428 
PROFESSOR J. CASEY ON A NEW 
Fig. 20. 
Let PQ be the tangent at the point P fixed in the revolving circle ; then, since B is 
the centre of instantaneous rotation, Q, the foot of the perpendicular from B on PQ, will 
be the point of contact of PQ with its envelope. From I let fall the perpendicular IE 
on BQ ; then it is easy to see that the locus of the point E will be a cycloid whose gene- 
rating circle will have IB for diameter, and EB will be the normal at E to the cycloid. 
Now, since EQ=IP —radius of generating circle, we see that the locus of Q is a 
parallel to the cycloid. 
The same result may be shown thus: — It is evident that OL = «<p-J-« tan \<p ; the 
equation of the curve which is the envelope of LP is 
v—aq>-\-a tan ^<p, 
and the intrinsic equation of this is 
s=a<p-{-2a sin <p, (259) 
which is a parallel to a cycloid. 
Cor. 1. The envelope of any line in rigid connexion with the generating circle of a 
cycloid is a parallel to another cycloid. 
Cor. 2. In like manner the envelope of a fixed tangent to, or of any line in rigid con- 
nexion with, the generating circle of an epicycloid is a parallel to another epicycloid. 
126. From the equation (254) we infer that if s=F(<p) be the intrinsic equation 
of a curve, the equation of the parallel to it is 
s=F(<p)+£(<p) ; 
and hence (see art. 64, equation (182)) the polar equation of the reciprocal of the 
parallel curve is 
7 = { 1 + (^)} {F'(<p)±£}, ....... (260) 
or 
~ — sin <p J cos <p(F'<p ± k) d<f > — cos <p J sin <p(F'( <p) + k) dq> 
•+■ Q cos <p + C 2 sin <p. 
In this equation we have used a 2 as the numerator to g on the left side of the equation 
instead of k 2 of recent articles in order to avoid confusion of notation. 
