Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/155

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102.]
MINIMUM VALUE OF .
115

found for a particular position of . To find the form of the function when the form of the surface is given and the position of is arbitrary, is a problem of far greater difficulty, though, as we have proved, it is mathematically possible.

Let us suppose the problem solved, and that the point is taken within the surface. Then for all external points the potential of the superficial distribution is equal and opposite to that of . The superficial distribution is therefore centrobaric[1], and its action on all external points is the same as that of a unit of negative electricity placed at .


Method of Approximating to the Values of Coefficients of Capacity, &c.

102.] Let a region be completely bounded by a number of surfaces , &c., and let be a quantity, positive or zero but not negative, given at every point of this region. Let be a function subject to the conditions that its values at the surfaces , &c. are the constant quantities , &c., and that at the surface

(1)

where is a normal to the surface . Then the integral

(2)

taken over the whole region, has a unique minimum when satisfies the equation

(3)

throughout the region, as well as the original conditions.

We have already shewn that a function exists which fulfils the conditions (1) and (3), and that it is determinate in value. We have next to shew that of all functions fulfilling the surface-conditions it makes a minimum.

Let be the function which satisfies (1) and (3), and let

(4)

be a function which satisfies (1).

It follows from this that at the surfaces , &c. .

The value of becomes

(5)
  1. Thomson and Tait’s Natural Philosophy, § 526.