FOEM OF TANGENTIAL EQUATION. 
413 
. ■ \/(0 ) = a sin 1 1 sin 0(<? cos 6 + Ad) [ + ^ A ^ in g C ° S 
Hence the tangential equation of ah intern or extern cycloid is 
v=a sin“ sin 0(c cos $ + A(0)) (208) 
88. In the triangle IBP', art. 83, we have 
sin 2 <p : sin Q : : BP' : b, 
.*. sin2<p=sin0(ccos0 + A((3)) ; 
and from the equation of the cycloid described by the point P we have 
v—2 a<p ; 
.•. eliminating <p, we have for the envelope of the line BP' 
<'=«sin“ 1 -|sin0(ccos 0 + A(0))f ; . (209) 
and this is the tangential equation of the evolute of an intern or extern cycloid. 
89. The tangential equation (208) can be expressed very simply as follows. For if 
we take the conic 
oft 
jp=p+& =1 . ( 21 °) 
we easily find its tangential equation to be 
b{ A(0) — cos 9) 
'= sin 9 
( 211 ) 
Hence we have the following theorem : — 
If j,=F(0) be the tangential equation of the evolute (see equation (209)), and 
v = G(0) the tangential equation of the ellipse (210), then the tangential equation of 
the intern or extern cycloid is 
»=F(0) + G(0). (212) 
Cor. 1. The intrinsic equation of the evolute of an intern or extern cycloid is 
b(c cos 9 + A(9)) 2 
A(9) 
(213) 
Cor. 2. If <r, a ', a" denote the lengths of an extern or intern cycloid, its evolute, and 
its auxiliary conic (see equation (210)), taken on the three curves from points whose 
tangents are parallel to other three points whose tangents also are parallel, then 
(214) 
Cor. 3. If g, g>', be the radii of curvature of the same three curves at points whose 
tangents are parallel, 
e=i+t ( 215 ) 
Cor. 4. § will be infinite when either §' or g>" is infinite ; but will be infinite when 
