We consider shallow elastic membrane caps that are rotationally symmetric in their undeformed state, and investigate their deformation under small uniform vertical pressure and a given boundary stress or boundary displacement. To do this we use the small-strain theory developed by Bromberg and Stoker, Reissner, and Dickey. We deal with the two-parameter family of membranes whose undeformed configuration is given in cylindrical coordinates as
\[
z
(
x
)
=
C
(
1
−
x
γ
)
,
(
1
)
z\left ( x \right ) = C\left ( {1 - {x^\gamma }} \right ), \qquad \left ( 1 \right )
\]
which includes the spherical cap as a special case (
γ
=
2
\gamma = 2
and
C
C
small). We show that if
γ
>
4
/
3
\gamma > 4/3
then a circularly symmetric deformation is possible for any positive boundary stress (or any boundary displacement) and any positive pressure, but if
1
>
γ
>
4
/
3
1 > \gamma > 4/3
then no circularly symmetric deformation is possible if the stress and pressure are positive and small (or for non-positive boundary displacement and small positive pressure).