scholarly journals Note on the electric capacity coefficients of spheres

1912 ◽  
Vol 87 (598) ◽  
pp. 485-487
Keyword(s):  
General Solution ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Electric Capacity ◽  
Physical Problems

In the 'Proceedings' of the Society, vol. 87, p. 109, Mr. Jeffery obtains a general solution of Laplace’s equation in a form suitable for physical problems in connection with two spheres. As an illustration he applies his solution to the problem of finding the capacity coefficients of two equal spheres, obtaining a result which he shows to be equivalent to one of Maxwell’s series formulæ. He then computes a table of the numerical values of these coefficients.

An Electrooptic Analog for the Solution of the Laplace Equation

1967 ◽  
Vol 34 (2) ◽  
pp. 452-456
Author(s):  
R. O’Regan
Keyword(s):  
Boundary Conditions ◽  
Potential Gradient ◽  
Mathematical Physics ◽  
Experimental Methods ◽  
Organic Dye ◽  
Voltage Gradient ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Physical Problems ◽  
Flow Heat

Laplace’s equation occurs frequently in mathematical physics for problems relating to fluid flow, heat transfer, and so on. For some simple cases, the boundary-value problem can be solved; but more often, the differential equation proves intractable, and numerical analysis or experimental methods are used. The electrooptic analog is an experimental method based upon the fact that an organic dye solution becomes birefringent in an electric field. This effect enables one to determine voltage gradient throughout a two-dimensional field. The boundary conditions most readily applied are prescribed constant values of the electric potential φ on conducting segments of a boundary and ∂φ/∂n = 0 on insulated segments of a boundary. With the known conditions on the boundary and the potential gradient found from experiment, the problem is solved. This analog can be used for all physical problems for which the boundary conditions are applicable, and which satisfy Laplace’s equation.

V.—On Whittaker's Solution of Laplace's Equation

1944 ◽  
Vol 62 (1) ◽  
pp. 31-36
Author(s):  
E. T. Copson
Keyword(s):  
Harmonic Function ◽  
General Solution ◽  
Arbitrary Function ◽  
Harmonic Functions ◽  
Singular Points ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the formwhere f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

scholarly journals Note on Whittaker's Solution of Laplace's Equation

1925 ◽  
Vol 44 ◽  
pp. 22-25
Author(s):  
E. T. Copson
Keyword(s):  
General Solution ◽  
Arbitrary Function ◽  
Regular Point ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

§1. Whittaker has shewn that a general solution of Laplace's Equationmay be put in the formwhere f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.

scholarly journals An Integral Formula For Qn (cos θ)

1945 ◽  
Vol 7 (2) ◽  
pp. 81-82
Author(s):  
E. T. Copson
Keyword(s):  
General Solution ◽  
Present Note ◽  
Integral Formula ◽  
Simple Expression ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Particular Solution ◽  
Image Position

The functionis, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

scholarly journals VI. Two-dimensional potential problems concerning a single closed boundary

1929 ◽  
Vol 228 (659-669) ◽  
pp. 235-274 ◽  
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Steady Flow ◽  
Applied Mathematics ◽  
Physical Significance ◽  
Two Dimensional ◽  
Laplace’S Equation ◽  
Boundary Values ◽  
Laplace's Equation ◽  
Potential Problems

Solutions of Laplace’s equation, ∂ 2 V/∂ x 2 + ∂ 2 V/∂ y 2 + ∂ 2 V/∂ z 2 = 0 . . . . . (1. 11) are required in many branches of Applied Mathematics, such as hydrodynamics, electro-and magneto-statics, steady flow of heat or electricity, etc. The two-dimensional form of the equation, ∂ 2 V/∂ x 2 + ∂ 2 V/∂ y 2 = 0, . . . . (1. 12) has a general solution V = f ( x + ɩy ) + F ( x – ɩy ), . . . (1. 21) f and F being arbitrary functions of their complex arguments. In the applications, one function alone is usually sufficient, and it is customary to write w = ϕ + ɩψ = f ( z ). . . . . (1. 22) with z = x + ɩ y , when ϕ and ψ usually have each some physical significance. Moreover, in most cases, the boundary conditions which have to be satisfied either are, or can be reduced to, the prescription of the boundary values of ϕ or ψ, of their derivatives.

scholarly journals On Spheroidal Harmonics and Allied Functions

1914 ◽  
Vol 33 ◽  
pp. 118-121 ◽  
Author(s):  
G. B. Jeffery
Keyword(s):  
General Solution ◽  
General Formula ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Spheroidal Harmonics ◽  
Image Position ◽  
Spheroidal Coordinates

In a paper recently read before this Society, Mr E. Blades obtained a general formula for spheroidal harmonics in the form of the general solution of Laplace's equation given by Professor Whittaker,If spheroidal coordinates r, θ, φ are defined bythe result obtained is

scholarly journals V. On toroidal functions

1881 ◽  
Vol 31 (206-211) ◽  
pp. 504-505
Keyword(s):  
Vortex Rings ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Physical Problems

This paper contains the development of a theory for functions which satisfy Laplace’s equation, and are suitable for conditions given over the surface of a circular anchor ring, and which therefore seem im portant in the possibility of their application to the theory of vortex rings, as well as other physical problems. From the nature of the case, it will not be easy to give an intelligent and full abstract of the results without making it unduly long, but it may be possible to give some idea of its scope and the method of development

scholarly journals On a form of the solution of Laplace's equation suitable for problems relating to two spheres

1912 ◽  
Vol 87 (593) ◽  
pp. 109-120 ◽  
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Fixed Points ◽  
Simple Expression ◽  
Laplace’S Equation ◽  
Current Function ◽  
Laplace's Equation ◽  
Spherical Surfaces ◽  
Method Of Solution ◽  
Solid Spheres

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 the z axis the system of circles, in any plane, through two fixed points on the axis and the orthogonal system of circles. Thus, if x, y, z , are the Cartesian co-ordinates and ρ = √( x 2 + y 2 ), and the distance between the fixed points is 2 a , we have a system of orthogonal curvilinear co-ordinates u, v, w , where u + iv =log ρ + i ( z + a )/ ρ + i ( z - a ), w = tan -1 y / x .

scholarly journals Generalised Solutions of Laplace's Equation

1931 ◽  
Vol 2 (4) ◽  
pp. 181-188
Author(s):  
H. S. Ruse
Keyword(s):  
General Solution ◽  
Riemannian Space ◽  
Present Theory ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Special Case

The present paper contains solutions of the tensor generalisation of Laplace's Equation. The results obtained are summarised in the two theorems enunciated in § 1. They apply only to the case when the Riemannian space forming the background of the theory is flat. In the concluding paragraph a special case is considered, and it is shown that the present theory is closely connected with Whittaker's well known general solution of the ordinary Laplace's Equation.

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