Hence, from (2), 



KADIUS OF CURVATURE. 





S39 



,(4). 



Therefore 



da? 



By (1) and (5) we have 



da? 



\dx J 



.(5). 





Hence the values of a, b, p, are found, and thus the position 

 and the radius of the circle of curvature at any point of a 

 curve are determined. 



In the value of p it will be proper in any particular 

 example to give to the radical in the numerator the same 



d z y 

 sign as ~ z has, so as to make p positive. Hence if y be 



positive and the curve concave to the axis of x we should put 



dx 



From the first of equations (4) we see that the point (a, &) 

 is on the normal to the given curve at the point (x, y). 

 The centre of the circle of curvature at any point is called 



Z2 



