A General Geomotry and Calculus

A General Geomotry and Calculus

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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. 1871 edition. Excerpt: ...the equation of the circle which has the three parameters m, n, andr. We will now give the demonstration. Dem.---Let y =f(x), and y' = p(x') be the equations of the loci. In order that we may make y = y' for some value of os = x' we must have liberty to impose one arbitrary condition (i e., to vary the second locus in at least one respect), but this requires one parameter. If, in addition to this parameter, there is a second (i. e., if we can vary the curve in another respect) we can impose another arbitrary eondi Price's Infinitesimal Calculus. tion, as-==: -, and so on for any number of parameters. Hence we see that dx dx' one parameter makes intersection possible; two make tangency or contact of the first order possible, three contact of the second order, etc. 204 Cor. 1.--The right line can have in general no higher order of contact than the first (tangency), since its equation y = ax-f-b has but hvo parameters a and b. 205 Cor. 2.--As the equation of the circle in its general form has but three parameters, it can in general have no higher order of contact than the second, 206 Cor. 3.--The parabola can have contact of the third order, and the ellipse and hyperbola of the fourth. 207 Sch.--This discussion assumes that y =f(x), which is given in all respects, is of such a character as to allow of any degree of contact. Of course the possibilities of contact are limited as much by one of the loci as by the other. Thus, if the first locus were a circle and the second an ellipse, the contact could not in general be above the second order, although the ellipse has a possible contact of the fourth order with other curves. Again, in this discussion we have said "in general," since exceptions occur at certain singular points. Some of...show more

Product details

  • Paperback
  • 189 x 246 x 6mm | 209g
  • Rarebooksclub.com
  • United States
  • English
  • black & white illustrations
  • 1236762657
  • 9781236762658