An introduction to the mathematical theory of attraction ...

124 The Potential. If v be selected in such a manner as to be zero at each of the boundaries, and so as to make lI v v2u d5 positive, and if a be positive and so small that its square is negligible compared with its first power, we have Q+av < Qu. Hence, if,t' = tu + av, the function u' satisfies the same boundary conditions as u, but Qu, < QU. From this it follows that if a function u satisfy the boundary conditions, we can always find another function u' satisfying the same conditions, and such that Qu, < Q, unless V2a = O throughout the field of integration. Now Q, is essentially positive and cannot therefore be diminished without limit. Hence a function < exists which satisfies the boundary conditions, and is such that Q&, cannot be made less, but if this be so we must have v29 = O throughout the field 5. Again there is only one such function. For if there were two, p and 0', and if we put < - p' = X, we should have X = 0 on the boundary, and V2X = throughout the field, and therefore have X zero. Hence < = <'. A similar theorem holds good for a plane. It is to be observed that the validity of the proof given above for the first part of Thomson's theorem is not admitted by Weierstrass and other eminent mathematicians. 71. Surface Distribution.-It is always possible to distribute mass over a closed surface S so as to produce the same potential in external space as a given distribution of mass M[ which is inside the surface, or the same potential in internal space as a given distribution M which is outside. To prove this, let ~i be the region outside S, and ~2 the region inside, v, the normal to S drawn into @i, and v, the normal drawn into 2; then, if V be the potential due to M, by Art. 70, there is a function < such that < = V at S, and is of the order at a point P at infinity, where R is the distance of P from the origin, and that v2< = O throughout 5i, and a function 4 such that = V at S, and V24 = 0 throughout ~2.

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Title
An introduction to the mathematical theory of attraction ...
Author
Tarleton, Francis Alexander.
Canvas
Page 122
Publication
London:: Longmans, Green & Co.,
1899-1913.
Subject terms
Attractions

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"An introduction to the mathematical theory of attraction ..." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abr3212.0001.001. University of Michigan Library Digital Collections. Accessed June 23, 2025.
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