4 J 6 
MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
with systems on, whose coordinates and velocities lie within the limits B, B', by the 
quantity 
dp x . . . dp n _ l dp 1 . . . dp „_i c/R (F f — F f) R per unit of time, 
and is increased per unit of time by collisions with systems on for all values of 
p r + 1 . . . p n by the quantity 
dpi . . ■ dp r dpi . . . dp r ||. . . (Ff — F/) R dp r+l . . . dp n _ Y eZR, 
all values of p r+ i, &c., being included in the integration. 
We will now assume (see p. 418, post ) that the velocities of M and on are not on 
average altered except by collision between M and on. And so the above-mentioned 
increments are the only increments by which the class of M systems within the limits 
A, A' is affected. 
In that case 
(1 f f dpi . . . dp r = dpi . . . dp r ||. . . (Ff' — F f) R dp r+1 . . . dp n _ x r/R, 
and, therefore, 
r r fZF 
I j ... — log F dpi . . . dp,, (over all values of p x . . . p r ) 
= || • • • (F/' — F f) R log F dpi . . . dp n _i dR. 
By symmetry, as the right-hand member includes all possible collision between 
M and on, 
ff • ■ • fh°g fdp r+l . . . dp, 
= {[... (F’f - Vf) E log f dp, . . . dp,., <£R, 
and, therefore, 
dF 
dt 
r- /» ^ 
log F d Pl . . . dp r + j j . . . j t lo gfdp r+l .. . dp n 
= || • • • (W' ~ F /) Tt log (F/) dp l . . . dp n _j dR. 
By symmetry, as we may interchange the accents, 
jj • • • f- lo g Fc ¥i • . ■ dp, + . . . d £ log/ dp, + , 
= {[{... (Ff- Ff) R log (Ff) dp, . . . dp,!., dB, 
