Small Vibratory 3Iotions of Elastic Fluids. 297 



propagation travels in the time t, and a'c' = ac, to represent the 

 velocity of the diaphragm; so that the area a'c'ca will represent 

 the condensations generated in the time r. Take fq = 2 x aa, 

 draw the ordinate pq, and pt parallel to a'q. Let qs make the 

 same angle with a'/ as e/ does, produce fe to b, and join pc'. 

 Since qa' + c'c = 2 x aa' + aq = of, and that ec = ea, it follows that 

 c'b = as. But the triangle pbt is in every respect equal to 

 qa's; therefore also bt = a's, and c't is bisected in b. Hence c'p is 

 parallel to cf, and wv is bisected in e. As the condensed and 

 rarefied waves travel with equal velocities in opposite directions, 

 their extremities in the time t will have separated by qf, and 

 qa's will be the portion of the rarefied wave not reflected, c'ceb 

 the reflected portion. Hence the condensations cc'a'a are di- 

 minished by cc'be, and also by arsa', or by its equal evtb, and 

 the remaining condensations are avta'. Again, the condensations 

 peaq are diminished by qar, or by its equal pve, which leaves 

 remaining pqva. Therefore at the given time the condensations 

 are given by the thick \ine fpt. Also since the direction of the 

 velocities in the rarefied wave is changed by reflection, the 

 velocities a'c'ca will be diminished by c'bec; but they will be 

 increased by arsa', or by its equal ewc'b; and the velocities on 

 the whole will be awc'd. Again, the velocities qpea are increased 

 by qar, or by its equal pew, becoming on the whole qpwa. Hence 

 at the given time the velocities are given by the thick line fpc'. 

 Because c'p is always parallel to cf, it follows that the particles 

 at any distance from the diaphragm move uniformly with the 

 velocity first impressed on them, till they acquire the conden- 

 sation corresponding to their velocity. When the motion has 

 continued long enough for the rarefied wave to be totally re- 

 flected, that is, during a time — , it will become such as Fig. 9. 



represents, c'ef being the line which gives both the condensations 

 Vol. III. FuTt I. P p 



