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VOL. XLII.J PHILOSOPHICAL TRANSACTIONS. 64 1 



curvature is greater than in any circle^ or the ray of curvature vanishes; and in 

 this case the curvature is said to be infinitely great. Of this kind is the curva- 

 ture at the cuspids of the Hnes of the third order. 



As lines which pass through the same point have the same tangent when the 

 first fluxions of the ordinate are equal, so they have the same curvature when 

 the second fluxions of the ordinate are likewise equal ; and half the chord of 

 the circle of curvature that is intercepted between the points wherein it inter- 

 sects the ordinate, is a third proportional to the right lines that measure the 

 second fluxion of the ordinate, and first fluxion of the curve, the base being 

 supposed to flow uniformly. When a ray revolving about a given point, and 

 terminated by the curve, becomes perpendicular to it, the first fluxion of the 

 ray vanishes; and if its second fluxion vanishes at the same time, that point 

 must be the centre of curvature. The same is to be said when the angular mo- 

 tion of the ray about that point is equal to the angular motion of the tangent 

 of the curve; as the angular motion of the radius of a circle about its centre 

 is always equal to the angular motion of the tangent of the circle. Thus the 

 various properties of the circle suggest various theorems for determining the 

 centre of the curvature. 



Because figures are often supposed to be described by the intersections of 

 right lines revolving about given poles, three theorems are given in prop. 18, 

 26, and 35, for determining the tangents, asymptotes, and curvature of such 

 lines, from the description, which are illustrated by examples. A new pro- 

 perty of lines of the third order is subjoined to prop. 35. The evolution of 

 lines is considered in prop. 36. The tangents of the evoluta are the rays of 

 curvature of the line which is described by its evolution; and the variation of 

 curvature in the latter is measured by the ratio of the ray of curvature of the 

 former to the ray of curvature of the latter. 



Sir Isaac Newton, in a treatise lately published, measures the variation of 

 the curvature by the ratio of the fluxion of the ray of curvature to the fluxion 

 of the curve; and is followed by the author, to avoid the perplexity which a 

 difference in definitions occasions to readers, though he hints in art. 386, tiiat 

 this ratio gives rather the variation of the ray of curvature, and that it might 

 have been proper to have measured the variation of curvature rather by the ratio 

 of the fluxion of the curvature itself to the fluxion of the curve; so that the 

 curvature being inversely as the ray of curvature, and consequently its fluxion 

 as the fluxion of the ray itself directly, and the square of the ray inversely, its 

 variation would have been directly as the measure of it, according to Sir Isaac 

 Newton's definition, and inversely as the square of the ray of curvature; ac- 

 cording to this explication, it would have been measured by the angle of con- 



VOL. VIII. 4N 



