BOLTtMAmft THWRW ON THE AVERAGE DISTRIBUTION 



PART II. A Free system. 



In a material system not acted on by external forces the motion satisfies 

 x equation* besides the equation of energy, so tout we must not include in 

 KIT integration ail the phases which satisfy the equation of energy, but only 



of them which also satisfy these six equations. 



In what follows, we shall suppose the system to consist of n particles, 

 m are i,...ro., and whose co-ordinates x, y, z, and velocity-components 

 , , v, are distinguished by the same suffix as the particle to which they belong. 



Let us now consider a system consisting of s of these particles, and write 



m l + m t + &c.+m t = M t (58), 



tn.x, + W + Ac. + mjc. = M t X t , ' 



m 1 y l +my 1 + &c. + my. = M t Y,, (59), 



j,2, + wiA +&c. + V, = M t Z t , 



then M, will be the mass of the minor system and X,, Y,, Z, the co-ordinates 

 of its centre of mass. If we also write 



m,w, + &c. + m t u, = M, U t , 1 



T/W + &c. + m.v. = M, V,, I (60), 



mjMJ, + &c. + m t w t = M, W, , J 



+ m.(y.v>. ~ av.) = F.+M.( Y.W, - Z. V.), 1 

 + m.(z.u.-x l w' l ) = G,+ M.(Z,U,-X.W t ), I ...(61), 

 &c. + m t (x.v. - y,u.) = H,+ M,(X t V, - Y. U,) , j 



then U tt V tt W t will be the velocity-components of the centre of mass, and 

 /' (J,, If, the components of angular momentum round this point. 



We shall also write 



!,(,' + ,* + !) + &c.+ii(tt. i + u. l + w. t ) = r. (62). 



The seven conditions satisfied by the whole system are that the seven 

 quantities U nt V n , W n , F n , G n , H n and E are 'constant during the motion. 



Under these conditions the 3n momentum-components are not independent. 

 We shall therefore transform equation (17) into one in which the differentials 



