358 ENVELOPS. 



Let F(x, y, a) = be the equation which gives the series 

 of curves by varying continuously the quantity a. Then the 

 locus of the ultimate intersections is found by eliminating a 

 between 



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



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



Suppose from (2) we obtain a in terms of x and y, say 

 a = (j> (x, y] ; then if we substitute in (1) we have 



F{x,y,<f>(x,y)} = ............... (3), 



which is therefore the equation to the locus of the ultimate 

 intersections. Now if for any assigned value of a the equa- 

 tions (1) and (2) give possible values to x and y, then the 

 curve represented by (1) when a has this assigned value, will 

 meet the curve represented by (3). 



The value of -& for the curve (1) is found by the equation 



dF(x, y, a) dF(x, y, a) dy _ ,. } 



dx dy dx~ 



d'u 



The value of for the curve (3) is found by the equation 



dF(x, y, <f>) dF(x, y, ft) dy 

 dx dy dx 



} 



777 /77?* 



But -TV only differs from -j- in having <f>(x, y} in the 



place of a; hence by (2) we have at the point where (1) 

 jjfi 



and (3) meet, -jr = Q- Thus (5) becomes at that point 



Ct(p 



dF(x, y, 



dx dy dx 



Since at the point of intersection of (1) and (3) we have 



a=<j>(x, y}, equation (6) gives for -^ at that point the same 



dx 



