280 NEWTON S PKINCIPIA. 



wave which forms it, all the notes that can be sounded 

 from the same closed tube will form the series 



,1111 



1 3 5 7 9 &C * 



for we have merely to give n all its values from nothing to 

 infinity, to get all the possible values of A. 



Suppose now the tube to be open at the end opposite to 

 that by which the air is set in motion. It is assumed that 

 at the open end the density of the air must be the same as 

 that of the surrounding air, a fact that is not exactly true. 

 Taking it for granted, it will follow that both ends of tube 

 must be loops. The length of the tube must therefore 



be an even multiple of -, that is 



The fundamental note corresponds to n = 1 or 



A = 2Z, 



and all the notes that can be sounded from the same open 

 tube will form the series 



1111 

 1 2 3 4 5&amp;gt; &C 



We also learn that the note of the pipe being measured 

 by the value of A, it will be proportional to the length of 

 the pipe, and that to get the same note from two pipes, 

 one closed and the other open at the end opposite to that 

 at which the air is set in motion, the second must be double 

 the length of the first. 



The theory of stringed instruments is in many instances 

 remarkably similar to that of wind ones. The equations of 

 the oscillations of a stretched string are exactly the same as 

 those of a sounding pipe. 



Much progress has been made by the experimental re 

 searches of Hopkins, Chaldni, Savart, Willis, and others. 



