DIFFERENTIAL AND INTEGRAL CALCULUS. 97 



curvature of a small arc As at the end of the former arc 

 will be - , and if we diminish ,As indefinitely, we have, 

 finally, the curvature at the end of an arc s of any curve 



is j-. or the radius of curvature, i.e. the radius of the 



ds ' 



circle which has the same curvature as the curve is - r . 



difr 



This can, of course, be expressed in terms of (#, ?/), since 

 dy , ds 



r dx dx' 

 hence the curvature is 



cty _ d^r _ ds_ _ tfy ^ f /^A 2 H 

 ds ~~ dx ' dx dx* ' ( \&) j ' 



In forming this we have assumed that - 7 - , -77- are 



dx ' d-fy 



positive, which will apply to a curve with its convexity 



ds 

 towards the axis of #; if -y- be negative, as in a curve 



whose concavity is towards the axis of #, the curvature is 



d' 2 y ( fdy\*)i 

 -7-^ -r- j 1 -f ( -j- J j- . Some prefer giving algebraic sign 



to the curvature and consider it positive when it turns 

 from the axis of #, and negative when it turns towards 

 the axis of x. This is however of slight importance. 



Another convenient expression for the radius of curva- 



ture is P + -TJ-V , where p is the perpendicular from a fixed 



point on the tangent, and ty as before. In fig. 22 OF is 

 perpendicular on the tangent PT at P, ON=x, NP=y, 

 AW, NZ perpendiculars on the tangent and on OY, then 



p= OY= OZ- YZ= OZ-MN=x 



