1894.] MaxiuelVs Laiv of Partition of Energy. 253 



6. We now proceed to show that other distributions exist 

 which satisfy the condition of permanency and in which the 

 energy is not distributed according to Maxwell's law. 



For the equations of motion have tvm integrals, one of these 

 expressing the constancy of energy being 



A(o^ + B(o^ + C<o^ = 2T= const, 



the other expressing the constancy of angular momentum being 



A'co^' + B'co^' + G'co^' =G' = const. 



If now, we assume the frequency factor to be a function of G 

 instead of T, say /((?), so that the number of bodies whose angular 

 velocities lie between co^ and w^ + dM^, co^ and (o.^ + dco^, co^ and 

 Wj + do)^ is 



f{G)d(o^.d(o^.dco^, 



then precisely the same argument as before shows that this distri- 

 bution is also a permanent one, being independent of the time. 



Also precisely the same argument as before shows that in this 

 case the mean values of the quantities 



A'<o,\ B'a>;\ G'ay^' 



are equal to one another, each being equal to 



I r f{G')G'dG-rr f{G')G'dG. 



O Jo ^0 



Therefore the mean values of the portions of the energy 



Aq>;\ B<o^, (7(y/ 



are inversely 'proportional to A, B, C and are unequal, so that 

 Maxwell's law does not hold good. 



7. If we were to take the frequency factor to be a function 

 both of T and (T^ we should have the most general distributions 

 consistent with permanency, in none of which would the kinetic 

 energy be distributed according to Maxwell's law. 



The present investigation shows that in order that Maxwell's 

 law may hold, the frequency factor f must he a function of the 

 energy only, at least so far as concerns the generalized velocities, 

 momenta, or momentoids which enter into it. 



Where, as in the present instance, other forms of the frequency 

 factor are consistent with permanency Maxwell's law of partition 

 although a possible law is not unique. 



19—2 



