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 + {i' (i + 1) − m (m + 1)} [P ̧P„dμ = 0. Hence, transforming the first two integrals by integration by parts, and remarking that we get i (i + 1) − m (m + 1) = (i − m) (i + m + 1), or i m + (i − m) (i + m + 1) [P ̧P„dμ = 0, (1 − μ") ( P2 dP ; — P, P) + (i − m) (i + m + 1) [P,P_du=0, m άμ -P、 since the second term vanishes identically. F. H. 2 Hence, taking the integral between the limits 1 and +1, we remark that the factor 1-3 vanishes at both limits, and therefore, except when i— m, or i+m+1=0, We may remark, also that we have in general dP dP Pm du i αμ 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_(4) the determination of the value of Pd will also P=P_ -(i+1)9 ‚: (1 − 2μh+h3) ̄1 = (P。+P ̧h+ 2 ... + Ph' + ...)* =P2 + P2h2 + ... + P2h2 + ... +2P ̧P ̧h+2P ̧Ph3 + ... + 2P ̧P2h3 +..... Integrate both sides with respect to μ; then since we get, taking this integral between the limits - 1 and +1, all the other terms vanishing, by the theorem just proved. 12. From the equation [P.Pd = 0, combined with -1 the fact that, when μ = 1, P ̧=1, and that P, is a rational integral function of u, 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 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, μ of P, and zonal harmonics of m m-1 m-1 lower orders, and so on. Hence "Pdu will be the sum of a series of multiples of quantities of the form PPdμ‚ 1 m being less than i, and therefore "P=0, if m be any integer less than i. Again, since -1 (1 − 2μh + h2)−1 – P ̧+P ̧h+ ... + P ̧h2 + ... it follows, writing - h for h, that (1 + 2μh+h2)- § = P ̧− P ̧h + ... + ( − 1)' P ̧l' + ..... P ̧−P ̧h+ And writingμ for μ in the first equation, ... (1 + 2μh + h3) ̄* = P' + P2h + ... + P'h' + ... P', P... P.. denoting the values which P., respectively assume, when -μ is written for u. PP or -P, according as i is P involves only odd, or only even, as i is odd or even*. Assume then even or odd. 1 P,... P1, powers of i, according P1 = '‚μ3 + A¡-¿ μ1? +... Our object is to determine A,, A-9.... i-2 i-4 Then, multiplying successively by μ, μ,... and integrating from 1 to +1, we obtain the following system of equations: 2i-1 2i-3 2i-2s -1 And lastly, since P1 = 1, when μ = 1, A+ A12 + ... + A12+ ..... = 1; the last terms of the first members of these several equa tions being 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. d' αμι * This is also evident, from the fact that P, is a constant multiple of (u2 - 1)'. From this system of equations we deduce the following, O being any quantity whatever, + y 1 (0-a) (0-ẞ) (a+w) (b+w) (c+w) + a + Ꮎ b+0 c+0 w (w−a) (w−ß) (a+0) (b+0) (c+0)* For this expression is of -1 dimension in a, b, c, a, B, Y, e, w; it vanishes when = a, or 0=ß, and for no other finite value of 0, and it becomes 1 == W , when e = w. 1 (a + a) (a + B) (a + w) (b + w) (c + w) w (a - b) (a−c) with similar values for y and z. (w-a) (w-B) ? And, if o be infinitely great, in which case the last equation assumes the form x+y+2=1, we have with similar values for y and z. 14. Now consider the general system |