ENVELOPS. 359 



value as equation (4). Hence (1) and (3) touch at their 

 common point. 



From this property the locus of the ultimate intersections 

 of a series of curves is called the envelop of the series of 

 curves. 



336. Example. Required the locus of the ultimate inter- 

 sections of a series of parabolas found by varying a in the 

 equation 



1+a 2 t 



y =ax 2j- x ' 



Here F(x, y, a)=y ax + - x 2 = Q (1), 



2p 



F'(x,y,a}= x = (2). 



From (2) a =2. 



x 



Substitute in (1) and we have 



, 



or a?+2py-p*=0, 



which is the equation to a parabola. 



337. Required the locus of the ultimate intersections of a 

 series of normals drawn at different points of a given curve. 



Let x, y be co-ordinates of a point in the given curve, then 



rf-s+tf-tfjl-o (i) 



is the equation to the normal at that point ; x, y', being the 

 variable co-ordinates. From the equation to the given curve 



y and -j- can be expressed as functions of x ; thus x is the 



parameter in (1), by varying which the series of normals 

 is obtained. Hence the required locus is to be found by 



