FOUNDATIONS OF THE KINETIC THEORY OF GASES. 1031 



speed in the " special " state ; — the other involving a second approximation to the 

 estimates of Viscosity and Thermal Conductivity already given in Part II. 



XV. Special Assumption as to Molecular Force. 



§ 57. To simplify the treatment of the molecular attraction between two particles, 

 let us make the assumption that the kinetic energy of their relative motion changes by 

 a constant (finite) amount at the instant when their centres are at a distance a apart. 

 This will be called an Encounter. There will be a refraction of the direction of their 

 relative path, exactly analogous to that of the path of a refracted particle on the corpus- 

 cular theory of light. To calculate the term of the virial (§ 30) which corresponds to 

 this, we must find 



(a) The probability that the relative speed before encounter lies between u and u + du. 



(b) The probability that its direction is inclined from 6 to 6 + dd to the line of centres 

 at encounter. 



(c) The magnitude of the encounter under these conditions, and its average value. 

 Next, to find the (altered) circumstances of impact, we must calculate 



(d) The probability that an encounter, defined as above, shall be followed by an impact. 



(e) The circumstances of the impact. 



(f) The magnitude of the impact, and its average value per encounter. 



In addition to these, we should also calculate the number of encounters per second, 

 and the average duration of the period from encounter to final disentanglement, in order 

 to obtain (from the actual speeds before encounter) the correction for the length of the free 

 path of each. This, however, is not easy. But it is to be observed that, in all probability, 

 this correction is not so serious as in the case when no molecular force is assumed. For, 

 in that case the free path is always shortened; whereas, in the present case it depends 

 upon circumstances whether it be shortened or lengthened. Thus, if the diameters of the 

 particles be nearly equal to the encounter distance, there will in general be shortening of 

 the paths, and consequent diminution of the time between successive impacts : — if the 

 diameters be small in comparison with the encounter distance, the whole of the paths will 

 be lengthened and the interval between two encounters may be lengthened or shortened. 

 Thus if we assume an intermediate relation of magnitude, there will be (on the average) 

 but little change in the intervals between successive impacts. Hence also the time 

 during which a particle is wholly free will be nearly that calculated as in § 14, with the 

 substitution, of course, of a for s. 



XVI. Average Values of Encounter and of Impact, 



§ 58. The number of encounters of a v, with a %, in directions making an angle fi with 

 one another, is by § 21 proportional to 



w^Dq sin /3i/3 , 

 where % 2 = v 2 + v^ — 2vv x cos /3 



