A treatise on analytical geometry of tree dimensions; containing the theory of curve surfaces and of curves of double curvature

 
9781130295856: A treatise on analytical geometry of tree dimensions; containing the theory of curve surfaces and of curves of double curvature

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. 1848 Excerpt: ...of the above system a sufficient number of equations to eliminate a, and the functions £,, Tt and their derived functions; the result is a complicated equation of the third order. When in the two preceding cases, the directrices all become straight lines, the surfaces generated become respectively the hyperbolic paraboloid, and the hyperboloid of one sheet, as is seen in the Appendix; they are the only twisted surfaces whose equations do not rise above the second degree. Developable Surfaces. 178. We next come to the consideration of the second class of surfaces which admit of being generated by a straight line, the characteristic law of the motion being that two consecutive positions are always in the same plane. Before proceeding to point out the various modes in which this condition may be satisfied, we shall shew that surfaces generated in this manner are developable; that is, supposing them flexible but inextensible, they may without rumpling or tearing be made to coincide with a plane in all their points. Let fig. 57 represent a surface of this sort, and let AN, A'N', A'N", &c. be positions of the generating line indefinitely near to one another; then from the definition of the surface, AN will be intersected by A'N' in some point m, A'N' by A"N" in m, and the latter by the next generating line in m", and so on; so that these successive points of intersection will form a polygon mm'm"..;, or rather a continuous curve to which all the generating lines are tangents. Also AN and A'N', and similarly every pair of consecutive generating lines, will include a sectorial area AmA' of indefinite length, but infinitely small angle, which may be regarded as a plane element of the surface. If now the first of these elements be tu...

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