Small Vibratory J\Iotions of Elastic Fluids. 285- 



for an indefinite length of time. AVe shall first consider the 

 case in which this line is continuous. 



The propagation being supposed in the positive direction only, 

 if also c = c'=c"=&c., V will be o for any given value of < at a 

 certain point, namely, that for which x = al-c. There will not be 

 another point at which r=o, unless X, X, X ", &c. have a common 



multiple. Let x'= -, x" = - , &c., and suppose also c = o, as its 

 value is arbitrary: hence, 



/''=«5 = Xwj]sin— ^- '- + fi sin !— - + m sin + &c. . 



(, X X \ ) 



Here f"' and aS become o as often as x = at + n\, and the curve 

 which gives the velocities and condensations at a given instant, 

 cuts the axis of x at points separated by the constant interval X. 

 Suppose ^ = 0; then the equation of the curve is 



( . ttX . 2TrX , . SttX a ■) 



y = m\ \s\n -— + m sin -— \- m sin — h &c. > . 



C X X X J 



The general form of it is given in Fig. 2. The loop cd is exactly 

 equal to ab, and symmetrically equal to be. Also equal ordi- 

 nates occur at points separated by the constant interval X. The 

 preceding is the general equation of the waves which produce 

 musical sounds, when the disturbance arises out of the action 

 of the parts of the fluid on one another, and follows the law. of 

 continuity: for the sole condition required in these waves is, 

 that they recur at regular intervals. It is possible that this 

 condition may be fulfilled at the same time that the loops 

 ab, be, cd, make up a discontinuous line; but in such cases, 

 the disturbance which causes the waves must also be of a dis- 

 continuous nature. 



6. Conceive two motions exactly alike to be propagated in 



