ': — '-yo. — S N' 
_L_ V V' - _ 
303 T - - • • • 
Again, finm the multiplication theorem of the preceding paragraph, we have, by a 
similar expansion, 
log ro(7rM) = — {in + m')'lTTL{i — ?>r)oS/(c) + (1 — ?a~) log po(wi, lo.;^ 
log m 
ySi'(o) + mz yS/“-’(o) + “ iSf 3>(o) 
I/i—] a? —1 
+ log n IT' To 
2) = 0 (^=0 
pcoj + yWo 
—1 ))l—1 
ru 
+ s 
y) = 0 (/=0 l_ 
lxfj.2 
( 1 ) 
+ log 
»' J 
m ' 
+ V'b 
Combine these two theorems, and equate coefficients of various powers of s in the 
resulting identity. We tind 
//f 
y.,(wi,wA = -1 0^0 (o) log ~ N N 
J nl — L ,, = o r, = 0 
Ill 
■III 
p = 0 5 = 
.E s' it a t"- + 
111 " - 1 y) = 0 5 = 0 
m 
and, when r is greater 
9 
"y* ”y'\yy) /TBbAlV": 
p = 0 '/ = 0 \ 
= 0 „!., = 0 
