Rotational-translational addition theorems for the scalar spheroidal wave function
ψ
m
n
(
i
)
(
h
;
η
,
ξ
,
ϕ
)
\psi _{mn}^{\left ( i \right )}\left ( {h;\eta ,\xi ,\phi } \right )
, with
i
=
1
,
3
,
4
i = 1,3,4
, are deduced. This permits one to represent the
m
n
t
h
m{n^{th}}
scalar spheroidal wave function, associated with one spheroidal coordinate system
(
h
q
;
η
q
,
ξ
q
,
ϕ
q
)
\left ( {{h_q};{\eta _q},{\xi _q},{\phi _q}} \right )
centered at its local origin
O
q
{O_q}
, by an addition series of spheroidal wave functions associated with a second rotated and translated system
(
h
r
;
η
r
,
ξ
r
,
ϕ
r
)
\left ( {{h_r};{\eta _r},{\xi _r},{\phi _r}} \right )
, centered at
O
r
{O_r}
. Such theorems are necessary in the rigorous analysis of radiation and scattering by spheroids with arbitrary spacings and orientations.