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CHAPTER XXV. 



ENVELOPS. 



334. SUPPOSE 



F(x,y,a)=0 (1) 



to be the equation to a curve, a being some constant quantity. 

 By changing a into a + k, we obtain another curve of the 

 same species as (1), the equation to which is 



F(x,y, a + h)=Q (2). 



The point of intersection of (1) and (2) will be found by 

 combining the equations. Now (2) may be written 



F(x,y,a)+hF'(x,y, a+6>A)=0 (3), 



the accent denoting that F(x, y, a) is to be differentiated 

 with respect to a, and in the result a changed into a + Oh. 

 Hence, combining (3) and (1), we have the point of inter- 

 section determined by 



F(x, y, a] = 0, and F' (x, y, a + dh) = (4). 



If we diminish A indefinitely, the equations (4) become 

 F(x, y, a) =0, and F' (x, y, o) = (5). 



The point determined by equations (5) is the limit of the 

 intersection of (1) and (2). 



If between equations (5) we eliminate a, we obtain the 

 equation to a curve which is called the locus of the ultimate 

 intersections of the curves formed by varying a continuously in 

 the equation F(x, y, a) = 0. 



The quantity a is called the parameter of the curve. 



335. The locus of the ultimate intersections of a series of 

 curves touches each of the series of intersecting curves. 



