FOEM OF TANGENTIAL EQUATION. 
407 
will be equal to the line BC ; and the point P may be considered as fixed in the circle 
LPB, and the locus of P will be the curve described by a fixed point in the circle LPB 
rolling on the line AC. In other words the locus of P is a cycloid. 
71. Since /‘(<p)=2a<p, the equations (1 91) denote a right line, and the equations (192) 
the circle x l -\-y‘ l =4a i . Hence the cycloid v=2a<p is the envelope of a fixed diameter of 
the circle # 2 +y 2 =4 a 2 , which rolls along the line y~ — 2 a. Therefore we have two 
methods of generating the same cycloid, either as a locus or an envelope. 
72. The coordinates of the point P are, from equations (46), (47), the system 
j^=c(2<p + sin2<p), (193) 
\y——2a sin 2 <p (194) 
From equation (62) we have the intrinsic equation 
II. s=4asin<p, (195) 
and from (61) 
III. q=ia cjs <p (196) 
If we differentiate the equation v=2 a<p we have the differential equation of the cycloid 
IV. ^=2a=constant (197) 
73. From equation (93), art. 39, the intrinsic equation of the evolute is 
s=4a cos <p=4a sin (198) 
Hence the evolute is another cycloid. 
We can show the same thing geometrically; for we have seen that the arc PB = the 
line BC. Hence denoting BC by v, and the angle PBC by 0, we have v=2 aQ, and 
therefore the envelope of PB is a cycloid. 
Cor. If the line PB be produced to R, making BR = BP, then R is the centre of 
curvature. 
74. From L let fall the perpendicular LQ on the diameter VP of the revolving circle, 
Fig. 9. 
then it is evident that the angle XLQ=2XLP=2<p ; and denoting this angle by \p, we 
have 
v=a\}/, 
MDCCCLXXVII. 3 M 
