By Ferrers, N. M. (Norman Macleod)
This quantity is made out of electronic photographs from the Cornell collage Library old arithmetic Monographs assortment.
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Extra info for An elementary treatise on spherical harmonics and subjects connected with them
Let OE=c, OQ = z. line } Then EL = 2 Cc + ^' TO THE THEORY OF ATTRACTION. And 49 the solid angle subtended by the circle at ON 27 = 27T II (c* +)*] To obtain the general expression for the solid angle subtended at any point, distant r from the centre, we first develope this expression in a converging by powers of z. 4c 6 proceeding 2. 4.. 2i and if be if z be greater than less series, c, c. r P,r J__1 _ b P ___ + 1 * 2 c c 3 . 4 _ 4- 5 c 1 / ^ IV ^) Q /'0/_1 v- 4 "-- 9 . - r be greater than 4. '+!
Within the bounding surface and suppose the potential, at any point of the axis within this surface, to be interior Then the ; potential at any point lying within the inner bounding surface will be For this expression, when substituted for V, satisfies the 2 of equation V F=0; it also agrees with the given value the potential for every point of the axis, lying within the inner bounding surface, and does not become infinite at any point within that surface. Again, suppose the potential at any point of the axis without the outer bounding surface to be TO THE THEORY OF ATTRACTION.
A . 4... 3... 3... 3... 4... 1J. - r^ _. 3... (i-l) "~2 2 . 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.
An elementary treatise on spherical harmonics and subjects connected with them by Ferrers, N. M. (Norman Macleod)