NEWTON S PRINCIPIA. 273 



As an example of a case in which the motion is actually 

 represented by this law, let us take an infinite tube, and 

 suppose the air in a small part of it to be set in motion, so 

 that it will begin to move according to the form we have 

 assumed for f. Let the extremities of this part of the tube 

 be A and B, the particles A and B and all the remainder 

 of the air in the tube is supposed to be at rest. Let us 

 now consider the motion during a small time 8 1. The 

 particles between A and B will, by the above reasoning, 

 continue to move according to the law assumed for . The 

 velocity and condensation, therefore, being represented by 



-7 and -r will be respectivelv 

 at d x 



a n sin (nt mx) and a m sin (n t m x). 

 Let x be measured in the direction from A to B, which we 

 shall suppose from left to right. The particle A is at rest, 

 the condensation at that point and for all points on the left 

 of A is zero. Take a particle a on the right of A, where 

 A a = $ x, the condensation at that point is indefinitely 

 small and is decreasing. The particle A will, therefore, 

 remain at rest. The particle a will come to rest, that is, 

 be in the same situation that A is in, at the end of the 

 time 8 1, where 



n 8 t = m 8 .?, 

 that is, the left end of the pulse travels onward with a 



velocity , leaving the air behind it undisturbed. By 



similar reasoning, it can be shown that the right end of 

 the pulse travels onward with the same velocity. At the 

 end of the time 8 t, the pulse will merely have advanced a 

 space 8 x, and the circumstances of the motion will be the 

 same as before. The same motion will, therefore, be 



T 



