304 
MR. E. AV. BARNES ON THE THEORY OF THE 
for we may suppose that the terms are represented by the corners of the small 
squares into which the positive quadrant is divided—in the new grouping we take 
together all terms lyijig on a line equally inclined to the two axes. 
O 
1 
\ 
A 
\ 
A 
\ 
/ 
A 
A 
Thus 
1 « /I 1 
- -= X ( - + -) + —^ 
0 0 (wq + e=i\e e’V 0*'+ 1) 
o + • • • + 
-f 
+ 
n 
_ o 
{n + 1)2 ' (/t + 2) 
+ 
So that when n is very large 
1 
71 n 
o H 
- = log n + y + ^ + I ^ 
1 
+ 
(w + nj- 
n — n 
{n + n)- 
:dx 
0 0 ^ ^ . o(l T x)~ 
+ terms which vanish when n becomes infinite 
7F • • 
= log n + y + + 1 — log 2 fi- simdar terms. 
Hence 
rn ") = hi 
log 6J - -- - 1 
y 
which is the first relation, 
In the second place we have, when and Wo are equal. 
ajy 23 (a), w) = Li 
0 0 5?q + ?/q 
and by the same method of reasoning as before 
1 " e -r i . n 
N N'-— 2n loo’ 2 — loe’ 2?? 
iwj , 
71 n 
_ _ V J: _p 
0 0 ''“'q + '^'q e = i £ n -\r \ 
V V 
1 n — ^ 
+ + • • ■ + 
n + H 
“1 1 1 
= X - + n H-- + . . . q- —X_ 
e=i e ?i- + 1 n + n 
+ « i 2 
1 +-— 
n + 
\ii + 1 
.^ + • • • + 
1 
11 + n 
— 71 -{■ log n y log 2 + ' 2 n log 2 — ^ — n, 
neglecting terms wliich vanish when n is infinite, and thus 
yoo (<y, oj) = - \y — h — log 
0) - c. - 
