EOEM OF TANGENTIAL EQUATION. 
401 
'Examples. 
Find the intrinsic equation of the first negative pedal of an ellipse. 
The polar equation of the ellipse, 
S=e where A 0P)=>/ 1 sin2 <?• 
Hence by art. 56, equation (ICO), the equation of its first negative pedal is 
s=b 
be 2 sin <p cos <p 
A 3 (p 
fe sin <p cos <p 
{ A»<p 
+¥(e,<p)[ 9 (165) 
and the intrinsic equation of the evolute of the pedal is 
b e 2 cos2<p b /e 2 sin 2 <p\ 2 
-A®i 1 + 
A 2 <p 4 
/ e 2 sin 2 <p\ 2 ) 
\±*t) /■ 
(166) 
58. The converse of the problem solved in art. 56 is, being given the intrinsic equation 
of a curve to find the polar equation of its first positive pedal. 
Let s=f(<p) be the given intrinsic equation, then we have, from equation (151), 
s4+s = /W’ 
••• S={ 1 + (4)T^ W (167) 
Cor. 1 . The polar equation of the positive pedal of the evolute is 
?={ 1 +(iYYm ( 168 > 
and of the involute 
« = { 1+ (4)T^ > • (169) 
Cor. 2. The equation (167) may be written 
g =sin <p J cos <pf'(<p)d<p — cos <p J sin <pf '(<p)d<? 
+ C, cos <p+C 2 sin (170) 
See Boole’s ‘ Differential Equations,’ where the reader will find illustrations of the 
cases in which the symbol + ^ on the right-hand side of equation (167) may 
be usefully expanded in ascending powers of , and thus the integration on the 
