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



These are the equations * of motion of the etherial particles, omitting 

 the terms depending on the action of the material particles, conse- 

 quently these are the equations of motion of the ether as it exists in 

 vacuum. 



$ 14. To compare the action of the material upon the etherial par- 

 ticles with what it would be if the former were absolutely fixed. 



In the equation (2?) the part which arises from the action of the 

 material particles, i. e. the part under the sign 2, may be written 

 thus, putting for A a, A/3, A a, their values a, — a, /3 ( — /3, <y, - 7), 

 viz. 



^m^cp(r')(u, - a) + ~.<p'(r')Ax\Ax(a - a) + Ay(/3, - /3) + A* (7, - 7 )}J. 



Now let a 2 /3 2 72 be the greatest values which a, /3, y t respect- 

 ively admit of at the time t, and a 3 /3 3 73 the least; then observing, 

 that a /3 7, a 2 /3 2 72, a 3 /3 3 73, may be brought outside the sign 



2,, and that 2, m t — <j>' (r) Aa;Ay, and '2m l — (p'(r')AxAx are zero in 



consequence of the symmetry, it is manifest that the above expres- 

 sion lies between 



(a 2 - a) 2m, U(r') + -,$' (r')AxA and (a 3 - a) 2m, (<p(r') + -,(p'(r') AxA , 



or between m t C{ai — a) and m,C(a 3 — a), 



if we assume C to denote ^U(r') + -^'(r')Af|. 



Now in the case of common luminous waves passing through trans- 

 parent bodies, the particles of matter are put in motion solely by the 

 vibrations of the particles of ether : if we consider how extremely 



* These equations are obtained by M. Cauchy in his Exercices, Vol. III. by a complicated 

 method, different from that made use of in the present paper. Mr. Green, also, has obtained 

 the same equations, in the Cambridge Philosophical Transactions, Vol. VII. Part 1. by a very 

 general but complicated method. 



