Open AccessOn a form of the solution of Laplace's equation suitable for problems relating to two spheres
Author(s) -
G. B. Jeffery
Publication year - 1912
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1912.0063
Subject(s) - laplace's equation , spheres , laplace transform , curvilinear coordinates , mathematical analysis , mathematics , plane (geometry) , cartesian coordinate system , function (biology) , boundary value problem , geometry , physics , astronomy , evolutionary biology , biology
The problems presented by the motion of two solid spheres in a perfect fluid have been attacked by various writers. In each case the method has been that of approximation by successive images, and it appears that no general analytical method of solution has been developed as in the case of the analogous problems for the sphere, ellipsoid and anchor-ring. In this paper a general solution of Laplace's equation is obtained in a form suitable for problems in which the boundary conditions are given over any two spherical surfaces. A similar solution is obtained of the differential equation of Stokes’ current function. With the aid of these results it is theoretically possible to determine completely a potential function when its value is specified over any two spheres. The method is illustrated by an application to the electrostatic field of two charged conducting spheres. In this case the method leads to a simple expression for the capacity of either of the spheres. The co-ordinates employed are defined by rotating about thez axis the system of circles, in any plane, through two fixed points on the axis and the orthogonal system of circles. Thus, ifx, y, z , are the Cartesian co-ordinates and ρ = √(x 2 +y 2 ), and the distance between the fixed points is 2a , we have a system of orthogonal curvilinear co-ordinatesu, v, w , whereu +iv =logρ +i (z +a )/ρ +i (z -a ),w = tan-1 y /x .