pressure; in the opposite phase-relation there is a slow change before and after the time of maximum pressure, and a rapid change before and after the time of minimum pressure. In the former case the difference of maximum from mean exceeds the difference of minimum from mean; in the latter the difference of the minimum from mean exceeds the difference of maximum from mean. If the ear could not distinguish between two such sounds, but could distinguish between either of them and the sounds represented by the first and third quarter-period curves, the number of beats would be twice the error of frequency of the higher note. But I find that the number of beats is quite distinctly equal to the error of frequency of the higher note. I have found that the beats on the 1:2 harmony are most easily perceived when the intensity of the higher note is ccmparatively faint, as would be the case if they were explained by Helmholtz's theory that they are the beats on the approximation to unison which there is between the higher note and the first overtone of the lower note. But the simple-harmonic character of the two constituent tones at the entrance of the ear precludes the acceptance of this theory unless extended, as it has actually been by its author, to the interior mechanism of the ear. Whatever may be the physiological theory by which the beats are to be explained, it is an interesting fact that the ear does distinguish, as it were, between push and pull on the tympanum in the manner illustrated by the preceding curves, not only for the case of approximation to the harmony 1:2, but for every other even binary harmony. I have heard distinctly the beats on approximations to each of the harmonies 2:3, 3:4, 4:5, 5:6, 6:7, 7:8, 1:3, and 3:5. The two last mentioned, though sometimes less easily heard than the beats on most of the others, are unmistakably distinct; and by counting the numbers of them in ten seconds or in twenty seconds, I have ascertained that they, as do all the others, fulfil the condition of having the whole period of the imperfection, and not any sub-multiple of it for their period of audible beat. They are interesting as being cases of odd binary harmonies. Before making the experiments, I thought it possible that what is heard in the beat might not make distinction between the configuration II. and IV. (first quarter phase and third quarter phase): but a revolving character which I perceive in the beat seems to me distinct enough to prove that the ear does distinguish between these configurations, which are one of them the same as the other taken in the reverse order of time. In every instance except the octave, the beat on the approximation to a binary harmony is less distinct than the beat on an approximation to a ternary or higher multiple harmony with only one note false. It is not because of the comparative slowness of the beat on the multiple harmony; for by taking alternately beats with one note slightly false in a binary harmony, and the same note made more false in a ternary or multiple harmony to such a degree as to give the same number of beats, I have always found the beats in the latter case much more prominent than in the former. Thus by taking first the perfect harmony C E G (4, 5, 6), and the three binary harmonies C G (2:3), C E (4:5), E G (5: 6), and flattening slightly any one of the three notes by screwing on a small mass of brass to either or to each prong of the tuning-fork producing it, it is easy after a little practice to count the beats on each of the binary harmonies. Thus, for example (supposing E, to designate a note of a slightly lower pitch than E), after a little practice it is easy to count the beats on CE, and on the E, G, and to verify that their frequencies are, the first of them four times, and the second of them six times the error of frequency of the E,, and then to verify that the much louder. beats on the ternary harmony C E, G, are of half the frequency of the former, and of one-third of the frequency of the latter, and to verify absolutely that they are of twice the frequency of the error of E. If when the approximate harmony CE, is being sounded, the faintest sound of G is produced by a very gentle excitation of the fork by the bow, instantly a loud beat at half speed is heard. The phenomenon is rendered very striking by alternately touching the top of the G fork by the bow so as to stop its vibrations, and then drawing the bow very gently for a fraction of a second* along one side to re-excite them. It is marvellous how small an intensity of the sound G is required to give a smooth unbroken loud beat in the double period. I have found it difficult to excite In every case, to obtain regular beats, each tuning-fork, after being set in vibration by the bow, must be left to itself. The sound is sensibly graver as long as the bow is applied to augment or sustain the vibration than when the fork is left free. Thus, if two tuning-forks nearly, but not quite, in unison, are alternately acted on by the bow and left free, the beats are less rapid during the time the bow is applied to the higher fork, and more rapid while to the lower, than when both forks are vibrating freely. the G gently enough to give the gradual transition from, let us say for example, four uniform beats per second, through the case of four beats per second with every alternate beat somewhat louder, to the case of only every