FORM OF TANGENTIAL EQUATION. 399 
Cor. 4. If § denote the radius vector of the first positive pedal of a curve, 
s =%+Se d * ( 161 ) 
Section II . — Negative Pedals. 
55. We have seen in art. 52 that the polar equation of the first positive pedal of a 
curve is obtained from its tangential equation by changing v into g, and multiplying the 
function on the right-hand side by sin <p. Hence, conversely, we have the following 
theorem: — If g = F(<p) be the polar equation of a curve, the tangential equation of its 
first negative pedal is 
, = ae> 
sm<p' 
(152) 
Thus the polar equation of a parabola is 
g=4a tan <p sec<p ; 
.•. the tangential equation of its first negative pedal is 
i/=4«sec 2 <p, (153) 
or, in Cartesian coordinates, 
(x-4a) 3 =27af, (154) 
showing that it is in the semicubical parabola. 
56. The equation of the line whose envelope is the negative pedal is 
x sin $-\-y cos <p— F(<p)=0. 
Hence the points where this line meets its envelope are given by the equations 
x=F(<p) sin <p+.F'(<p) cos <p, (155) 
g=F(<p) cos <p — F'(<p) sin <p ; (156) 
and by eliminating <p between these equations, we get the equation of the pedal. 
Cor. 
^+/=(F(cp)) 2 +(F'(<p)) 2 ] 
or ! (157) 
*■+*■=»■+ $)‘ 
Hence the distance from the extremity of § to where the perpendicular to it meets 
its envelope is . 
Examples. 
(1) Find the first negative pedal of the cardioide. 
The polar equation of this curve is, taking the perpendicular to the cuspidal tangent 
as the initial line, 
mdccClxxvii. 
£=a(lq-sin <p); 
3 L 
