40 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



2.231 The envelope to a family of curves, 



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



where a is a parameter, is obtained by eliminating a between (i) and 



dF 



2. = o. 



da 



2.232 If the curve is given in the form, 



1. x =/i(/, a) 



2. y=M*, a), 



the envelope is obtained by eliminating / and a between (i), (2) and the func- 

 tional determinant, 



g$-. (see 1.370) 



2.233 Pedal Curves. The locus of the foot of the perpendicular from a fixed 

 point upon the tangent to a given curve is the pedal of the given curve with 

 reference to the fixed point. 



2.240 Asymptotes. The line 



y = ax + b 



is an asymptote to the curve y = f(x) if 



limit 



fi/\ 



/'(*) 



2.241 If the curve is 



* 



and if for a value of /, d, /i or / 2 becomes infinite, there will be an asymptote if 

 for that value of / the direction of the tangent to the curve approaches a limit 

 and the distance of the tangent from a fixed point approaches a limit. 



2.242 An asymptote may sometimes be determined by expanding the equation 

 of the curve in a series, 



k = 



if Z 



the equation of the asymptote is 



y = 



