An elementary course of infinitesimal calculus . follows that This is identical, except as to notation, with the formula(2) of Art. 152. 406 INFINITESIMAL CALCULUS. [CH. X 154 Osculating Circle. A slightly different way of treating the matter is basedon the notion of the osculating circle. If Q and R be twoneighbouring points of the curve, one on each side of P, weconsider the limiting value of the radius of the circle PQB,when Q and R are taken infinitely close to P. We can shew that if the curvature of the given curve becontinuous at P, this circle coincides in the limit with the circle of c
An elementary course of infinitesimal calculus . follows that This is identical, except as to notation, with the formula(2) of Art. 152. 406 INFINITESIMAL CALCULUS. [CH. X 154 Osculating Circle. A slightly different way of treating the matter is basedon the notion of the osculating circle. If Q and R be twoneighbouring points of the curve, one on each side of P, weconsider the limiting value of the radius of the circle PQB,when Q and R are taken infinitely close to P. We can shew that if the curvature of the given curve becontinuous at P, this circle coincides in the limit with the circle of curvature. For if G be the centre of the circlePQR, there will be a point P, between P and Q, such thatOP is normal to the given curve, and a point P, between Pand R, such that GP is normal to the curve. Let PG andPG meet the normal at P in the points G and G,respectively. Under the condition stated, G and G willultimately coincide with the centre of curvature at P, and,since GG < GG, G will d fortiori ultimately coincide withthe same Fig. 119. Since, before the limit, the circle PQR crosses the givencurve three times in the neighbourhood of P, it appears thatthe osculating circle will in general cross the curve at thepoint of contact. See Fig. 123, p. 422. 154] CURVATURE. 407 Ex. If in Fig. 117, p. 404, the circle PQR meet PVinW, we have QV. VR = PV. VW, and therefore VW=iSP. Hence the chord of curvature parallel to the axis of the parabolais iSP. A similar argument may be used to find the chord ofcurvature through the centre, in the case of the ellipse (Fig. 118,p. 404). If in Fig. 42, p. 153, Q V meet the circle through P, Q, Pagain in W, we have VW = PVIQV, and therefore, for the chord of curvature of the curve2/ = (/) («), parallel to the axis of y, 1 QV QV - = lim ^, = lira ^gj cos f-=i^ (a) cos^ f, as in Art. 153 (9). EXAMPLES. XLIV. 1. Prove that the circle is the only curve whose curvature isconstant. 2. Prove that the intrinsic equation of an equiangular sp
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