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**Example text**

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.