300 



ENVELOPS. 



eliminating x between (1) and the equation obtained from (1) 

 by differentiating it with respect to x, which is 



.(2). 



It appears from (1) and (2), compared with Art. 320, that 

 x, y will be the co-ordinates of the centre of curvature at 

 the point (x, y) of the given curve. Hence the locus of the 

 ultimate intersections of the normals to a curve is the evolute 

 of that curve. 



338. It may happen that the envelop does not touch all 

 the curves of the series, as will appear from an example. 



Suppose the centre of a circle of variable radius to move 

 along the axis of x, so that the 

 abscissa OP of its centre and its 

 radius PM are the abscissa and 

 ordinate of an ellipse AMB which 

 has for its equation 



2 2 



2 * 2 * 



m n 



required the envelop of the system of circles. 

 If OP= a, the equation to the circle will be 



Hence differentiating with respect to a, we have 



na * 



x a , =0: 



m 



therefore 



a = 



mx 



Substitute in (1) and we obtain 



.(2). 



m a +n^n*- 

 which is the equation to the envelop. 



(3), 



