SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SECTORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXES IN ANY PAGE 69 ib. 1. Spherical Harmonics in general 2. Relation between the potentials of a spherical shell at an inter- 3. Relation between the density and the potential of a spherical 90 19. Potential of homogeneous solid nearly spherical in form 20. Potential of a solid composed of homogeneous spherical strata ib. 27. Potential varying as the distance from a principal plane 28. Potential varying as the product of the distances from two prin- 150 CHAPTER I. INTRODUCTORY, DEFINITION OF SPHERICAL HARMONICS. 1. IF V be the potential of an attracting mass, at any point x, y, z, not forming a part of the mass itself, it is known that V must satisfy the differential equation or, as we shall write it for shortness, VV=0. The general solution of this equation cannot be obtained in finite terms. We can, however, determine an expression which we shall call V, an homogeneous function of x, y, z of the degree i, i being any positive integer, which will satisfy the equation; and we may prove that to every such solution V, there corresponds another, of the degree — (i + 1), expressed by, where »2 = x2+ y2+z2. For the equation (1) when transformed to polar co-ordinates by writing x = r sin cos 0, y = r sin 0 sin 4, z = r cos 0, becomes And since V satisfies this equation, and is an homogeneous function of the degree i, V, must satisfy the equa |