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

μ

unity.

dU

Then

(1 − v2)

(1 - μ3)

αμ

d U

+

=

μι

2

27

μυ

=2

be expanded in a series of zonal harmonics

P。(μ), P1(μ).....P, (μ), so that

= $。(v) Po(μ) + $2 (μ) P1 (μ) + ... + $; (v) P ̧ (μ)+.....

Ο

by the definition of P (μ).

And these two expressions are equal. Hence, equating

the coefficients of P(u),

Hence

(v) satisfies the same differential equation as P and Q. But since U=0 when v∞, it follows that 4, (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

and also = 4, (v) Po(u) +4, (v) P ̧ (u) +...+$, (v) P ̧(u)+.....

Now, by Chap. II. Art. 17, we see that, if m be any integer greater than i, the coefficient of P, in " is

(2i+1)

-

m

(m − i + 2) (m − i + 4) ... (m − 1)
(m + i + 1) (m + i − 1) ... (m + 4) (m +2)

and (2i+1)

(m-i + 2) (m-i + 4) ... m
(m+i+1) (m+i−1)... (m+3) (m+1)

m-i being always even.

if i be odd,

if i be even,

Hence,' writing for m successively i, i + 2, i + 4, ... we get

হ

2.4...(-1)

1

4. (v) = (2i + 1) { (2i + 1) (2i − 1).....(i+2) ‚1

we see that, if Q.(v) be developed in a series of ascending

1

powers of, the first term will be

is the coefficient of μ in the development of P (μ);

1

where C

C (2i+1) vi

Hence the first term in the development of Q. (v) is

which is the same as the first term of the development of

1

· Qo (v) P2 (u) + 3 Q1 (v) P ̧ (μ) + 5Q ̧ (v) P2(μ) +.....

2

3. The expression for Q, may be thrown into a more 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

1.2.3...i 1 3.4.5...(+2) 1
+
i+1
1. 3. 5... (2i + 1) v2 3. 5. 7... (2i + 3) vits

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

Now we have seen (Chap. II. Art. 22) that

And therefore the coefficient of Pu) in the expansion

d 1

of

is

αμν-μ

(2i+1) {(2i+3) Qi+1 (v) +(2i+7) Qits (v)+(2i+11) Qi+5 (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