the number of equations, and therefore of letters of the forms & and a, being 11 if i be odd,+ 1 if i be even; and i+1 i 1 the number of letters of the form a being if i be odd, 2 == and, multiplying by a-+0, and then putting =— A¡-21' 1 (α;-2, +a¿) (α¿-28 +αi-2) • • • (αi-2 +¤¿-20) • • • Xi-28 = i-28 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; and Hence (2¿ — 2s − 1) (2i — 2s — 3).....{2 (i — 2s) — 1}... - (− 2s) { — (2s — 2)}...{(¿ — 2s − 1) or (i −28)} (2i2s-1) (2i — 2s — 3)...{2 (i — 28) — 1}... =(-1) 25 (28-2)...2 × 2.4... (i-2s -1) or (i - 2s) * 2s We give the values of the several zonal harmonics, from P to P inclusive, calculated by this formula, 10 128 15.13.11.9 13.11.9.7 2.4 x 2.4 8 11.9.7.5 9.7.5.3 2x2.4.6 2.4.6.8μ 12155μ- 25740μ+18018μ- 4620μ3 +315μ 19.17.15.13.11 128 2.4.6.8.10 46189μ10-109395μ3+90090μ3 — 30030μ*+ 3465μ3 — 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 P1, is that which enters as a factor into the continued product 1.2...i. •1 m 16. We have seen that [μ" P,.dμ = 0, if m be any integer less than i. μ It will easily be seen that if m+i be an odd number, the values of fuTM P..du are the same, whether μ be put=1 or Jum -1; but if m+i be an even number, the values of μm P1.dμ corresponding to these limits are equal and opposite. Hence, (m + i being even) and then "P,. dμ = 0, if m = i – 2, i – 4......... We may proceed to investigate the value of " P..du, μη if m have any other value. For this purpose, resuming the notation of the equations of Art. 13, we see that, putting 0 = m + 1, and w∞, we have = (m +1 − a) (m + 1 − x_2) ... (m + 1 − 01-28) (a + m + 1)(a12 + m + 1) ... (α-28 + m + 1) ... and therefore, putting x, A...., a=i..., α= i − 1 ..., = = and = and . αμ if i be odd, if i be even. (m + i + 1)(m + i − 1) ... (m + 4) (m +2) (m − i + 2) (m − i + 4) ... m (m + i + 1) (m + i − 1)... (m + 3) (m + 1) In the particular case in which m = i, we get 17. We may apply these formulæ to develope any positive integral power of μ in a series of zonal harmonics, as we proceed to shew. Suppose that m is a positive integer, and that μm is developed in such a series, the coefficient of P being C, so that then, multiplying both sides of this equation by P, and integrating between the limits - 1 and 1, all the terms on the right-hand side will disappear except C, P, du, which will which is equal to 0, if m+ be odd. Hence no terms appear unless m + i be even. In this case we have |