Astronomical and Nautical Collections. 119 



even if the amplitude of these oscillations were not in the first 

 instance so wholly inconsiderable, it would be sufficient to 

 consider an undulation at a greater distance from the centre 

 of agitation, in order that their extent might be diminished in 

 any required proportion. 



In the second, or retrograde oscillation, the plane, return- 

 ing through the same space, must communicate to the stratum 

 of fluid in contact with it, and to the rest in succession, a mo- 

 tion in a direction contrary to that of the first oscillation ; 

 for when the plane recedes, the stratum in contact with it, 

 urged against the plane by the elasticity or the expansive 

 force of the fluid, necessarily follows it, and fills up the vacuum 

 which its retrograde motion tends to produce. For the same 

 reason, the second stratum is urged against the first, the third 

 against the second, and so forth. It is thus that the retro- 

 grade motion is communicated, step by step, to the most dis- 

 tant strata : its propagation is efiected according to the same 

 law that governs the direct motion ; the only diff*erence is in 

 the direction of the motions, or, in the language of mathe- 

 matics, in the sign of the velocities which are imparted to the 

 molecules of the fluid. We see then that the different velo- 

 cities which have existed in the solid plane, during its second 

 oscillation, must exist at the moment which we are considering, 

 in the different strata comprehended in the other half of o?, but 

 with contrary signs. Thus the velocity, for example, which 

 the plane had in the middle of the second oscillation, which 

 is its maximum of retrograde velocity, must now be found in 

 the fluid stratum situated at the distance | d from the centre 

 of agitation, while the maximum of direct velocity is found, 

 at the same instant, in the stratum which is at the distance | d 

 from the centre of agitation. 



The extent of the fluid, agitated by the two opposite oscil- 

 lations of the solid plane, is what we call the breadth of 

 an entire undulation, and we may consequently give the 

 name of semiundulation to each of the parts actuated by the 

 opposite undulations ; the whole constituting a complete oscil- 

 lation, since it comprehends the return of the vibrating plane 

 to the initial situation. It is obvious, that the two semiundu- 

 lations, which compose the complete undulation, exhibit, in 



