^SiJ I.ECTUIIE XXXIl. 



inflecting any long chord near one of its ends, having first drawn a line 

 under its natural position, and it will then be evident that the chord returns 

 in each vibration nearly to the point of inflection, and passes at that end 

 but to a much shorter distance on the opposite side of the line, while at the 

 other end its excursions are greatest on the opposite side Of the line. The 

 result of the calculation of the frequency of vibration agrees also perfectly 

 with experiment, nor is the coincidence materially affected by the inflexi- 

 bility or elasticity of the string, by the resistance of the air, nor by the 

 slight increase of the tension occasioned by the elongation of the string 

 when it is inflected: thus, if the weight or force causing the tension of a 

 string were equal or equivalent to the weight of 200 feet of the same string, 

 that is, if the modulus of tension were 200 feet long, the velocity corres- 

 ponding to half this height would be 80 feet in a second; and every impulse 

 would be conveyed with th's velocity from one end of the string to the 

 other, so that if the string were 1 foot long, it would vibrate 40 times in a 

 second, if 6 inches, 80 times, and if it were 40 feet long, only once in a 

 second. Hence, it is obvious that the time of vibration of any chord is 

 simply proportional to the length ; and this may be shown either by means 

 of such vibrations as are slow enough to be reckoned, or by a comparison with 

 the sounds of pipes, or with other musical sounds. But if the tension of a 

 chord of given length were changed, it would require to be quadrupled in 

 order to double the frequency of vibration; and if the tension and length 

 remained unaltered, and the weight of the chord were caused to vary, it 

 would also be necessary to make the weight four times as great in order to 

 reduce the frequency of vibration to one half. 



It appears from the mode of tracing the progress of a vibration, which 

 lias already been laid down, that every chord vibrates in the same manner 

 us if it were a part of a longer chord, composed of any number of such 

 chords, continually repeated in positions alternately inverted; consequently 

 if a long chord be initially divided into any number of such equal portions, 

 its parts will continue to vibrate in the same manner as if they Avere sepa- 

 rate chords; the points of division only remaining always at rest. Such 

 subordinate sounds are called harmonics: they are often produced in violins 

 by lightly touching one of the points of division with the finger, when the 

 bow is applied, and in all such cases it may be shown, by putting small 



