constant radial force $g{\bf e}_r$ included in the time-periodic force and without surface tension. Our simulations of this case exhibit a periodic subharmonic translational motion of the sphere about its original position, as shown in figure \[fig:l=1\] and supplementary movie 1. This explains why the ${\ell}= 1$ case is prohibited for capillary waves: translational motion does not involve deformation of the interface, and so surface tension cannot act as a restoring force. (Note the factor of ($\ell-1$) in formulas and .) Although the drop arrives quite close to the bounding sphere in this simulation, it does not touch it. As stated by [@Busse1975], there is no competition between different patterns in this case. All solutions are obtained by rotation of a single axisymmetric pattern, which is expected to be stable. (The three spherical harmonics $Y_1^1,Y_1^0,Y_1^{-1}$ are also related by rotation.)

![Above: visualisation of ${\ell}=1$ mode for spherical drop. The drop is displaced alternately to the left and the right. Length and colors of arrows indicate the velocity magnitude (in m/s) of the surrounding air. Below: center of the drop as a function of time. Dots indicate the instants at which the visualisations are drawn. See also supplementary movie 1.[]{data-label="fig:l=1"}](fig7.eps){width="\textwidth"}

Case 