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A Treatise on Infinitesimal Calculus Volume N . 3; Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics

A Treatise on Infinitesimal Calculus Volume N . 3; Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics


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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. 1856 Excerpt: ...the lines of force of a straight bar: and spheroids are the surfaces of equilibrium. We shall hereafter see that of an ellipsoid attracting an external particle the confocal ellipsoid passing through the particle is the surface of equilibrium. For the same surface of equilibrium the constant in the right-hand member of (94) is the same; it changes however as we pass from one such surface to another. If 6 is the angle between ds and the resultant attraction, by ( E cos dds =--mdv, where v0 and vt are the values of v for the limiting points of; and therefore if s is a closed line, and if the integration is extended through the whole line, 191. For a simple application of this theory of the potential, let us consider a sphere consisting of homogeneous concentric shells, the density of each of which varies as some function of the distance from the centre, and let the attracted particle (m) be on the axis of z at a distance y from its centre. Let the radius of the sphere be a, and let the density of a shell, the internal radius of which is r, be expressed by f(r) then ma rif(r) Bin Q Q M (y2--2yrcos d + r)i = 2-/Y. (95) Jo Jo (y2--2yrcos0+r2) Here we have two cases, according as the attracted particle is external or internal; for an external particle y is greater than r, so that (y2--2yr + r2) = y--r, if M is the mass of the sphere: taking the y-differential of which, we have, by reason of equation (92), the attraction along the axis of z: and thus (dv Mm Z = mTy) = y (96) and which therefore varies inversely as the square of the distance. For an internal particle, the integral (95) must be divided into two parts, the former of which will correspond to the shell on the interior surface of which the attracted particle is, and the latter to the sphere on the ...


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Product Details
  • ISBN-13: 9781235990335
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 166
  • Spine Width: 9 mm
  • Width: 189 mm
  • ISBN-10: 1235990338
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 308 gr


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A Treatise on Infinitesimal Calculus Volume N . 3; Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics
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