On the attractions of homogeneous ellipsoids / [Sir James Ivory].

  • Ivory, James, Sir, 1765-1842.
Date:
1809
Catalogue details

Licence: Public Domain Mark

Credit: On the attractions of homogeneous ellipsoids / [Sir James Ivory]. Source: Wellcome Collection.

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    position that one of the indeterminate angles is constant; thus, if be constant, then dy = k cos. <p cos. ij>. d<p: and, becausey must be constant when z varies, we must make dz = k cos. <p sin. i]/. dcp -|- k sin. (p cos. i[/. dty o s— k! cos. <p cos. ij/. d(p — k' sin. <p sin. i[/. dtyt and, by exterminating dp, we get dz = k£j-. di>. Thus, by substitution, the formula (2) will become A = k'kJf sin. (p cos. . dp . dty . jT- — ; (3) and, , A={ (0-—& cos. <p )*-{-( 6—sin. <p cos. 40“+ (c~£sin. <p sin. 4')' A'= | (a-\-k cos. (p )*+(6—k' sin. <p cos. i[/)®+ (c—k sin. <p sin. xj/ )• | \ the double fluent must be taken from <p = 0, to (p = ~ [w de¬ noting half the pheriphery of the circle, whose radius is 1), and from i|/ = 0, to t]y = 223-. To obtain a further transformation of the last expression of A, we are now to determine the semi-axes of an ellipsoid, whose surface shall pass through the attracted point, and which shall have the same excentricities, and its principal sections in the same planes, as the given ellipsoid. Let h, h'} li be the semi-axes required: then, because the attracted point is to be in the surface of the solid, £ \ £ \ £ _ bx * b'x “+* bz — 1: and, because the excentricities must be equal to those of the given ellipsoid, therefore h!x — h• = k'z — k? = ez, and h% — hx = knz — k* = e/a: hence £ 4. b* J. bz T = l; b*+ e'z an equation which now contains only one unknown quantity, C2