FORM OF TANGENTIAL EQUATION. 40-3 
§ must be infinite when <p =0, we have C x =c and C 2 = 0, and the required equation is 
k 2 
~ =c sin l°g (sec <p+tan <p) — 2 c sin 2 ^<p 
(2) Find the reciprocal of the curve 
= a m sin m<p. 
_1 
^(<p) = a (sin m<p)™; 
the tangential equation of its reciprocal is 
A 2 
(186) 
a sin <p (sin mf)'> 
(3) Find the reciprocal of the cycloid. 
The intrinsic equation is 
s=z4«cos<p. 
Hence, from equations (182) and (170), 
(187) 
A 2 
■ 2 a<p sin <p — a cos (p+C! cos <p-j-C 2 sin <p. 
Now it is evident that § must be infinite when <p vanishes, and that — must be equal 
to an when <p=^. Hence C \=a, C 2 =0, and therefore the required reciprocal curve is 
y=2«<psin<p (188) 
(4) The reciprocal of the logarithmic curve 
v=a log tan <p 
is 
k 2 
~ = a sin <P l°g t an 
or, in Cartesian coordinates, 
ay 
x—ye * 2 (189) 
(5) Find the reciprocal of the curve 
v=(l + cot*<p) 3 . 
Here we have, from equation (178), 
k 2 
— = (l+cot*<p ) 3 sin <p, 
+r ) 3 ; 
( 190 ) 
