1888 .] Professor Burnside on MaxweWs Theorem. 
107 
If y is the angle between the line of centres and the direction 
of Vq , and the angle between the plane parallel to these 
directions and the plane parallel to the directions of v and , the 
above quantity is 
cos 
and since in the process of averaging the terms involving ^ will 
obviously vanish, they may be omitted from the first. 
Hence, the probability of a collision being proportional to the 
relative velocity, the average energy exchanged between a P and a 
Q at impact 
2PQ - / /) 
P + y COS y + Sin y 
2PQ 
^P + Q 
in r ^7 -f*«2-&Do2-2ci5yoCOS^-o n • n • _ o 
dv / di'Q / d9 / dye sm 6 , Vq sin y cos y . vVq cos-y 
0 y 0 y 0 y 0 
7T 
y <*°° f'^i -ay2-6t>o2_2cOToCOs0 o o • /, 
dv / dv^ / dd / dye sm 6 . Vq sm y cos y . 
0 y 0 y 0 -'0 
0 ^ 0 ^ 0 
Performing the integrations with respect to 0 and y , this becomes 
PQ 
/•oo r" 
T ^-av'^-lvA 
0 y 0 L 
-av'^-hvM 3 cosh 2cWn~ ^^sinh 
2c 
^0 ] dvdvQ 
P+ Q 
y ^OO /-•GO 
y ^ vVq^ smh 2cvV(f . dvdvQ 
[No element of the integral in the numerator can be negative 
hence it can only vanish if 
1 
cosh 2cvvn 
2cVVn 
sinh 2cvVn = 0 
always, i.e. if c = 0 .] 
The integrals involved all depend on 
^00 /^oo 
0 Q 
which when ah — is positive, as it is in this case, can at once be 
shown to be equal to 
i{ah - c^) * 
Finally then, the average energy exchanged at an impact 
