the limits of r being the two values of r which satisfy the equation of the surface of the sphere, viz. Hence, if r1, r, be the two limiting values of r, we have Now, if V be the potential at P, we have (see Chap. I. Art. 1) Assume then, as the complete solution of the equation, 2 προ V = − 3 mpc2 + ( 4, + B.) P. + (4,r + B) P, +. 3 p It remains to determine the coefficients A, A,...A....B1, B...B, so that this expression may not become infinite for any value of r corresponding to a point within the sphere, and that at any point P on the surface of the sphere it may be equal to the surface, M where O'P: OP :: a: c, and therefore, at whence we obtain, as the expression for the potential at any internal point, 6. We shall next proceed to establish the proposition that if the density of a spherical shell, of indefinitely small thickness, be a zonal surface harmonic, its potential at any internal point will be proportional to the corresponding solid harmonic of positive degree, and its potential at any external point will be proportional to the corresponding solid harmonic of negative degree. Take the centre of the sphere as origin, and the axis of the system of zonal harmonics as the axis of z. Let b be the radius of the sphere, 8b its thickness, U its volume, so that U=4b28b. Let CP, be the density of the sphere, P, being the zonal surface harmonic of the degree i, and C any constant. +d Draw two planes cutting the sphere perpendicular to the axis of 2, at distances from the centre equal to respectively. The volume of the strip of the sphere intercepted between these planes will be dr U, and its mass will be 26 Now bu, hence d=bdμ, and this mass becomes = CU Hence the potential of this strip at a point on the axis of z, distantz from the centre, will be To obtain the potential of the whole shell, we must integrate these expressions with respect to μ between the limits 1 and +1. Hence by the fundamental property of Zonal Harmonics, proved in Chap. II. Art. 10, we get for the potential of the whole shell From these expressions for the potential at a point on the axis we deduce, by the method of Art. 1 of the present Chapter, the following expressions for the potential at any point whatever : From hence we deduce the following expressions for the normal component of the attraction on the point. Normal component of the attraction on an internal point, measured towards the centre of the sphere, Normal component of the attraction on an external point, measured towards the sphere, In the immediate neighbourhood of the sphere, where r is indefinitely nearly equal to b, these normal component attractions become respectively And writing for U its value, 476283, this expression be comes 4πδ . CP Or, the density may be obtained by dividing the algebraic sum of the normal component attractions on two points, one external and the other internal, indefinitely near the sphere, and situated on the same normal, by 47 x thickness of the shell. 7. It follows from this that if the density of a spherical shell be expressed by the series 2 C1, C1, C2 ... C... being any constants, its potential (V1) at an internal point will be and its potential (V1) at an external point will be 2i+1 pi+1 In the last two Articles, by the word "density" is meant "volume density," i.e. the mass of an indefinitely small element of the attracting sphere, divided by the volume of |