296 LECTURE XXXII. 



substances may be either fluids or solids, or instruments composed of a com- 

 bination of fluids with solids. The resonance of a room or passage is on? of 

 the simplest sources of a musical sound ; the walls being parallel, the impulse 

 is reflected backwards and forwards continually at equal intervals of time, 

 so as to agree with the definition, and to produce the effect, of a musical 

 sound. When we blow obliquely and uniformly into a cylindrical pipe 

 closed at one end, it is probable that the impulse or condensation must 

 travel to the bottom and back before the resistance is increased ; the cur- 

 rent of our breath will then be diverted from the mouth of the pipe for an 

 equal time, which will be required for the diminution of the resistance by 

 the discharge of the condensed air, so that the whole time of a vibration 

 will be equal to the time occupied by an impulse of any kind in passing 

 through four times the length of the pipe. An open pipe may be considered 

 nearly as if it consisted of two such pipes, united at their closed ends, the 

 portions of air contained by them being agitated by contrary motions, so as 

 always to afford each other a resistance similar to that which the bottom of 

 the stopped pipe would have furnished. It is probable that when an 

 open pipe is once filled with air a little condensed, the oblique current 

 is diverted, until the effect of the discharge, beginning at the remoter end, 

 has returned to the inflated orifice, and allowed the current to re-enter the 

 pipe. Where the diameter of the pipe is different at different parts of its 

 length, the investigation of the sound becomes much more intricate ; but it 

 has been pursued by Daniel Bernoulli* with considerable success, although 

 upon suppositions not strictly consistent with the actual state of the motions 

 concerned. 



In the same manner as an open pipe is divided by an imaginary basis, 

 so as to produce the same sound with a stopped pipe of half the length, a 

 pipe of any kind is capable of being subdivided into any number of such 

 pipes, supposed to meet each other's corresponding ends only ; and, in 

 general, the more violently the pipe is inflated, the greater is the number 

 of parts into which it subdivides itself, the frequency of the vibrations being 

 always proportional to that number. Thus, an open pipe may be divided 

 not only into two, but also into four, six, eight, or more portions, producing 

 the same sounds as a pipe of one half, one third, one fourth, or any other 

 aliquot part of the length ; but a stopped pipe cannot be divided into any 

 even number of similar parts, its secondary sounds being only those of a 

 pipe of which the proportion is determined by the odd numbers, its length 

 being, for example, one third, one fi/th, or one seventh of the original 

 length. These secondary notes are sometimes called harmonics ; they are 

 not only produced in succession from the same pipe, but they are also often 

 faintly heard together, while the fundamental note of the pipe continues to 

 sound. When the pipe has a large cavity connected with it, or consists 

 principally of such a cavity, with a small opening, its vibrations are usually 

 much less frequent, and it is generally incapable of producing a regular 

 series of harmonics. 



It is obvious from this statement of the analogy between the velocity of 



* Hist, et Mem. de 1'Acad. 1762, p. 431, H. 170. See Euler, Nov. Com. Petr. 

 xvi. 281. Hauy, Traite de Physique, i. 316. Biot, do. ii. 111. 



