410 MATHEMATICAL APPENDICES 27* 



the value of which may also be interpreted as the probability of 

 an encounter in unit time. 



To this simple problem another, which better corresponds to 

 reality, may be reduced. 



Suppose a multitude of particles in motion, and all with the 

 same velocity in the same direction, so that all the particles have 

 the same velocity-components U, V,W; assume further that the 

 particles fill the space with equal density on the average, and that 

 there are n of these particles per unit volume. Into this swarm 

 let another, or even a number of other particles, enter, which move 

 with a different velocity in a different direction ; let the velocity 

 of this second group when resolved in the same three directions 

 have the components u, v, w. We have to find the probability 

 of an encounter, and the probable time that elapses before an 

 encounter occurs. 



The probability of an encounter in this case is the same as if, 

 instead of allowing both systems to move in two different 

 directions, we had, more simply, assumed that the one swarm was 

 at rest and the other moved relatively to it with the relative 

 velocity 



r = N/ {(u - U) 2 + (v - F) 2 + (w - W)*} . 



The probability, therefore, that a given particle of the one 

 system should collide with any particle of the other in the unit of 

 time is to be represented by the same formula as before when for 

 the absolute velocity w the relative velocity r is substituted. Thus 

 the probability sought is 



28*. Number of Encounters 



From this simple formula we obtain that which holds for the 

 case of a real gas by simply finding the mean value of the relative 

 velocity of two of its molecules. In this calculation we first of all 

 assume that all the particles are moving with the same speed. 

 This assumption is certainly not quite true, as we know from our 

 former investigations; since, however, it has shown itself very 

 serviceable in the calculation of the pressure and in other 

 problems, we may here, too, expect by its help to obtain formulae 

 that are approximately correct. 



If, as before, we denote by G the velocity which all the 



