SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SECTORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXES IN ANY 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 5. Derivation of successive harmonics from the zonal harmonic by 6. Tesseral and sectorial harmonics 7. Expression of tesseral and sectorial harmonics in a completely 90 19. Potential of homogeneous solid nearly spherical in form 20. Potential of a solid composed of homogeneous spherical strata 19. Analogy with Spherical Harmonics 20. Modification of equations when the ellipsoid is one of revolution 22. Special examples. Density varying as P.(μ) 23. External potential varying inversely as distance from focus 27. Potential varying as the distance from a principal plane 28. Potential varying as the product of the distances from two prin- ib. 30. Application to the case of the Earth considered as an ellipsoid 150 31, 32. Expression of any rational integral function of x, y, z, in a 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, V2 V = 0. .(1), 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. Vi 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 |