400 
PKOFESSOR J. CASEY ON A NEW 
F(<p)=a(l + sin <p) ; 
x=2a-j-a sin <p, 
y=a cos <p, 
.\(x — 2 af +y 2 = a 2 . . 
Therefore the pedal is a circle. 
(2) Find the negative pedal of g=fctari 2 <psec(p. 
?=2 tan <p-f-4 tan 3 <p, 
( 158 ) 
*^= — tan 2 <p — 3 tan 4 <p. 
Then if we put 
the result of eliminating <p will be 
Z-4 r 2 TST 
N+3 ; ^p. . 
(159) 
By differentiating the values of x and y given in equations (155), (156), then 
squaring and adding &c., we get the length of the first negative pedal, 
S =F(p)+jF($)*p (ICO) 
an equation which agrees with equation (151), but expressed in a different notation. 
57. If in art. 39 we substitute for we get, from equation (93), 
s ={ 1+ (^) }*w 
Hence we have the following theorem : — 
If %= F(<p) be the polar equation of a curve , the intrinsic equation of the evolute of 
its first negative pedal is 

(161) 
In like manner, from art. 50, the intrinsic equation of the 
negative pedal is 
involute of the first 
*={ 1+ (!)'>)• • • • 
(162) 
Cor. The intrinsic equation of the nth evolute is 
4 -{U) + (sp) } F(<p) ’ ■ ■ 
(163) 
and of the nth involute is 
(164) 
