378 
MR. E. ^V. BARNES ON THE THEORY OF THE 
log 
= [L («) s +fi («)] log Z + 4 ., (a) z + 4 , (a) + S 
^2 (*) 
, = i r(r + 
where / («) and </>(«) are algebraical polynomials of degree indicated by their 
suffixes, and X'-(f') so long as r is finite, likewise an algebraical function. 
Now Ijy the fundamental difference equation 
1 Eol^ + + &)j) Fc, (3; + ft) 
mo' —=-= ioo’ —=- 
FoCO ^ r,(o 
— lo 
O’ 
Fj ( i + ft I CO.i) 
Pi (^ 2 ) 
-f 277l7ri (s “h (7 I CO.iJ, 
Again (“ Theory of the Gamma Function,” § 41), we have the asymptotic 
expansion 
log 
F-] (.? + ft I (Do) 
Pi (Wo) 
— 8/(2 + a 1 CO.,) lo 
O' 
^10-2 
(z + a I oj^) + 
where log„, 2 is that natural logarithm of s which has its principal value with respect 
to the axis of — co^. It is thus equal to 
log 2 — ' 2 m'TTL, 
the latter logarithm having its principal value with respect to the axis of —(wiff co.,). 
We have then, if log z have its principal value with respect to the axis of 
— (Wj + CO.,), 
[/i(« + +yo(a + wi) -fi{a)z -/o(«)]log 2 
+ + wj) — + f/>o(a + Wi) — (^o(«) + :£ ( —)'■ 
y*=l 
= — 8/(2 + a 1 (X).,) [log 2 — 2(m + Pi')7n] + 2 8 ['’(2 + a I w,,) + 
Xr (ft + Ml) — Xr{('- ) 
r (r + 1 ) 2 '- 
” ( —)'• S',+ifft[wb 
Ti r ( /• +1)^ 
If we equate corresponding powers of 2 on both sides of this result, we find 
X,(a + wi) — x.(«) = 8b+i(«lw.), 
and similar relations among the f’s and (^’s. 
We shall get, in like manner, another set of relations in which 00 ^ and co., are 
interchanged. Ptemembering that the_/’’s, x}/s, and y’s are all algebraical polynomials 
which vanish with «, we thus prove that 
Ji{a)z +./o(n) = 08 ( 1 ( 2 ) — 08 , 1(2 + a) 
(/q(a)2 + r/),(a) = 2 ( 08 , 1 ' (2 + a) - 38 o'( 2 )} 
Xr(a) = o8'„;^i(«) — 08T+1 (0). 
By tills process, which may appro})riately be called a ])rocess of finite integration, 
we olitain the asynq)totic expansion 
