  We study the free energy landscapes of a pair of submicron spherical
particles floating at the surface of a sessile droplet. The particles are
subjected to radial external forces resulting in a deformation of the droplet
shape relative to the reference shape of a spherical cap. This deformation
leads to tangential forces on the particles. For small deformations and for the
contact angle $\theta_0$ at the substrate being equal to $\pi/2$, the
corresponding linearized Young-Laplace equation is solved analytically. The
solution is constructed by employing the method of images from electrostatics,
where each of the particles plays the role of a capillary monopole and the
substrate is replaced by a virtual drop with image charges and by imposing the
conditions of fixed droplet volume and vanishing total force on the droplet.
The substrate boundary conditions determine the signs of the image capillary
charges and therefore also the strength of the tangential forces on the
particles. In the cases of an arbitrary contact angle $\theta_0$ these forces
are calculated numerically by employing a finite element method to find the
equilibrium shape of the droplet for those configurations in which the
particles are close to the local free energy minima.
