on the principles of Hj/drodynamics. 359 



=4-1) 



■■mcos 



= m cos < n{z—ai) {z+at)+c X 



Let, now, 



so that 

 Then 





e /4>v'^ , \\ 



f = w cos — ( ;2-a^ y/ 1 + -^ +Cj- 



Hence it appears that the velocity of propagation of the 

 wave, or series of waves, defined by the above form of (p, is 





the propagation taking place along a straight line parallel 

 to the axis of ^, or, if we please, along the axis of z itself. 

 Here it is important to remark, that the particular expres- 

 sion obtained for (p, and the consequent velocity of pro- 

 pagation, have been arrived at by a strict induction. The 

 course of the investigation leads to these results and to no 

 others of a like kind. Hence as there is at present no case 

 of disturbance under consideration, these results have, with 

 regard to vibratory motion, a general significance. The in- 

 ference from them is, that whatever be the disturbance, the 

 motion consists of vibrations defined by a circular function of 

 the above form, and that the velocity of propagation exceeds 

 the value a by a quantity depending on the numerical value 



of-2- 



If the investigation be conducted on the hypothesis of plane- 

 waves, the solution to the first order of approximation is 

 rJ; = G(2f— rt/), and the velocity of propagation is «, the form 

 of G remaining arbitrary. I have already argued that these 

 results have no significance, because the exact integral, of 

 which this is the first approximation, conducts to results in- 



