Probabilities to the Movement of Gas-Molecules. 255 



premisses, for F(U,V)/(w,u) substitute -f(U,Y, u,v) ; and, 

 putting x £° r l°g^ solve the functional equation 

 %(U,V,^)- % (U',W,*/) 



+ X{M(U 2 4-V 2 ) + ^(^ 2 +^ 2 ) 



-M(U /2 + V' 2 ) -m{u' 2 + v' 2 )} = 0. 

 Differentiating twice with regard to each of the variables 

 U, V, u, i\ we have 



(»).--«--(fli). ©)--»-&. 



Whence #= Const. - \{M(U 2 + Y 2 ) + m{u 2 + <)} + a linear 

 function of the variables, which disappears, if, as here 

 generally, the mass-centre of the system is assumed to be at 

 rest ; or otherwise can be adapted to uniform motion of the 

 mass-centre (I. c. pp. 253, 257). 



Next suppose that in addition to the motion of the masses 

 there are internal movements designated by generalized co- 

 ordinates. As above, the species and varieties of couples 

 may be defined by the velocities of the mass-centres and the 

 place of the point of impact on the contour (of one of the 

 molecules). But that point does not now in general move 

 in a right line parallel to the mass-centre ; not like the 

 point of a lance at a tournament, but, rather, like a point on 

 the edge of a sabre which the dragoon whirls as he charges. 

 Collisions therefore will now be divided not only according 

 to the velocities, the classes of the colliding molecules, but' 

 also according to the values of their generalized co-ordinates, 

 their genera (I. c. p. 263). If now stability is defined as 

 steady distribution of velocities for each particular set of 

 values assumed by the co-ordinates, then we may reason as 

 before that each variety of each .class of each genus must 

 remain equal in content to its reciprocal ; and therefore that 

 the velocities must be distributed according to a normal law 

 of error of which the variables are (squares and products of) 

 velocities and the coefficients are functions of the co-ordinates. 

 But if the steady distribution of the classes in each genus is 

 not implied in the definition of stability, it may be deduced 

 as before (I. c. p. 263) from the definition of stability as 

 steady distribution of classes on an average of all the distri- 

 butions in the medley. 



When we pass to encounters the argument is not materially 

 affected by the circumstances that the change of velocities is 



