298 LECTURE XXXII. 



of the tension. If the cord be at first bent into a figure of any kind, and 

 then set at liberty, the place of any part of it at every subsequent time will 

 be such, that it will always be in a right line between two points moving 

 along the figure each way with the appropriate velocity ; but in order to 

 pursue this determination, we must repeat the figure of the cord on each 

 side of the fixed points in an inverted position, changing the ends as well 

 as the sides. Hence it appears that, at the end of a single vibration, the 

 whole cord will assume a similar figure on the opposite side of its natural 

 place, but in an inverted position, and after a complete or double vibration, 

 it will return precisely to the form which it had in the beginning. The 

 truth of this result is easily shown by inflecting any long cord near one of 

 its ends, having first drawn a line under its natural position, and it will 

 then be evident that the cord 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 fre- 

 quency of vibration agrees also perfectly with experiment, nor is the 

 coincidence materially affected by the inflexibility 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 corresponding to half 

 this height would be 80 feet in a second ; and every impulse would be con- 

 veyed with this velocity from one end of the string to the other, so that if 

 the string were 1 foot long, if 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 cord 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 cord 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 cord 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 

 has already been laid down, that every cord vibrates in the same manner 

 as if it were a part of a longer cord, composed of any number of such 

 cords, continually repeated in positions alternately inverted ; consequently 

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

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

 rate cords ; 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 sma.ll 

 feathers or pieces of paper on the string, that the remaining points of 



