OF THE PROPERTIES Of CURVES. 45 



volute is that portion of the flexible line which is unbent, 

 when any part of it is described. 



193. Theorem. The radius of- curvature always 

 touches the evolute, and is perpendicular to the involute. 



If the radius of curvature did not touch the evolute, it would make 

 an angle with it, and would, therefore, not be unbent ; and if the 

 evolute were a polygon composed of right lines, each part of the in- 

 volute would be a portion of a circle, and its tangent, therefore, per- 

 pendicular to the radius : but the number of sides is of no conse- 

 quence, and if it became infinite, the curvature would be continued, 

 and the curve would still at each point be perpendicular to the 

 radius of curvature. 



194. Theorem. The chord, cut ofFin the ordinate by 

 the circle of curvature, is directly as the square of the 

 fluxion of the curve, and inversely as the second fluxion 

 of the ordinate, that is, as the fluxion of its fluxion. 



The constant fluxion of the absciss being equal 



to AB, the fluxion of the ordinate, at A, is BC, 



at D, DE, consequently its increment is CD -f y{ 



BE, or CD+AF, twice the sagitta of the arc -"^^i 



DV 



AD : and the chord is equal to the square of AC divided by CD, 

 and it is, therefore, always in the direct ratio of the square of the 

 fluxion of the curve, and the inverse ratio of the second fluxion of 

 the ordinate. See 268. 



195. Theorem. When the curve approaches infi- 

 nitely near to the absciss, the curvature is simply as the 

 second fluxion of the ordinate. 



For the fluxion of the curve becomes equal to that of the absciss, 

 and the perpendicular chord to the diameter. 



196. Definition. If the sum of two right lines, 

 drawn from each point of a curve to two given points, is 

 constant, the curve is an ellipsis, and the two points are 

 its foci. 



