336 
MR. E. W. BARXES OX THE THEORY OF THE 
But when 'p — q — 1, the expression on the left-hand side may be written 
lA 
/? = CO 
>\ ■/» 1 
S' V' _ 
n ^ 
0 0 
log n + log(wi + W,) - log coi - log O), 
“^1^3 
— (?i + 1) ( ^ log {wy + log loo' OJ 
\a)y COy 
(Oi 
01.1 
o 
Since the principal values of the logarithms with resj^ect to the axis of — (w^ + wn) 
are in all cases to be taken, we see from § E3 that this expression is ecjual to 
y.2o(coi, ojn) + 2^S/^'(o) 2 (m + m'jin. 
For denoting by a capital letter the logarithm which lias its principal value with 
res])ect to the axis of — 1, we have 
log + COn) = Log (Wj 4- (d^), 
log CU^ = Log OJ^ + 2mTTL, 
log Cdo = Log Wo + 2771 7TL. 
And therefore 
Id 
a = 0 
s = 1 
(.9, a I (Dy, Wo) + - 
oSpA'^) 1' 
= u 
^ - o ~ l^^g '' + ALAT ("1 + " 2 ) - log coi -- log Wo 
00 -^*^1^3 -W^ Wo J 
“ " 1 “ 1) ( “t" ) log ("1 d" ^ 2 ) d-- log ^2 d" log 
Wi Wo 
I ^ O I ^ o ' 
~r 271177 L -{- 2 111 77L 
W, 
Wo 
Wi 
Wo 
Wj -f Wo 
2wiWo 
l[m -F - 111)77771 
= yoi(wj, Wo) + oS/^^(o) ' 2(711 -h 111 ) 771 . 
We notice that this formula agrees with our previous results. For by § 47 
7 n("n " 1 ) d- 28 ^( 0 ) [(w + ?;f)27rt] 
~ _ L/ f f ^ , 1 
a = oL-'^-h (1— « 
JMttioSj ( 0 ), 
and by § 42 this last expression 
== L^ 
a = 0 
5 = 1 
o 1 w^, w 
1 1‘ 
s - 1 a. 
It is worth noticing that incidentally yoo (wj, Wo) has been obtained as a limit in the 
more general form 
yoij(wi, Wo) = lut 
% 1 Wi -t Wo o 
S X „-: - log n 
fi 2wiWo 
0 0 
d- d- log?M - log< 7 ^ 2 } 
^ " [over] 
