338 CIRCLE OF CURVATURE. 



given curve be y <f> (x), and the given point that whose 

 co-ordinates are x a, y = </> (a), then we must have 



ma + c = <f> (a), 



and m = <f>'(d). 



Hence m and c are determined. 



If y = $ (x) be the equation to a curve, then 



y = * (a) + (x-a} f (a) + ^-=-^V - + ^T^ *"() 



Li L5 



is the equation to a curve which has a contact of the w th order 

 with the given curve at the point x = a. This may be easily 

 verified. 



320. Circle of curvature. The general equation to a circle 

 involves three constants ; hence at any point of a curve a circle 

 may be found which has contact of the second order with the 

 curve at that point. We proceed to determine the radius and 

 the centre of such a circle. 



DEFINITION. The circle of curvature at any point of a 

 curve is a circle which has at that point a contact of the 

 second order with the curve. 



Let (X-a?+(Y-by = p* .................. (1) 



be the equation to a circle, so that a, b, are the co-ordinates 

 of its centre and p its radius. From (1) by differentiating 

 we have 



(2). 



_ 



If this circle is the circle of curvature at the point (x, y} 

 of a given curve, we must have 



JL .......... (3). 



dX dx 



