248 Mb KELLAND, ON THE MOTION OF 



tlierefore the following solutions 



a = a cos (ut — kx), 

 a = — (I cos {ut — /e.v), 



appear to be the only possible way of satisfying the equations so as to 

 retain the form. 



12. Had we, however, taken the more general expression for a, viz. 



a = a cos (lit - /,\v) + h sin {ut — kx), 



it is not impossible that we should have obtained other modes of 

 solution. The above is sufficiently general for my present purpose. 

 Recurring again to the differential equations it will be found that they 

 become 



-TTj = - {'iPaA sm' -— + 2Q^B sni^ -— 



-7— = - 2PaA sm' — r- + 2 QEB sm' -— - « , 



«f \ 2 2 / 



SO that when the forces, which the particles of the same kind exert 

 on each other, are repulsive, we have vibrations in the direction of the 

 motion ; when attractive, transverse to that direction. 



13. This conclusion is not altogether uninteresting, as it leads us 

 to the possibility of a transfer of particles, owing to vibratory motion, 

 which we shall discuss in the sequel. Suppose, for instance, the solution 

 of one equation made it appear, that the particles, of which it re- 

 presented the motion, had an uniform velocity of transmission along 

 a certain line, then the corresponding equation for the other series of 

 particles would shew us, that these particles were transmitted with the 

 same velocity, along the same line, but in an opposite direction : thus 

 giving us a transfer forwards of one series of particles to supply the 

 places of the other series of which the transfer would be backwards. 



For the present then we may confine ourselves to the consideration 

 of one medium, as none of its results, so far as regards its own motion, 

 have a different form on account of the interposition of another. 



