( 134 ) 



1 st . The molecules are electrical double points with a constant 

 moment m. 



2 nd . The mean distance of the molecule is so great that we may 

 neglect the cases in which more than two molecules interact. 



3rd The velocities and the accelerations of the molecules have a 

 value relatively so small, that we may assume that their field of 

 force does not differ appreciably from the electrostatic field of the 

 double points. A consequence of this supposition is that the energy 

 of the system may be represented by : 



L — 1 '\ 2 m (2 cos <p -f C 



where L represents the kinetic energy of the system, £ the electric 

 force, <p the angle between the axis of a molecule and the electric 

 force at that place, and C a constant which does not depend on the 

 velocity and the mutual position of the molecules. 



If this last condition is satisfied the statistical mechanical conside- 

 rations of Boltzmann and of Gibbs are directly applicable to our 

 problem. If on the other hand it is not satisfied these considerations 

 cease to be applicable and then it is impossible to solve the problem 

 before a statistical treatment of a continuum as the electromagnetic 

 field has been worked out, which is analogous to Gibbs's treatment 

 of systems with a finite number of degrees of freedom ; this however 

 is not the case as yet. A rigorous discussion of the case that the 

 molecules are vibrators is therefore as yet impossible and so we 

 shall have to confine ourselves to the supposition of constant double 

 points. 



Let us imagine a molecule A and at a distance r another mole- 

 cule B. The angle between the axis of A and the radius vector will 

 be called &, then the electric force exerted by A at the point where 

 the molecule B is found, will be : 



£ = — V\ cos 7 & + sin* # = — |/ 3 cos- # -f 1. 



If again <p is the angle between (£ and the axis of B then the 

 potential energy of B is: 



m 1 



|/3 cos' & + l.cosif 



3r 8 



According to the well-known theory of Boltzmann and Gibbs the 

 probability that the angle <p will fall between the limits <p and 

 (f -J- chp and the angle & between the limits & and # -f- dd- is : 



m 2 J/3 cos" S -j- 1 . cos f 

 3 r s t 



V 4 sin <f df . sin & dd- . e , 



