of LiiminoHS Waves in Transparent Bodies. 213 



The third ciise we shall apply our equations of motion to is 

 that in which the particles of matter are free to move, and 

 do actually vibrate along with the particles of aether. 



In this case we must not neglect the equations (2.), nor the 

 part of our equations under the sign S. Now in the equation 

 (1.) the sign X has reference to the particles of matter, and 

 denotes the sum of a number of quantities in which a^ /3, y^ 

 have different values; but if the sphere of attraction or repul- 

 sion of the particles of matter on the particles of aether be 

 very small, as we assume it to be, it is evident that «, /S^ •// 

 will vary very little for all the different particles of matter to 

 which the sign X has reference. 



Hence for a first approximation we shall suppose a^ /3, y^ 

 the same for all the particles of matter to which the sign Z 

 has reference ; and therefore a^ ^^ y, may be brought outside 

 the sign Z, and then the part affected by the sign ^ becomes 



a^ Xm^ {$ (?•') + — cf) (?') {x^—xf) + a part which is zero 



in consequence of the symmetry. 

 Hencesince2?« r^ (Z) + ~ ^' {r^){ + x^- xf\= C, we 



have ^ = A-^ + B(^-^+_j+(A-B) 



Kdxdy^l^^dr^) -<-^ + <-«M 



and similar equations with reference to —^ — ^ • and in 



dt^ dfi ' '" 



exactly the same manner the equation (2.) becomes 



and similar equations for — ^ lL 



^ de^ dt'^ 



We shall now suppose, as in the previous case, thit the vi- 

 brations of the aether constitute plane waves propao-ated with 

 a uniform velocity; and it is evident that this being the case 

 the vibrations of the particles of matter will also^onstitute 

 plane waves propagated with a unilbrm velocity; hence ma- 

 king the same simplifications in our equation as in the pre- 

 ceding case, we have 



^""'~^^^ 7^ + Ca - Ca^ = (1.) 



