352 
5IE. E. AV. BARNES ON THE THEORY OF THE 
We thus have 
//l —1 7/(—1 
/ (1) / \ K' ^ ; (1) / I 1 I ("l + "A 1 
m xjj,, (mz) = N 2 (a + ---) - log m + m log rn. 
j)=o q=o 
Integrate again with respect to 2 :, determine the constant by making 2 = 0, 
and remember that : we obtain the formula 
;=0 i 
"ff n‘ r., (. + + *“■' 
m r, (mz) = — 
~ ’ ,„-l rn-l 
'III 
™ * _ / rW] + sco.z 
11 W T, ( 
r=0 s=0 ~ V 
g-.,So (;«;) log ,n 
the principal value of the logarithm Ijeing taken. 
§ Go. By means of the extension of Stirlexg’s theorem to two parameters (§ 50), 
w-i i.i-i 
it is possible to express the form 11 If' F., ( ' ^ ) which has thus arisen in terms 
r = 0 
Nt 
of the double Stirling function oi.z)- 
For, in the limit when n is infinite. 
( 2 ) = 
11 rr 
0 0 
1 + 
n 
•) n m -i 
and therefore, under the same condition, we oljtain from the result of the previous 
jjaragraph 
/it 'd -r 7/1 —• 1 
II n' 
0 
Td 1 
//}: 
mz , rd'^: 
1 +”-) 
/ rwj + 5W2\2 
V_ rd _ } 
'nie log 'll 
Hi — 1 nl — I 
m 
' /'Wi + .SO)., 
- 1 - n , v-ft), + scooA rn 
n n 2 + —^1 c 
r = 0 s = 0 ’ ' 
TO)^ + SCO:,-. 
z + 
'd n 
X n II ) \ 1 + 
0 0 
-/I, W2 
m. 
o 
■N' 
Tojl -h / '/’Wj + 
}il y" Yd / 
_ - j 1 -i - 
r* T c r\> 
Make now 2 = 0 , and we find 
tn — 1 //< — 1 
r[ II' T-^ 
* "* * _i /rwj + scoA ^ J /rtOj + '' '' 
,■ = 0 3 = 0 ~ \ 
X Exp. 
, /'I'co-i -|- SO)., 
where ' ~ ■ 
= n 11 
,• = 0 S = 0 
III 
—A n n'li + 
0 0 
?'&)) + SO)^ 
m 
o 
- ^ _ yA +. (v%. 
■III 
0 
//l] Uln 
/ \ 0 
Hi) «i.i 
711 
\ - .. 7‘ = 0] . 
j denotes that in the product the term for which _ q| simul¬ 
taneously is to be excluded. 
