A:_gt w forms x and a, being : + 1 if i be odd, 3 +1 if i be even ; and X; *1-24 1 + i ai-2st w the number of equations, and therefore of letters of the i , 2 2 i-1 the number of letters of the form a being if i be odd, 2 1 if i be even. -28 1 (0-a)(0-di_2)... (0–21-98) ... (az +w) (ai-g+w)... (Qi-+w)... w (w-a;)(W__)...(w-i-2)... (a;+0) (01-2+0)... (Q:_28+)... and, multiplying by dz-+0, and then putting 0 =-01-20 · 1 (az_2 + a.) (Q:-20 +0,-2)... (Q:_2 + (x-2)... X ;-28 (w-Q;) (w – Q:-) ... (w – Qz-28)... (a; +w) (az-,+ w)...(Q:_28+w)... (0:-, - a.) (03-2 - As-2)... (0:-, -a, or a.) 15. To apply this to the case of zonal harmonics, we see, by comparing the equations for x with the equations for A, that we must suppose w = c; and a; = i, diz=i – 2....Az-20 = – 2s ... Q; = 1-1, diz = 1 – 3,...Az_2=i – 28 – 1... (2i – 28 – 1) (2i – 28 – 3)...{2 (i – 28) – 1}... ( } 2s (2s – 2)...2 x 2.4... (i – 2s - 1) or (i – 2s) 1-28 = -28 2 Or, generally, if i be odd, (2i – 1) (2i – 3)...(i +2) 2.4...(i – 1) – 2.4...(i – 5) x 2.4 Av=- – 5 = 4. =(-1); (i – 1) (2 – 3 ... 1 10 – ) 2.4...i P=1, 3 1 2 5 3 P= 24 5. – 3 2 2 II = M ? 8 Р. ,6 M + & 15.13.11.9 13.11.9.7 11.9.7.5 9.7.5.3 7.5.3.1 2.4.6.8 643548 – 120124 + 69304* – 1260j? + 35 128 2 * 2.4.64+ & 2.4.6.84 11.9.7.5 9.7.5.3 2x2.4.6 + 121554 – 25740u? + 180184 - 4620' + 315u 128 4 2 ? 13.11.9.7.5 11.9.7.5.3 9.7.5.3.1 ut 2.4.6.8.10 46189410 – 109395m® + 900904® – 300304* + 34654-63 256 It will be observed that, when these fractions are reduced to their lowest terms, the denominators are in all cases powers of 2, the other factors being cancelled by corresponding factors in the numerator. The power of 2, in the denominator of Pi is that which enters as a factor into the continued product 1.2.... We have seen that sem Pr. du = 0, if m be any . 16. Mein 0, m integer less than i. It will easily be seen that if m+ i be an odd number, the , M values of sum P..du are the same, whether he be put=1 or Swm -1; but if m +i be an even number, the values of sum P.dp corresponding to these limits are equal and opposite. Hence, (m + i being even) 1 1 m 0 LPd25*2.01 Sw, and then sow" 2. dp = 0, if m=i–2, 6–4..... may proceed to investigate the value of new P.. du, m = 1 um i We if m have any other value. For this purpose, resuming the notation of the equations of Art. 13, we see that, putting O = m +1, and w=00, we have X + + + 0-2 + m +1 ) ; (as + m + 1)(arx+ m + 1) ... (Q;-88 + m + 1). X 3-28 + Sowa Podu m and therefore, putting x:= A..., Q, = i..., Q;=i – 1 ..., we get A to.. 2-28+m+1 (m - i+ 2)(m - 2+4) ... (m – 1) if i be odd, (m + 3 + 1)(m + 3 – 1) ... (m + 4)(m +2) (m - i+2) (m - i+ 4)... m and if i be even. (m + 3+ 1)(m +i – 1) ... (m + 3) (m + 1) In the particular case in which m=i, we get 2.4 ...(i-1) (i odd), (2i + 1) (21-1) ... (i + 4) (0+2) (+ 2.4 ...i and (i even). (2i + 1) (2i – 1) ... (i + 3) (i+1) 17. We may apply these formulæ to develope any positive integral power of u in a series of zonal harmonics, as we proceed to shew. Suppose that m is a positive integer, and that he is developed in such a series, the coefficient of P, being Có, so that μ* = ΣCP; then, multiplying both sides of this equation by P, and integrating between the limits – 1 and 1, all the terms on the So w Pdp = m 1 right-hand side will disappear except | C, P, dw, which will -1 2 become equal to C. 2i +1 2i + 1 Hence Ci u 2 which is equal to 0, if m +i be odd. Hence no terms appear unless m + i be even. In this case we have 2i +1 Ci rem P, du 2 -1 -1 |