must be expressible in terms of 22. It will be seen that i-1 function of μ, ..., P-1 P. To determine this expression, assume dP άμ = · C1- Pi- + Cps Prs + ... + CmPm + ... 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 Pm or P1 = −1, when μ = −1; = 2m + 1 · (2i − 1) P12 + (2i — 5) P¡-3 + (2i — 9) Pi-s + ... 23. From this equation we deduce P1 — P12 = — (2i — 1) [ P1, dp, the limits μ and 1 being taken, in order that P.-P., may be equal to 0 at the superior limit. Now, recurring to the fundamental equation for a zonal 24. We have already seen that PP du=0, i and m being different positive integers. -1 Suppose now that it is 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 1 f П S 0 z-√-1pcos (22 +p2)3° We may remark, in passing, that Soz-v-1 pcose ρ = = and is therefore wholly real. 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—μ3)3r for 26. Again, we have 1 (a2 - b2) In this write 1 0 a-b cos y μh for a, and ± (μ2 – 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, .. 1+P ̧h + ... + P ̧h' + ... − 1 ), dy [1 + [μ ± (u2 − 1) + cos ↓} & +...' = h ; Hence, equating coefficients of h', P1 = 1 √" {μ ± (μ2 — 1)1 cos y}' dy. π 0 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. Q 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. Then CP2=a+b2-2ab cos, C'Q2=a2+b2 2ab cos . Hence (a2 + b2 — 2ab cos D) (a2 + b2 — 2ab cos y) = (a2 — b2)2; |