Memoirs and Proceedings - Manchester Literary and Philosophical Society Volume 27

Memoirs and Proceedings - Manchester Literary and Philosophical Society Volume 27

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1882 edition. Excerpt: ...of the element. This method leads to a more general integral form than that of Weber, and finally exhibits some of the known properties of fluid-motion in a novel manner. A similar method with the ordinary coordinates has previously J. of Math. Feb. I880) been used by Mr. Hill to obtain some similar results, which will be referred to later. If qs be a scalar function of the position of a point in the fluid, its total differential after time 3! is D, centsp3t; and if D, cents=o, the property of the point of which rp is the analytical expression is unchanged during the motion. Now let cents=;4. be the equation to a surface all points on which enjoy the same property; such a surface will, if D, centsp=o, move with the fluid. The number of independent surfaces of this kind which can pass through any point, or the number of independent integrals of the equation D, cents= o, is three, or the dimensions of the space considered; for the Jacobian of a higher number would vanish. The position of a point in the fluid in motion may be represented in terms of the three independent sets of surfaces cents, =, u., q0, =p., q.', =p., where, u-, p., and p., are parameters. I have avoided the use of theorems and terms such as those of " circulation," as I think they are apt sometimes to override and hide the more important facts which their discoverers intended them to express; and I have endeavoured to bring the fundamental properties to the surface. Imagine an element of the fluid separated from the rest of the fluid by the surfaces which we shall denote by We will investigate expressions for the parts of this element. First, V9, Vcents, V9, represent vectors in the directions of the normals to the three surfac
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Product details

  • Paperback | 66 pages
  • 189 x 246 x 4mm | 136g
  • United States
  • English
  • black & white illustrations
  • 123679236X
  • 9781236792365