FORM OF TANGENTIAL EQUATION. 
381 
where 
Hence 
7 _ V3±l 
k ~ 2 V2 ' 
(See Dueege, ‘Theorie der elliptischen Functionen.’) 
F(£, 0)+2 . 3*E(£, d) + 2 . 3* cot ±0 . A(0). 
(67) 
Examples. 
(1) Let the tangential equation be v — k tan”$, to find the intrinsic equation. 
Here we have f(<p)=k tan" <p ; 
.*. /"T'P) sin 2 ( P =n fc tan" +1 <p. 
ds 
Hence from equation (64) ~^=n(n-\-\)k tan™ -1 <p sec 3 <p ; 
.-. s=w(?i+l)#Jtan“ -1 <p sec 3 <p dtp (68) 
We can get a formula of reduction for this integral as follows : — Put P=tan” -2 <p sec 3 <p, 
then by differentiation and reduction, 
^=(n- j-1) tan” -1 (<p) sec 3 <p-\-(n— 2) tan” -3 <p sec 2 <p ; 
— 2 (p scc^ (D 7i 2 C* 
tan” -1 (<p) sec 3 <p dq> = tan” -3 <p sec 3 <p d<p, . (69) 
which is the required formula ; and the integral will ultimately depend on known forms. 
(2) Let the tangential equation be that of the evolute of an ellipse, 
V « 2 + tan 2 <p 
We have 
/'(?>) sin 2 <?= 
— 6 2 c 2 tan 3 <p 
(« 2 + 6 2 tan 2 <p)t 
Hence, from equation (64), 
where 
ds 3 b 2 sin<pcos<p 
dtp a ' A 5 (<p) ’ 
A(<p) =\/ 1 — e 2 sin 2 cp ; 
, 
a A 3 (<p) ’ 
• • (70) 
and this is the intrinsic equation of the evolute of an ellipse. 
(3) To find a curve in which the radius of curvature bears a constant ratio to the 
normal, the given condition is expressed by the equation 
