Again, since the angles OPC, OQC are equal to one another, sin sin OPC sin 0QC sin whence (a2+b2 — 2ab cos)1= (a+b2 - 2ab cos )1' y) dy (a2 + b2 — 2ab cos 9)+(a+b2 - 2ab cos ↓) (a3 — b2) 2i+1 And = (a2+b2 - 2ab cos)1+; In this, write a2+b2 =μ, 2ab = = (μ2 − 1), which gives a2-b21, and we get We also see, by reference to the figure, that as increases from 0 to π, diminishes from π to 0. Hence 28. From the last definite integral, we may obtain an expansion of P, in terms of cos 0 and sin 0. Putting μ = cos 0, we get And i (i − 1)... (i — 2m + 1) (2m − 1) (2m — 3)........1 1.2...2m i (i − 1)... (i — 2m+1) CHAPTER III. APPLICATION OF ZONAL HARMONICS TO THE THEORY OF REPRESENTATION OF DISCONTINUOUS ATTRACTION. FUNCTIONS BY SERIES OF ZONAL HARMONICS, 1. WE shall, in this chapter, give some applications of Zonal Harmonics to the determination of the potential of a solid of revolution, symmetrical about an axis. When the value of this potential, at every point of the axis, is known, we can obtain, by means of these functions, an expression for the potential at any point which can be reached from the axis without passing through the attracting mass. The simplest case of this kind is that in which the attracting mass is an uniform circular wire, of indefinitely small transverse section. Let c be the radius of such a wire, p its density, k its transverse section. Then its mass, M, will be equal to 2πрck, and if its centre be taken as the origin, its potential at any point of its axis, distant z from its centre, will be M (c2 + z2) Now, this expression may be developed into either of the following series: We must employ the series (1) if z be less than c, or if the attracted point lie within the sphere of which the ring is a great circle, and the series (2) if z be greater than c, or if the attracted point lie without this sphere. Now, take any point whose distance from the centre is r, and let the inclination of this distance to the axis of the ring be 0. In accordance with the notation already employed, let cos 0μ. Then, the potential at this point will be given by one of the following series: 1 For each of these expressions, when substituted for V, satisfies the equation ▼V=0, and they become respectively equal to (1) and (2) when is put = 0, and consequently 1'= 2. The expression (2′) also vanishes when r is infinitely great, and must therefore be employed for values of r greater than c, while (1') becomes equal to (2') when r = c, and will therefore denote the required potential for all values of r less than c. These expressions may be reduced to other forms by means of the expressions investigated in Chap. 2, Art. 25, viz. Substitute the first of these in (1′) and (observing that urz) we see that it assumes the form = The substitution of the last form of P, in the series (2') brings it into the form 2. Suppose next that the attracting mass is a hollow shell of uniform density, 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 suppose the potential, at any point of the axis within this surface, to be A ̧ + Â ̧≈ + Â ̧≈2 + ... + Â ̧1⁄23 + ..... Then the potential at any point lying within the inner bounding surface will be AP+AP+ Â ̧Ð ̧r2+...+Â ̧Ð ̧2 +... 2 2 For this expression, when substituted for V, satisfies the equation V2V=0; it also agrees with the given value of the potential for every point of the axis, lying within the inner bounding surface, and does not become infinite at any point within that surface. Again, suppose the potential at any point of the axis without the outer bounding surface to be |