FIFTH REPORT— 1835 . 
302 
7) X n 
deflection x would be ——, and its pressure upwards would be 
p x 
-- {iv 4- q ); and the inertia being w 4- r, 
g 
. •. retarding force = 
w + t 
But by the laws of motion, 
f p x n 
\—-r - (» + y) 
d v 
g 
1 dx w + r 
p x 
— - (w + q) 
Integrating 
to 
lp O' 
Q ___ __ & 
2 w 4- r 
p x 
n + 1 
{?i + 1) e 
- - ( w + q) x- 
The body w will, by falling through h, have acquired a velocity 
V 2 g h ; and after impact on the mass, whose inertia is they 
will commence going together with the diminished velocity 
w 
_ _ _ 
V 2 g h. Therefore, when x — ev = -—*/2 g h } and 
w 4 - r 
w + r 
these substituted give 
C 
w 2 a' h 
(7^+7)* + j (fri)T» ~ ( “’ + q) e ' 
p e 
'n + 1 
>9 
substituting for C, in the general equation, the value just ob- 
id t 
2 O' Ji 
tained gives 
W‘ 
V A 
g 
(iv + r)* 2 w + r 
p (x n + 1 — e' n + l ) 
- - 7 -—— -— (tv + q) (x — d) 
(n + 1) e n . ” K 
> . 
But when the deflection is completed v = o, and x = e •, and at 
that time, dividing by the coefficient of //, we obtain 
h = 
w + r 
iv 2 
p (e n +1 — d n + l ) 
(n + 1) e n 
{iv + q) (e — d) 
(A.) 
If the flexure of the beam is not so great as to injure its elas¬ 
ticity, n = 1, and 
h = 
(w + r) [e — d) 
w < 
JL( e + e i) _ ( w + q) 
w G 
. . . . (B.) 
Or since q is the pressure which accompanies the deflection 
e' } we may substitute for q in terms of e, by supposing that 
