In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval
0
⩽
x
1
⩽
1
0\, \leqslant \,{x_1}\, \leqslant \,1
, the second for the interval
0
⩽
x
2
⩽
1
0\, \leqslant \,{x_{2\,}}\, \leqslant \,1
, and each containing the parameters
λ
\lambda
and
μ
\mu
. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by
(
λ
j
,
k
,
μ
j
,
k
)
({\lambda _{j,k}},{\mu _{j,k}})
and
ψ
j
,
k
(
x
1
,
x
2
)
{\psi _{j,k}}({x_{1,}}{x_2})
, respectively,
j
,
k
=
0
,
1
,
…
j,\,k\, = \,0,\,1,\, \ldots \,
, asymptotic methods are employed to derive asymptotic formulae for these expressions, as
j
+
k
→
∞
j + k \to \infty
when
(
j
,
k
)
(j,\,k)
is restricted to lie in a certain sector of the
(
x
,
y
)
(x,\,y)
-plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the
ψ
j
,
k
(
x
1
,
x
2
)
{\psi _{j,k}}\,({x_1},\,{x_2})
.