varying function of x, y, z, which vanishes at an infinite distance, and satisfies the partial differential equation (1), V is the potential of the attraction of the mass whose density at the point (x, y, z) is p; or, in other words,

.(2),

where r is the distance between the points (x, y, z) and (x', y', z′), p' the density at (x', y', z'), and the limits are -- to, is the complete integral of (1) subject to the condition that V shall vanish at an infinite distance.

This may be proved in different ways; most directly perhaps by taking the expression for the potential (U suppose) which forms the right-hand member of (2), substituting for p' its equivalent VV', V' being the same function of x', y', z' that V is of

1

477

x, y, z, and transforming the integral in the manner done by Green*, when we readily find U = V.

0

Suppose now that we have a given closed surface S containing within it all the attracting matter, and that the potential has a given, in general variable, value V at the surface. For the portion of space external to S, V is to be determined by the general equation VV = 0, subject to the conditions VV, at the surface, and V0 at an infinite distance. We know that the problem of determining V under these circumstances admits of one and but one solution, though it is only for a very limited number of forms of the surface S that the solution can actually be effected. Conceive the problem, however, solved, and from the solution let the value of dV/dv at the surface be found, v being measured outwards along the normal. Now complete V for infinite space by assigning to the space within S any arbitrary but continuous† function we

*

Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism," Nottingham, 1828, Art. 3; or the reprint in Crelle's Journal, Vol. XLIV, p. 360.

To avoid prolixity, I include in "continuous" the requirement that the differential coefficients of the function, to any order required, shall vary continuously. What that order may be it is perfectly easy in any case to see. We may of course imagine distributions in which the density becomes infinite at one or more points, lines, or surfaces, but so that a finite volume contains only a finite mass. But such distributions may be regarded as limiting, and therefore particular, cases of a distribution in which the density is finite; and therefore the supposition that p is finite does not in effect limit the generality of our results.

0

please, subject to the two conditions, 1st, that at the surface it is equal to the given function V.; 2ndly, that it gives for the value of dV/dv at the surface that already got from the solution of the problem referred to in this paragraph. This of course may be done in an infinite number of ways, just as we may in an infinite number of ways join two points in a plane by a continuous curve starting from the two points respectively in given directions, which curve may be either expressed by some algebraical or transcendental equation, or conceived as drawn liberâ manu, and thought of independently of any idea of algebraical expression. The function V having been thus assigned to the space internal to S, the equation (1) gives, according to what we have seen, the most general expression for the density of the internal matter.

There is, however, no distinction made in this between positive and negative matter, and if we wish to avoid introducing negative matter we must restrict the function V for the space internal to S to satisfy the imparity

It is easy from the general expression to show, what is already known, that the matter may be distributed in an infinitely thin, and consequently infinitely dense stratum over the surface S, and that such a distribution is determinate.

0

We know that there exists one and but one continuous function applying to the space within S which satisfies the equation VV=0, and is equal to V, at the surface. Call this function V1. It is to be remarked that the value of dV1/dv at the surface is not the same as that of dV/dv, V being the external potential, though V, and V are there each equal to V. The argument, it is to be observed, does not assume that the two are different; it merely avoids assuming that they are the same; the result will prove that they cannot be the same all over S unless the density, and consequently the potential, be everywhere null, and therefore V0. Now attribute to the interior of S a function V which is equal to V1 except over a narrow stratum adjacent to S, the thickness of which will in the end be supposed to vanish, within which V is made to deviate from V1 in such a manner as to render the variation of dV/dv continuous and rapid instead of abrupt. On applying equation (1), we see that the density is everywhere null except within

1

1

this stratum, in which it is very great, and in the limit infinite. For the total quantity of matter contained in any portion of the stratum, we have from (1)

the integration extending over that portion. Let the portion in question be that corresponding to a very small area A of the surface S; we may suppose it bounded laterally by the ultimately cylindrical surface generated by a normal to S which travels round the perimeter of A. Taking now rectangular coordinates λ, μ, v, of which the last is parallel to the normal at one point of A, since V is not changed in form by referring it to a new set of rectangular axes, we have for the mass required

Of the differential coefficients within brackets, the last alone becomes infinite when the thickness of the stratum, and consequently the range of integration relatively to λ, becomes infinitely small. We have in the limit

both differential coefficients having their values belonging to the surface. Hence we have ultimately for the mass

Hence, if a be the superficial density, defined as the limit of the mass corresponding to any small portion of the surface divided by the area of that portion,

In assigning arbitrarily a function V to the interior of S, in order to get the internal density by the application of the formula (1), we may if we please discard the second of the conditions

which V had to satisfy at the surface, namely, that

dV,

1

dV
; but

dv dv

in that case to the mass, of finite density, determined

by (1) must

be added an infinitely dense and infinitely thin stratum extending over the surface, the finite superficial density of this stratum being given by (3).

We have seen that the determination of the most general internal arrangement requires the solution of the problem, To determine the potential for space external to S, supposed free from attracting matter, in terms of the given potential at the surface; and the determination of that particular arrangement in which the matter is wholly distributed over the surface, requires further the solution of the same problem for space internal to S. If, however, instead of having merely the potential given at the surface S we had given a particular arrangement of matter within S, and sought the most general rearrangement which should not alter the potential at S, there would have been no preliminary problem to solve, since V, and therefore its differential coefficients, are known for space generally, and therefore for the surface S, being expressed by triple integrals.

Instead of having the attracting matter contained within a closed surface S, and the attraction considered for space external to S, it might have been the reverse, and the same methods would still have been applicable. The problem in this form is more interesting with reference to electricity than gravitation.

SUPPLEMENT TO A PAPER ON THE DISCONTINUITY OF ARBITRARY CONSTANTS WHICH APPEAR IN DIVERGENT DEVELOPMENTS.

[From the Transactions of the Cambridge Philosophical Society, Vol. XI, Part II. Read May 25, 1868.]

IN a paper "On the Numerical Calculation of a Class of Definite Integrals and Infinite Series," printed in the IXth volume of the Transactions of this Society*, I gave a method by which a definite integral, to which Mr Airy was led in calculating the intensity of light in the neighbourhood of a caustic, may be readily calculated for large values, whether positive or negative, of a certain variable which appears as a constant under the sign of integration. The method consists in forming a differential equation of which the definite integral is a particular solution, obtaining the complete integral of the equation under a form, indicated by the equation itself, involving series according to descending powers of the variable, and determining the arbitrary constants. The equation admits also of integration by means of ascending series multiplied by other arbitrary constants. The ascending series are always convergent, but when the variable is large begin by diverging rapidly: the descending series, on the other hand, are always divergent, but when the variable is large begin by converging rapidly.

The same method was found to apply to several other definite integrals which occur in physical investigations, as well as to differential equations of frequent occurrence. The ascending and descending series are usually both required, the one for application to small, the other to large values of the variable; and it is necessary to connect the arbitrary constants in the descending with those in the ascending series. The analytical determination of the arbitrary constants by which the divergent series are multiplied forms the chief difficulty, a difficulty only partially [* Ante, Vol. 11, p. 329.]