The properties in question are as follows: The following is a proof of the first property. Multiplying the first of these equations by Pm, the second by P1, 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 i (i + 1) − m (m + 1) = (i − m) (i + m + 1), (1 − μ) (P. T; − P, T-2) + (i − m) (i +m+1) [PP„&u=0, i άμ άμ since the second term vanishes identically. F. H. 2 2/ Hence, taking the integral between the limits 1 and +1, we remark that the factor 1-2 vanishes at both limits, and therefore, except when i - m, 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 of f* P; du will also the determination of the value of P du will also +... +2P ̧Ph+2P ̧Ph2 + ... + 2P ̧Ph3 +.... 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 με -1 ทเ the fact that, when u = 1, P=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 m-1 ... μ"Ραμ = 0. For as Pm, Pm may all be expressed as rational integral functions of μ, of the degrees m, m-1... respectively, it follows that μm will be a linear function of Pm and zonal harmonics of lower orders, μ-1 of P, and zonal harmonics of m-1 lower orders, and so on. Hence "Pu will be the sum of a series of multiples of quantities of the form PPdμ, m being less than i, and therefore integer less than i. Again, since "Pdμ=0, if m be any (1 − 2μh + h2)− = P ̧+P ̧h + ... + P ̧h2+... it follows, writing - h for h, that (1 + 2μh+h2) = P ̧-Ph+...+(− 1)' Ph' + ... And writing for μ in the first equation, - μ μ 1 (1 + 2μh + h2) ̄ = P' + Ph + ... + P'h' + ... Hence P', P... P... denoting the values which P, P,... P, respectively assume, when -μ is written for μ. 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¡-..... Then, multiplying successively by μ-2, μ grating from 1 to +1, we obtain the following system of equations: 2i-1 2i-3 + + And lastly, since P1 = 1, when μ = 1, A+A12+...+A12+..... = 1; -28 the last terms of the first members of these several equa tions being 13. i-1' i-3*** 1 A, A, A1 i-2' i-42 4,, if i be odd. 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 From this system of equations we deduce the following, O being any quantity whatever, + y + 2 1 (0–a) (0–B) (a+w) (b+w) (c+w) 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 0=a, or 0=ß, and for no other finite value of 0, and it becomes x= = 1 =- when e = W. 1 (0–a) (0–ẞ) (a+w) (b+w) (c+w) w (w-a) (w-B) (b+0) (c+U). == α, 1 (a + a) (a + B) (a + w) (b + w) (c + w) with similar values for y and z. (wa) (w - B) And, if o be infinitely great, in which case the last equation assumes the form x+y+z = 1, we have |