The properties in question are as follows: The following is a proof of the first property. dPill +i (i+1) P;=0, du dul d +m (m +1) Pm=0. du du Multiplying the first of these equations by Pm, the second by P, subtracting and integrating, we get dP:) d du dus {(1 – wey ) e (1 – wed) 2 Hence, transforming the first two integrals by integration by parts, and remarking that i(i+1) - m (m +1)= (i – m) (i + m + 1), we get dP dP Pm) 1 ) ( du du du du dP dP du du 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 - u vanishes at both limits, and therefore, except when i - m, or i + m + 1 = 0, dP, m Pi m P=P of | r; du will also SP -(i+1) -(i+1) -1 We may remark also that we have in general dP. P. du du (i-m) (i+m+1) a result which will be useful hereafter, 11. We will now consider the cases in which ¿ – m, or ¿ + 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 the determination of the value of give the value of S,P.Pd. Р. du. off",pidu may be calculated as follows: + P.h' =P+ P.,?? + ... +Pik" + ... Pifh' * 1 2h we get, taking this integral between the limits – 1 and + 1, h log h L Pidu 1=, Pdu* -h all the other terms vanishing, by the theorem just pioved. -1 = 0 0 2i ' -1 2i+1 1+h Now log 1 h 2 h hi + 3 + + 21 +1 h? Hence 2(1+ + 3 + 2i +1+...) h? log **-? (+ ...... 2 (1+ - Pu+a", p?d4 + ... + L', + + ** *, pidu +.. L., = 12. From the equation | P.Pudp= 0, combined with -1 Hence, equating coefficients of ha, 2 Pidu = 2i +1' m -1 1 ท -1 1 1 -1 the fact that, when p=1, P1=1, and that Pi 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 integer less than i, s il uim Pidu = 0. For as Pm, Pm-, ... may all be expressed as rational integral functions of p, of the degrees m, m-1... respectively, it follows that rem will be a linear function of Pm and zonal harmonics of lower orders, fum- of Prm, and zonal harmonics of , a series of multiples of quantities of the form (P.Pidp, m being less than i, and therefore 1,m pe" P.,du= 0, if m be any integer less than i. Again, since (1 – 2uh + ho)"}= P.+P/h + ... +Pik+... it follows, writing - h for h, that (1 + 2uh +h)-=P.-Ph+ ... + (-1)'PK + ... lower orders, and so on. Hence JumPdy will be the sum of 1 -2 And writing - u for p in the first equation, (1+2+h+h)-3= P0'+P,'h + + Pidhi +. P', P,' ... P' ... denoting the values which P., P,... P., Pi respectively assume, when - M is written for jo Hence P = P, or - Pi, according as i is even or odd. That is, P,= Aix' + Ai-Mi? +... Then, multiplying successively by hi-?, uts, ... and integrating from - 1 to +1, we obtain the following system of equations: A 2i - 2s -1 A+ + 2i - 3 2i - 5 21 - 2s 3 2 Ai-2 Ai-23 + + + Ai Ai-2 + + - 0, ... And lastly, since Pi=1, when j = 1, A:+Ang+...+ 41-+...=1; the last terms of the first members of these several equations being A 4., 4., if i be even, A, A 2-2 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 Pi is a constant multiple of :(-1). + w y Z 0, 1 cto From this system of equations we deduce the following, e being any quantity whatever, y 1 (0-a) (0-1) (a+w) (6+w) (c+w) + + b+o'c+ 0 w (w-a)(w-B)(a+) (6+0)(c+0). For this expression is of – 1 dimension in a, b, c, d, B, Y, 0, w; it vanishes when 0=0, or 0=B, and for no other 1 finite value of 0, and it becomes when e: ato C = We hence obtain y 1 (0–a) (0-1) (a+w)(b+w)(c+w) +0 c+) w w-a)(w-B) (6+0) (c+0) ( and therefore, putting 0=-a, , w (a - b) (a-c) (w-a) (w –B) with similar values for y and z. And, if w be infinitely great, in which case the last equation assumes the form x+y+z=1, we have (a + c)(a +B) (a - b) (a -c)' with similar values for y and z. 14. Now consider the general system X= X; + t.. ...+ ar tai-8 |