410 
PROFESSOR J. CASEY ON A NEW 
Fig. 12. 
/. PA . AT=PA' . A'T'. Hence the radical axis of the two circles is parallel to AA', 
and therefore CC' is perpendicular to AA'. 
Again, the radical axis of the two circles 
passes through P ; hence, by a known pro- 
perty of coaxal circles, the rectangle 
PA . AT=2CC'. PD ; 
that is, 
PA . AT=2CC'. PA . sin <p, 
AT=2CC' sin <p ; 
but 
AT =2 a sin<p, 
.-. CC '—a. Q. E. D. 
81. If the angle TPT' be constant, the 
locus of C', the centre of the circle described 
about the triangle formed by the two tangents and the chord of contact , is a right line. 
This is evident, since CC=a and perpendicular to AA', and the locus of C is a right 
line. (See art. 79.) 
Section II. — Intern and Extern Cycloids. 
82. Definition. — When the extremity of the revolving radius of the generating circle 
describes a cycloid, a fixed point in the radius describes a curve, which, according, as 
the point is inside or outside the circle, I shall call the intern or extern cycloid. 
These curves are usually called the prolate and the curtate cycloid; but the names I 
have adopted are more suggestive. 
83. To find the intrinsic equation of an extern cycloid. 
Let BPL be the generating circle 
the point which describes it, and P 
describes the extern cycloid ; then cl 
a , b, and the angles as in the diagri 
v=2a<f>, the coordinates of the poi 
equations 
x=2a$-\-b sin 2 <p, 
y= a -f£cos2<p. 
ot the cycloid, P 
11 the point which 
enoting IP, IP' by 
»m, we have, since 
nt P' given by the 
. . ( 199 ) 
Fig. 13. 
If we differentiate these equations with respect to <p, then square and add, we get 
(^) =4(a + bf—16ab sin 2 <p ; 
