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By Ferrers, N. M. (Norman Macleod)

ISBN-10: 1429700521

ISBN-13: 9781429700528

This quantity is made from electronic photographs from the Cornell college Library ancient arithmetic Monographs assortment.

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Extra resources for An elementary treatise on spherical harmonics and subjects connected with them

Sample text

Pj 3 (2t- 1) 9) P^ P^. rift this equation we deduce and 1 being taken, in order that at the superior limit. P f P ( _2 may ZONAL HAKMONICS. 38 to the fundamental equation for a zonal see that Now, recurring harmonic, we 1 24. We have already seen that r P Pm dp = 0, I t i and ra 1 Suppose now that being different positive integers. required to find the value of JI n We have P Pm t it is d/j,. already seen (Art. 10) that And, from above, ' i+1 -P V _ p . ZONAL HARMONICS. 39 "We "will next proceed to give two modes of expressing Zonal Harmonics, by means of Definite Integrals.

F- + ... + -^^- + ... = o, last ZONAL HARMONICS. 22 X o* /y the /v ^f-2 i number of equations, i> forms x and the a, number and ^ ^j We 1 if ^V i and therefore of 4~ 1 i be odd, = if i being 2> 1 if i be even 2i of letters of the form a being i i\ be odd, if i 2 obtain, as before, *-. ^, t) (ca + ^, and then putting 6 - Gi _ 2) (a, (i-2 , To apply + e) . . (to - a.. J 4 2 . . a^, . . (a4 _ 2 +&>)... _ 2J ~ ^) (a^ - a _ ) = + &>)... _ a = = * - 2* ... 2 _ = i-2s- 1... i _ 2t i 2, Hence .

To - a.. J 4 2 . . a^, . . (a4 _ 2 +&>)... _ 2J ~ ^) (a^ - a _ ) = + &>)... _ a = = * - 2* ... 2 _ = i-2s- 1... i _ 2t i 2, Hence . ^i-8, and ; be even. ^ 15. 9 - 1) (2i - 2g - 3). {2 (^ - 2g) - 1|... {(- 2s - 1) or (t - 2s) ( | ^ } ' aj we see, for A, ZONAL HARMONICS. 4 We give the values of the several zonal harmonics, P inclusive, calculated by this formula, to 10 P_~ 8 ^ 2 "~ 1 2 2 2 ' ' from 24 ZONAL HARMONICS. 1 . 4^ 8 _ 231/I. - 315//, + lOo/i - 5 ~~ 4 p _ 13. 11. 4x2^ + 2x 2. 4^ 15. 8 ^ .

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


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