21)8 
iMR. E. W. BARNES ON THE THEORY OF THE 
Since m or m always vanishes, we see, on combining these results, that in all cases 
73,3 (cOj, ^3) = 
'rt=co L 0 0 
( 2 yi, 4 1 ) , (2/1 + 1 ) Wj — W, [ni 
+ -^ log nco. s--- log nwi 4-2-1-) (?i + 1) 77t 
NS- (n + t) ----- log n(a/i -H C03) 
^CO^O) 
= Lt 
1^-2 
^ V/ 1 "1 + 1 
S N — — V-- log n 
0 12 2u >^, ° 
-COjO)^ 
\a>o ! 
0 
1 
— (/i 4- 1) ~ (log (Wi 4 - "2) ~ log Wi — 27/1771} 
Wo 
— {ix -{- 1} — {log “h ^>3) — log cJo — 2/)/ 77 t} 
w, 
H-o {log ("l + — log Wi — log W3} 
2a)^a)o 
Wo 
But log(wi + W3) — logwj — 2///7r(, = log ^1 + 
log ("1 + “ log Wo — "2111 m = log [^1 + “j 5 
the principal values of the logarithms being always taken. Hence 
733 {w^. Wo} — Lt 
A 1 Wi + Wo , 
S S ^-o ^ log n 
_o 0 22 “W^W^ 
’^ + iiog(i+fk\_^L{;iiog(i + ^i 
Wo 
W, 
W. 
lo 
g Wi — 
log Wo) 
As a coi'ollarij, we see that, when n is very large. 
tl 71 
V s'' 
1 
n 
_ _ - is intinite like Iqo- — — log 
= 0 7 ,i, = o7//iWj + 7//oWo 2 W 3 W 0 Wo ^ 
Wi 
^logf] 
w 
1 
+ ~ 
Wo 
■ 
1 
fi ''' '.’n^ 
§ 24 . We now determine the constant C in the expression 
IV' ( 21 Wi, W 3 ) = Ce"^. c. n ir 
///, r= 0 REj = ^ 
by the condition assigned in § 19 , tluit 
\jt {z I fo)j, Wo) — 1. 
c = 0 ~ 
This at once gives us 0 = 1. 
It is evident that, witli the conditions of ^ 19 , one and only one function can be 
constructed, and this is the doul/le gamma function ro(2;|ajj, Wo), which is such that 
