ILLUSTRATIONS OF THE DYNAMICAL THEORY OF GASES. 379 



Prop. II. To find the probability of the direction of the velocity after 

 impact lying between given limits. 



In order that a collision may take place, the line of motion of one of the 

 balls must pass the centre of the other at a distance less than the sum of 

 their radii ; that is, it must pass through a circle whose centre is that of the 

 other ball, and radius (s) the sum of the radii of the balls. Within this circle 

 every position is equally probable, and therefore the probability of the distance 

 from the centre being between r and r + dr is 



2rdr 



~7~' 



Now let <f be the angle A Pa between the original direction and the direction 

 after impact, then APN =%<$>, and r = s sin ^<f>, and the probability becomes 



sn 



The area of a spherical zone between the angles of polar distance <f> and tf> + d<f> is 



ZTT sin <f>d<f> ; 



therefore if 01 be any small area on the surface of a sphere, radius unity, the 

 probability of the direction of rebound passing through this area is 



CM 



4^ ; 



so that the probability is independent of <f>, that is, all directions of rebound 

 are equally likely. 



Prop. III. Given the direction and magnitude of the velocities of two 

 spheres before impact, and the line of centres at impact ; to find the velocities 

 after impact. 



Let OA, OB represent the velocities before impact, so that if there had been 

 no action between the bodies they would 

 have been at A and B at the end of a 

 second. Join AB, and let G be their centre 

 of gravity, the position of which is not 

 affected by their mutual action. Draw GN 

 parallel to the line of centres at impact (not 

 necessarily in the plane AOB). Draw aGb 



in the plane AGN, making NGa^NGA, and Ga=GA and Gb = GB; then by 



482 



