Prof. Tyndall on the Vibrations of Strings. 75 



length by laying four strands of the first side by side. I attach this 

 compound thread to b, and, keeping the tension the same as in the 

 last experiment, set b in vibration. The compound thread synchro- 

 nizes with b, and swings as a whole. Now by quadrupling the ori- 

 ginal thread, I obtained a string of twice the diameter of the original 

 one ; for the transverse section of any string is as the square of its 

 diameter. Hence, as the fork b vibrates with half the rapidity of a, 

 by doubling the diameter of the string, I halved its rapidity of vibra- 

 tion. In the same simple way it might be proved that by trebling 

 the diameter of the string we reduce the number of its vibrations to 

 one third. In general terms : — 



42. The rapidity of vibration is inversely proportional to the dia- 

 meter of the string. 



43. A beautiful confirmation of this result is thus obtained: — 

 Attached to this tuning-fork is a silk string 6 feet long, Two feet 

 of this string are composed of four strands of the single thread placed 

 side by side, the remaining 4 feet are a single thread. I apply a 

 tension which causes the string to divide into two ventral segments. 

 But how does it divide ? Not at its centre, as is the case when the 

 string is of uniform thickness throughout, but at the point where the 

 thick string terminates. This thick segment 2 feet long is now 

 vibrating at the same rate as the thin segment 4 feet long, a result 

 which must manifestly follow from the combination of the two laws 

 which we have already established. I need hardly say that if the 

 lengths were in any other ratio than 1:2, the node would not be 

 formed at the point of union of the two strings. 



44. We have now to establish the third law .of vibrating chords. 

 Here are two strings of the same length and thickness. One of 

 them is attached to the fork b, the other to the fork a, which vibrates 

 with twice the rapidity of b. Stretched by a weight of 20 grains 

 placed on this balanced scale-pan, the string attached to b vibrates 

 as a whole. Substituting the fork a for b, I find that a weight of 

 80 grains causes the string to vibrate as a whole. Hence to double 

 the rapidity of vibration we must quadruple the stretching weight. 

 In the same way it might be proved that to treble the rapidity of 

 vibration we should have to make the stretching weight nine-fold. 

 In general terms : — 



45. The rapidity of vibration is proportional to the square root of 

 the tension. 



46. In the foregoing experiment both the tension and the rate of 

 vibration were caused to vary. We will now cause the tension 

 alone to vary, and observe the effect upon the entire string. I carry 

 this silk chord from this tuning-fork to the scale-pan, and stretch it 

 by a weight of 80 grains. The string vibrates as a whole. By di- 

 minishing the weight I relax the string, which finally divides sharply 

 into two ventral segments. What is now the stretching weight ? 

 20 grains. With a stretching weight of almost exactly 9 grains it 

 divides into three segments, while with a stretching weight of 5 grains 

 it divides into four segments. Thus then a tension of one-fourth 

 doubles, a tension of one-ninth trebles, a tension of one-sixteenth 

 quadruples the number of ventral segments. In general terms, the 



