114 On the Longitudinal Vibrations 



Rods. A firing can extend ^ind eontra<£t itfelf either wholly 

 in the direftion of its length, or divide itfelf into two, three, 

 four, or more parts, each of which is fubjeft to a movement 

 of the like kind. The tones, which in comparifon of the 

 tranfverfal tones are exceedingly high, are to each other as 

 the numbers i, 3, 3, 4, &c,, or as the number of the vi- 

 brating parts. They fo far correfpond with the tranfverfal 

 tones, that in the fame kind of vibration they are in the in- 

 verfe ratio of the length of the firings ; but totally differ 

 from them in this, that the thicknefs and extenfiori' of the 

 firing are of no account, and that, for this reafon, more de- 

 pends upon the nature of the fubftance. Thus the tone of 

 a brafs firing is about a fixth or a feventh higher than the 

 tone of a cat-gut firing ; and the tone of a fteel firing is a 

 fourth or nearly a fifth higher than the tone of a brafs firing. 

 On the other hand, in the tranfverfal vibrations of a firing, 

 the tone, as is well known, does not depend on the quality 

 of the fubfiance, but on the length, weight, and degree of 

 tenfion. A firing, therefore, which vibrates longitudinally^ 

 is not to be confidered fo much as a thread-like body ren- 

 dered elafiic by tenfion, but as a piece of matter extended 

 in length, and which vibrates exaftly in tlie fame manner 

 as a rod or ftaff fafiened at both ends. When a flaff is free 

 at both ends, in its firfi manner of longitudinal vibration, 

 there is a certain .point in the middle at which the vibra- 

 tions of each half fiop ; in the next there are two, each at 

 the diftance of a fourth part from the end j and in the fol- 

 lowing there are three, &c. The tones correfpond with the 

 natural feries of the numbers i, a, 3, 4, &c. If a rod be^ 

 faftened at one end during the firfl kind of longitudinal vi- 

 bration, the alternate expanfion and contraftion of the whole 

 rod take place in fuch a manner that they fiop at the fixed 

 end ; in the next tone there is a refiing point at the difiance 

 of one-third from the free end;, and in the following there^ 

 are two, 8cc. The tones correfpond with the numbers i, 3^ 

 5, 7, Sec, and the firft of thefe tones is an o6lave lower than 



the 



