And, in a similar manner, the values of Q, Q,,... may be calculated. 2. But there is another manner of arriving at these functions, which will enable us to express them, when the variable is greater than unity, in a converging series, without the necessity of integration. This we shall do in the following manner. v being not less, and μ not greater, than = = = (1 − v2) dv (-1 = 1 2μ 2 1-μ ע + 2 = 2 - μ 1 - μν = 2 1- μν = $。 (v) P2 (u) + $1 (μ) P1 (μ) + ... + $, (v) P1 (u)+... 0 And these two expressions are equal. Hence, equating the coefficients of P¡(μ), d dv Hence do.(v) {(1-20) '}} = − i (i + 1) 4, (v). (v) satisfies the same differential equation as P and Q. But since U=0 when v∞, it follows that ; (v) =0) when v = ∞. Hence (v) is some multiple of Q. (v)=AQ,(v) suppose. It remains to determine A. Now, (v) may be developed in a series proceeding by ascending powers of We have and also 1 ע = as follows. 1 μ V = 4。(v) Po(u) +4, (v) P1 (μ) +.....+$; (v) P¡ (μ) +....... Now, by Chap. II. Art. 17, we see that, if m be any integer greater than i, the coefficient of P, in u m (m+i+1) (m + i −1)... (m+3) (m+1) m-i being always even. is if i be odd, if i be even, Hence,' writing for m successively i, i + 2, i + 4, ... we get + 1 (2i+1) (2i −1)... (i + 1) v ̧* 1 (2i+3) (2i + 1)...(i+3) vits 1 +1 6.8... (i+4) if i be even. (2i+ 5) (2i+3)..... (i + 5) x2+s+.... Now, recurring to the equation we see that, if Q.(v) be developed in a series of ascending 1 powers of the first term will be ע is the coefficient of μ in the development of P(μ); Hence the first term in the development of Q, (v) is which is the same as the first term of the development of Qo (v) Po(u)+3Q2 (v) P ̧ (μ) + 5Q, (v) P2 (μ) + · 1 3. The expression for Q, may be thrown into a convenient form, by introducing into the numerator and de nominator of the coefficient of each term, the factor necessary to make the numerator the product of i consecutive integers. We shall thus make the denominator the product of i consecutive odd integers, and may write + (2k + 1) (2k + 2).....(i + 2k) 1 (2k + 1) (2k + 3)... (2i+ 2k+1) 2+2-1+.... whether i be odd or even. 4. We shall not enter into a full discussion of the properties of Zonal Harmonics of the Second Kind. They will be found very completely treated by Heine, in his Handbuch der Kugelfunctionen. We will however, as an example, investigate the expression for in terms of Q1, Qi+s••• d Qi dv we see that 1 + (2i+1) 5Q, (v) P. (μ) + ... Now we have seen (Chap. II. Art. 22) that i+1 άμ dP,(u) = (2i — î) P1., (μ) + (2i − 5) P ̧-s (μ) + ... du Hence i) dP1+1 (μ) = άμ i-1 (2i + 1) P ̧ (μ) + (2i − 3) Pi-2 (μ) +..... And therefore the coefficient of Pu) in the expansion d 1 of is άμν-μ (2i+1) {(2i+3) Qi+1 (v) +(2i+7) Qits (v)+(2i+11) Qi+s(v) +.....}. 5. By similar reasoning to that by which the existence of Tesseral Harmonics was established, we may prove that there is a system of functions, which may be called Tesseral Harmonics of the Second Kind, derived from T) in the same |