POEM OP TANGENTIAL EQUATION. 
379 
as the Cartesian equation of the curve 
v=c |l-j-(tan <p) 7i j . 
c 2 
(5) Let v= ^ fl2+6 2 tan 2 <p)* ’ w -^ ere c 2 =a 2 —h 2 . This curve is the evolute of the ellipse. 
The Cartesian equation is 
(axf+(hyf=c i (56) 
(6) To find a curve in which the subnormal is constant, let the constant be 2 a; then 
from equation (49) we have 
f r (<p) sin 2 <p tan (p = 2a ; 
. ‘ ■ • f( P) — ~~ a cosec 2 <p, 
which is the common parabola (see art. 25). 
Cor. In like manner the curve in which the subtangent is constant is 
or 
v—a\ og tan <p 
(58) 
(59) 
(7) If in fig. 4 (art. 26) PL be produced to meet OS in T, required to find the 
curve in which PL : LT in a given ratio, say n : 1 . Plere we have evidently 
f'(<p) sin <p cos <p 
W) ~ n ’ 
.•. log/(<p)=C+rclogtanp, 
/. f(<p)=htan n <p, if h=e c ; 
.-. the required curve is 
v=#tan”<p (60) 
or 
fo* +1 = 0 (61) 
Section II. — Transformation of the Tangential into the Intrinsic Equation. 
30. If we differentiate the value of x given in art. 26, we get 
<|= 2 /0) cos 2 <P+f"(<P) sin <P cos <p ; 
Lt 
dx dx ds ds 
dp ds^ dtp C0S ^ dp 1 
ds 
^= 2 / , (^)cos<p+/"((p)sin<p, (62) 
«=/'(<P) sin <P+S/'(<P) cos <P 
