412 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



In any system of particles arranged symmetrically, suppose that one 

 of the particles is slightly displaced from its position of rest, the others 

 remaining undisturbed ; then if we make use of the notation in Article 

 (1), putting a — 0, /3' = 0, y = 0, and therefore $a = - a, 5/3 = — /3, 

 Sy = — y ; the forces which act on the particle parallel to the axes in 

 consequence of its displacement, are evidently - Co., — C/3, — Cy, 



where C = Vm {/(r) + \f(r) $x*}. 



Hence if the law of molecular force be such that C is zero in all 

 cases, it is evident that in a symmetrical system any particle may be 

 slightly disturbed from its position of rest, without bringing any force 

 into action upon it, i. e. the system is in neutral equilibrium*. 



Now it is very improbable that substances in nature are so held 

 together, that a particle may be slightly displaced from its position of 

 rest without bringing any force into action upon it : therefore it is not 

 likely that the molecular force is of such a nature as to make C zero 

 in all cases. 



If the system consist of two or more sets of different particles 

 exercising different kinds of molecular forces, the same is evidently 

 true; for then the forces which act on the particle parallel to the 

 axes, in consequence of its displacement, are — alC — /32C — ylC, 



where 



2C = {2m [f(r) + -f(r) Ja*] +X,», [0 (r,) 4- \ f'(r,) A<] + similar terms}, 



» 



IS 



Hence it is evident that the condition of stability of the equilibrium of the ether in vacuum 



2{y(r)+ -y'(r)Sx*} = a positive quantity, 



and in the interior of a transparent body 



m 2 {/(r) + -f'(r) Sx*} + m l 'Z l { (p{r f ) + —, <t>'(r r ) Ax* } = a positive quantity. 



As in § 11, the condition of stability in vacuum may be put in the form 

 _/ 1 d(Rr*)\ 



I r 3 ~d — I = a P 0Sltlve quantity, 



R being the law of molecular force. 



