MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
409 
« (Pi + Pi) + £ (Pi + P a)» &c - = ffi 
or 
a-Pi + fat + • • • = R = — («p'i + ftp 2 • • .)» 
where a, /3, &c. are functions of the coordinates and constants. 
That is, we have in all n — 1 linear functions of the velocities, namely, S 1 . . . S„_ l3 
which remain unaltered by the collision, and one other linear function, R, which 
remains unaltered in value, but changes sign. That must be the case on every 
collision of elastic bodies. 
3. The kinetic energy may be expressed in terms of the n variables S x . . . S„_ x , 
and It, in lieu of the n velocities p x . . . p n , and since it is not altered by the collision, 
which changes the sign of R, leaving S x . . . S, { _! unaltered, it must be of the form 
2E=/(S 1 ...S, i _ 1 ) + XR 2 , 
in which f( Sj . . . S„_j) is a quadratic function of S : . . . S„_! with coefficients functions 
of the coordinates and constants of the system, and X is a function of the coordinates 
and constants. 
4. The system, after collision, has velocities p\, &c., which we will call the 
second state. We may conceive a system with the same coordinates having- 
velocities — p\, — p\, &c., and call this the second state with reversed velocities. In 
this state, S 2 . . . S«_j will have opposite signs to those they have in the first state, 
and R has the same sign as in the first state. The system retraces its course, and a 
collision occurs changing — p\ into — p x , See., leaving S x . . . S„_ x unaltered, and 
changing R into — R. 
5. To define a collision, we may suppose that a certain function, xfj, of the coordinates 
and constants is prevented by the physical conditions of the system from becoming 
positive. When t fj becomes zero, dxjj/dt from being positive becomes discontinuously 
negative, and a collision is said to take place. We may take \p for one of our 
generalised coordinates in lieu of p n , and i fj, or d\}j/dt, for the corresponding component of 
velocity. The kinetic energy is a function ofyq .. . p n _ h \p, and we may express it as 
a function of S x .. . S„_ l3 and xjj, where S x . . . S„_j are the constants found above. 
Since the kinetic energy is not altered by the discontinuous change in ip, whatever the 
values of S x . . . it must be of the form f(S x .. . S„_j) + -gXi/r. That is ip is 
reversed in sign, but unaltered in magnitude by the collision, and is, therefore, equal 
or proportional to R found above. 
6. What we have proved for a system is of course true if for system we write pair 
of systems. For instance, let there be two sets of systems : (1) systems M defined by 
coordinates p x . . . p r , and velocities p x . . . p r , and (2) systems m defined by coordi 
nates and velocities p r + \ • • . p n and p r + l . . , p n . If xjj is a function of p x . . . p n such 
MDCCCXCII.—A. 3 G 
