## Get An elementary treatise on spherical harmonics and subjects PDF

By Ferrers, N. M. (Norman Macleod)

ISBN-10: 1429700521

ISBN-13: 9781429700528

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

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

Example text

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?

4... 19. be even, and . 4... (i cos 6, if i be odd. 1) Let us next proceed to investigate the value of Clt P cos m0 sin t Jo 6 1 d0. ZONAL HARMONICS. SO Tins might be done, by direct integration, from the above Or we may proceed as follows. The above value of P when multiplied by expression. i is (that by ~ (sin Zt (m+ 1) 6 sin cos mO sin 1) 9}} will consist of (m a form {i 2n + (m I)} 0, that odd multiples of 0, as i + m is odd or even. and TT it Therefore, when integrated between the limits We may therefore limit ourwill vanish, if i + m be odd.

ZONAL HARMONICS. = 11 _Aii p* . 7 p ^T p 10 p 4 p 21 p l 2 ' '39 , p 3 -19 8 6 495 2 4 - - 6 - ^L8 q ' 143 p4 ' 99^ 2 _ 21 9 _ ' 192 " 2431 . 8 T-. 16 18 7 6- 8 4 - 10 * 9 0> 8 56 p ""^ 7 -r-. 6 n , p 7 15. 13. 11. 9^'"*" 429 ? 6 . p 143 187 6 4 + 10 ^ 143 pip ^ 2 11 ZONAL HARMONICS. 18. Any 29 zonal harmonic P< may be expressed in a finite of multiples of 6, these multiples being series of cosines i0,(i-2)0.... Thus therefore, writing cos 6 for and observing that - 2 cos 6h + h* = (1 - Ae v~i e) 1 we /-i, - he- v ~^), (1 obtain or 1 whence, equating coefficients of A , ....