REFLECTED AT THE SURFACE OF A CRYSTAL. 43 



To obtain the equations of motion of a particle in the outer medium, we 

 adopt the follow notation : r (f> (r) represents the force which one particle exerts 

 on another at the distance r ; x, y, z are the co-ordinates of the particle under 

 consideration ; x + 8x, y+Sy, z+8 z, those of another particle without the crystal ; 

 x + d xf, y + $/, 2 + d z', those of a particle within the crystal : r the distance between 

 the particle under consideration and another particle in the upper medium, r' the 

 corresponding distance for a particle in the lower medium ; all taken when the 

 system is in a state of rest : / + p the value which r acquires at the end of the 

 time t ; / + g' the corresponding value of r' ; (a,), (/?,), (7,) are the displacements of 

 a particle within the medium at the time t. Thus, by the usual process (See Me- 

 moir on Dispersion in Trans. Camb. Philos. Soc., vol. vi. p. 158), we have, for the 

 force on the particle resolved parallel to the axis of x, 



v ( 

 =Il< 



^ 



But (a,) = a, + 8 a, ; therefore the force parallel to the axis of x is ~ = 



t* t 



(5.) 



It may be remarked that we have, in deducing the last equation from the 

 preceding one, suffered ourselves to imagine that a, has a value when the particle 

 is without the medium. Although it can hardly be doubted that the value of (a,) 

 so obtained is quite correct, we do not purpose to insist on it, but shall obviate 

 the objection at once by restricting our discussion to the particles situated at the 

 common surface of the two media. The values of 2 are the same on both sides 

 of the surface, each extending over half an infinite space, the one upwards, the 

 other downwards. 



In order to find the value of the force, or of -^ , all that remains to be done 



Ct f 



is to substitute in equation (5) the values of 8 a, 8/3, &c. from equation (4). 



