FOEM OF TAN GENTIAL EQUATION. 
391 
Similarly, from equation (99), we get 
fry 
d<p n 
=y n —ux n _. 
71 . 71—1 , 71 . 71 — 1.77 — 2 
— { 2 — Vn - 2 + |3 
(103) 
47. The Intrinsic Equation of a Curve being given, we can find the Tangential Equa- 
tion of the Evolute thus : — 
Let s=f(q>) be the given intrinsic equation, then the intrinsic equation of the 
evolute is 
*=/«>). 
(see Whewell, “ On the Intrinsic Equation of Curves,” Phil. Trans, vol. viii. p. 659) ; 
and therefore, by art. 38, equation (85), 
v=§ cosec 2 ${§/"($) sin <p d<f 5 }d<p (104) 
Cor. The tangential equation of the second evolute is 
cosec 2 <p{jf"'((p) sin <p d<p}d<p, ....... (105) 
and, in general, of the nth evolute 
i>=j cosec 2 <p{jf (n+1) (<p) sin <p d<p}d<£> (106) 
Examples. 
(1) Find the tangential equation of the evolute of the catenary. 
Here we ha ve/'(<p) = c tan <p ; 
.’. J/"(<p) sin <p d<p=c \ sec <p tan <p— log(sec <p-f tan <p)(, 
.-.j cosec 2 J/"(<p) sin <p d<p}d<p 
=c|sec <p + cot <p . log (sec <p-t~tan <p) [, (107) 
which is the required equation. 
The following three examples are illustrations of art. 39. 
(2) To find the intrinsic equation of the evolute of the curve v=(l + cot' <p ; 3 : — 
/(<P)=(l + cot *<p) 3 ; 
.*. s . sin <p= — § {coPcp-l-cot 1 <p}sec 2 <p 
(see example 6, Section II., Chapter II.), 
sins p.coss p^sins <p cost 
1 
( 108 ) 
(3) If the curve be the lemniscate, 
§ ' 
ccseccp (see art. 25), 
3 K 
MDCCCLXXVn. 
