14. We have hitherto considered the Zonal Harmonic under its simplest form, that of a "Legendre's Coefficient" in which the axis of z, i.e. the line from which is measured, is the axis of the system. We shall now proceed to consider it under the more general form of a "Laplace's Coefficient," in which the axis of the system of zonal harmonics is in any position whatever, and shall shew how this general form may be expressed in terms of P(u) and of the system of Tesseral and Sectorial Harmonics deduced from it.

Suppose that ', ' are the angular co-ordinates of the axis of the Zonal Harmonic, i.e. that the angle between this axis and the axis of z is e, and that the plane containing these two axes is inclined to a fixed plane through the axis of z which we may consider as that of zx, at the angle '. In accordance with the notation already employed, we shall represent cos' by μ'.

The rectangular equations of the axis of this system will be

Hence the Solid Zonal Harmonic of which this is the axis is deduced from the ordinary form of the solid zonal harmonic expressed as a function of z and r by writing, in place of z, x sin cos p'+ y sin e' sin +z cos e'.

To deduce the Surface Zonal Harmonic, transform the solid zonal harmonic to polar co-ordinates, by writing r sin cos & for x, rsin e sin & for y, r cos 0 for z, and divide by r3.

The transformation from the special to the general form of surface zonal harmonic may be at once effected, by substituting for μ, or cos e, cose cose'+sin @sin e cos(4—4′). Now, in order to develope

P1{cos e cos + sin 0 sin cos (p-')}

in the manner already pointed out, assume

P{cos cos + sin 0 sin e' cos (4-(')}

= AP ̧ (μ) + (C(1) cos +S1 sin p) Tα1)

=

the letters A, ... C), S)... denoting functions of μ and p', to be determined.

To determine C), multiply both sides of this equation by cos ooT) and integrate all over the surface of the sphere, i.e. between the limits - 1 and 1 of μ, and 0 and 27 of p. We then get

2π

[P, {cos e cos ' + sin @ sin ơ cos (4 – $′)] cos σ¿T() dμdp

It remains to find the value of the left-hand member of this equation.

Now cos op T is a surface harmonic of the degree i, and therefore a function of the kind denoted by Y, in Art. 12.

And we have shewn, in that Article, that

that is, that if any surface harmonic of the degree i be multiplied by the zonal harmonic of the same degree, and the product integrated all over the surface of the sphere, the integral is into the value which the surface harmonic

equal to

4π 2i+ 1 assumes at the pole of the zonal harmonic.

Hence

2π

[P. (cos e cose + sin @ sin ơ cos ($ – $)} Y, (μ, $) dμđộ

and therefore

[P, {cos e cose + sin @ sin @′ cos († — $')} cos σ¶Ã¶›dμdp

0

2π

[P, {cos e cose + sin @ sin ơ′ cos ($ — $')} P, (μ) dμđô

0

Hence, P {cos e cose + sin 0 sin cos (4 – p')}

1

= P; (u') P; (μ) + 2 cos (p-p') T(1) (μ') T, (1) (μ)

2+1

15. We have already seen (Chap. II. Art. 20) how any rational integral function of μ can be expressed by a finite series of zonal harmonics. We shall now shew how any rational integral function of cose, sin cos o, sin 0 sin &, can be expressed by a finite series of zonal, tesseral, and sectorial harmonics.

For any power of cos o or sin o, or any product of such powers, may be expressed as the sum of a series of terms of the form cos oo, or sin op, the greatest value of σ being the sum of the indices of cos and sin o, and the other values diminishing by 2 in each successive term. Hence any rational integral function of cos 0, sin 0 cos &, sin ✪ sin &, will consist of a series of terms of the form

cos" e sin" e cos op or cos" e sin" 0 sin op,

where n is not less than σ.

If n be greater than σ, no must be an even integer. Let n-σ = 2s, then writing sin" under the form (1- cos20)o sino0, we reduce cos 0 sin" e cos σp to the sum of a series of terms of the form cos" e sin cos σp, or, writing cos 0 = μ, of the

m

σ

Similarly cos" sin" e sin op is reduced to a series of terms of the form μ" (1-μ) sin op.

Now МД =

1

do

(p+o) (p + o− 1) ... (p + 1) duo μp to

and up can be developed in a series of terms of the form (Chap. II. Art. 17.)

of multiples of Ppto, Ppto-2....

Hence μ can be expressed in a series of the form

do

αμσ

· (A。 Ppto + A2 Ppto-2 + ...),

A., A, representing known numerical constants, and therefore

consequently multiplying these series by cos op or sin op, we obtain the developments of

μ3 (1 — μ2)3 cos σp and μ3 (1 – μ2) sin op

in series of tesseral harmonics.

16. We will give two illustrations of this transformation. First, suppose it is required to express cos2 0 sin20 sin & cos in a series of Spherical Harmonics.