Distribution e~ 2hX and van der Waals' Theorem. 405 



and x t° be the potential of the force exerted by it on any 

 molecule, and no other forces to act. Then the law applies 

 with A=€~ 2A *. Now let m' be set in motion with velocities 



dx' , dy' , dz 1 , 

 dt ' dt ' dt 



What is now necessary to make the motion stationary ? 



Let A be a function of x' y' z, as well as of the coordinates 

 xy z &c. of the molecules, and let Q be a function of v! V it*', 

 as well as of the velocities u v w of the molecules. Then 



~Ae-^ = leads to 



\ax ax J \ ax aw at J 



du f 

 We might assume -r? =0 &c, or u, v r , w' are constants, 



and 



Q = 2w((M-tt / )'+(«-w') 2 +(w-^)0» A = e_2 * x > 



and this is a solution provided that for each molecule 



dk _ _dA. „ 

 dx~~ dx~ n ^ C ' ; 



that is, provided that A is a function of the coordinates, only 

 as they are contained in the r's. This solution merely states 

 that the motion being stationary with A = e -2A * and m fixed, is 

 none the less so if m and all the molecules have the common 

 velocity u' v' w f in space. It is of no use or interest for us. 



Again, if the velocities u 1 v r w f are constrained in any 

 way, solutions may be devised for the constrained system. 

 They do not here concern us. If, however, the motion of m' 

 be unconstrained, we have by the law of action and reaction 



,aW__dx 

 m dt~ dx 1 ' 



And we now find a solution in the form A=e~ 2A x, and 



Q = Sm(u 2 + v* + w*) + m'{u'* + v'* + w ! *) . 



That is, we get a solution by making m itself a molecule. 

 We see then that if one molecule of the system is connected 

 with each of the others by intermolecular forces of the kind 

 assumed, A = e~ 2A * is a solution. Then it is also a solution 

 PUl* Mag. 8. 6. Vol. 2. No. 10. Oct. 1901. 2 E 



