. A new treatise on the elements of the differential and integral calculus . / dii — IrttjAy — ny^Ay^^:hence, ostabHshing tlu^ iiunjualilios. and dividing through byAX, wo have 280 DIFFERENTIAL CALCULUS. :-!<-(-*+iU^+-(i)-(2)- dy Ml At the limit, the second member of each of these inequalities becomes equal to ^ny A^ -\- (y^) • hence ,. ^S dS ^ I /dy\^ ^ ds dS = 27iyds. SECTION IV. DIFFERENT ORDERS OP CONTACT OF PLANE CURVES. OSCULA- TORY CURVES. — OSCULATORY CIRCLE. RADIUS OF CURVA-TURE. 168. Suppose y =: i^(x), y =/(x)j to be the equatioiis ofthe two curves RFJSf, RPN^which have a common
. A new treatise on the elements of the differential and integral calculus . / dii — IrttjAy — ny^Ay^^:hence, ostabHshing tlu^ iiunjualilios. and dividing through byAX, wo have 280 DIFFERENTIAL CALCULUS. :-!<-(-*+iU^+-(i)-(2)- dy Ml At the limit, the second member of each of these inequalities becomes equal to ^ny A^ -\- (y^) • hence ,. ^S dS ^ I /dy\^ ^ ds dS = 27iyds. SECTION IV. DIFFERENT ORDERS OP CONTACT OF PLANE CURVES. OSCULA- TORY CURVES. — OSCULATORY CIRCLE. RADIUS OF CURVA-TURE. 168. Suppose y =: i^(x), y =/(x)j to be the equatioiis ofthe two curves RFJSf, RPN^which have a common point P;and let ns compare the ordi-nates MN, IIN, of thesecurves corresponding to thesame abscissa OM =: x ~\- h,differing but Httle from theabscissa 031 =:x of the point have 3rJSr= F(x + h), MN =f{x + 70:NJSf =z F(x + h) —f{x 4- A). Developing each term in the vahio oi NN by the formula oiArt. Gl, observing that F{x) =/(x*) by hypothesis, we find. dx dx 1. ■Li dx- dF d\f^ ^ \(/.f dx) ^ ^...;/-}- 1 \dx + dx + / the hist term of which may he writton.— 30 2S1 282 DIFFERENTIAL CALCULUS, + 1 n-^2 ^ + lH being a quantity that vanishes witli h: hence \dx dxj^ \dx dxy^ j^n + l /gn + lj^ ^n + y ^,.ni-l\dx^ + ~ dx- + ^If, in addition to F(x) = fix), we have —-=—-, the curves have a common tangent, PT, at the point P, and are said tohave a contact of the J77^st order: and if, at the same time, d^F d^f -T—^ ■=■ y^T,, the contact is of the second order; and, generally, the contact is of the n^^ order if n denotes the highest orderof the differential co-efficients of the ordinates of the twocurves that become equal when in them the co-ordinates of thecommon point are substituted. 169, When two curves have a contact of the n^^ order, nothird curve can pass between them in the vicinity of theircommon point, unless it have, with each of the two curves, acontact of an order at least equal to the ?^
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