22. It will be seen that i-1 i-3 αμ function of μ, ..., must be expressible in terms of P-1 P. To determine this expression, assume then multiplying by P, and integrating with respect to μ from 1 to +1, since either m or i must be odd, and therefore either P or P1=-1, when μ = −1; = (2i − 1) P12 + (2i − 5) P¡-3 + (2i − 9) Pi-5 + ... 1-1 dp αμ dP. αμ Hence i-2 the limits μ and 1 being taken, in order that P、- P1-, may be equal to 0 at the superior limit. Now, recurring to the fundamental equation for a zonal harmonic, we see that SPP i-1 άμ 24. We have already seen that PP du=0, i and m -1 being different positive integers. Suppose now that it is required to find the value of PP dμ. SPP d We have already seen (Art. 10) that 25. We will next proceed to give two modes of expressing Zonal Harmonics, by means of Definite Integrals. The two expressions are as follows: The only limitation upon the quantities denoted by a and b in this equation is that b2 should not be greater than a2. For, if b2 be not greater than a2, cos I cannot become α equal to while increases from 0 to π, and therefore the b expression under the integral sign cannot become infinite. Supposing then that we write z for a, and √-1p for b, we get ρ Supposing that p2=x2+ y2, and that x2+ y2+ z2 = r2, we thus obtain Differentiate i times with respect to z, and there results In this, write ur for z, and (1 — μ2)3r for 1 Π = o {μ- (u2-1) cos} it' which, writing π-9 for 9, gives In this write 1 μh for a, and ± (-1) h for b, which is admissible for all values of h from 0 up to μ-(2-1), and we obtain, since a2 - b2 becomes 1 – 2μh + h3, π) 。 1 - {μ ± (u2 — 1) cos } h 0 ..1+Ph+...+P ̧h' + ... — ["* dy [1 + {μ ± (μ3 — 1) 13 cos ¥} k+.....· = π The equality of the two expressions thus obtained for P, is in harmony with the fact to which attention has already been directed, that the value of P is unaltered if — (i + 1) be written for i. 27. The equality of the two definite integrals which thus present themselves may be illustrated by the following geometrical considerations. Let O be the centre of a circle, radius a, C any point within the circle, PCQ any chord drawn through C, and let OC=b, COP=9, COQ=y. Then CP= a+b2-2ab cos, CQ2 = a2 + b2 — 2ab cos . Hence (a2 + b2 — 2ab cos D) (a2 + b2 — 2ab cos ¥) = (a2 — b2)2; sind de + sin dy = 0. |