From a rod or bar fixed at one end, we will now pass to rods or bars free at both ends; for such an arrangement has also been employed in music. By a method afterward to be described, Chladni, the father of modern acoustics, determined experimentally the modes of vibration possible to such bars. The simplest mode of division in this case occurs when the rod is divided by two nodes into three vibrating parts. This division is easily illustrated by a flexible box ruler, six feet long. Holding it at about twelve inches from its two ends between the forefinger and thumb of each hand, and shaking it, or causing its centre to be struck, it vibrates, the middle segment forming a shadowy spindle, and the two ends forming fans. The shadow of the ruler on the screen renders the mode of vibration very evident. In this case the distance of each node from the end of the ruler is about one-fourth of the distance between the two nodes. In its second mode of vibration the rod or ruler is divided into four vibrating parts by three nodes. In Fig. 60, 1 and 2, these respective modes of division are shown. Looking at the edge of the ruler 1, the dotted lines cutting _a a′_, _b b′_, show the manner in which the segments bend up and down when the first division occurs, while _c c′_, _d d′_, show the mode of vibration corresponding to the second division. The deepest tone of a rod free at both ends is higher than the deepest tone of a rod fixed at one end in the proportion of 4:25. Beginning with the first two nodes, the rates of vibration of the free bar rise in the following proportion: Number of nodes 2, 3, 4, 5, 6, 7 Numbers to the squares of which the} 3, 5, 7, 9, 11, 13 pitch is approximately proportional} Here, also, we have a similarly rapid rise of pitch to that noticed in the last two cases. [Illustration: FIG. 61.] For musical purposes the first division only of a free rod has been employed. When bars of wood of different lengths, widths, and depths, are strung along a cord which passes through the nodes, we have the _claque-bois_ of the French, an instrument now before you, A B, Fig. 61. Supporting the cord at one end by a hook _k_ and holding it at the other in the left hand, I run the hammer _h_ along the series of bars, and produce an agreeable succession of musical tones. Instead of using the cord, the bars may rest at their nodes on cylinders of twisted straw; hence the name “straw-fiddle,” sometimes applied to this instrument. Chladni informs us that it is introduced as a play of bells (Glockenspiel) into Mozart’s opera of “Die Zauberflöte.” If, instead of bars of wood, we employ strips of glass, we have the glass harmonica. § 5. _Vibrations of a Tuning-fork_