is > and <π, its first term must be taken as - 2πp instead
of 2πp.

of potential

547. If two systems, or distributions of matter, M and M', Exhaustion given in spaces each finite, but infinitely far asunder, be allowed energy." to approach one another, a certain amount of work is obtained by mutual gravitation: and their mutual potential energy loses, or as we may say suffers exhaustion, to this amount: which amount will (§ 486) be the same by whatever paths the changes of position are effected, provided the relative initial positions and the relative final positions of all the particles are given. Hence if m1, m,,... be particles of M; m'1, m'„,... particles of M'; v', ',... the potentials due to M' at the points occupied by m1, m,,...; v1, v,,... those due to M at the points occupied by m',, m',,...; and E the exhaustion of mutual potential energy between the two systems in any actual configurations; we have

E=Σmv' =Σm'v.

This may be otherwise written, if p denote a discontinuous function, expressing the density at any point, (x, y, z) of the mass M, and vanishing at all points not occupied by matter of this distribution, and if p' be taken to specify similarly the other mass M'. Thus we have

E = [[[pv'dx dy dz=[[[p'vdx dyde,

the integrals being extended through all space. The equality of
the second and third members here is verified by remarking that

if D denote the distance between (x, y, z) and (x, y, z), the
latter being any point of space, and the value of
P
P at it. A
corresponding expression of course gives v': and thus we find one
sextuple integral to express identically the second and third
members, or the value of E, as follows:-

548. It is remarkable that it was on the consideration of Green's an analytical formula which, when properly interpreted with method. reference to two masses, has precisely the same signification as

Green's

method.

Exhaustion

of potential energy,

in allowing

the preceding expressions for E, that Green founded his whole structure of general theorems regarding attraction.

In App. A. (a) let a be constant, and let U, U' be the potentials at (x, y, z) of two finite masses, M, M', finitely distant from one another: so that if p and p' denote the densities of M and M' respectively at the point (x, y, z), we have [§ 491 (c)]

It must be remembered that p vanishes at every point not forming part of the mass M: and so for p' and M'. In the present merely abstract investigation the two masses may, in part or in whole, jointly occupy the same space: or they may be merely imagined subdivisions of the density of one real mass. Then, supposing S to be infinitely distant in all directions, and observing that UU' and U'U are small quantities of the order of the inverse cube of the distance of any point of S from M and M', whereas the whole area of S over which the surface integrals of App. A. (a) (1) are taken as infinitely great, only of the order of the square of the same distance, we have

ffdSU'ǝU=0, and ƒƒdSUǝU' = 0.

Hence (a) (1) becomes

'dU dU' dU dU'

dU dU

+

+

dr dr dy dy

dz dz) dxdydz=4π fffpU'dxdydz=4#fffp'Udxdydz ;

showing that the first member divided by 4π is equal to the exhaustion of potential energy accompanying the approach of the two masses from an infinite mutual distance to the relative position which they actually occupy.

Without supposing S infinite, we see that the second member of (a) (1), divided by 4, is the direct expression for the exhaustion of mutual energy between M' and a distribution consisting of the part of M within S and a distribution over S, of density U'; and the third member the corresponding expression for M and derivations from M'.

549. If, instead of two distributions, M and M', two particles, m,, m, alone be given; the exhaustion of mutual

potential energy in allowing them to come together from in- condensafinity, to any distance D (1, 2) asunder, is

mm

D (1, 2) *

If now a third particle m, be allowed to come into their neighbourhood, there is a further exhaustion of potential energy amounting to

By considering any number of particles coming thus necessarily into position in a group, we find for the whole exhaustion of potential energy

Ε = ΣΣ

mm'
D

tion of diffused matter.

of potential

where m, m' denote the masses of any two of the particles, D Exhaustion the distance between them, and Σ the sum of the expressions energy. for all the pairs, each pair taken only once. If v denote the potential at the point occupied by m, of all the other masses, the expression becomes a simple sum, with as many terms as there are masses, which we may write thus

E = Σmv;

the factor being necessary, because Σmv takes each such term If the particles form an ultimately con

as

mm2 twice over. D (1, 2) tinuous mass, with density p at any point (x, y, z), we have only to write the sum as an integral; and thus we have

E= pvdx dy dz

as the exhaustion of potential energy of gravitation accompanying the condensation of a quantity of matter from a state of infinite diffusion (that is to say, a state in which the density is everywhere infinitely small) to its actual condition in any finite body.

An important analytical transformation of this expression is suggested by the preceding interpretation of App. A. (a); by

Exhaustion

of potential energy.

Gauss's method.

if R denote the resultant force at (x, y, z), the integration being extended through all space.

Detailed interpretations in connexion with the theory of energy, of the remainder of App. A., with a constant, and of its more general propositions and formulæ not involving this restriction, especially of the minimum problems with which it deals, are of importance with reference to the dynamics of incompressible fluids, and to the physical theory of the propagation of electric and magnetic force through space occupied by homogeneous or heterogeneous matter; and we intend to return to it when we shall be specially occupied with these subjects.

550. The beautiful and instructive manner in which Gauss independently proved Green's theorems is more immediately and easily interpretable in terms of energy, according to the commonly-accepted idea of forces acting simply between particles at a distance without any assistance or influence of interposed matter. Thus, to prove that a given quantity, Q, of matter is distributable in one and only one way over a given single finite surface S (whether a closed or an open shell), so as to produce equal potential over the whole of this surface, he shows (1) that the integral

Iep'da

pp'do do'
PP'

has a minimum value, subject to the condition

SJpdo = Q,

where Ρ is a function of the position of a point, P, on S, p' its value at P', and do and do' elements of S at these points: and (2) that this minimum is produced by only one determinate distribution of values of p. By what we have just seen (§ 549) the first of these integrals is double the potential energy of a

* Nichol's Encyclopædia, 2d Ed. 1860. Magnetism, Dynamical Relations of.

method.

distribution over S of an infinite number of infinitely small Gauss's mutually repelling particles: and hence this minimum problem is (§ 292) merely an analytical statement of the problem to find how these particles must be distributed to be in stable equilibrium.

brium of

Similarly, Gauss's second minimum problem, of which the Equilipreceding is a particular case, and which is, to find p so as to repelling make

where is any given arbitrary function of the position of P,

and

particles enclosed in a rigid smooth

surface.

is merely an analytical statement of the question:-how must a given quantity of repelling particles confined to a surface S be distributed so as to make the whole potential energy due to their mutual forces, and to the forces exerted on them by a given fixed attracting or repelling body (of which is the potential at P), be a minimum? In other words (§ 292), to find how the movable particles will place themselves, under the influence of the acting forces.

VOL. II.

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7