ee 
The Persistence of Molecular Velocities. 701 
OQ represent the two velocities before collision, so that PQ 
represents the relative velocity. If we suppose the two 
molecules to be of equal mass, the velocity of the centre of 
_ gravity will be represented by OR, where R is the middle 
point of PQ. Thus if the two molecules have velocities 
represented by OQ, OP before collision, the expectation of 
velocity of the former after collision is represented by OR. 
If the magnitude, but not the direction, of the velocity of 
the second molecule is known, we can find the expectation of 
the tinal velocity of the first molecule by averaging the 
components of the velocity OR over all possible directions for 
OQ. We must not take all directions for OQ to be equally 
probable, for the number of collisions occurring between 
molecules having specified resultant velocities is proportional 
to the relative velocity of the two molecules. 
It is at once clear on averaging the components of the 
velocity OR, that the averaged components perpendicular to 
OP vanish. We are therefore left with the single component 
in the direction OP of which the average value is 
ONOPO da Y HE 
2s <a bs 
PQ sin 0d0 
J0 
where @ is the angle POQ. 
To evaluate this fraction, let us write OP=a, OQ=8, 
PR=r, so that 7?>=a? + b? — 2ab cos 8, and 
ON=4(0P+0M) 
=(a+l cos @) 
Lge Dh) 
We also have rdr=ab sin @d6, so that the fraction (i.) 
becomes 
\(3a? + 0? =7?)r2dr 
Terie 
the limits of integration being from r=a+b to r=a~b. 
When a>), the fraction is found to be equal to 
cele 
10a(3a? + 2): 
and when a</J, it is equal to 
a(db* + 3a?) 
5( 30? +a?) ° 
Phil. Mag. 8. 6. Vol. 8. No. 48. Dec. 1904. 3C 
