^oS IIADIUS OP CUnVAWRE. 



rpectkig [he '^*"" ordinates : we must therefore in order to adapt it io 

 radius of cur- practical purposes find a value for v, in such terms as shall 

 i':)vo[ve'^seco"°d ^^ <^onsonant to the characters in which the equations of 

 fluxions, curves are generally written, viz. those expressing the ab- 



sciss and ordinate : in order to which, let the above expres- 

 sion be put again into fluxions, and made equal to nothing 



(it being a constant quantity) and we shall have — _ — — 



•*■ 



'!ff=Q;andt;= --'^'^^'' --T T^'^^' ; therefore, B D r: 



t'z yxz^ ' . . „ , 



— — — T-n r, another general expression in terms oi the 



X X Z XX 



fluxions of the arc, absciss and ordinate of the Involute — = 

 but this, like the other is still inconvenient for practice; 

 yet the difficulty may be removed very easily by expunging 

 x: for put x^-\-j'zzz* into fluxions, and we get xx-{-yy 



:z:zzi and g ~ ■ ' . — so that our last expression becomes 



BD=-r#^ = _=ii4'-^. = t:?i^!42ll; the 



X z — xz x^'x+jyx'^ — Ar« X x^-yyx — xz 

 general value for the radius of curvature, when both the absciss 

 and ordinate flow inconstantly : but as all curves may be gene- 

 rated, either by the uniform increase of the absciss and incon- 

 stant variation of the ordinate; or by the uniform flow of the 

 ordinate and variable flux of vheabsciss ; we are at liberty to as- 

 sume the ilvst fluxion of either constant as it may suit our con- 

 venience ; and thus simplifying the expression, avoid the 

 trouble which would otherwise arise. — Thus, if i- be sup- 



x^-^-y"^ * 

 posed constant, the expression will be — ;-;; — ; and if^ 



V , ^ X -^ 1 — y-^ ' x^'+y'^ —yJ: 



be made constant it becomes, > — ^ ^ :t~ — X 



XX — X ji—y X X 



It is to be remarked, that all those expressions for the 

 radius of curvature are strictly true and geueral ; yet being 

 in terms of quantities whose values would be extremely 

 difiicult to find, are not so applicable to practice as that 

 containing only the fluxions of the absciss and ordinate* 

 The entry of second fluxions into the definitive expression, 



does 



