'Small Vibratory Motions of Elastic Fluids. 291 



number of portions of one or more of these curves be taken 

 such, that being translated to a common axis, and maintaining 

 relatively to it the positions they had with respect to their own 

 axes, they are capable of uniting at their extremities so as to 

 form an irregular line, the ends of which are situated in the 

 axis, this line will designate in the most general manner, the 

 state of particles subject to motion propagated in a single direc- 

 tion. It will be seen that the equation V = aS always obtains, 

 and that the particles pass consecutively through the same states. 

 Two such lines moving with the uniform velocity a in opposite 

 directions, will represent the most general motion of which the 

 fluid is susceptible. 



9. It has been shewn in the preceding Article, that the 

 effect of suddenly stopping the diaphragm, was to change the 

 velocities proportional to the ordinates of en, into velocities pro- 

 portional to the difference of the ordinates of en and ef, (Fig. 5.). 

 Hence, describing a curve cf, the ordinates of which are double 

 the corresponding ordinates of ef, the effect was to diminish the 

 velocities of the particles included in a certain extent df, by 

 quantities proportional to the ordinates of cf. Thei'efore if the 

 fluid be at rest, and at its mean density, and the diaphragm be 

 suddenly made to move vdth the velocity ac, it will shake the 

 particles to an extent af, and cause them to commence moving 

 with velocities, the law of which will be given without sensible 

 error by the curve cf. We have now to find the nature of this 

 curve. Suppose ac, which represents the velocity given by the 

 impulse to the particles on which it immediately acts, to be 

 some function of af, the distance to which its effect extends; 



ac=Y, af=X, Y =f{X). 



But as the particles at p are affected just as if a dia- 

 phragm were placed there, and suddenly made to move with 



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