FOEM OF TANGENTIAL EQUATION. 
415 
tional between the radius of the circle APA' and half the chord of curvature at P, 
passing through the centre of APA'. 
Demonstration . — Let the radii of the circles be r, r', and the angle CPC' be u > ; then, 
by art. 85, we have PK : PC' : : PC' : g. But PK=r cos u , .\r cos air 1 :: r 1 : g ; 
.*. r' 2 =rxg cos<y. Q. E. D. 
CHAPTER VI. 
Section I. — Epicycloids. 
93. The form of tangential equation employed in the previous portion of this memoir 
may be usefully generalized as follows : — Thus, instead of taking a directing line OX 
(see art. 1) and a variable line LP, making an angle <p 
with OX at the distance v from the origin, let us take a 
directing curve OX, and a variable line LP, making an 
angle Q with the curve at L, and denoting the arc OL 
by <r; then any relation between a and 0, such as <7 =f(Q), 
may be called the tangential equation of the curve which 
the line LP envelopes. 
Let us take a consecutive position, L'P of LP, then P 
is the point where LP touches its envelope, and LL' = t?<r. 
Let the intrinsic equation of the directing curve be 
<r==/’(<p), then the angle LPL' is easily seen to be dq>-\-dQ ; 
and if S denote the diameter of the circle described about the infinitesimal triangle 
LPL', we have 
1 _ dtp + d8 
8 da 
Fisr. 14. 
Hence if § denotes the radius of curvature of the directing curve at L, we have 
1 1 
S 
~- + /m 
Hence 
• S— i®- 
'?+/(«)• 
rp g/'ffl S1 *n 6 
L ^~ §+/'(«) • 
If s denotes the length of the curve which is the envelope of LP from some fixed 
point in it up to P, then (see art. 31) 
ds=UJ cos $+(Z(LP) 
Hence 
= da cosQ +^(LP). 
s=LP-[-j’ cos Qda ; 
3 N 
MDCCCLXXV1I. 
