MK. E. W. BAKXES OX THE THEOEY OF THE 
358 
In the relation which has been obtained, change s into s + oijp and subtract the 
first residt from the f)ne so formed. We find by § 23, 
log 
fti.iX 
G (a 
9 / 
Pi 
CO-: 
2mp,q in S/ ( 2 ; ~~) = {^) “ 
+ log 1 / -j ^ ^ ^ y ! "0 f” — 2 ot m 
Pi ("j) 
-I 
,<=o 
7 
.<j = 0 
' 9. / 
f ^ 
does 
here m has the value asinned in ^ 22 , and where m,,,,. = U, unless y 
o O' I ^ 1 Wt o)., , 
-—-■ does not 
V ■ 9 
lie between — t and ——. in which case = fo 1 as is positive or negative 
Now (“ Theory of the Gamma Function,” § 7) 
log 11 Fj ( 2 + — log Fj ( 2 
s = 0 ' 9 
+ 7 log Pi (old ~ log Pi ( , 
and by § 18 of the same paper. 
8/(2 Sj (2 + ^ 
=0 9 
01.1 
We therefore have 
Similarly, 
t) ^ - m) S/ 
X2 “ X2 (•) = (p 
oil 
9 
op 
P j 
where m' has the value assigned in § 21 , and difiers from it in that poii + yoi^ 
must in the definition be substituted for oi^ -j- 01 . 1 . 
Now we have seen that (§ 22 ) m — vi = dz 1 , the upper or lower sign being taken 
0 ): 
as I j is negative or positive ; and, since p and 7 are positive integers, the same 
is true of — m' 
Thus 
m 
'p^q 
— m = in'p^j — m = pp,g (say), 
and now y.i ( 2 ) satisfies the two difference relations 
X2(" + ^') - X2 i^) = P/i,? 
i / 
1 
ft) 
