Attractive and Repulsive Forces. 207 



the same. Consequently 



^p - ^ l^y + ^/J- 



For taking the next step in the argument reference is to be made 

 to a law ah-eady admitted (art. 27), according to which the vi- 

 brations of an unhmited elastic fluid must be such that for every 

 acceleration and movement of a particle in a given direction 

 there must be an equal acceleration and movement of the same 

 in the opposite direction. Let therefore o-j and cr/ be conden- 

 sations corresponding to movements so related. Then we shall 

 have 



a'^da, a'^do-l 



Consequently 



(l-^ai)da; (l + aDda; 



or 



d.{(Tj-\-(7l) +d.(T^al=0. 

 Hence, by integration, 



no arbitrary constant being added, because plainly cr^ and cr/ 

 must vanish together. This result shows that G-f and cri must 

 be one positive and the other negative. Let cr/ be negative, so 

 that it belongs to the rarefied portion of a wave. Then, since 



cr/= — - — - — , apart from sign crj is less than a, which belongs 



to the condensed portion; as plainly should be the case. Also, 

 from the above equation, 



2 



cr. + o-/ = — a, (7 1 = y-^ — = (Tf^ nearly. 

 Therefore 



^^2 """ A '^'• 



This equation gives the residual acceleration of the atom when 

 the actions upon it at any two corresponding positions in the 

 condensed and rarefied portions of an undulation are taken into 

 account simultaneously; whence it may be inferred that the 

 total condensations and rarefactions accompanying every com- 

 plete vibration of the fluid backwards and forwards produce on 

 the whole an acceleration of the sphere in the direction of the 

 propagation of the waves if H be negative, and in the opposite 

 direction if that quantity be positive. These effects correspond 

 respectively to repulsive and attractive forces. 



