An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them |
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A₁ a²+v axis become infinite bounding surface c²+v centre Chap coefficients component attraction confocal ellipsoid corresponding cos² cose deduce degree denoted determine differential equation distance distribution of density dV dV dx² dy dz dµdp equal expression external point factor follows Hence homogeneous function internal lamina monics multiplying obtain P₁ P₁₂ P₂ perpendicular pi+1 plane positive integer potential putting radius rational integral function respectively satisfies the equation series of zonal shew shewn sin² sin³ solid angle solid harmonic solid zonal solutions sphere spherical harmonic spherical shell suppose surface harmonic theorem thickness V₁ w₁ whole shell writing Y₁ zonal harmonics αλ αμ αμσ μ² µ³ προ
Popular passages
Page 67 - V(a2 + c2) cft = í \/(а" + c2) + (7. 126. When polar co-ordinates are used to determine the position of a point in space, we have the following equations connecting the rectangular and polar co-ordinates of any point, x = r sin в cos ф, y = r sin в sin ф, z = r cos в. And as a curve in space is determined by two equations between x, y, and z, it may also be determined by two equations between r, в, and ф.
Page 90 - In these* the variation of temperature with latitude is expressed as the sum of a series of terms of the form...
Page 3 - Weber makes the equilibrium of electricity unstable in any conductor of moderate dimen282 sions, and renders possible the development of infinite quantities of work from finite bodies. I do not find that the objections, brought forward at first by Sir W. Thomson and Professor Tait in their Treatise on Natural Philosophy, and discussed and specialised afterwards by myself, have been invalidated by the discussions going on about this question. The hypothesis of...
Page 155 - Let a spherical portion of an infinite quiescent liquid be separated from the liquid round it by an infinitely thin flexible membrane, and let this membrane be suddenly set in motion, every part of it in the direction of the radius and with velocity equal to 8„ a harmonic function of position on the surface.
Page 44 - FES, and HBC DARLING. (Received April 8, 1918,— Revised December, 1918.) Introduction. The object of this paper is to make a contribution to the theory of attraction when the force is proportional to any given power of the distance. In the case of the law of nature, it is well known that when the attracting mass is a hollow shell of uniform density, whose exterior and interior bounding surfaces are both surfaces of revolution about the axis of z, the potential at any point exterior to the exterior...
Page 45 - ... consecutive level surfaces extending from /S to infinity be drawn, prove that the potential continually decreases outwards from each to the next until it vanishes at an infinite distance. 117. If the potential is constant throughout any finite space, it is also constant throughout all external space which can be reached without passing through any portion, of the attracting mass. [Stokes.] The external boundary of the space is necessarily a level surface. If possible let A be a point outside...
Page 158 - ST is the length of the perpendicular from the centre on the tangent plane to the confocal at E'.
Page 109 - Thus a2 4- e, 52+ e, c2 + e are the squares on the semiaxes of the confocal ellipsoid passing through the point x, y, z.
Page 44 - M, of uniform density, is a hollow shell whose exterior and interior bounding surfaces are both surfaces of revolution, their common axis being the axis of z ; let the origin be taken within the .interior bounding surface, and...
Page 45 - We must first prove the following theorem. The solid angle, subtended by a closed plane curve at any point...