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.

25/36 page 419

$Application of the preceding General Theory to the Determination of the Attractions of Ellipsoids. 13. Suppose it is required to determine the attractions exerted by an ellipsoid whose semi-axes are a', b', c' whether the attracted point j) is situated within the ellipsoid or not, the law of the attraction being inversely as the power of the distance. Then it is well known that the required attractions may always be deduced from the function p dx' dy d% {{x~xy + {y- y'f f p being the density of the element dx' dy' d%' of the ellipsoid, and X, y, z being the rectangular co-ordinates of p. We may avoid the breach of the law of continuity which takes place in the value of V, when the point p passes from the interior of the ellipsoid into the exterior space, by adding the positive quantity to that inclosed in the braces, and may afterwards suppose ti eva- nescent in the final result. Let us therefore now consider the function, p dx' dy dz' {(^ - x’y + (y- y'f + (z — z'f + u^] * this triple integral like the preceding including all the values of x', y', z', admitted by the condition X'^ y'2 g,/2 — -F — + — Z 1 o', -t- ^ ^=/ If now we suppose the density p’ is of the form '2s n'-2 S''.*') (84), which will simplify f {x', y, z') when p is constant and n' = 2, and then compare this value with the one immediately deducible from the general expression (28) by supposing for a moment »' = w, viz. p — ^^2 ^'2 j T {x, y, z). VoL, V. Part III. 3l$