ACOUSTICS. 79 
occupied themselves with the investigations of this relation; Lagrange, 
however, was the first fully to elucidate it. The propositions established by 
him are the following: 1. The number of vibrations of a string is inversely 
as the length, that is, half the string makes twice the vibrations of the whole, 
&c. 2. The number of vibrations is proportional to the square root of the 
stretching weights, that is, four times the weight produces twice the number 
of vibrations. 3. The number of vibrations of cords of the same material 
is inversely as their thickness, that is, a string half as thick as another 
makes twice the number of vibrations in the same time. 4. The number 
of vibrations of strings of different material is inversely as the square roots 
of their densities: thus, taking a string of copper whose density is 9, and a 
string of catgut whose density is 1, their diameters and lengths being equal, 
the latter will make three vibrations in the same time that the first makes 
one. 
The Monochord, invented by Savart, and represented in pl. 19, fig. 50, is 
used for determining the laws of oscillation of stretched strings, and their 
tones. It consists principally of a hollow box, ss’. At c is a bridge with 
slits in which the strings are fixed, which then pass over the two bridges, 
f and m, and beyond m may be stretched by weights. A third bridge, h, 
may be moved along under the strings without touching them, and any 
point of the string may be pressed down upon it by means of a binding 
screw. By moving along this bridge, all the notes of an octave may be 
produced, and we shall find that the lengths for a fundamental note c = 1 are 
mm, the following proportion;,c=1,.d=>5, ¢=4 f= g=sh @=% 
b= *,c=4H, the same ratio that is found to exist in organ pipes. These 
ratios confirm, at the same time, Nos. 1 and 2 of the propositions just 
adduced ; for to obtain, for instance, the octave of the fundamental note by 
tension, it is necessary to attach four times, and for the fifth, nine times the 
weight, &c. 
e. Of Longitudinal Vibrations. 
Strings and rods have not only transverse vibrations. such as we have 
already considered, but they also vibrate longitudinally, like the air inclosed 
in a tube. This is shown by rubbing a glass tube longitudinally with a 
damp finger, or drawing a fiddle-bow across it at a very acute angle. The 
same takes place in massive rods of glass, metal, or wood, although here it 
becomes necessary to make use of a piece of rag, sprinkled with powdered 
rosin. It is, however, more convenient to make use of a so-called sounding 
rod, namely, a short glass tube whose axis is made a continuation of that of 
the body to be set into vibration. Vibrations produced in the first by 
rubbing with a damp cloth, will then be communicated to the second, and 
the two will vibrate together. “traight rods held in the middle and free 
at the extremities vibrate like open tubes; and all rods of equal length, 
whatever be their thickness, give the same tones. Nodal lines are also 
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