108 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Hence, to determine directly the position and dimen- 

 sions of the circle of curvature of a given curve at a given 

 point, we have to find a circle (1) passing through the 



point (a;, #), (2) having the same value of -j- , (3) the same 



ctx 



7*3 



value of ~, as the given curve has at (a>, y). 



Let, then, (a, b) be the coordinates of the centre (fig. 31), 

 and p the radius of this circle, then if (XY) be any point 

 on this circle (X- a) 2 + (Y- &) 2 = p 2 , also the values of 



jz -y 



~7 -jri at the point (XY) are found from the equations 



Hence, to determine the circle of curvature of the given 

 curve at (a-y), we shall Lave the three equations 



, 1) x a 1 



80 that 



/ - dj( (djtf\ - ' 



\3*) dx\ + \dx)} dx< 



and therefore 



so that a, 5, p are completely determined. 



These equations are more symmetrical if we take s the 

 arc of the curve as independent variable. We shall then 

 have 



(*-)+ (y-6) = p- ................ (1),. 



14 (*-) J +(*,-&) g=.0 ........ (3); 



