The rate of vibration of the rod, as a whole, is to the rate corresponding to its first division nearly as the square of 2 is to the square of 5, or as 4:25. From the first division onward the rates of vibration are approximately proportional to the squares of the series of odd numbers 3, 5, 7, 9, 11, etc. Supposing the vibrations of the rod as a whole to number 36, then the vibrations corresponding to this and to its successive divisions would be expressed approximately by the following series of number’s: 36, 225, 625, 1225, 2025, etc. In Fig. 58, _a_, _b_, _c_, _d_, _e_, are shown the modes of division corresponding to this series of numbers. You will not fail to observe that these overtones of a vibrating rod rise far more rapidly in pitch than the harmonics of a string. [Illustration: FIG. 58.] Other forms of vibration may be obtained by smartly striking the rod with the finger near its fixed end. In fact, an almost infinite variety of luminous scrolls can be thus produced, the beauty of which may be inferred from the subjoined figures (see next page) first obtained by Sir C. Wheatstone. They may be produced by illuminating the bead with sunlight, or with the light of a lamp or candle. The scrolls, moreover, may be doubled by employing two candles instead of one. Two spots of light then appear, each of which describes its own luminous line when the knitting-needle is set in vibration. In a subsequent lecture we shall become acquainted with Wheatstone’s application of his method to the study of rectangular vibrations. [Illustration: FIG. 59.] § 4. _Transverse Vibrations of a Rod free at Both Ends. The Claque-bois and Glass Harmonica_ [Illustration: FIG. 60.]