412 MATHEMATICAL APPENDICES 28* 



of its motion. The number of particles with a given direction, 

 therefore, when all directions occur equally, bears the same ratio to 

 the whole number of particles as the surface of the zone inter- 

 cepted between these two cones on a sphere constructed with the 

 colliding molecule as centre bears to the whole surface of the 



sphere, viz. 



2?r sin s ds : 4?r. 



The number, therefore, n of the particles in the unit of volume 

 which move in the direction denned by the angle s is 



n = yV sin s ds, 



where N is the whole number contained in unit volume. 



It is now easy to find, in the way required above, the total 

 number of the collisions. Since the angle s can increase from 

 to 180, the value of this sum is 



A = Tr^NG-r sin is sin s ds = Zn^NG f 71 " sin 2 s cos ^s ds ; 



o 



and the evaluation of this integral gives the value 



A = 



for the number of collisions which a particle undergoes in unit 

 time in a large group of other similar particles, when all the 

 particles have the same velocity G, and there are on the average 

 N particles in unit volume. 



Compare this number with that first found 



which holds for the case of a particle when it moves with the 

 speed w among a multitude of particles at rest, of which there are 

 n in the unit of volume. If we assume the speed and the number 

 of particles to be the same in both cases, or w = G and n = N, we 

 see that the number of collisions denoted by A is greater than the 

 other in the ratio 4:3. A gaseous particle, therefore, as Clausius 1 

 first perceived, meets with others more frequently when they are 

 all in motion than when one only is in motion and the others are 

 at rest. 



Inversely, the mean length of the straight path which a 

 particle traverses between two successive collisions is smaller in 



1 Phil. Mag. [4] xix. 1860, p. 434 ; Abhandl, iiber die mech. Wcirmetheorie, 

 2. Abth. 1867, Note on p. 265. 



