33* MOLECULAR FREE PATHS 423 



as follows at once from the formulae of 29*, by employing his law 

 of distribution of speeds. 



34*. Correction of the Formula "by allowing for 

 the Dimensions of the Molecules 



The formula obtained for the free path may be further im- 

 proved by a correction which van der Waals 1 first attempted 

 to apply. The length as given by the formula is somewhat too 

 great, since the calculation of the probability of collision in 27* 

 was conducted as if the space occupied by the sphere of action of 

 a molecule between two collisions (or the fourfold volume of the 

 path, as in 69) were equal to a cylinder whose base is a central 

 section of the sphere of action, and whose height is the free path 

 in a strict calculation we must remember that this space has 

 hemispherical ends. Owing to this circumstance, the free path L 

 which the centre of a molecule traverses between two collisions is 

 diminished by a magnitude of the order of s, the radius of the 

 sphere of action. 



In the case of central collisions the diameter of a molecule or 

 the radius of its sphere of action would have to be subtracted, and 

 a smaller amount in every other case. The mean value of the 

 correction may be found by a simple integration over the hemi- 

 sphere. The correction corresponding to an angle of impact s 

 made by the direction of the relative motion with that of the line 

 of centres is s cos s. The probability of this value of the angle of 

 impact is obtained by projecting the ring-shaped element 2w sin s ds 

 of the spherical surface on the plane of the section and dividing 

 by TT, the area of this section ; its value is 2 cos s sin s ds, and conse- 

 quently the mean value of the correction is 



2? f cos 2 s sin s ds = f s. 



Jo 



Van der Waals, who instead of this had found the value 

 ^s, has given a more exact calculation in a later memoir. 2 He 

 pointed out that the correction thus found does not refer to the 

 absolute value of the molecular free path ; it is rather the path of 



1 Over de continuiteit van den gas- en vloeistoftoestand, Leiden 1873, 

 p. 48. Abstract in Beiblatter, 1877, i. p. 10. An English translation of F. 

 Both's version has been published by the Physical Society of London. 



2 Arch. NterL xii. p. 201, 1877. 



