•294 
MR. E. W. BARNES OX THE THEORY OF THE 
Substitute now the value of oj^) obtained in § 21, and we find 
+ cod 
r.-i (z) 
= exp. 
ZZCOi + CO 
'-CO,CO 
~ {lo^' (0.1 -}“ — loo’ (<^1 + C0.1 )1 + "ir-2 
X u 
exp. 
+ COi 
2.i(Oj + co^~ 
2(o^(Ui 
loo- 'll 
o 
ji n' + (?/p + 1) + 
nj=0»i2=e 1 ~ "T //pcOj + 'in.2CO^ J _ 
exp. 
2 rco, E (0.1 
-COiCO., 
floo’ (Oj (^1 ~b ^■■’)} ~b cojy.i.i 
. r/21 CO.) 
X 
r 2,iC0j + cop 
[ 2coiCOi 
71 d 
log n — tl' 
r^[;g + {ii + 1) • {cOi + <^3) I _ 
Ih [" + ('/2 + 1) COj I C0|>] + (?t + l)cOn I C0i]_ 
by the employment of the identity of § 18. Their principal values must throughout 
be assigmed to the logarithms. 
But, as has fjeen seen in §21, from the formula obtained in the “ Theoiy of the 
Gamma Function,” §41, we have when \z\ is large and 2 : not in the vicinity of the 
axis of —6J, 
log Ih (2 + a I co) = ^ '' — i) {log 2 + -ZIcttl] - - + log pi((o) 
CO ' CO 
+ terms which vanish when is infinite, 
where 1: = 0, unless 2 lies within the reo’ion between the axes of —1 and — co, in 
which case = d: 1, tbe upper or lower sign being taken as I(co) is positive or 
negative. I’he principal value of log 2 is to be taken, and the prescription to fie 
given to log ri (2 + «| w) is left indeterminate ; it is obvious that we only get additive 
terms involving 27rc, which vanish in the secpiel. 
Inasmuch as when n is large, none of the points 
2 “F (d “h l)( 6 jj “h ^•’)) ~ "k (d “k 1) and 2 -F ~F 1 ) cj.i 
lie in tlie vicinity of the negative direction of the axis of co., we may substitute the 
Ik^kr + a)j) 
values given liy the asymptotic expansion in the expression for 
We shall find 
F.,-kd 
G ^ (2 + Ml) 
rw (2)l\(,da)i) 
exp. Lt 
+ COi" 
71 = cc '•COiCO^ 
log' /ico. “F log cjj — (^1 ~k “k cj2'y.i.i 
^ ^ - + ( --ijlog + ^h) - - 
0 0 
o 
CO, 
1 , 1 / .2 + (li + 1) &)., , , , 
log p-,{co.) — 1 -- — A I loG' 7cCO. 
CO. 
/z + (n;+l)coi i\i 
(---- i ) log ’^"1 + 
V CUo / 
iljcOi + CO.) 
CO. 
-k 
+ 0 ^' + 1) («1 + f«o) 1 \ 
-- — A ~ 2 
CO. “ / 
o / ^ "t ( 'll + 1 ) Wi 
- 1F1 
TTl 
CO. 
