1. The boundary value problem of Laplace's equation for two spheres is a classical one, and has been the subject of discussion by many mathematicians (1). The earliest attempt to solve a boundary value problem of this type is due to Poisson (2), but his analysis is applicable only to the electrostatic problem. The first of the methods which can be successfully applied to both electrostatic arid hydrodynamical problems was developed later by Lord Kelvin (3); this procedure, which is known as the ‘method of images’, was first applied to the problem of the motion of two spheres in a perfect fluid by Hicks (4). Another method of great generality, that of transforming Laplace's equation to bipolar coordinates and studying the solutions in these coordinates, was developed about the same time by Neumann (5) and much later by Jeffery (6). More recently a new method has been developed by Mitra (7) for the solution of the problem of two spheres in a potential field. It makes use of two sets of spherical polar coordinate systems; the solution is expressed in terms of infinite series whose coefficients satisfy an infinite set of linear algebraic equations. The chief interest of Mitra's method lies in the fact that he has found it possible to derive exact solutions of this infinite set of equations. All of these methods suffer from the disadvantage that the potential function is obtained in the form of an infinite series so that any numerical calculations are rendered cumbersome.