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PROCESSOR J. CASEY ON A NEW 
the auxiliary conic is a hyperbola and the point of contact at infinity. Now if b is less 
than a, we have the following theorem : — 
An intern cycloid has two points of inflection, the tangents at which are parallel to 
the asymptotes of the auxiliary conic. 
Section III. 
90. If two tangents to a cycloid intersect at a constant angle , the locus of their point 
of intersection is an extern cycloid. 
Demonstration. — Since the angle APA' (see fig. art. 79) between the tangents 1 1' is 
constant, <p + <p' is constant, .\ AA' = 2ff(<p-|-<p') is constant; but the diameter of the 
circle about the triangle APA' 
A A' 2 a(<p + <p') _ 
sin P sin (<p + <p') ’ 
. np_ a (?+<?') 
’ ■ ' sin(<p + <p')' 
Hence CP is constant. Again we have (see art. 79) 
= — a— constant. 
ay 
Hence if CP were equal to a, the locus of P would be a cycloid ; but since <p + <f>' is 
always greater than sin (<?5 -f- <p'), CP is greater than a, and therefore the locus of P is an 
extern cycloid. 
Lemma. If two tangents, PT, PT', to any given curve be inclined at a constant angle, 
the circle described about the triangle formed by the two tangents and the chord of 
contact touches the locus of P. 
Demonstration. — Let P' be a consecutive point on the locus, then the tangents from 
P' touch the curve in the points T, V. Hence, since the angle TPT'==TP'T', the quadri- 
lateral is inscribed in a circle, and the line joining the consecutive points is a tangent to 
the circle. Hence the proposition is proved. 
91. If two tangents to a given cycloid make a given angle, the centre of the circle 
described about the triangle formed by the two tangents and the chord of contact is the 
centre of instantaneous rotation for the extern cycloid, which is the locus of the inter- 
section of the tangents. 
Demonstration. — Since the angle TPT' is given, the locus of P is an extern cycloid, 
and therefore, by the preceding lemma (see fig. art. 80), C'P is normal to the locus 
of P. 
Again, since P is a point in the revolving radius of a circle whose centre is C and 
radius a, and we have proved CC'=a, the circle rolls on the locus of C'. Hence the 
proposition is proved. 
92. If the angle TPT' be constant, the radius of the circle TPT' is a mean propor- 
