408 
MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
momenta be denoted by p\ . . . p„, q\ . . . q n . Let U be the potential. In the con¬ 
figuration Jh ■ • • Pn there are n — 1 independent linear functions of the n forces, 
— 7 ^-... — each equal to zero. We might then find n new coordinates c, . . . c„, 
dpy dp n 1 ° 
. . = 0 , and therefore if S x , S 2 . . . S„ he the components 
dc n —\ 
_ a& H _ x 
dt 
dU 
such that 7 - = 0 
dt] 
of momentum corresponding to c 1 . . . c -y 1 = ~ &c., and 1 
are finite or 
dt dcj “ dt dt 
zero. In the limit when the remaining force becomes infinite and acts for an infinitely 
ft dS 
short time t, -- 1 dt = 0 or S'j — S x = 0 ... S '„_ 1 — S „_ 1 = 0 , and restoring the 
j o ci t 
original coordinates. 
«1 (Pi ~ P'l) + h l (P2 ~P\) + •■•+*! (Pn -Pn) = 01 
a APi-p\) + • • • = 0 ' 
> 
= 0 
(A), 
««- (Pi ~ P 1 ) + • • • 
in which the coefficients are functions of the coordinates. 
And therefore n — 1 li near functions of the p’s are unaltered by the collision, namely 
a iPi + b±p 2 + • ■ . + hP» = a \Pi + b } p' 2 + . . . + k x p n = S x supposel 
a zPi “h baP* -}"••• + k 2 p /t = a 2 p 1 -f* a 2P 2 + • • • + k 2 p n = So 
Ctn-lPl + t> n -\P 2 + • • • + k>i-\P>i = Ctn-lP l 4" ^ u-lP o + • • • + k n _^p n — S„_j J 
Again, since the kinetic energy is unchanged, 
tpq= tpq, 
and by the properties of generalised coordinates 
Therefore 
tp’q = tpq. 
- (q + q)(P ~p) = 0 
(B). 
. . . (C). 
The last equation forms, with the n — 1 equations (A), a system of n equations, 
all linear as regards p>\ ~ Pi > &c. Since p x — p\, &c., are not all zero, we must 
equate the determinant of the system (A) and (C) to zero. That gives us a linear 
equation in q Y + q\, q 2 + q z , &c. We can now substitute for the q s their values in 
terms of p l . . . p„, and so obtain a linear equation 
