17* MAXWELL'S LAW 383 



only ones which must be taken into account in an exhaustive 

 treatment of the subject. In a system of bodies free fromi 

 external action, the sum of the energy and the velocity of the 

 centroid of the whole system are not the only invariable magni- 

 tudes ; but there are others, too, that always have the same value. 

 According to the theorem of the conservation of areas the 

 moments of the momenta of the system about th^ftf fl.vp. a.t. right. 

 angles to each other have also constant values. By the method 

 here employed it is easily possible to take this proposition of 

 mechanics also into account in the calculation. 



If we denote by x, y, z the rectangular coordinates of a 

 particle of mass m whose velocity is made up of the components 

 u, v, w in the directions of the system of coordinate axes, this 

 general proposition of mechanics is expressed by the equations 



GI = ^.m(yw zv) 

 c 2 = ^.m(zu xw) 

 C 3 = 2.m(xv yu), 



in which the magnitudes c are independent of the time and 

 position, and the summations 3 are taken over all the particles 

 of the whole mass. We may either refer the coordinates x, y, z 

 to a fixed system of axes, or choose as origin of coordinates the 

 centroid of the whole gaseous mass, which moves on with un- 

 changeable velocity : we will do the latter in order to gain the 

 advantage of obtaining formulae which refer only to rotations 

 about the centroid. 



The new formulae differ from those established earlier in 12* 

 by containing the coordinates as well as the components of 

 velocity, yet they can be introduced as equations of condition in 

 the same way as those which refer to the energy and the motion of 

 the centroid. 



For, to express the fact that the occurrence of other values 

 u + &u, v + &v, w + Sw of the velocities has the same degree 

 of probability, provided that they satisfy the laws of mechanics, 

 we have to add to the formulae before developed the conditions 



c t = 2.ra [y(w + 8w) - z(v 

 c 2 = $.m{z(u + $u) x(w 

 c 3 = 2t.m{x(v + Sv) y(u 



in these formulae only the velocities u, v, w, and not the co- 

 ordinates x, y, z, are varied, since among the changed values 



