CUR
C U R
only of the fecond degree : and this gradation of more and more intimate contact, or of approximation towards coinci- dence, may be continued indefinitely, the contact of EM and ER at E being always of an order two degrees clofer than that of BK and B Q_ at B. There is alfo an indefi-1 nite variety comprehended under each order: thus, when E M and E R have the fame curvature, the angle formed by the tangents of BK. and BQ_ admits of indefinite variety, and the contact of EM and ER is the clofer, the lefs that angle is. And when that angle is of the fame magnitude, the contact of E M and E R is the clofer the greater the cir- cle < f curvature is. When BK and BQ_ touch at B, they may touch on the fame or on different fides of their common tangent; and the ancle of contact KBQ_may admit of the fame variety with the anrrle of contact MER ; hut as there is feldom occafion for confidering thofe higher degrees of more intimate contact of the curve EMH and circle of cur- vature ERB, Mr. Mac Laurin calls the contact or ofcula- tion of the fame kind, when the chord EB and angle BET being given, the angle contained by the tangents of BK and BQ^is of the fame magnitude. Lib. cit. Art. 368. When the curvature of EMH increafes from E towards H, and confequcntly correfponds to that of a circle gradually lefs and lefs, the arch EM falls within ER, the arch of the circle of curvature, and BK is within BQ. The contrary happens when the curvature of EM decreafes from E to- wards H, and confequently correfponds to that of a circle that is gradually greater and greater, the arch EM falls with- out ER, the arch of the circle of curvature, and BK is without BQ^ And according as the curvature of EM va- ries more or lefs, it is more or lefs unlike to the uniform j curvature of a circle j the arch of the curve EMH feparates j more or lefs from the arch of the circle of curvature ERB, i and the angle contained by the tangents of BKK and BQ_E at B, is greater or lefs. Thus the quality of curvature, as it is called by Sir Ifaac Newton 3 , depends on the angle con- tained by the tangents of BK and BQ_at B ; and the mea- sure of the inequability or variation of curvature, is as the tangent of this angle, the radius being given and the angle BET being right b . — [» Method of Flux, and Inf. Ser. Prob. vi. p. 75. b Mac Laurin, lib. cit. Art. 369. J The rays of curvature of fimilar arches in fimilar figures, are in the fame ratio as any homologous lines of thefe figures j and the variation of curvature is the fame. See Mac Laurin, Jib. cit. feet. 370.
When the propofed curve EMH, Is a conic-fection, the new line BKF is alfo a conic-fection ; and it is a right line v/hen EMH is a parabola, to the axis of which the ordinates TK are parallel. BKF is alfo a right line when EMH is an hyperbola, to one afymptote of which the ordinate T K is parallel. Mac Laurin, lib. cit. Art. 371, 37?-. "When the ordinate EB, at the point of contact E, inftead of meeting the new curve B K, is an afiymptote to it, the cur- vature of EM will be lefs than in any circle ; and this is the cafe in which it is faid to be infinitely little, or that the ra- dius of curvature is infinitely great. Of this kind is the cur- vature at the points of contrary flexure in lines of the third order. See Mac Laurin, lib. cit. Art. 377 — 379. When the curve B K pafTes thro' the point of contact E, the curvature is greater than in any circle, or the radius of cur- vature vanifhes j and In this cafe the curvature is faid to he infinitely great. Of this kind is the curvature at the cufpids of the lines of the third order. See Mac Laurin, lib. cit. Art. 378, 379.
As to the circles of curvature for lines of the third or higher orders, fee lib. cit. Art. 379 ; and Art. 380, when the pro- pofed curve is mechanical.
As lines which pafs thro' the fame point have the fame tangent when the firft fluxions of the ordinates are equal, fo they have the fame curvature when the fecond fluxions of the ordinate are likewife equal ; and half the chord of the cir- cle of curvature that is intercepted between the points where- in it interfects the ordinate, is a third proportional to the right lines that meafure the fecond fluxion of the ordinate and firft fluxion of the curve, the bafe being fuppofed to flow uniformly. When a ray revolving about a given point, and terminated by the curve, becomes perpendicular to it, the firft fluxion of the radius vanifhes ; and if its fecond fluxion vanifhes at the fame time, that point muff be the center of curvature 3 -. The fame may be faid, when the angular mo- tion of the ray about that point is equal to the angular mo- tion of the tangent of the curve ; as the angular motion of the radius of a circle about its center is always equal to the angular motion of the tangent of the circle. Hence the va- rious properties of the circle may fuggeft feveral theorems for determining the center of curvature b . — [ a Mac Laurin, lib. cit. Art. 382, &c. b Mi. Art. 389, &c] See alfo Art. 396, of the faid treatife, and the following, concerning the curvature of lines that are defcribed by means of right lines revolving about given poles, or of angles that either revolve about inch poles, or are carried along fixed lines.
