420 
MR. S. H. BUREURY ON THE COLLISION OF ELASTIC BODIES. 
The terms independent of D and cl are equivalent to H 1? the minimum value of H 
when there is no disturbance. 
And so 
K = H — Hj = |N log (I -4- D) -f£nlog(l +d). 
That is 
Now since 
and 
K = | re ( c Z-f) + |N( , D- 1,i 
N , n , T . 
+ 4 AH = N + n, 
1 + D 1 1 4 d 
D = - 
nd 
K = |w (d-- 0 -|N. 
N + (N + n ) cl 
nd 
X + (X + n ) d 4 ‘ {X 4 - (X 4 n ) cl }~ 
= | n ( cl — 7 -) — # nd. {1 
X 4 n A 3 n - d 2 
X 7 — T {X 4 (X 4 n ) ciy 
= I N - (N + ») d 3 . 
In order to find dK/dt we will transform our coordinates thus : Let V denote the 
velocity of the centre of inertia of a pair of spheres M and m, p their relative velocity, 
6 the angle between Y and p. Then 
2 m 
m 
U ' + Ul 4- m P ) + M 4 m 
Y p cos 6 
3 _™ . ( M 
— V ~ -f - ( vv- p 
\M + m ' 
2M 
, T Yp cos 6 
M 4 m r 
and 
and so 
MU~ + mid = (M + m) V 2 + 
Mm 
X M + in 
F/= Nn (~J (~~f (1 + D)*(l + d)U- h< d u + m)Y1 + w^/ 2 } (1 - D//MU- - clhmvd) 
Yf = Nw (^J (1 + D)* (1 + df e~ h { (M + “) V2+ W ! } (1 - DAMU ' 2 - dhmu'*) 
where U', u' are the values of U, u after collision. 
