FORM OF TANGENTIAL EQUATION. 
419 
the involute of the locus of P, then from the last article we have 
P"L 8 -< 9 , , Q7 
pl=t=7-i from art - 97 ; 
P»L-LP 8 LP ( , „„ 
pffp - =2^Ig~PC from art 99 ‘ 
Hence the points P", P are harmonic conjugates to the points L and C. 
Cor. 1. Every radius of curvature of an epicycloid is divided harmonically by the base 
of the epicycloid. 
Cor. 2. If in the figure (art. 100) B", C", D'' be the points on the involute of the 
three epicycloids which touch the sides of the triangle BCD, corresponding to the points 
of contact of these epicycloids , then the points B", C", D" are collinear. 
105. From art. 99 we see that the arc AL : arc AC : : : 2g»+S_ 1 . Hence, letting 
fall the perpendicular OE, and denoting the angle AOE by <p, 
the angle LOE will be 5 ; if m=—A— , and if we denote the 
& r g + 8_i’ 
radius of the circle by a , we have OE=acosw<p. Hence the 
equation of the tangent to the epicycloid is 
^sin<p+ycos<j5=«cosw2<p .... (227) 
For examples of the case in which the envelope of this line is 
an algebraic curve, see Salmon’s ‘ Higher Curves,’ p. 270. 
106. The equation (227) may be written in the form x-\-y cot <p=.a cos m <p cosec <p. 
Hence if the line OX be taken as the director line, the tangential equation of the epi- 
cycloid is 
»=a cos m<p . cosec <p (228) 
107. In order to find the intrinsic equation we have f(<p)=a cos m<p . cosec <p. 
Hence from equation (64), art. 30, we find 
ds 
-^=a(l—m 2 ) cos m<p ; 
a(l— m 2 ) . 
•*• *= — m — - sin m<p, (229) 
which is the required intrinsic equation. 
The same result may be obtained from art. 93, equation (216). 
Cor. If we substitute for m its value we find, from art. 97, equation (222) combined 
with (229), that 
s—(b+b_ i)sin0 (230) 
Hence putting and doubling, we have the whole length of the epicycloid from 
cusp to cusp = twice the sum of the diameters of the generating circles of the curve and 
its involute. 
Fig. 17. 
