Proceedings of the Manchester Literary and Philosophical Society Volume 16-18

Proceedings of the Manchester Literary and Philosophical Society Volume 16-18

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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. 1877 edition. Excerpt: ...instead of the axes being rectangular, the primitive equation is referred to a pair of conjugate diameters as axes, providing, however, that the tubular ordinates be of elliptical transverse section such that their section by the plane of the two other coordinates is circular. The extension of the method so as to represent an equation between three variables, referred to rectangular axes, presents no difficulties. It will be observed that the envelope of the circular ordinates is the same central conic, that we should obtain by taking, in the proper plane, ordinates affected with the sign V--1 as normals instead of ordinates and measured from the extremities of the respective abscissae which yield them. One interesting consequence of taking account of both the representative curves of an equation, is that it affords a geometrical illustration of the four-point intersection of pairs of conies in cases which are unintelligible without it: in the case of similar and similarly situated concentric ellipses, for example, the absence of any points of intersection on the ellipses is compensated for by the touching of the pair of supplementary hyperbolas on the coincident asymptotes. The writer is under the impression that he will find that, in the general case of elimination between the equations of a pair of conies, if Years ago, the writer became acquainted with the method in analytical geometry which regards a plane and a straight line--not as the pure abstractions of the mathematician, but--as small portions of the surface of a sphere and of the periphery of a circle respectively in the limits in which the radii of the sphere and circle become infinite; and, whilst noting that these definitions were open to serious objections (chiefly more

Product details

  • Paperback | 170 pages
  • 189 x 246 x 9mm | 313g
  • United States
  • English
  • black & white illustrations
  • 1236894685
  • 9781236894687