. An elementary treatise on the differential and integral calculus. circle of curvature of that point of the curve. Theradius of curvature is the radius of the osculating centre of curvature is the centre of the osculating circle. For example, let ABAB be an ellipse. If differentcircles be passed through B with their centres on BB, it is c Fig. 36, ORDER OF CONTACT OF CURVES. 217 clear that they will coincide with the ellipse in very differ-ent degrees, some falling within and others without. Now,that one which coincides with the ellipse the most nearlyof all of them, as in this cas
. An elementary treatise on the differential and integral calculus. circle of curvature of that point of the curve. Theradius of curvature is the radius of the osculating centre of curvature is the centre of the osculating circle. For example, let ABAB be an ellipse. If differentcircles be passed through B with their centres on BB, it is c Fig. 36, ORDER OF CONTACT OF CURVES. 217 clear that they will coincide with the ellipse in very differ-ent degrees, some falling within and others without. Now,that one which coincides with the ellipse the most nearlyof all of them, as in this case MN, is the osculating circleof the ellipse at B, and is entirely exterior to the osculating circle at A or A, is entirely within theellipse ; while at any other point, as P, it cuts the ellipse,as will be shown hereafter. 117. Order of Contact of Curves. — Let y — f (x) and y=z 0 (./•) be the equations of thetwo curves, AB and ab, referred tothe axes OX and OY. Giving toscan infinitesimal increment ft, andexpanding by Taylors theorem, wehave,. Fig. 37. lr yx =f(x + h) = f(x) + / (*) h + / (*) g- + /(z)^33 + etc. y, = (x + h) = (*) + (x) h + f («) j etc. (1) (2) Now if, when x — a — OM, we have / (a) = 0 (a), thetwo curves intersect at P, i. e., have one point in in addition we have / (a) = {a), the curves have acommon tangent at P, i. e., have two consecutive points incommon ; in this case they arc said to have a contact of thefirst order. If also we have, not only/(a) = {a) and/ (a)= 0 (a), but/ (a) — <p (a); i. e., in passing along one of cfiiithe curves to the next consecutive point, -~ (i. e., the curva-ture), remains the same in both curves, and the new point10 218 CONTACT OF THE SECOND ORDER. is also a point of the second curve; i. e., the curves havethree consecutive points in common ; in this case the curvesare said to have a contact of the second order. If f(a)= (a),f (a) = f (a),f (a) = <p (a),/ («) = (a),the contact is o
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