An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . l Harmonic Series withdevelopment in Fouriers Series we give on page 185 a cut representing thefirst seven Surface Zonal Harmonics Pi(cos 6), P2(cos 6), • • •P7(cos 0), whichare of course somewhat complicated Trigonometric curves resembling roughlycos^, cos2^, ••-cos 7^; and on page 186, the first four successive approxi-mations to the Zonal Harmonic Series 1 + ^ P,(cos 6) P3(cos ^) + j|- II JsCcos 6) [I] [v. (1) Art. 93], and I [Po(cos 0
An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics, with applications to problems in mathematical physics . l Harmonic Series withdevelopment in Fouriers Series we give on page 185 a cut representing thefirst seven Surface Zonal Harmonics Pi(cos 6), P2(cos 6), • • •P7(cos 0), whichare of course somewhat complicated Trigonometric curves resembling roughlycos^, cos2^, ••-cos 7^; and on page 186, the first four successive approxi-mations to the Zonal Harmonic Series 1 + ^ P,(cos 6) P3(cos ^) + j|- II JsCcos 6) [I] [v. (1) Art. 93], and I [Po(cos 0) - 3 Q Vi(cos 0) - 7 (^)^3(cos 0) -iK2sy^^(^°^^)--]M (v. Ex. 3 Art. 97). 7j- 7j- [i] is equal to 1 from ^ = 0 to 0 = jj, and to 0 from 6 = -^ to 0 = 7r; and [ii] is equal to 6 from 6 = 0 to 6---^=7r. The figures on page 186 are constructed on precisely the same principle asthose on pages 63 and 64, with which they should be carefully compared. 98. By applying Gausss Theorem (B. 0. Peirce, Newt. Pot. Func. § 31) orthe special Form of Greens Theorem. ( ( f^^ ^dx dij dz = Cd,^ Vds = — Cpdx dy dz, Chap. V.] ZONAL HARMONIC CURVES,. 186 ZONAL HARMONICS. [Art. 9/ -] ^ ^^ ^ ~--,~ \ ^\ ,— \-. » / > <: 0 / ?^^ V .^. It __,. v_ V
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