A Treatise on Analytical Statics; Attractions. the Bending of Rods. Astatics Volume 2

A Treatise on Analytical Statics; Attractions. the Bending of Rods. Astatics Volume 2

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 edition. Excerpt: ...The case in which the homoeoid is heterogeneous is not discussed. confocal ellipsoid be described passing through the point (f', rj', 2f) so that its major axis is given by the equation fc'2 '2 V'2 a'2 + a'2-(a2-62)+ a'2-(a2-c2) k ' Let this ellipsoid be the inner boundary of a second thin homoeoid whose volume' is equal to that of the former. Let its density at any point (#', y', z') be p = f(-, x, f, The potential of (X, 0 Of this second homoeoid at the internal point fj', if, % is equal to the potential required. 193. Taking the case in which the two thin homoeoids are homogeneous, the potential of the outer has been proved constant for all internal points, Art. 56. It immediately follows that the potential of the inner is the same at all external points which lie on the same confocal. We therefore infer that the level surfaces of any thin homogeneous homoeoid are confocal ellipsoids. 194. Since two thin confocal homoeoids have the same level surfaces, their potentials can be made equal over any level surface enclosing both by properly adjusting their masses. It immediately follows that their potentials are also equal throughout all external space, Art. 106. Since the potentials of finite bodies vanish at infinity in the ratio of their masses, it is evident that the masses of the two homoeoids must be equal. We have therefore the following theorem, the potentials, and therefore also the resolved attractions, of two confocal thin homoeoids of equal masses are equal throughout all space external to both. 195. Lines of force. The lines of force of a homogeneous thin homoeoid are the orthogonal trajectories of all the confocal ellipsoids. Let a', b', c'), a," b," c"), (a," b'," c") be the semiaxes of the...
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Product details

  • Paperback | 88 pages
  • 189 x 246 x 5mm | 172g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236504216
  • 9781236504210