412 
PROFESSOR J. CASEY ON A NEW 
In the figure, art. 82, we have evidently 
FK=#(A(0)), P'B=£(c cos fl+ A(0)) ; 
P'B 2 
PK ' * ' • 
(205) 
Hence we have the following elegant construction for the centre of curvature of an 
intern or extern cycloid : — - 
From the centre I of the generating circle let fall the perpendicular IK on the normal 
to the curve , and then the third proportional P'Q to P'K and P'B will he the radius of 
curvature. 
86. Since 
xyr\_ h ( c cos0 + A(0)) 2 
A 9 
and 
we have 
P'B=%cos04-A(0)), 
BQ= 
a cos 0(ccos 0 + A'0)) 
~ A(d)“ 
(206) 
Again, if the line BP', that is, the normal to the curve, meet the polar of the point 
P', with respect to the generating circle, in the point N, then the line BN is divided 
harmonically, and we have 
1 1 2 
BP + BN — BM 
or 
1 1 1 
b(c cos, 0 + A(0))^~BN be cos 0 ' 
.-. BN= 
a cos 0(e cos 0 + A(0)) 
A(0) ’ 
(207) 
.*. BQ=BN. 
Hence w r e have the following theorem : — The portion of the normal to an intern or extern 
cycloid at any point l v of the curve included between the polar of the point P' with 
respect to the centre of the generating circle and the corresponding centre of curvature 
is bisected by the centre of instantaneous rotation. 
87. By art. 30, equation (64), if v=f{b) be the tangential equation of a curve, 
* . 
</0 sin 0 ’ 
but by art. 85, 
ds b{c cos 0 + A (0)) 2 
70 = A0 ' 
Hence 
(6) sin 2 Q=a sin 2 Q — b cos Q . A(0)-f-& ; 
