We have now to inquire into the relation of these successive notes to each other. The space from node to node has been called all through “a ventral segment”; hence the space between the middle of a ventral segment and a node is a semi-ventral segment. You will readily bear in mind the law that _the number of vibrations is directly proportional to the number of semi-ventral segments_ into which the column of air within the tube is divided. Thus, when the fundamental note is sounded, we have but a single semi-ventral segment, as at _a_ and _b_. The bottom here is a node, and the open end of the tube, where the air is agitated, is the middle of a ventral segment. The mode of division represented in _c_ and _d_ yields three semi-ventral segments; in _e_ and _f_ we have five. The vibrations, therefore, corresponding to this series of notes, augment in the proportion of the series of odd numbers 1:3:5. Could we obtain still higher notes, their relative rates of vibration would continue to be represented by the odd numbers 7, 9, 11, 13, etc. It is evident that this _must_ be the law of succession. For the time of vibration in _c_ or _d_ is that of a stopped tube of the length _x y_; but this length is one-third of the length of the whole tube, consequently its vibrations must be three times as rapid. The time of vibration in _e_ or _f_ is that of a stopped tube of the length _x′ y′_, and inasmuch as this length is one-fifth that of the whole tube, its vibrations must be five times as rapid. We thus obtain the succession 1, 3, 5; if we pushed matters further we should obtain the continuation of the series of odd numbers. [Illustration: FIG. 97.] And here it is once more in your power to subject my statements to an experimental test. Here are two tubes, one of which is three times the length of the other. I sound the fundamental note of the longest tube, and then the next note above the fundamental. The vibrations of these two notes are stated to be in the ratio of 1:3. This latter note, therefore, ought to be of precisely the same pitch as the fundamental note of the shorter of the two tubes. When both tubes are sounded their notes are identical. It is only necessary to place a series of such tubes of different lengths thus together to form that ancient instrument, Pan’s pipes, P P′, Fig. 97 (page 223), with which we are so well acquainted. The successive divisions, and the relation of the overtones of a rod fixed at one end (described in page 205), are plainly identical with those of a column of air in a tube stopped at one end, which we have been here considering. § 14. _Vibrations of Open Pipes: Modes of Division: Overtones_