418 
PKOFESSOR J. CASEY ON A NEW 
Eig. 16 . 
ratio in the point L, the envelope of the chord LC is an epicycloid , which it touches in 
a point P, determined by the proportion LP : PC : : arc AL : arc AC. 
100. If A be a fixed point, and B, C, D variable points, and if 
the ratios be given arc AB : arc BC : arc CD, then, from the 
last article, the envelope of each side of the triangle BCD is an 
epicycloid touching the circle in the point A. Hence we have 
the folloAving theorem : — If a variable polygon be inscribed in 
a circle , and if the envelopes of all the sides but one be epicy- 
cloids which have a common point of contact with the circle , 
then the envelope of the remaining side is another epicycloid , 
having the same point of contact with the circle. 
Cor. If the points of contact of the sides of the triangle BCD, with their respective 
envelopes, be B', C', D', the three lines BB', CC', DD', are concurrent. This is evident 
from art. 99. 
101. In the figure (art. 97), since the arc XD = arc PD, then denoting XD by a' and 
the angle which PD makes with the circle XDE by S', we have 
a>=W; (224) 
therefore the envelope of PD is an epicycloid, whose directing circle is the circle XDE. 
Hence , the evolute of an epicycloid is another epicycloid , and the director circle of 
one is the base of the other. 
102. If denote the diameter of the generating circle of the evolute, we have, as in 
art. 97, 
>,=^+1 ( 225 ) 
But and h denote respectively the diameters of the generating circles of an epicycloid 
and its involute. Hence, the difference between the reciprocals of the diameters of the 
generating circles of an epicycloid and its involute equals reciprocal of radius of directing 
circle of the epicycloid. 
Cor. In the equation a—'h_ y b, the constant is the diameter of the generating circle 
of the involute. 
This follows from the present article combined with equation (222). It was on this 
account that the negative suffix was put to S. 
103. In the figure (art. 97), if PD meet its envelope in P', then P' is the centre, and 
PP' the radius of curvature at P; but PD=fccos0, and P'D=S, cos 6, 
PD : P'D (226) 
That is, the base of an epicycloid divides its radii of curvature in the constant ratio of 
the diameters of the generating circles of the epicycloid and its evolute. 
104. Let P'/ (see fig., art. 97) be the point where LC meets the epicycloid, which is 
