. Differential and integral calculus, an introductory course for colleges and engineering schools. The figure is drawn for &=H% CHAPTER XIV THE DERIVATIVE OF THE ARC. THE METHOD OFINFINITESIMALS 100. A Theorem of Geometry. Let PQ be an arc of any curve, and suppose this arc to be everywhere concave towards itschord. We shall prove arcPQ lira 1. q=p chord PQDraw PT tangent at P, and QT perpendicular to PT. We sup- pose Q taken near enough to P sothat QT does not meet the arc in anypoint other than Q. LetZ QPT = + TQ> arc PQ > chord PQ. The first inequality is a case of theprincip
. Differential and integral calculus, an introductory course for colleges and engineering schools. The figure is drawn for &=H% CHAPTER XIV THE DERIVATIVE OF THE ARC. THE METHOD OFINFINITESIMALS 100. A Theorem of Geometry. Let PQ be an arc of any curve, and suppose this arc to be everywhere concave towards itschord. We shall prove arcPQ lira 1. q=p chord PQDraw PT tangent at P, and QT perpendicular to PT. We sup- pose Q taken near enough to P sothat QT does not meet the arc in anypoint other than Q. LetZ QPT = + TQ> arc PQ > chord PQ. The first inequality is a case of theprinciple of plane geometry that thelength of any arc that is everywhereconcave towards its chord is lessthan the length of any broken lineinclosing it and having the same extremities. The second in-equality holds because a straight line is the shortest distancebetween two points. Now PT = PQ cos a and TQ = PQ sin <*,whence. cos a + sin a > arc PQ > 1. chord PQ As Q =P, a = 0, sin a = 0, cos a — 1, and consequently arc PQ (a) q±p chord PQ140 = 1. Q. E. D. §101 THE DERIVATIVE OF THE ARC 141 sin $ 1, which was proved in Art. 11, is obvi- The theorem lim 0=0 9 ously a special case of this theorem (a). 101. The Derivative of the Arc: Cartesian Coordinates. Let P and Q be any two points of a curve, and let their coordinatesbe x, y and x + Ax, y + s be the length of the arcAP measured from any con-venient point A. Then s is afunction of x, and arc PQ isthe increment of s due to theincrement Ax of x. We there-fore set arc PQ = As. Then lim -T— = Dxs, and we seek to Ax=oA£ express Dxs in terms of Dxy. Let c be the length of the chord PQ. We may suppose Q taken at the start so near P that arc PQ is everywhere concave towards its chord. Then theorem (a) of the preceding article applies, and we have As
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