434 Mr. OBRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 

 Now let us assume, as we evidently may, that 



5f-*w* 



and similar expressions for -y— -, -^ 



(M), 



and then we shall have 



a„ dstaihr e Xpre ss io »s f o r ^,gj 



(iV). 



The equations (iW) being linear differential equations with constant 

 coefficients, similar to the equations (D), in § (15), we may deduce 

 values of a /3 7 from them similar to those of a /3 y obtained in the 

 previous part of this paper. It is easy to see that F x (a (1 y) is of exactly 

 the same form as the second members of the equations (D), differing 

 only in having different values of A, B, and C. 



The equations (N) are the same as the equations (J3), having the 

 term F 2 (a /3 7) added to the first, and similar terms to the other two, 

 and having e »; £ in place of « /3 7. 



I shall now shew that these equations are satisfied by such values 

 of e tj £, that a /3 7 are the mean values of a /3 7. 



To make myself better understood, I shall suppose the material par- 

 ticles to be placed at the corners of cubes, and call the set of etherial 

 particles which lie in the cube formed by any eight contiguous particles 

 of matter, a cluster of particles. 



Then, in a state of equilibrium, it is evident that the particles which 

 compose any cluster are arranged symmetrically with respect to the middle 

 point of their cube. Moreover, in any two clusters the arrangement of 

 the particles is the same in all respects, so that one cluster is perfectly 

 similar to another. 



