A SYSTEM OF PARTICLES. 239 



hut P'^.(x(pr.Sx + 31. ^<pR . d,x is the force which acts on the particle 

 at rest; and consequently is identically equal to zero; 



assume a = a cos {ct — hoc), 

 then la — a cos (ct — kx — kSx) — a cos {ct — kx) 



= a {cos {ct — kx) cos kSx + sin {ct — kx) sin kSx — cos {ct — kx)\ 



= a sin {ct — Aa;) . sin kSx —2a. sin' . 



' 2 



In the same manner, if both particles vibrate in the direction of 

 transmission, 



Aa = a sin {ct - kx) sin kAx — 2a sin^ , 



and, by the hypothesis that each medium is a medium of symmetry, 

 we shall, by reasoning precisely the same as that which I adopted on 

 a previous occasion, [Part i. p. 156. of this Vol.], arrive at the following 

 result : 



— = }^2P.{<pr + ^Sx')sm^~ 



^M:^,^ „ FR ^ ,, . ,kAi 



2M FR^„^. ^kAx\ , ^ 



+ -jp-I.{<pR + -^Ax')sin'-^\.a (1). 



= — c-a suppose, 

 where a = a cos {ct — kx). 



4. It is clear that, in order to effect this, we must suppose 



Fr 



a {(jir + — Sx')Sx' 



a negative quantity. Now if <pr = ^^ , or the force vary as tlu' in- 

 verse m"" power of the distance, this can be accomplished ; 



