Now, when m is very large as compared with i, this be

since C+ C+...=1, as may be seen by putting 0 = 0.

Hence ["P, cos me sin e de tends to the limit

is indefinitely increased.

2

m2,

as m

The value of the factor involving m has been shewn above to be

{m — (i − 2)} {m — (i − 4)} ... (m − 2) m2 (m + 2) ... (m + i − 2)

{m − (i + 1)} {m — (i − 1)}

if m be even, and

(m + i + 1)

{m — (i − 2)} {m — (i — 4)} ... (m − 1) (m + 1) ... (m + i− 2)

{m − (i + 1)} {m — (i − 1)}

if m be odd.

Each of these factors contains in its numerator two factors less than in its denominator. It approaches, therefore, when

m is indefinitely increased, to the value

{m — (i—2)} {m— (i—4)} ... (m −2) m2 (m+2)... {m+ (i −2)} ́{m−(i+1)} {m—(i −1)}... (m − 1) (m + 1) ... {m + (i + 1)} if m and i be even, and

In each of these expressions i may be any integer such that mi is even, i being not greater than m. Hence they will always be negative, except when i is equal to m.

20. We may apply these expressions to develop cos me in a series of zonal harmonics..

Assume

cos me = BmPm+ Bm-2P,

+

... m-2

+B,P+...

Multiply by P, sin 0, and integrate between the limits 0

and π, and we get

- 2

{m — (i − 2)} {m — (i − 4)} ... {m + (i − 2)}

' {m − (i + 1)} {m − (i − 1)}

Hence

B1 = − (2i + 1)

·

...

{m + (i + 1)} ̄ ̄ 2i+1

{m — (i − 2)} {m — (i − 4)} ... {m + (i −2)}
{m − (i + 1)} {m — (¿ − 1)} ... {m + (i + 1)}

-

Hence, putting m successively = 0, 1, 2, ... 10,

21. The present will be a convenient opportunity for investigating the development of sin in a series of zonal harmonics. Since sin = (1-2), it will be seen that the series must be infinite, and that no zonal harmonic of an odd order can enter. Assume then

Multiplying by P, and integrating with respect to μ between the limits - 1 and + 1, we get

supposing P expressed in terms of the cosines of 0 and its

multiples

For values of i exceeding 2, we observe, that if we write for P; the expression investigated in Art. 18, the only part

of the expression [P. (1 - cos 20) de which does not vanish

i

will arise either from the terms in P; which involve cos 20, or from those which are independent of 0. We have therefore

=

2+1 1.3... (¿ − 1) 1. 3 ... (¿– 3)

4 2.4...

π

—

г

2.4... (2)

-1, +1.

2

+ 2 cos 20) (1-cos 20) de

i+2

П

i+2/

4 2.4... 2 2.4...(-2) 2
2i+11.3... (1) 1.3... (i-3)
π 2 2.4...i (i + 2) 2. 4 ... (i − 2) i'

(2i + 1) π 1 . 3 ... (¿ −1) 1.3 ... (i − 3)
2 2.4...i (+2) 2.4... (i − 2) i

i being any even integer.