316 
ME. E. W. BAENES ON THE THEOEY OF THE 
the series will, lu the language of M. Borel,* * be summable, that Is to say, ^^uthln 
this region (which Is conveniently bounded by straight lines from the points , 
— which pass to infinity through the remaining poles) it is possible from the 
^2 
values of terms of the series at any point to obtain, by the employment of 
intermediate functions, a magnitude, independent of these particular intermediate 
functions, which is the value of the function at the point. 
If then on any term of‘such a series we jierform the operation which is 
expressed by 
we shall expect to obtain a cpiantity which is the i;-th term of a possibly, and even 
probably, divergent sequence, which in turn is, by suitable operations, summable to 
the value which results from the performance of the fundamental operation on the 
function of which the original series is the expression. 
Such considerations being understood to underlie the operations, we liave 
_,s)r ^ “h t’” 
iir 
,r (1 - 6-) r (- z) 
-yV-3 
COi&Jo 
(1 — (1 — 
[-g-iOio,,. _j_ 
(- 
+ 
+ 
ir(i - .S-) 
tr(i - s) 
ilT 
tF (1 + s) - 
-TT ,„=I 
< + •'«— 2 
J Wj + Wo 
~ "2 
W^Wo 
^dz 
1 2w^W3 
[ Wj -f Wo 
C ~ Wol 
dz 
[ 2wiWo 
WjW2 J 
m ; 
|oS -|- a)[)e ’I- oS ,n {ci “b <3 ‘^'^\dz 
1 f 1 , 1 
(.s — l)(.s — 2 )&)i &)3 
1 ct)| "I- ct)o '2cc 
S — 1 2(0^0),2 
*-l + 
—L—1 
(qnoj.y ij 
+ 
1 r 1 
1 1 
l[£Uj ’ ^ ' Wo b 
X ( _ ) • (g + 1) • • • (g + r oSb(c + wd + Wo) I 
Now it may be readily deduced from the results obtained in Part L, tliat 
oS/„ (a) = ?n[oS„,_j((( -j- Wj) -f- oB„J 
* Borel, ‘ Liotiville,’5 Ser,, ^ol. 2, pp. 103 sey. ‘ Annales de Ecole Normale Superieure,’3 Ser., 
vol. 6, pp, 1 et seq. 
