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

Sample text

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?