98 
FIFTH REPORT— 1835 . 
tr 
when x — b, 
= .■■v = h = the 
e (w + ry V e (w + r) 
greatest velocity of recoil of the ball. 
To find the value of the chord c. We have, from the nature of 
c 2 
the circle, —j = the versed sine, or height ascended. Whence 
Z L 
the velocity due to that height is c 
Putting this for v gives 
Whence 
v/f = ‘ v4# 
= * \/ , 
P l 
+ r) 
— the chord of the arc 
e (iv + r) 
through which the ball would recoil if the 
beam were perfectly elastic. 
To find the time of straightening the beam. Since, from 
above, 
v_ 
2 
g'P 
e (tv + r) 
v = \/ 8 p 
e (tv + r) 
\/(2 b x — x 9 ). 
d x 
And since d t = —, we have 
v 
d t 
e (tv + r) 
d x 
whence 
V/ gp * a/( 2 b x — x 2 ) 7 
(w + r) 
gP 
l • x \ 
. arc ( ver sin = J . 
And when x = 
t = .A («?+jO 
V 
x 1-57079. 
The time is therefore constant, when w -f r and —are con- 
P 
stant, whatever the deflection may be; which is analogous to 
the case of a vibrating chord. 