It is to be obferved, that as when a right line interfects an arch of a curve in two points, if by varying the pofition of
that line the two interfections unite in one point, it then be- comes the tangent of the arch ; fo when a circle touches a curve in one point and interfects it in another, if, by varying the center, this interfection joins the point of contact, the circle then has the clofeft contact with the arch, and becomes the circle of curvature ; but it ftill continues to iuterfect the curve at the fame point where it touches it, that is, where the fame right line is their common tangent, unlefs another interfection join that point at the fame time. In general, the circle of curvature interfects the curve at the point of of- culation, only, when the number of the fucceflive orders of fluxions of the radius of curvature, that vanifhes at the term of time that this radius comes to the point of ofculation, is an even number. Mac Laurin, Flux. Art. 493. It has been fuppofed by fome, that two points of contact, or four interfections of the curve and circle of curvature, muft join to form an ofculation. But Mr. James Bernoulli infift- ed juftly, that the coalition of one point of contact and one interfection, or of three interfections, was fufficient. In Which cafe, and in general, when an odd number of inter- fections only join each other, the point where they coincide continues to be an interfection of the curve and circle of cur- vature, as well as a point of their mutual contact and ofcu- lation. See Mac Laurin's Fluxions, Art. 493. From thefe principles the circle of curvature at any point of a conic-fection may be determined. SuppofeEMH (fig. III.) to be any conic-fection, ET the tangent at E, HI a tangent pa- rallel to E B, (a chord of the circle of curvature) that meets ET in I, and let EMH meet EB in G. Take E B to E G, in the fame ratio as the fquare of EI is to the fquare of HI; or, when the lection has a center as in the ellipfis and hyperbola, as the fquare of the femi-diameter Oa pa- rallel to ET, is to the fquare of the femi-diameter O A pa- rallel to EB ; and a circle defcribed upon the chord EB that touches ET, will be the circle of curvature.
Fie. III.
When BET is a right angle, or EB is the diameter of the circle of curvature, EG will be the axis of the conic-fec- tion, and EB will be the parameter of this axis ; and when the point G where the conic-fection cuts EB, and B are on the fame fide of the point E, EMG will be an ellipfis, and EG the greater or lefier axis, according as EG is neater or lefs than EB. &
The propofitions relating to the curvature of the conic-fec- tions, commonly given by authors, follow without much dif- ficulty from this conhxuetion.
1. When the chord'of curvature thus found panes thro' the center of the conic-fection, it will then be equal to the pa- rameter of the diameter that pafTes thro' the point of con- tact.
2. The fquare of the femi-diameter O a, is to the rectangle of half the tranverfe and half the conjugate axis, as the ray of curvature CE is to O a. And therefore the cube of the ferni-diameter Qa, parallel to the tangent ET, is equal to the folid contained by the radius of curvature CE, and the rectangle of the two axes. See De Moivre, Mi feel. Analyt. P- 2 35-
3. The perpendicular to cither axis bifedts the angle made by the chord of curvature, and the common tangent of the conic-fection and circle of curvature.
4. The chord of the circle of curvature that panes thro' the focus, the diameter conjugate to that which panes thro' the point of contact, and the tranfverfe axis of the figure, are in continued proportion.
5. When the fection is an ellipfis, if the circle of curvature at E meet O a in d, the fquare of E d will be equal to twice the fquare of O a. Hence E(/:Oa::^2:i. Which gives an eafy method of determining the circle of curvature to any point E, when the femi-diameter O a is given in magnitude and pofition.
Several other properties of the circle of curvature, and me- thods of determining 'It when the fection is given; or vica verfa, of determining the fection when the circle of curva- ture is given, may be feen in Mr. Mac Laurin's Fluxions, Art. 375. Variation of Curvature. See Variation, SuppL
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