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9, 10, 11. Expression of EHH' in terms of x, y, 2
28. Potential varying as the product of the distances from two prin.
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
d'y d'V dV
dz2 or, as we shall write it for shortness, V*V = 0.
The general solution of this equation cannot be obtained in finite terms. We can, however, determine an expression which we shall call Vi, 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),
V. expressed by tti, where r*=x*+y*+z".
For the equation (1) when transformed to polar co-ordinates by writing x=r sin 0 cos 0, y=r sin sin , z=r cos 0, becomes do (V) 1 d
= 0...(2). dr2 sin Ꮎ dᎾ do
sino do And since V satisfies this equation, and is an homogeneous function of the degree i, Vi must satisfy the equation 1 d
1 d'V; ii + 1) Vit
siu o do