A Treatise on Differential Equations

A Treatise on Differential Equations

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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. 1865 edition. Excerpt: ...of which I propose to consider in this paper, is remarkable from its connexion with Geometry. If the equation of a surface contain three constants which vary as parameters in subjection to any two conditions connecting them, the generated envelope will satisfy a partial differential equation of the above form. In other words any envelope of the surface F(x, y, z, a, b, c) = 0 formed by the variation of a, b, c in subjection to two connecting conditions & (, h c) = 0, 02 (a, b, c) = 0 is necessarily an integral of a partial differential equation of the form given above. Now this theorem is the more important, because it is only when three parameters in the equation of a surface vary in subjection to two relations that the envelope possesses, irrespectively of the form of the connecting relations, any definite character. If there be but one connecting relation it is possible to determine that relation so as to make the envelope assume the form of any surface whatever, and therefore the possible system of envelopes is in such case R.D.E. II. 10 unlimited. If there be three connecting relations the parameters become absolutely constant and no envelope exists. The partial differential equation Br + Ss + Tt + U(a2-rt) = V is remarkable also as including all the cases in which a partial differential equation of the second order admits a first integral of the form u=f(v), u and v being definite functions of x, y, z, p, q, and/(r) arbitrary in form. Neither of these statements is sufficiently general to constitute a theory of the genesis of the partial differential equation under consideration, but the second one is more general than the first, and is indeed sufficiently so to serve as the ground of an investigation which connects the..show more

Product details

  • Paperback
  • 188.98 x 246.13 x 3.3mm | 131.54g
  • Rarebooksclub.com
  • English
  • Illustrations, black and white
  • 1236984765
  • 9781236984760