second beat perceptible, or, in all, two beats per second; but it can be done, and the result is an interesting and instructive illustration of the slowing down from the quick beat of the binary harmony to half speed, or to one-third speed, or to onefifth speed, as the case may be, by the introduction of a third note. In the several cases I have foundthat I can, by making the added note faint enough, produce a succession of beats of which every second, or every third, or every fifth, as the case may be, is louder than the others, and that, as the intensity of the added note is gradually increased, the fainter beats become imperceptible, and a regular unbroken slow beat is heard distinctly alone, always in the theoretical time of the whole imperfection of the harmony. I have verified this distinctly in the cases of 1, 2, 3; 2, 3, 4; 3, 4, 5; 4, 5, 6 (as stated above); 5, 6, 7; and 6, 7, 8. I have not succeeded in hearing the beats on the approximations to the harmonies 8:9 and 9:10. But the slow beat on the 8, 9, 10 (with vibrational frequencies 256, 288, 320), with any one of the three notes slightly flattened, is very remarkable. The sound is like that of a wheel going round with decided roughness of motion in every part of its revolution, but much rougher in one part than another, with a loudly perceptible periodic return of the roughness in the theoretical period of the approximate harmony. The beats on the harmony C E G (vibrational frequencies 256, 320, 384), with any one of the three notes slightly flattened, are very perceptible: untrained ears hear them instantly the first time without any education, and the beat is heard almost to the very end of the sound if three of Koenig's forks, one of them, the C, for example, being slightly flattened by a brass sliding piece screwed to it, be caused to sound. The sound dies beating, the beats being distinctly heard all through a large room as long as the faintest breath of the sound is perceptible. The smooth melodious periodic moaning of the beat is particularly beautiful when the beat is slow (at the rate, for instance, of one beat in two seconds or thereabouts), being, in fact, sometimes the very last sound heard when the intensities of the three notes chance at the end to be suitably proportioned. Monday, 15th April 1878. SIR WILLIAM THOMSON, President, in the Chair. The following Communications were read :— 1. On Vortex Vibrations, and on Instability of Vortex Motions. By Sir William Thomson. 2. On the Theory of Vowel Sounds. By Professor 3. Preliminary Note on a Method of Detecting Fire-Damp in Coal Mines. By Professor George Forbes. The author exhibited two instruments, both founded upon the same principles, for measuring the quantity of fire-damp present in a coal mine. The first instrument consists of a tuning-fork fixed above the open end of a resonating tube, whose other end is closed by a piston whose position (read off on a scale) regulates the length of the resonating tube. The length of the tube, which resounds to the definite pitch of the tuning-fork, depends upon the nature of the gas with which it is filled. The more fire-damp, the longer is the tube. Barometric pressure has no effect upon this instrument. The correction for temperature is made by reading off, not a fixed mark upon the piston, but the top of the mercury of a thermo VOL. IX. 4 M meter attached thereto, of dimensions determined by actual experiment. The only source of error to which the instrument seems liable is the counteracting influence of dense carbonic acid gas in choke-damp. But it is found that the presence of choke-damp destroys the explosive character of fire-damp; and, so far as experiments go, it seems certain that, in all cases when the presence of choke-damp prevents the instrument from indicating the presence of fire-damp, the fire-damp is denuded of its explosive character. The second instrument is a combination of a harmonium reed and an organ pipe, through which the air is driven. They are arranged so as to sound the same note when pure air is used, so that when there is a lighter gas present the organ pipe sounds a higher note, thus producing beats. So far as the experiments have gone hitherto, the first form is by far the most accurate, being capable of detecting the presence of 1 or 2 per cent. of fire-damp. 4. Note on Electrolytic Conduction. By Professor Tait. It is commonly said that there is a resistance to a current at the surface of contact of a solid conductor and an electrolyte. Some good authorities, however, say that we have as yet no proof of this, as the effects observed may be due to polarisation. It is obvious that, if the reverse electromotive force due to polarisation contain a term directly proportional to the strength of the current, the ordinary methods of measurement would not enable us to distinguish this from the surface resistance above mentioned. For, in the expression if the numerator contain a term of the form el, it may be expunged, provided e be added to the denominator. To clear up this point I have recently made a number of experiments. These have led me to some curious results bearing on the theory of electrolysis, which I propose to bring before the Society on a future occasion. At present I refer to them merely so far as to say that they establish fully the existence of the surface resistance above mentioned. Thus I was led to see that if a slip of platinum |