286 G. H. BRYAN. 



[b) If further the potentlal energij u 'whollij nmfital, tlien l'\ =^ J\ = 

 and V = Fj 2 giving in tliis case 



dr__fIT d'^V _ rpr 



(]x^ d.v^ dx^"^ dx.^ 



Heuce 



or Maxwelt/s Law of Partition of Energy liolds good. 



Tliis will not^ liowever^ be tlie case as a gênerai resuit if the potential 

 energy of tlie iield be différent from zéro. 



Tliere are, however, certain conditions under whicli the resuit still 

 liolds good. One case in which thèse conditions are satisfied is when 

 /'(;!•,, .;•.,) = ,/'(''"2j --^'i) 3.nd }\ is the same function of .;-j tliat V.^ is of 

 X., ; in other words when the probability of any given position of the 

 particles is equal to the probability of the position obtained by inter- 

 changing the particles, and the forces ou the two particles are eqnal for 

 e(]ual values of their coordiuates. 



In the tields of force conimonly considered in Theoretical Dynamics, 

 the force on a particle for given values of its coordinates is proportional 

 to the mass of the particle, not iudependent of its mass as . the last 

 conditions assume. 



But the question now arises, Does the mean product [tj ^2] neces- 

 sarily vauish? To answer tliis question, we must form the second difFe- 

 rential coefficient of [y?, y'2] witli respect to the time. We tind 



dfi ^ ^ m^ni.^ dx^ dx.-, Kw.^ m^J dx^dx^ 



1 d'^V , 1 d'^r^ 



Xi d^l^ i d^y\ 

 ^ ^ \w.j dxy ^ w,^ dx-i V 



In order tliat \i\ v.^ may always vanish, its second ditierential 

 coefficient with respect to / must vanish, and therefore also 



\_dx^ dx.^\ L ' ' ' ' J \.dxy dx.^A 



