I 



24 MAXWELL'S LAW 45 



If 10,000 molecules move with such velocities that their 

 components in any given direction lie in 'magnitude between 

 and a certain value W (see 2 7), then there are only 1,811 

 for which this component lies between W and 2TF, and but 

 fifty-five with a value between 2TF and 3JF for the compo- 

 nent. The small values therefore predominate in remark- 

 ably large proportion, and the probability of larger values of 

 a component of the molecular velocity, just as that of large 

 errors of observation, is vanishingly small. 



In this form the law expresses the frequency of occur- 

 rence of the values which the three components of the 

 velocity assume. We shall show later on, in 26, how 

 the probability of a particular value of the resultant velocity 

 of a molecule can be deduced from it. 



As has been already mentioned, Maxwell's law can be 



employed in two ways. First of all it tells us how many of 



a certain number of molecules move with a given velocity 



at the same moment ; but, secondly, it serves equally well 



> to give the frequency with which one and the same particle 



> attains a given velocity in consequence of its encounters 



with other particles. 



26. Proof of Maxwell's Law 



Several demonstrations resting on different footings have 

 been tentatively given for this law of distribution of mole- 

 cular speeds. 



Its discoverer, J. Cl. Maxwell, first 1 proved it by the 

 assumption of a principle which, though true, itself needs 

 proof. 2 Since Maxwell himself recognised this defect, he\J/ 

 later gave a second proof, 3 the basis of which is subject to '' 

 no doubt. Since the state of equilibrium with which the 

 law is concerned is not disturbed by encounters between the 

 molecules, but is continuously maintained, every change 

 produced by collision must at once be cancelled by other 

 collisions. A velocity of a particular magnitude and direction 



1 Phil Mag. [4] xix. 1860, p. 22 ; Scientific Papers, i. p. 377. 



2 See the end of 14 * of the Mathematical Appendices. 



3 Phil. Trans, clvii. 1867, p. 49; Phil. Mag. [4] xxxv. 1868, p. 185; 

 Scientific Papers, ii. p. 43. 



