406 
PEOFESSOE J. CASEY ON A NEW 
CHAPTEE Y. 
Section I. — The Cycloid. 
67. There is one curve which, though we have very seldom mentioned hitherto in 
our memoir, was the one which led to the discovery of its methods. This curve is the 
cycloid ; and the reason it has not been more frequently used in our illustrations is that 
we consider its importance demands a chapter to itself. The novelty of the methods 
and of most of the results is our apology for devoting so much space to its investigation. 
68. In the figure, art. 26, it is evident that the point Q is the centre of instantaneous 
rotation for the line LP, because the motion of the points L and P are respectively at 
right angles to the lines LQ and PQ respectively, and since the coordinates of the point 
Q are OL and LQ. Hence the locus of the centres of instantaneous rotation of the 
line LP, whose position is given at any time by the quantities v and <p, where i 
is the curve obtained by eliminating <p between the equations 
(y=fW-> 
(191) 
69. In the same fig., if LV, VQ be at right angles to LP, PQ, and since LQ 
the values of LV, VQ will be /'(<¥>) cos <p, f'(<p) sin <p ; and therefore the motion of the 
line LP will be given by supposing a curve whose equation is the system 
x=f'(q>) cos <p,~i 
y=f ($) sin ’ 
(192) 
to roll on the curve whose equation is the system (191), and the line LP will be the 
axis of y with respect to this rolling curve. 
70. Let /(<p)=2a<p, then v=2 a<p; let O be the origin, OL=j», the angle XLP = <p; 
then if P be the point of contact of LP with its envelope we have, by art. 26, the diameter 
Fig. 8. 
of the circle touching OX at L and passing through the point P=/’ , (<p) = 2a. Hence 
if we erect LB at right angles to OX, and PB to LP, the diameter LB of the circle LPB 
will be constant and equal to 2a, and the arc LP of the same circle will be equal 2a<p ; 
the arc LP=the line OL=the line AB. Hence if we make AC =xa, the arc PB 
