ito + (Coco + Soso) + ...+[2i (Cc,+$$)} • 2 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 ', and that the plane containing these two axes is inclined to a fixed plane through the axis of which we may consider as that of zx, at the angle '. In accordance with the notation already employed, we shall represent cos' by u. The rectangular equations of the axis of this system will be У 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 sine cos p'+ y sin e' sin +z cos 0. To deduce the Surface Zonal Harmonic, transform the solid zonal harmonic to polar co-ordinates, by writing r sin cos & for x, r sin 0 sin for y, r cos e for z, and divide by r. 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(-). Now, in order to develope P{cos e cos + sin 0 sin e cos (6-p')} in the manner already pointed out, assume P1 {cos e cos + sin @ sin e' cos (p −¤')} =AP (u)+(C(1) cos +S) sin p) T1) +(C(2) cos 24+ S(2) sin 24) T2)+... +(C) cos σ$+S) sin op) T()+... + (C(i) cos ip + S«) sin ip) T↔, the letters A, ... C), S()... denoting functions of μ and p', to be determined. To determine C), multiply both sides of this equation by cos opT) and integrate all over the surface of the sphere, i.e. between the limits - 1 and 1 of μ, and 0 and 2 of 4. We then get 2T [["P, {cos e cos ' + sin é sin &′ cos ($ — $')} cos σ& T ̧« dμdó It remains to find the value of the left-hand member of this equation. Now cos ooT) 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 [__[** P. (u) Y, dμdo 0 = 2i+1 Y, (1), 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 4πT 2i+1 assumes at the pole of the zonal harmonic. = sin 0 sin e' cos (4-p')} Y¿ (μ, 6) dμdo 4π [* P. (cos e cos + sin @ sin @′ cos (¿ — ¿')} cos σ¿T("dμd& 27 P(cos e cos + sin sin ơ′ cos (4 – 4')} P, (μ) dμdo Ꮎ or A = P1 (μ'). Hence, P {cos e cos 0' + sin ✪ sin ✔ cos (p − 6')} 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 cos 0, sin 0 cos o, sin @ 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 op, or sin op, the greatest value of a being the sum of the indices of cos o 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 p, sin @ sin &, will consist of a series of terms of the form cos" sin" e cos op or cos" e sin” 0 sin op, where n is not less than σ. п If n be greater than σ, no must be an even integer. Let n-σ = 2s, then writing sin" under the form (1 - cos20)3 sin ́0, we reduce cos" sin" 0 cos op to the sum of a series of terms of the form cos e sine cos op, or, writing cos 0 = μ, of the m Similarly cos sin" e sin op is reduced to a series of terms of the form μ" (1-μ) sin op. and up can be developed in a series of terms of the form of multiples of Ppto, Ppto-2.... (Chap. II. Art. 17.) Hence μ can be expressed in a series of the form A, A, representing known numerical constants, and therefore μ3 (1 − μ3) assumes the form T() (A。 Toto+A, T40-2 + ...), consequently multiplying these series by cos op or sin op, we obtain the developments of μ3 (1 — μ3) 3 cos σp and μ3 (1 — μ3)3 sin in series of tesseral harmonics. σφ 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. Comparing this with cos" 0 sin" 0 sin op, we see that n is not greater than σ. Hence cos2 ◊ sin2 @ sin & cos p = 1⁄2, μ2 (1 — μ2) sin 24. 1 2 d2 4 1/8 d'P1+ 2 d2P 105 αμε (1 — μ3) sin 24 + 2 29 09 2 |