FOEM OF TANGENTIAL EQUATION. 
417 
but 
**• (f — ; 
that is, since LD=S, the arc ED = arc LP ; and making the arc EX = semicircle LPD, we 
have the arc XD=arc DP. Hence the locus of P is an epicycloid described by the 
rolling of the circle DPL on the circle EDX. 
98. Def. We shall call the circle DPL the generating circle of the epicycloid, and 
the circle EDX, on which it rolls, the base. 
It is evident that the motion of the circle DPL with respect to the director circle 
ALC is that of pure sliding, and its motion with respect to EDX is that of pure rolling. 
99. Since L is the centre of similitude of the circles DPL and ALC, we have 
but, from equation, (222) 
Again, since 
and arc 
we have 
Hence 
LP S_ 
PC 2g — § ’ 
S _ 
2g-S‘"2g + 8_ - ,’ 
LP 8_, 
PC 2g + 8_ 1 * 
AL=M, 
LC=2$ 
arc AL 
arc AC 2g + 8_ 1 " 
LP : PC : : arc AL : arc AC (223) 
Hence, if a variable arc , AC, has one extremity , A, fixed, and be divided in a given 
3 n 2 
