106 Proceedings of Royal Society of EdinhurgJi. [jan. 6, 
4. On a Simplified Proof of Maxwell’s Theorem. By Pro- 
fessor Burnside. Communicated hy Professor Tait. 
In the course of verifying some of the mathematical work in the 
third instalment of Professor Tait’s paper on the Kinetic Theory, the 
following simplification of his proof of Maxwell’s theorem occurred 
to me. 
The number of pairs of particles, one from a set (P, h)^ the 
other from a set (Q, ^), for which the velocities parallel to the 
axes lie between 
X and x-\-dx ^ y and y + dy^ z and z-vdzm the one case, 
and between 
a?' and x + dx, y' and y + dy\ z' and z' 4- dz' in the other, 
oc e~^^^'^'^y^+^‘^^~^^^"^'^y"^^^'‘^'>dxdydzdx dy dz' 
Write 
Pa? + Qa?' = (P + Q)a 
x-x' =d 
and similar substitutions for y,f and z,z. 
Then 
lix^ + hx"^ = {li + h)o? + 
Q2/z+P2^ ,2 , oQ^^-P^ 
(P + Q)^"" "pW 
aa 
= aa? + ba!^ -1- 2caa! say . 
d^az e-«(a24-^2+Y2)_&(a'2+^'2+y2)_2c(aa'+i3^'-l-yy') 
X dad(^dyda! d(^' dy 
Hence if a? + +y^ = r? 
a'2 +/5'2 +y'2 
aa! + f^f^' -4- yy = VV^ COS 6 , 
( 1 -) 
the numbers of pairs of particles one from each system for which 
the velocity of the centre of inertia lies between v and v + dv, the 
relative velocity between and + dv ^^ , and the angle included 
by the directions of v and between 6 and 6+dQ 
Qc Q-av^-hvo^-cvvQco?,d vHf^dvdv^ sin 0 dO . 
The energy exchanged between a P and a Q at impact is 
X product of components of v and in the line of 
centres at impact. 
