332 Theoretical Proof of Avogadro's Law. 



later paper* Maxwell himself speaks of this assumption as 

 precarious ; and therefore gives a proof resting on a quite 

 different foundation. In fact, we should expect that greater 

 velocities in the direction of the axis of abscissae in comparison 

 with the smaller ones would be so much the more improbable 

 the greater the component of velocity of the molecule at right 

 angles to the axis of abscissae. If, for example, 



then the quotient just mentioned would be 



F(#, y, z)dx _ dx^ m^-x^ „2h(%2-x2) (y 2 +Z 2) 



The larger \/y 2 + z 2 , the more would small values in com- 

 parison with large ones gain in probability. Now, by means 

 of the law of distribution of velocities, which is to be proved, 

 we obtain the proof of the very remarkable theorem : that the 

 relative probability of the different values of x is altogether 

 independent of the value, supposed to be given, which sjy 2 + z 2 

 has for the same molecule; that therefore the quotient F(x,y,z) : 

 F(f, y, z) is independent of y and z ; or, what is the same, 

 since the three axes of coordinates must play the same part, 

 that F,(<#, y, z) may be represented as a product of three 

 functions of which the first contains only x, the second only 

 y, the third only z. 



It is therefore an altogether inadmissible cir cuius viliosus to 

 make use of this assumption to prove Maxwell's law of distribu- 

 tion of velocities. This therefore also holds good of the proof 

 which Prof. Tait has given (pp. 68 & 69 of the paper quoted), 

 and which is only a reproduction of Maxwell's first proof, 

 which he himself later rejected. For, from the circumstance 

 that the distribution of velocities must be independent of the 

 special system of coordinates chosen for its calculation, we can 

 never show that ¥(x,y, z) must have the form /(<£)<£(*/) ^(2), 

 only when this has been already proved. One might make use 

 of the circumstance to show the similarity of form of the three 

 functions/, </>, and i|r. I do not even need to enter upon known 

 geometrical investigations if the value of a function of three 

 rectangular coordinates x, y, z is independent of the choice of 

 the system of coordinates. For Prof. Tait has already shown 

 of the function denoted above by F, that it can only be a 

 function of Vx 2 -\-y 2 + z 2 \ but the value of the expression 

 V x 2 + y 2 + z 2 is already quite independent of the special posi- 



\Phil. Mag. [4] xxxv.p. 145 (1868). 





