IN THE INTERIOR OF TRANSPARENT BODIES. 435 



§ 42. To shew from the nature of the case, that the equations of motion 

 (B) are satisfied by values of a (Z y, whose mean values for all the particles 

 in any cluster are zero, and the same is true when we add to these equations 

 any terms whose mean values for any cluster are zero, and which are the 

 same, or not sensibly different, for similar particles in any two clusters within 

 the sphere of mutual action. 



Conceive the particles of each cluster to be placed a little out of 

 their positions of equilibrium, in such a manner that each cluster is 

 still symmetrically arranged with respect to the center of its cube, and 

 that all the clusters are still perfectly similar to each other ; then it 

 is evident, that if the particles be let go, an oscillatory motion will take 

 place (supposing the equilibrium stable), such that each cluster is always 

 symmetrically arranged with respect to the center of its cube, and all 

 the clusters are always perfectly similar to each other. 



Hence we may conclude that the equations of motion (2?) are satis- 

 fied by such values of a y3 7, that the center of gravity of each cluster 

 has no motion ; i.e. that the mean values of a /3 7 for all the particles 

 in any cluster are zero. 



The same is true if we add to the second members of the three 

 equations (JB) any terms being functions of x, y, %, t, whose mean 

 values for all the particles of any cluster are zero, and which are the 

 same for similar particles in any two clusters within the sphere of 

 mutual action. 



This is evident from the fact that the terms represent forces which 

 clearly have no effect on the center of gravity of any cluster, and which, 

 being the same for similar particles in any two clusters within the 

 sphere of mutual action, do not derange the similarity of the motion 

 in these two clusters. 



It is only necessary that these terms should be the same for similar 

 particles in any two clusters within the sphere of mutual action : hence 

 if there be a very gradual alteration in these terms as we pass from 

 cluster to cluster, so gradual as not to be perceptible within the sphere 

 of mutual action, it is still true that the equations are satisfied by 

 such values of a j3 7, the mean values of which for any cluster are zero. 



