Energy between Matter and JEther. 93 



and there is a similar relation for X 3 , Y 2 , Z 3 , and similarly 

 for ct 1} ft, 7i and a 2 , ft, y 2 . 



The quantities X l7 X 2 , Y l5 Y 2 , a lt a 2 , ft, ft can f now be 

 taken to be the independent Lagrangian coordinates f x f 2 . , . f m , 

 the vahies of Z„ Z 2 , 71, 7 2 being given by relations of the 

 type (2). Substituting from equations (2) we find that 



^rrr(X 2 +Y 2 +z 2 + a 2 +ft 2 + 7 2 )^r^^ 



becomes a sum of squares of these coordinates. 

 The total energy is accordingly of the form 



E = i m X(ul + vl + wl) + XaS 2 . 



The dynamical equations which regulate the variation of 

 fi» ?2 • • • are obtained from the electrodynamical equations 



As a typical equation we have 



Xl = Vcosfl ^ ^ sin2 ° sin2 * " C ° s2 ^ ^ + sin2 6 sin * cos * ' a 2 } • (3) 



If qi, pi are any pair of coordinates of position and 

 momentum of the material system, we have the variation of 

 Pi, gi determined by the usual Hamiltonian equations 



Pl= ~*9i> qi= ^Pi ■ •• • • (4) 



and equations (3) and (4) contain between them the dynamics 

 of the system. 



Let us represent the motion in a generalised space corre- 

 sponding to the 6N + 7W dimensions 



Pi, 9h • ■ ' &J &j • • • £»• 



If p is the density of a fluid moving in this space, the 

 equation of continuity is 



P Dt \d Pl B^i/ 3fi 



If the fluid moves in accordance with the dynamical equa- 

 tions, the first sum vanishes as usual by equations (4), and 

 the second sum vanishes by equations (3). Thus Dp/Dt=F0, 

 so that the fluid, if initially homogeneous, remains so through 

 all time. \ 



