The triple integrals
\[
W
1
(
z
1
)
=
1
π
3
∫
0
π
∫
0
π
∫
0
π
d
θ
1
d
θ
2
d
θ
3
1
−
z
1
3
(
cos
θ
1
cos
θ
2
+
cos
θ
2
cos
θ
3
+
cos
θ
3
cos
θ
1
)
W_1(z_1)=\frac {1}{\pi ^3}\int _0^\pi \int _0^\pi \int _0^\pi \frac {\mathrm {d}\theta _1\mathrm {d}\theta _2\mathrm {d}\theta _3}{1-\frac {z_1}{3} (\cos \theta _1\cos \theta _2+\cos \theta _2\cos \theta _3+\cos \theta _3\cos \theta _1)}
\]
and
\[
W
2
(
z
2
)
=
1
π
3
∫
0
π
∫
0
π
∫
0
π
d
θ
1
d
θ
2
d
θ
3
1
−
z
2
3
(
cos
θ
1
+
cos
θ
2
+
cos
θ
3
)
,
W_2(z_2)=\frac {1}{\pi ^3}\int _0^\pi \int _0^\pi \int _0^\pi \frac {\mathrm {d}\theta _1\mathrm {d}\theta _2\mathrm {d}\theta _3}{1-\frac {z_2}{3}(\cos \theta _1+\cos \theta _2+ \cos \theta _3)},
\]
where
z
1
z_1
and
z
2
z_2
are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases
z
1
=
1
z_1=1
and
z
2
=
1
z_2=1
, respectively. In the present paper simple direct methods are used to prove that
{
W
j
(
z
j
)
:
j
=
1
,
2
}
\{W_j(z_j)\colon j=1,2\}
can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of
z
1
z_1
and
z
2
z_2
. It is also shown that
W
1
(
z
1
)
W_1(z_1)
and
W
2
(
z
2
)
W_2(z_2)
are related by the transformation formula
\[
W
2
(
z
2
)
=
(
1
−
z
1
)
1
/
2
W
1
(
z
1
)
,
W_2(z_2)=(1-z_1)^{1/2}W_1(z_1),
\]
where
\[
z
2
2
=
−
z
1
(
3
+
z
1
1
−
z
1
)
.
z_2^2=-z_1\left (\frac {3+z_1}{1-z_1}\right ).
\]
Thus both of Watson’s results for
{
W
j
(
1
)
:
j
=
1
,
2
}
\{W_j(1)\colon j=1,2\}
are contained within a single formula for
W
1
(
z
1
)
W_1(z_1)
.