On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College.

George Green

Date:: [1833]

Credit: On the determination of the exterior and interior attractions of ellipsoids of variable densitites. / By George Green, Esq., Caius College. Source: Wellcome Collection.

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$To satisfy this equation, let us assume p = Then by substituting in the above and equating separately the coefficients of the various powers of n, we have in the first place from the highest == - e{e + r-l) (44), and afterwards generally . e — i — 2t.e—i—2t—l . “ 2^ + 2x 2e + r-2t-3 ’ But the equation (43) may evidently be made to coincide with (44), by writing for i, and for e, since then both will be comprised in i f^<'•> + r-2| (45). Hence we readily get for the general solution of the system (41), + 2 X + r — 3 - &c.]; 2.4 X {2/^'* + r — 3j {2^^'■^ + r - 5} where /u. = cos0,_r, and represents any positive integer whatever, pro- vided is never greater than Though we have thus the solution of every equation in the system (41), yet that of the first may be obtained under a simpler form by writing therein for X,_i its value deduced from (45). We shall then immediately perceive that it is satisfied by In consequence of the formula (45), the equation (42) becomes ^ f/’P s—l — np^ dP -t-^—2) k ] „ = IV Tinyf■ 1—?—^ + w-r- which is satisfied by making — X, - (/^'*-)-2a)) (/<’>-t-2a) + « — 1), and$