Radius of curvature. g^ijr 



An attempt to shoiv, that the nature of the problem re- Problem re- 

 Specting the radius of curvature does not involve in it <Ae s pectin gt!»era- 

 consideration of second fluxions; hut that they are made to tuxedoes not 

 enter into the deflnitive expression as a matter of mere con- involve secon4 

 vemence. uuaiuu». 



Definition. If one end, O, of the thread A L D O, PI. vii, 

 fig. 4, be first fixed to the point O, in a curve L D O, concave 

 the same way, and afterward the thread be put about the said 

 curve so as to touch it in every part : then if the other end. A, 

 of the thread be tightly moved in the same plane with the 

 curve L D O, the said end. A, will describe a curve A BI, 

 called the involute curve to L D O which is the evolute : 

 and the right lines D B, O I, are said to be radii of cur« 

 vature. 



It is evident from the method of generation of the curve 

 A B Ij that if at any point D, in the evolute L D O, the 

 string should cease to uiiwind itself and the radius DB con- 

 tinue to revolve about D, as a centre (see figure 5), the 

 circle thereby described would have the same degree of cur- 

 vature as the involute at the point B ; and that a tangent 

 drawn to either curve would be common to both. More- 

 over, because the said two curves, viz. the involute and 

 circle, have the same curvature at the point B, and their con- 

 tinuations one and the same curve, namely, the circle where 

 radius is D B ; the fluxions of the absciss and ordinate of 

 the one, will be equal to the fluxions of the absciss and or- 

 dinate of the other, and consequently the same will bold of 

 any other order of fluxions whatever. This being pren^ised, 

 let it be required to find a definitive expression for the ra- 

 dius of curvature of any curve, as A B I. For which pur- 

 pose let as usual the absciss and ordinate AC, B C, or the 

 curve A B I, be denoted by x and y ; also A Br:Z, and put 

 BE the corresponding ordinate of the circle MBN— i?. 

 Then the triangles Bum, BED, being similar, we have 



B u : B m : : BE : BD ; ov x:z ::v: BU-—. Now this 



X 



expression is general ; but being in terms of the ordinate of 

 the circle M B N and unequal to B C, the ordinate of the 

 circle ; it cannot with convenience be applied to curves 

 whose equations are generally expressed in terms of theif 

 Vol. XXL— Dec. 1808. S 



