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An Elementary Treatise on Solid Geometry

An Elementary Treatise on Solid Geometry


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About the Book

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. 1884 edition. Excerpt: ...Prove that the four lines of intersection are generating lines, of the same system, of a hyperboloid. CHAPTER X. Surfaces In General. 201. We shall in the present Chapter discuss some properties of surfaces of higher degree than the second. 202. Let F(x, y, z) = 0 he the equation of any surface. To find the points of intersection of the surface and the straight line whose equations are x--x'_y--yl_z--z' l m n' we have the equation Fx' ] Ir, y + mr, z + nr) = 0, or _, ., ., f, dF, dF, dF F(x, y, z)+r(l, + m, + n-d-F) If the equation of the surface be of the?ith degree, the equation (i) will be of the nth degree. Hence a straight line will meet a surface of the nth degree in n points, and any plane will cut the surface in a curve of the nth degree. 203. To find the equation of the tangent plane at any point of a surface. If (x y', z) be a point on F(x, y, z) = 0, one root of the equation for r, found in the preceding article, will be zero. Two roots will be zero if I, m, n satisfy the relation 7dF t dF dF. lM + md? + ncb' =-The line will in that case be a tangent line to the surface; and the locus of all the tangent lines is found by eliminating I, m, n by means of the equations of the straight line. We thus obtain the required equation of the tangent plane, dF, . dF, dF _ If the equation of the surface be z--fx, y) = 0, it is easy to deduce from the above, or to shew independently, that the equation of the tangent plane at (x't y, z) is 204. The two real or imaginary lines whose directioncosines satisfy both the relations.dF dF dF _ ldx' + md-y+ndz' =, and (ld%+mk+nT.)F=0' meet the surface in three coincident points. Hence two of the tangent lines at any point of a surface meet the surface in three coincident points. These are called the...



Product Details
  • ISBN-13: 9781154496758
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 72
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 1154496759
  • Publisher Date: 26 Jun 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 145 gr

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