where the arbitrary constants are supposed to be included in the signs of integration. Now V vanishes at an infinite distance, and does not become infinite at the centre, and therefore the second integral vanishes when r=0, and the first when r∞, or, which is the same, when r = R, since p=0 when r> R. We get therefore finally, = If F' be the required force of attraction, we have F=-dV/dr; and observing that the two terms arising from the variation of the limits destroy each other, we get Now 4π [" 0 pr3dr is the mass contained within a sphere de scribed about the centre with a radius r, and therefore the attraction is the same as if the mass within this sphere were collected at its centre, and the mass outside it were removed. The attraction of the sphere on an external particle may be considered as a particular case of the preceding, since we may first suppose the sphere to extend beyond the attracted particle, and then make o vanish when r> R. ρ Before concluding, one or two more known theorems may be noticed, which admit of being readily proved by the method employed in Prop. v. Prop. IX. If T be a space which contains none of the attracting matter, the potential V cannot be constant throughout any finite portion of T without having the same constant value throughout the whole of the space T and at its surface. For if possible let V have the constant value A throughout the space T1, which forms a portion of T, and a greater or less value at the portions of Tadjacent to T. Let R be a region of T adjacent to T, where V By what has been already remarked, V must is greater than A. 2 increase continuously in passing from T1 into R. Draw a closed surface o lying partly within T, and partly within R, and call the portions lying in T ̧ and R, σ ̧, σ respectively. Then if v be a normal to σ, drawn outwards, d V/dv will be positive throughout σ, if be drawn sufficiently close to the space T, (see Prop. v. and note), and d V/dv is equal to zero throughout the surface since 1 V is constant throughout the space T1; and therefore taken throughout the whole surface σ, will be positive, which is contrary to Prop. IV. Hence V cannot be greater than A in any portion of T adjacent to T1, and similarly it cannot be less, and therefore V must have the constant value A throughout T, and therefore, on account of the continuity of V, at the surface of T. Combining this with Prop. v. Cor. 1, we see that if V be constant throughout the whole surface of a space T which contains no attracting matter, it will have the same constant value throughout T; but if V be not constant throughout the whole surface, it cannot be constant throughout any finite portion of T, but only throughout a surface. Such a surface cannot be closed, but must abut upon the surface of T, since otherwise V would be constant within it. Prop. x. The potential V cannot admit of a maximum or minimum value in the space T. 1 It appears from the demonstration of Prop. v. that I cannot have a maximum or minimum value at a point, or throughout a line, surface, or space, which is isolated in T. But not even can V have the maximum or minimum value V, throughout T, if T reach up to the surface S of T; though the term maximum or minimum is not strictly applicable to this case. By Prop. IX. V cannot have the value V1 throughout a space, and therefore T, can only be a surface or a line. 1 1 If possible, let V have the maximum value V1 throughout a line Z which reaches up to S. Consider the loci of the points where V has the successive values V2, V,..., decreasing by infinitely small steps from 1'. In the immediate neighbourhood of L, these loci will evidently be tube-shaped surfaces, each lying outside the preceding, the first of which will ultimately coincide with L. Lets be an element of L not adjacent to S, nor reaching up to the extremity of L, in case L terminate abruptly. At each extremity of s draw an infinite number of lines of force, that is, lines traced from point to point in the direction of the force, and therefore perpendicular to the surfaces of equilibrium. The assemblage of these lines will evidently constitute two surfaces cutting the tubes, and perpendicular to s at its extremities. Call the space contained within the two surfaces and one of the tubes T,, and apply equation (5) to this space. Since V is a maximum at L, dV/dn is negative for the tube surface of T,, and it vanishes for the other surfaces, as readily follows from equation (4). Hence dds, taken throughout the whole surface T, is negative, which is contrary to equation (5). Hence V cannot have a maximum value at the line L; and similarly it cannot have a minimum value. dn 1 It may be proved in a similar manner that V cannot have a maximum or minimum value V1 throughout a surface S1 which reaches up to S. For this purpose it will be sufficient to draw a line of force through a point in S1, and make it travel round an elementary area σ which forms part of S1, and to apply equation (5) to the space contained between the surface generated by this line, and the two portions, one on each side of S1, of a surface of equilibrium corresponding to a value of V very little different from V1. It should be observed that the space T considered in this proposition and in the preceding need not be closed: all that is requisite is that it contain none of the attracting mass. Thus, for instance, T may be the infinite space surrounding an attracting mass or set of masses. It is to be observed also, that although attractive forces have been spoken of throughout, all that has been proved is equally true of repulsive forces, or of forces partly attractive and partly repulsive. In fact, nothing in the reasoning depends upon the sign of m; and by making m negative we pass to the case of repulsive forces. Prop. XI. If an isolated particle be in equilibrium under the action of forces varying inversely as the square of the distance, the equilibrium cannot be stable with reference to every possible S. II. 9 displacement, nor unstable, but must be stable with reference to some displacements and unstable with reference to others; and therefore the equilibrium of a free isolated particle in such circumstances must be unstable*. For we have seen that V cannot be a maximum or minimum, and therefore either V must be absolutely constant, (as for instance within a uniform spherical shell), in which case the particle may be in equilibrium at any point of the space in which it is situated, or else, if the particle be displaced along any straight line or curve, for some directions of the line or curve V will be increasing and for some decreasing. In the former case the force resolved along a tangent to the particle's path will be directed from the position of equilibrium, and will tend to remove the particle still farther from it, while in the latter case the reverse will take place. * This theorem was first given by Mr Earnshaw in his memoir on Molecular Forces read at the Cambridge Philosophical Society, March 18, 1839 (Trans. Vol. VII.). See also a paper by Professor Thomson in the first series of this Journal, Vol. IV. p. 223. [From the Transactions of the Cambridge Philosophical Society, Vol. VIII. p. 672.] ON THE VARIATION OF GRAVITY AT THE SURFACE OF THE EARTH. [Read April 23, 1849.] ON adopting the hypothesis of the earth's original fluidity, it has been shewn that the surface ought to be perpendicular to the direction of gravity, that it ought to be of the form of an oblate spheroid of small ellipticity, having its axis of figure coincident with the axis of rotation, and that gravity ought to vary along the surface according to a simple law, leading to the numerical relation between the ellipticity and the ratio between polar and equatorial gravity which is known by the name of Clairaut's Theorem. Without assuming the earth's original fluidity, but merely supposing that it consists of nearly spherical strata of equal density, and observing that its surface may be regarded as covered by a fluid, inasmuch as all observations relating to the earth's figure are reduced to the level of the sea, Laplace has established a connexion between the form of the surface and the variation of gravity, which in the particular case of an oblate spheroid agrees with the connexion which is found on the hypothesis of original fluidity. The object of the first portion of this paper is to establish this general connexion without making any hypothesis whatsoever respecting the distribution of matter in the interior of the earth, but merely assuming the theory of universal gravitation. It appears that if the form of the surface be given, gravity is determined throughout the whole surface, except so far as regards one arbitrary constant which is contained in its complete expression, and which |