104-107] System of Loaded Spheres 93 



whole motion, just as, in the example of 59, the values of u* for the various 

 molecules remained constant. It is, then, clear that the general result 

 obtained in the present chapter will not apply to the present system, without 

 some modification being made. 



106. The most obvious modification to make is to ignore the rotation w 3 

 altogether, just as, in finding the law of distribution for a system of sym- 

 metrical spheres in Chapter II., we ignored all the three rotations. In either 

 case the motion is the same as if the ignored rotations were non-existent. 



Let us, then, suppose the kinetic energy given by 



2L = m (u? + v- + w 2 } + mk 2 (is? + OT 2 2 ). 



There are now 'five degrees of freedom, and the separate terms on the right- 

 hand of the above equation may be regarded as contributions from separate 

 momentoids, in the sense required in 77. The analysis of this chapter 

 accordingly shews that in the steady state we must have 



mu 2 = mv 2 = mw' 2 = mk 2 ^ = mk 2 ta/ (222). 



We can, however, investigate the steady state by <jonsidering the effect 

 of individual collisions, somewhat after the method of Chapter II. It will 

 be of interest to do this, and so to verify the result expressed by equation 

 (222). The main importance of the problem, however, is that we shall obtain 

 information, which will subsequently be found useful, as to the rate at which 

 the gas, if disturbed from the steady state, returns to that state. This 

 information cannot be obtained by the general methods which have been 

 used to determine the steady state. 



The Transfer of Energy in a system of Loaded Spheres. 



107. Let us suppose the distance of the centre of gravity from the 

 geometrical centre to be r in each molecule, r being small in comparison 

 with a, the diameter of the molecule. In the final result it is obvious that 

 only even powers of r can occur, for we can replace r by r without altering 

 the nature of the gas. The solution of the whole problem when r = is 

 known, for the problem then reduces to that of the symmetrical spheres 

 of Chapter II. For the present purpose we shall be content to find the 

 solution as far as r 2 only, neglecting r 4 , r 6 , etc. 



In addition to the rectangular axes x, y, z fixed in space, let us suppose 

 there to be a system of rectangular axes , rj, fixed in each molecule, 

 having the centre of gravity for origin, and coinciding with the three axes of 

 rotation already specified. The coordinates of the geometrical centre will 

 then be 0, 0, r, and the moments of inertia abou" the axe f, 77, will be 

 mk 2 , mk'' 2 respectively. 



