The order of the tones of a rod fixed at one end and vibrating longitudinally is that of the odd numbers 1, 3, 5, 7, etc. It is easy to see that this must be the case. For the time of vibration of _c_ or _d_ is that of the segment above the dotted line: and the length of this segment being only one-third that of the whole rod, its vibrations must be three times as rapid. The time of vibration in _e_ or _f_ is also that of its highest segment, and as this segment is one-fifth of the length of the whole rod, its vibrations must be five times as rapid. Thus the order of the tones must be that of the odd numbers. Before you, Fig. 84, is a musical instrument, the sounds of which are due to the longitudinal vibrations of a number of deal rods of different lengths. Passing the resined fingers over the rods in succession, a series of notes of varying pitch is obtained. An expert performer might render the tones of this instrument very pleasant to you. § 4. _Vibrations of Rods free at Both Ends_ [Illustration: FIG. 85.] Rods _with both ends free_ are also capable of vibrating longitudinally, and producing musical tones. The investigation of this subject will lead us to exceedingly important results. Clasping a long glass tube exactly at its centre, and passing a wetted cloth over one of its halves, a clear musical sound is the result. A solid glass rod of the same length would yield the same note. In this case the centre of the tube is a node, and the two halves elongate and shorten in quick alternation. M. König, of Paris, has provided us with an instrument which will illustrate this action. A rod of brass, _a b_, Fig. 85, is held at its centre by the clamp _s_, while an ivory ball, suspended by two strings from the points, _m_ and _n_, of a wooden frame, is caused to rest against the end, _b_, of the brass rod. Drawing gently a bit of resined leather over the rod near _a_, it is thrown into longitudinal vibration. The centre, _s_, is a node; but the uneasiness of the ivory ball shows you that the end, _b_, is in a state of tremor. I apply the rubber still more briskly. The ball, _b_, rattles, and now the vibration is so intense that the ball is rejected with violence whenever it comes into contact with the end of the rod. § 5. _Fracture of Glass Tube by Sonorous Vibrations_