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By Ferrers, N. M. (Norman Macleod)

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

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

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4 being any even integer. "(i + 2) 1 . 3 ... 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.

L)'P /*'+. 4 22 . ZONAL HAKMONICS. ^, Hence when assume, /A is written for /A. respectively = or according as i is even or odd. That is, P. involves only odd, or only even, powers of i, according as i is odd or even*. P/... P P ' 4 , l l , Assume then Our to determine is A A^.... 4 2 Then, multiplying successively by ^~ /uT ... and inte1 to + 1, we obtain the following system of grating from object t , , equations : _^ + _^_ + 13 i_3 j nr /^ 2*? And , I o 2? er L 1*=! _ + i i o 2t-5 ' ...

U f? . 9;6 P 2< I _1_ ~ / "'I (f)'\ v-' y* For each of these expressions, when substituted for V, 2 the equation V V = 0, and they become respectively is put = 0, and equal to (1) and (2) when consequently satisfies r = z. ') becomes equal to (2') when r = c, and will great, therefore denote the required potential for all values of r less than c. These expressions means may be reduced to other forms by of the expressions investigated in Chap. 2, Art. 25, viz. Or P. 3 APPLICATION OF ZONAL HARMONICS 46 which is equivalent to M The brings it substitution of the last form of into the form 1 fa c*_ 2t{2 + (2?

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