436 Mb. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



§ 43. Now it is evident that in the quantity F 2 (afiy), ajiy vary 

 very slowly indeed as we pass from cluster to cluster, for a /3 7 are 

 displacements which constitute a common wave of light in vacuum, and 

 the length of a wave in vacuum must be very large compared with 

 the intervals between the particles of matter, and therefore the vari- 

 ation of a /3 7 must be extremely small when we pass over a distance 

 equal to the interval between two particles of matter. 



Moreover, the coefficients in F z (a /3 7) are periodical quantities whose 

 mean values for . any cluster are zero. Hence, F 2 (« /3 7) is a quantity 

 whose mean value for any cluster is zero, and in which the alteration 

 is very gradual as we pass from cluster to cluster, and not perceptible 

 within the sphere of mutual action, (remembering that the sphere of 

 mutual action must be extremely small compared with the length of 

 a wave in vacuum.) 



Therefore, if to the first of the equations of motion (B) we add 

 the terms F 2 (a fS 7), and similar terms to the other two, the equations 

 so formed will be satisfied by values of a /3 7, whose mean values for 

 any cluster are zero. And therefore the equations (N) (which only 

 differ from such equations in having e n £ instead of a fi 7,) are satis- 

 fied by values of ev^ whose mean values for any cluster are zero. 



§ 44. Now if the mean values of e tj £ for any cluster be zero, 

 it is evident that a (Z 7 are the mean values of a/37. Hence it appears 

 that the mean values of a /3 7 may be obtained by expanding the equa- 

 tions (B), as in § (8), neglecting all terms involving differential co- 

 efficients above the second, and putting for the coefficients of the retained 

 terms their mean values; which process will lead to equations exactly 

 the same in form as those in § (15). 



Hence, every thing that has been proved in the previous part of the 

 paper respecting a /3 7, on the supposition of perfect symmetry, is also true 

 of the mean values of ajiy for any cluster when the symmetry is dis- 

 turbed, as it must be, by the action of the material particles. 



