Download The mathematical theory of electricity and magnetism: by Henry William Watson PDF
By Henry William Watson
This quantity is made out of electronic photos from the Cornell collage Library historic arithmetic Monographs assortment.
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Additional info for The mathematical theory of electricity and magnetism:
Sample text
For by differentiation r 2i+l V i (i+ I), will be a spherical har- = (2 * + 1) r '- x V + r"+1 2 1 i Similar expressions hold for Adding these expressions, and remembering that we obtain V 1 2 (r** V) t = (2i+ 1) (2 1 + 2) r 2 *'- 1 V i j 4 -t- \ 24 SPHERICAL HARMONICS. and =-+>". 5+^-5 = V V 2 [23. 0. i Therefore = and and r* is We i+ ^ 0, a homogeneous function of as, y, z of degree therefore a spherical harmonic function of degree i. Fi is i: Y. have seen that -~j , as above defined, monic function of degree (i + is a spherical har- 1).
Make the angle $ Let a small cone with solid angle da> be described about from 8 in the neighbourhood of P the as axis, cutting off OP ele- mentary surface dS. The area of dS m from is , is equal to + ; in the VOL. two I. , also the repulsion at and the resolved part the direction of the normal to according as without Bm * OP is in S at sm P in . (f> or passing out of NdS = +mco>, cases respectively.
1, l-e Hence, if p = 1, P = t 1 for all values of 1 Hence, if ^ If u < 1 Hence the = 1, P =+ i == ' series i ; if IJL = 1, 1 or is 1 according as always P + P2 +... 1 is finite, and a convergent i is even or odd. is finite if e = 1 . series. It is evident from the formation of P^ as the coefficient of (V) i e in the expansion of P 4 but can contain no than //, and no powers of which the index higher powers of differs from i by an odd number. ] . same value ZONAL SPHERICAL HARMONICS. for -f /x as for and p, be odd the same value if i with opposite sign.