58-60] Historical Note 51 



Historical Note. 



60. The law of distribution of velocities which has been found both in 

 this chapter and the preceding one was discovered by Maxwell, and is 

 generally associated with his name. It first appears in the paper already 

 referred to ( 9), communicated to the British Association in 1859. The 

 original proof is now universally admitted to be unsatisfactory, but is of 

 interest from its historical importance. Except for a slight change of 

 notation, the form in which it was given is as follows*. 



"Let N be the whole number of particles. Let u, v,w be the components 

 of the velocity of each particle in three rectangular directions, and let the 

 number of particles for which u lies between u and u+du be Nf(u)du, 

 where f(u) is a function of u to be determined. 



" The number of particles for which v lies between v and v + dv will be 

 Nf(v) dv, and the number for which w lies between w and w + dw will be 

 Nf(w) dw, where /always stands for the same function. 



" Now the existence of the velocity u does not in any way affect that of 

 the velocities v or w, since these are all at right angles to each other and 

 independent, so that the number of particles whose velocity lies between 

 u and u + du, and also between v and v + dv and also between w and w 4- dw is 



Nf(u)f(v)f(w) dudvdw. 



If we suppose the N particles to start from the origin at the same instant, 

 then this will be the number in the element of volume dudvdw after unit of 

 time, and the number referred to unit of volume will be 



Nf(u)f(v}f(w}. 



" But the directions of the coordinates are perfectly arbitrary, and there- 

 fore this number must depend on the distance from the origin alone, that is 



Solving this functional equation, we find 



f(u) = Ce Au \ (-U 2 + t> 8 + w*) = C s 



This proof must be admitted to be unsatisfactory, because it assumes the 

 three velocity components to be independent. The velocities do not, however, 

 enter independently into the dynamical equations of collisions between 

 molecules, so that until the contrary has been proved, we should expect to 

 find correlation between these velocities. 



On account of this defect, Maxwell attempted a second proof f, which 

 after emendations by BoltzmannJ and Lorentz assumes the form given 



* J. C. Maxwell, Collected Works, i. p. 380. 

 t Collected Works, n. p. 43. 



J Wiener Sitzungsber., LVIII. p. 517 (1868), LXVI. p. 275 (1872), xcv. p. 153 (1887), Vorlesungen 

 fiber Gastheorie, i. p. 15. 



Wiener Sitzungsber., xcv. p. 115 (1887). 



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