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 ancient arithmetic Monographs assortment.

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4... (i-2)" ZONAL HARMONICS. 22. , must be expressible in terms To determine this expression, assume function of /uT PJ_ I} Pi-3 "' , of dP. -. '. 1, > m, ^^= or i when p = = 1 From the limits be, equal to - 5) P . + (2i - (2i 4 2 cZju, 23. 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.

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 ^ . 17. 15. 13. 6 x 6 2.

E. m) roots each 1, and m in other words that (x2 is a factor of I) ***"" J_iy 1, it roots ' daT"' We proceed to calculate the other factor. For (x this + crj purpose consider the expression (a; + 2) ... (a; +a ; ) (a; + ft) (a; + ft) ... ( m Conceive this differentiated (I) i times, (II) The two expressions thus obtained will consist of an times. ^i ... (a? + a<) (# + ,) (# + &)... e. the term in (II) is the product of all the factors omitted from the corresponding term in (I) and of those factors only.

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