Memoir on the Conservation of Force. 379 



provided that we neglect the friction of the sides of the tube 

 upon the air within it. 



This will be the case whatever be the form of disturbance, 

 provided only 



1. That the velocities and condensations are small, and follow 

 the law of continuity, 



2. That the condensation at any point bears a fixed ratio to 

 the particle-velocity at that point [Encyc. Met. Art. Sound, 

 No. 128). 



Suppose that we have a wave consisting of condensation only 

 which fulfils these conditions ; and suppose for simplicity, 

 though this is by no means essential, that the condensation is 

 distributed symmetrically about the middle point of the wave, 

 and that it has a single maximum, which will, of course, be that 

 of the middle point 6 The particle-velocity throughout will be 

 in the same direction — that, namely, of transmission. 



Suppose, further, that we have in a different part of the tube 

 a second condensed wave, equal in length to the former, having 

 also but one maximum, viz. at its middle point, where the con- 

 densation is equal to the condensation at the middle point of the 

 first wave, and having at equal distances from the middle point 

 on either side of it the same amount of condensation as the first 

 wave at the same distance from its middle point. 



I shall also suppose that at equal distances from their respec- 

 tive middle points the particle-velocity in each wave is the same, 

 in amount but opposite in direction. It follows that the waves 

 will move in opposite directions. 



Consider the waves, first, as they advance towards each other ; 

 next, as after the meeting they overlap ; finally, at the period of 

 complete occultation, when the middle point of the one wave 

 coincides with the middle point of the other. 



Before meeting, the vis viva of either wave will be constant, 

 and the vis viva of the system of two waves will be double that 

 of either taken singly. 



When the waves overlap, the condensation at any point of the 

 overlapping portions will be the sum of the condensations of the 

 portions superposed, but the particle-velocity at this point will 

 be the difference of the velocities of the superposed portions 

 taken singly. In this part of the system, therefore, i. e. in each 

 element of the overlapping portion of the waves, vis viva will be 

 lost; so that, as the waves after meeting will gradually more 

 and more overlap, the vis viva of the sytem will continually di- 

 minish, till, when the position of complete occultation is arrived 

 at, the velocity at each point, and consequently the vis viva of 

 the system, will have wholly vanished. 



As the waves emerge from occultation, velocity and conse- 



