of the Motion of Fluids, Sfc. 40''; 



satisfied by the same value of m: the two forms of the arbitrary 

 function they give are vii'tually the same. The form thus 

 determined, evidently possesses the jiroperties indicated by the 

 equations, 



f{l-z)-f{l + z) = o, 



f{l - z) +f{l +z) = 0. 

 The same properties are possessed by the function 



sin I + m sin 3l + rri sin 5^ + &c., 



which, on account of the unlimited number of terms, fulfils the 

 required conditions in the most general manner, and is inclusive 

 of every function that fulfils them. But the primary form of 

 the arbitrary function is sin I, and every other form points t<» 

 motion compounded of the motion indicated by this. As we 

 have arrived at this form by reasoning upon the arbitrary func- 

 tions, on the supposition that the origins of s and t are arbitrary, 

 it must refer to every elementary portion of the fluid, and indi- 

 cate the mode of action of the parts on each other. 



In general we shall have, 



a> = a hyp. log. p = m sm - . (s - a< - wf) ; 



and by supposing m and X to vary at a given in.stant from one 

 point to another, either continuously, or in a manner not subject 

 to the law of continuity, these equations will accommodate them- 

 selves to any mode of vibration we choose to impress on the 

 fluid. When m and X are constant whatever be s and t, the 

 vibrations are of a particular kind, which may be called pri- 

 mary, and are generated by the action of the parts of the fluid 

 on each other. In this ca.se, the velocity and density at a 

 given instant, have the .same values at points separated by the 

 constant interval 2X. Also at a given point, the velocity and 



