287-289] Persistence of Molecular Velocities 



237 



two molecules into (i) a motion of the centre of mass of the two, the velocity 

 of this motion being represented by OR, and (ii) two equal and opposite 

 velocities relative to the centre of mass, these being represented by RP 

 and R Q. 



Imagine a plane RTS drawn through R parallel to the common tangent 

 to the spheres at the moment of impact, and let P, Q' be the images of P, Q 

 in this plane. Then clearly RP' and RQ' represent the velocities relatively 

 to the centre of gravity after impact, so that OP' and OQ' represent the 

 actual velocities in space. 



In fig. 18, let the directions of motion relatively to the centre of gravity 

 before impact be AB, DE. and let those after impact be BC, EF. Then the 

 line of centres bisects each of the angles ABO, DEF. Let us call each of 



FIG. 18. 



these angles <, measured so as to be acute in the figure. Imagine the 

 point E surrounded by a circle of radius <r (the diameter of' a molecule) of 

 which the plane is perpendicular to the direction AB. Then in order that a 

 collision may take place, the line AB produced must cut the plane of this 

 circle at some point P inside the circle. Also, all positions of P inside this 

 circle are equally probable, so that the probability that the distance EP shall 



7 . 2rdr ' . rf> , . 



lie between r and r + dr is . oince r = <rsm^ ; this may be written 



O" *j 



sin ^ cos d<j>. 



This, then, is the probability that < shall lie between <f> and 



<J> -f d</>, and therefore that the angle which EF makes with DE shall 

 *be between $ and <f> + d(f>. The expression found is, however, equal to 

 sin <})d<f> and therefore to that part of the area of a unit sphere for which 

 the radius makes an angle between </> and < + d<J> with a given line. Hence 



