16. The same conclusion may be arrived at as follows: The potential of a spherical shell, whose density is p, and volume U, at any point on the axis of z, is which is equal to { pdx + pP,(x)dx + ... U 2c -1 It hence follows that the potential, at a point situated anywhere, is And these expressions are respectively equal to those for the potentials, at an internal and external point respectively, for matter distributed according to the following law of density: 1 { ["_pdλ + 3P ̧(4) [_pP ̧(x)dλ + ... + (2i+1)P,(u)(_ ̧ pP,(x)dx + ...}. It will be observed, in applying this formula, that if p be a discontinuous function of A, each of the expressions of the form pP(X)dx will be the sum of the results of a series of integrations, each integration being taken through a series of values of A, for which p varies continuously. CHAPTER IV. SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SEC- AXIS IN ANY POSITION. POTENTIAL OF A SOLID NEARLY 1. We have hitherto discussed those solutions of the equation VV=0 which are symmetrical about the axis of z, or in other words, those solutions of the equivalent equation in polar co-ordinates which are independent of p. We propose, in the present Chapter, to consider the forms of spherical harmonics in general, understanding by a Solid Spherical Harmonic of the th degree a rational integral homogeneous function of x, y, z, of the ith degree which satisfies the equation V=0, and by a Surface Spherical Harmonic of the ith degree the quotient obtained by dividing a Solid Spherical Harmonic by (∞2 + y2+2). Such an expression, as we see by writing xr sin cos 0, y =r sin 0 sin 4, z=r cos 0, will be of the 2th degree in sin cos o, sin 0 sin o, cos 0; and will satisfy the differential equation in Y, It will be convenient, before proceeding to investigate the algebraical forms of these expressions, to discuss some of their simpler physical properties. 2. We will then proceed to shew how spherical harmonics may be employed to determine the potential, and consequently the attraction, of a spherical shell of indefinitely small thickness. We will first establish an important theorem, connecting the potential of such a shell on an external point with that on a corresponding internal point. The theorem is as follows: in If O be the centre of such a shell, c its radius, P any ternal point, P' an external point, so situated that P' lies on OP produced, and that OP. OP' = c3, and if OP =r, OP′=r′, then the potential of the shell at P is to its potential at P' as c to r, or (which is the same thing) as r' to c. For, let A be the point where OP' meets the surface of the sphere, Q any other point of its surface. Then, by a known geometrical theorem, QP: QP :: AP: AP c-r: y — c. Again, considering the element of the shell in the immediate neighbourhood of Q, its potential at P is to its potential at P as QP is to QP, that is, as c to r, or (which is the same thing) as r' to c, which ratio, being independent of the position of Q, must be true for every element of the spherical shell, and therefore for the whole shell. Hence the proposition is proved. 3. Now, suppose the law of density of the shell to be such that its potential at any internal point is F(μ, 6). дов Then Fu, 4) must be a solid harmonic of the degree i. Cε Hence Fu, ) must be a surface harmonic of the degree i Let us represent it by Y. By the proposition just proved, the potential at any external point, distant r' from the centre, must be Hence, the component of the attraction of the sphere on the internal point measured in the direction from the point inwards, i.e. towards the centre of the sphere, is And the component in the same direction of the attraction on the external point, measured inwards, is Now suppose the two points to lie on the same line passing through the centre of the sphere, and to be both indefinitely close to the surface of the sphere, so that r and r' are each indefinitely nearly equal to c. And the attraction on the external point exceeds the attraction on the internal point by Now, supposing the shell to be divided into two parts, by a plane passing through the internal point perpendicular to the line joining it with the centre, we see that the attraction of the larger part of the shell on the two points will be ultimately the same, while the component attractions of the smaller portions, in the direction above considered, will be equal in magnitude and opposite in direction. Hence the difference between these components, viz. (2i+1) i will be equal to twice the component attraction of the smaller portion in the direction of the line joining the two points. But if p, be the density of the shell, dc its thickness, this component attraction is 2πp, Sc. Pi Y с |