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'V d'V dev
d2 V
+ + = 0.
dx2 + dy2 + dz2
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), where r2 = x2+ y2+z2. expressed by
For the equation (1) when transformed to polar co-ordinates by writing xr sin cos 0, y = r sin 0 sin p, z = r cos 0,
And since V satisfies this equation, and is an homogeneous function of the degree i, V, must satisfy the equa