The properties in question are as follows: The following is a proof of the first property. 2/ Multiplying the first of these equations by Pm, the second by P, subtracting and integrating, we get Hence, transforming the first two integrals by integration by parts, and remarking that i (i + 1) − m (m + 1) = (i − m) (i + m + 1), m άμ du du du du + (i − m) (i + m + 1) [ P‚P„dμ= 0, (1 − μ2) (P_ d) ( − P, D) + (i − m) (i + m + 1) [P ̧P_qu=0, (i m αμ i Hence, taking the integral between the limits - 1 and +1, we remark that the factor 1-μ2 vanishes at both limits, and therefore, except when im, or i+m+1=0, a result which will be useful hereafter. 11. We will now consider the cases in which i-m, or i+m+ 1 = 0. We see that i+m+1 cannot be equal to 0, if i and m are both positive integers. Hence we need only discuss the case in which m= i. We may remark, however, that since P1 = P_(i+1)› of f*P; du will also the determination of the value of P du will also .: (1 − 2μh + h2) ̄1 = (P2+ P ̧h + ... + Ph' + ...)2 +2P ̧P ̧h+2P ̧P ̧h2 + ... + 2P ̧Ph3 + ... Integrate both sides with respect to μ; then since [(1 − 2μh + h2)~1 dμ = − 1, log (1 — 2μh + h”), 2h we get, taking this integral between the limits −1 and + 1, 1+h 1 } log } ±h=[' ̧P;dμ +h2 [' ̧ P;dμ+ ... + h* [' ̧P¿dμ + ... 1 -1 -1 -1 all the other terms vanishing, by the theorem just proved. 12. From the equation [P.Pdμ = 0, combined with -1 the fact that, when μ = 1, P1 = 1, and that P is a rational integral function of μ, of the degree i, P, may be expressed in a series by the following method. We may observe in the first place that, if m be any 1 integer less than i, [_μ"Pdμ = 0. m μ"Ραμ For as Pm Pm-1... may all be expressed as rational integral functions of μ, of the degrees m, m-1... respectively, it follows that will be a linear function of P and zonal harmonics of lower orders, 1 of Pm, and zonal harmonics of lower orders, and so on. m-1 m-1 m Hence ("Pu will be the sum of a series of multiples of quantities of the form PPd, m being less than i, and therefore integer less than i. Again, since μ"P,dμ = 0, if m be any (1 − 2μh+h2)-1 = P2+ Ph + ... + P ̧h2+... it follows, writing - h for h, that (1 + 2μh+h2)=P-Ph+...+(− 1)' Ph' + ... (1 + 2μh + h2) ̄ = P' + P2'h + ... + P'h' + ... P, P... P... denoting the values which P., P1... P1, respectively assume, when -μ is written for u. Hence PP, or -P, according as i is even or odd. That is, P involves only odd, or only even, powers of i, according as i is odd or even*. Our object is to determine A,, A¡-2..... ,... and inte grating from 1 to +1, we obtain the following system of the last terms of the first members of these several equations being A ¿-1'-..., 4., if i be even, A A。 13. The mode of solving the class of systems of equations to which this system belongs will be best seen by considering a particular example. * This is also evident, from the fact that P; is a constant multiple of (-1). From this system of equations we deduce the following, O being any quantity whatever, + a + o y b + Ꮎ + 2 1 (0–a) (0–B) (a+w) (b+w) (c+w) с+6 = w (w-a) (w−B) (a+0) (b+0) (c+0) * For this expression is of -1 dimension in a, b, c, a, B, v, 0, w; it vanishes when = a, or 0=ß, and for no other And, if o be infinitely great, in which case the last equation assumes the form x + y + z = 1, we have with similar values for y and z. 14. Now consider the general system |