IN THE INTERIOR OF TRANSPARENT BODIES. 401 



m rf(r) the force of P' on P, 

 m^'fyi?) ... do. ... of P 4 on P. 



Then the force of P' on P at the time t resolved parallel to the 

 axis of x is 



3C -\- a — J]C • a 



m(r+ p)f{r + p) , which = m/{r + p) (x + a - x - a), 



T + p 



and we have similar expressions for the resolved parts of the force 

 of P, on P; hence we evidently have 



d* a. \ 



-^ = Zmf(r + P )(x' + a - x - a) + ^mrfir' + p) ( X/ + U/ - x - a) J 



d 2 8 d 2 y /•••(■«•) 



and similar expressions for —£ -~ \ 



The sign of summation 2 refers to all the particles of ether, and 2, 

 to all the particles of matter, which exert sensible forces on P. 



§ 6. To simplify these equations when the motion is a very small vibra- 

 tory motion. 



For the sake of brevity assume Sx, $a, to denote x — x, a — a, 

 respectively, and a similar notation with respect to the other co-ordinates ; 

 also assume Ax, Aa, to denote x 4 — x, a t — a, respectively, and a similar 

 notation with respect to the other co-ordinates. 



Since the motion is a very small vibratory motion, we may assume that 

 the relative displacement of any two particles is very small, compared 

 with the distance between them. 



This assumption, so far as I am aware, has been made by every 

 one who has written on the subject of undulations, whether in the 

 case of light or of sound ; indeed the equation of continuity in Hy- 

 drodynamics cannot be proved unless this assumption is made. But 

 it seems to me to be no assumption in the present investigation, at least 

 if we confine our attention to the case of light at some distance from 

 its source ; for example, solar light at the Earth : for suppose that there 

 is a considerable degree of condensation and rarefaction in the ether 

 in the immediate vicinity of the Sun, be it ever so considerable there, 



