FORM OF TANGENTIAL EQUATION. 409 
common tangent at infinity, three finite common tangents. Hence we have the following 
theorem : — 
Four lines being given, three cycloids can be described to touch them. 
79. If two variable tangents ( t ti) to a cycloid intersect at a constant angle, and a circle 
be described about the triangle formed 
by t f and the tangent at the vertex of 
the cycloid, then (1°) the envelope of the 
diameter of this circle passing through 
the points ti! is a cycloid-, (2°) the enve- 
lopes of the chords passing through the 
same point, and through the highest and 
lowest points of the circle, are cycloids. 
Let the tangents 1 1' intersect in P, 
and let C be the centre of the circle APA' ; 
then since the angle APA' is constant, 
<p + <p' is constant, AA'=2 a(<p-\-<f>') is 
constant. Hence the base and the ver- 
tical angle of the triangle APA' is con- 
stant ; the diameter of the circum- 
scribing circle is constant, and it is evident 
that the loci of the points E, C, F are right lines parallel to AA'. 
Again, since D is the middle point of AA', 
OH==A(OA-OA')=a(<p_<p') =a (FCP). 
Hence 
v=a 
dv 
dg>~~ a ~ con stant ; 
therefore (see art 72, equation (197)) the envelope of CP is a cycloid. 
(2) Since OD=a (FCP)=2# (FEP), we have, for finding the envelope of EP, 
„=2«(l-4'); 
therefore it is a cycloid. 
(3) The angle ~FF,P=yp", v=2a\p", and the envelope of FP is a cycloid. 
80. If C, C' be the centres of the circles APA' and TPT', then CC is perpendicular 
to the tangent at the vertex of the cycloid, and egual to the radius of its generating 
circle. 
Demonstration. — Since AT=2«sin<p, and A'T'=2« sin <p', we have 
AT sin <p PA' 
AT 7 sin <p' PA ’ 
3 M 2 
