# An elementary treatise on spherical harmonics and subjects by Ferrers, N. M. (Norman Macleod)

By Ferrers, N. M. (Norman Macleod)

This quantity is made out of electronic pictures from the Cornell collage Library historic arithmetic Monographs assortment.

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**Example text**

11. 17 1 cocQg- 5.! 19 J 4096 3405 p 6 256 1001 4 . 5 7 9 11 13 . . 11 64 693 _ " p_ T73T577 . 13 . "15". 17 _ 7 _ 3 . 16 -l. 3. 5. 7. 11. 13. 15. 17719 __ i^>jAi 10_- . S 9 6 8 1(K 12_ 5T77ii7l37T5 . 13 . 11 * ZONAL HARMONICS. 21 ,_ ~~ J. / "^ ~ 8. 10 4^6. i _ s . 6 . 15 5 32768 24453 12. 14. 11 ^00 4 1001 2 9009 ' 99 21. The present will be a convenient opportunity for investigating the development of sin# in a series of zonal 2 harmonics. Since sin 6 = (1 /A )^, it will be seen that the be and that no zonal harmonic of an odd series must infinite, order can enter.

06' 2 - 2a& cos*)* 2 (a + 6 2 (a J* , whence 2 (a + 6 2 - 2a6 cos , * &)* / 2 a 6 2 this, = 1, We write a also see, to From 28. //,, - i _ 2 7> Z> 2 (/A = 0. 9/ = - (a + 2 - 2a6cos 2a&= + P i by reference TT, 1)^, ^ ^f. which gives the last definite integral, and in terms of cos 6 1 ['{(cos ^) to the figure, that as S- in from TT to 0. Hence ty diminishes we may obtain an sin 0. 4 1 cos ^ sin #]'] d^r M (sin^)'+... 43 ZONAL HARMONICS. 2.. 2 (2. 4.. 2m) P = (cos fff - m ^) er (sm ^) + ff)* CHAPTER III.

Now, writing P = C cos id + C^ cos (i - 2) + P cos m0 sin dd will consist of a series we ... t t see that of angles of the forms ( { m _ { _ l) $ 5 coefficient of (m + i + 1) 0, (m + i I) ... of sines to down there being no term involving m0, since the Hence such a term must be zero. /' Joo < cos m0 sin dd, ' will consist of a series of fractions 1 i volve the factors i+l, m+ m+ Therefore if TO (m + i+1) (TO+i-1) even, and ... (m + (m-1) ... (m-i-V) + 2) (m-2) ... (m-i-T) 1) m be (wi if TO + + l)(m + i-l)...