A treatise on the application of analysis to solid geometry

Gregory, Duncan Farquharson; Walton, William

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Monograph

Persistent identifier:: 898898862

Author:: Gregory, Duncan Farquharson; Walton, William

Title:: A treatise on the application of analysis to solid geometry

Scope:: 1 Online-Ressource (XII, 276 Seiten, 3 gefaltete Blätter mit Bildtafeln)

Year of publication:: 1845

Place of publication:: Cambridge

Identifier (digital):: 898898862

Language:: English

Publisher of the original:: Deighton

Usage licence:: Public Domain Mark 1.0

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2017

Document type:: Monograph

Collection:: Mathematics

Chapter

Title:: CHAPTER XIV. ON THE CURVATURE OF CURVES IN SPACE.

Document type:: Monograph

Structure type:: Chapter

( 217 ) CHAPTER XIV. ON THE CURVATURE OF CURVES IN SPACE. (260) Let any number of points P, P', P", P", ... (fig. 29) be taken in a curve AB in space. Join PP', P'P, PP",... by straight lines : these lines will be chords of the curve, and when the number of the points is increased without limit, will ulti mately assume ratios of equality with the corresponding elements of the arc of the curve. Produce PP', P'P, indefinitely to points T, T' ; P T, P' T, will ultimately be tangents to consecu tive points of the curve : and it is evident that the amount of the curvature of the curve in the vicinity of the point P may be properly measured by the ratio of the angle TP'T to the chord or elemental arc PP', that is, by the rate at which the tangent changes its direction in passing from any point of a curve to a consecutive one. (261) In the osculating plane PP'P' di*aw two normals KO, K' 0, from the middle points K, K', of the chords PP', P'P' : these normals will include an angle KOK' equal to the angle TP'T' ; call this angle de; the point 0, in which the two normals intersect, will be the centre of a circle passing through the three points P, P', P . This circle is called the osculating circle to the curve at the point P, in consequence of having two consecu tive elements PP', P'P, in common with the curve. Each of the distances OP, OP', OP ", is a radius of this circle. Sup posing the elements PP', P'P, to have been taken equal, then the bent line KPK' will ultimately be an elemental arc ds, both of the curve and of the osculating circle : and, since it subtends at 0 an angle de, we shall have in the limit, p being the ultimate value of OP, or de ds 1 P (!)•

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