MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS.
165
Now put b=2, then
2a

7rg 2
a
cos2
Cot'
lsOia
and putting oa=n+?, we have
( G?)= v
4X cos'Q+0cotwOgOi(l+s)n
whsencewe find, from series IV.,
1?~ ~ ?73+
2
1i
G2
+ &c.
2
22
3j 'i
(7r5i
=2
I
4
cos
COS cot20g2~sgcos CsSin @ g2zxdpdO
gcosO ?i(6+0)
2~~rJ2
.
. c(since
2
f
4
serieswe
have
X j'wcosff cotO ? cosCO (sinp) COs6COB(0+dd cos{it cos Osin (O8+) + 2 + 2
}
O
73P

7
3 .4
2
72+
i2
(1g2
2 Vvp+)
.
Let us again consider the series
c7 r+1)
x.2
we(ez+1)(++2)
hav3
Then making use of the integrals
r( +n)
dz, and
2
+n)=+
2
where (h) is a constant quantity, we find as the sum of this series,
an~d
r~*2h ^a
2C
dvdz za1kiz
(l+
T"
~v;
also when i0 is an integer, we may find the following expressions as the sum of the
same series a
rl 10x00 'r
XZC
Ml
ddzZOC1~cos(csin0) ecosO ((31)iO e
r
a2r
(I
+r
vih0s
and also
ra
,

h dhdz
ki c (I+csin Io
(3l)o
MDcccLv.
2A
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