The south-western and southern margin of this great northern basin of the Old Red Sandstone can still be traced nearly continuously from the confines of Caithness to the borders of Aberdeenshire, its position being marked by a zone of littoral conglomerates. Beyond the edge of that zone, however, there occur some interesting outliers, which in some cases may represent long fjord-like indentations of the coast-line; in others may mark what were really independent basins lying at the base of the Grampian mountains. The author points out that probably most of the difficulty which has hitherto been experienced in understanding the sequence of beds along the southern shores of the Moray Firth, and their parallelism with those of Caithness and Orkney, is not to be attributed to the amount of detritus covering the country, but rather to the fact which has not heretofore been observed, that the Upper Old Red Sandstone, with Holoptychius and Pterichthys major, really overlaps unconformably upon the older nodular clays and conglomerates with Coccosteus, Cheirolepis, &c. This relation can be satisfactorily determined in Morayshire, and is now being worked out by Mr John Horne in the course of the Geological Survey. The author traces in great detail, from the Spey into Sutherlandshire, the development of the lower sandstones, conglomerates, and clays, which have been regarded as equivalents of the Caithness flagstones. He thinks that in no sense can this comparatively thin group of rocks (seldom 1400 feet in depth) be regarded as a mere southward attenuation of the great Caithness series, as suggested by Murchison, for that neither lithologically nor paleontologically can that view be sustained. He has been led to the conclusion that the whole of these rocks from the borders of Sutherlandshire to those of Aberdeenshire represent only the higher portions of the great Caithness series, and that they were formed during a gradual depression of the ancient high grounds whereby the waters of Lake Orcadie were allowed to creep southward over the descending land. This movement is indicated by the character of the strata, and that it took place about the time of deposit of the later flagstones of Caithness is shown by the cccurrence of the fossils of that division in the nodules, flags, and clays of the Moray Firth region, while those of the Lower division are absent. Allusion is likewise made to the discovery of two localities where contemporaneous volcanic action has recently been observed in the Moray Firth area, the whole of the basin of Lake Orcadie being otherwise remarkably free from any trace of such action except on the northern margin in Shetland. The history of the area embraced by Lake Caledonia will form the subject of the next paper. 3. On Beats of Imperfect Harmonies. By Sir William Thomson. 30 According to a usage which has been adopted from the German of Helmholtz by the best English scientific writers on sound, a sound is called a "simple tone,"* or without qualification a "tone," when the variation of pressure of the air in the neighbourhood of the ear which is the immediate excitant of the sense is according to a simple harmonic function of the time; that is to say, when the whole pressure of the air varies in simple proportion to the distance, from a fixed plane, of a point moving uniformly in a circle. Considering the actual sensibility of the human ear to musical sounds, we must introduce farther as a practical restriction that the period of the variation of the pressure must be less then of a second, and greater than 1000 or 20000 of a second. The vibrations of the air produced by a simple harmonic vibrator are either simple harmonic, or are in circular or elliptic orbits, resulting from the composition of two simple harmonic motions; and the consequent change of air-pressure in the neighbourhood of the ear follows the simple harmonic law, provided the maximum velocity of the vibrator and of the air in its neighbourhood be infinitely small in comparison with the velocity of sound. Hence the more nearly this condition is fulfilled the more nearly a simple tone is the sound heard; but it is far from being fulfilled when the vibrator, though itself performing simple harmonic motion, has sharp edges round which the * The old musical usage, according to which the word tone denotes an interval (the major tone or minor tone, or the mean tone of the tempe red scale), though it unfortunately clashes with this recent scientific use of the word tone, can scarcely be abandoned. air is forced to rush with great velocity, or when, as in the case of free-reed organ pipes or the reeds of a harmonium, the vibrator is an elastic solid moving to and fro in a very narrow aperture. (In the case of a slapping reed, as of trumpet stops in an organ, the motion of the vibrator itself is not simple harmonic, and the sound is excessively rich in overtones, giving it its peculiarly rich or harsh character.) A harmony is any sound of which the excitant change of airpressure is strictly periodic, and is not a simple tone. simple tone. According to Fourier's beautiful analysis* of periodic variations, to which the name of the harmonic analysis has been given, any periodically varying quantity may be regarded as the sum of quantities varying separately according to the simple harmonic law, in periods respectively equal to the main period, half the main period, a third of the main period, and so on. According to this analysis we see that the variation of air-pressure constituting a harmony may be regarded as the sum of variations constituting simple tones, one having its period equal to the period of the harmony; a second, half that of the harmony; a third, one-third that of the harmony, and so on; in other words, we may regard the harmony as compounded of these simple tones. Practically, in musical language the term harmony is not applied. when the tone of the main period predominates in the sensory impression, and in this case the sound is simply called a note; its pitch is reckoned according to the main period; and the effect of the other tones, now called overtones, which enter into its composition, are merely felt as giving it its character or quality of sound. Thus the name harmony is in musical practice restricted to cases in which there is either no tone of the main or fundamental period, or not enough to produce a predominating impression, and a sound compounded of two, three, four, or more simple tones, having commensurable periods, is heard. In ordinary musical language a harmony is not regarded as having any one pitch, but is thought of as compounded of its known constituents. The true period of Compare "Trans. R.S. E." April 30th, 1860, "Reduction of Observations of Underground Temperature," where a short description of Fourier's analysis is to be found. the harmony is, however, in every case the least common multiple of the period of its constituent tones. The number of times that the period of the harmony contains the period of any one of its constituent tones is called the harmonic number of that tone. This expression is only applicable to any particular tone when viewed as one constituent of a harmony. Following the usage of Lord Rayleigh and Professor Everett, I shall employ the word "frequency" to denote the number of periods per unit of time,-per second, let us say, generally in acoustical reckonings. Thus the "frequency" of a tone or of a harmony means the number of its periods per second. Similarly the frequency of any set of beats, according to the definitions and descriptions below, will mean the number of the beats per second, and in this application of the term it will designate sometimes a proper fraction, and sometimes a small whole number with fraction. The quality of a harmony, when the periods of its several constituent tones are given, depends upon the amplitudes of the different constituents, and on the relation of their phases. Thus, for example, consider a harmony of two tones. They may be so related in phase that at one of the instants of maximum pressure of one of the constituents there is also maximum pressure of the other constituent. The same phase-relation, if the harmonic numbers of the constituent tones be both odd, will give also coincident minimums. But when one of the harmonic numbers is even and the other odd the phase-relation of coincident maximums will also be such that there is a coincidence of minimum pressure due to one tone with maximum pressure due to the other; and again there will be an opposite phase in which there will be coincidence of minimums, and in this opposite phase there will also be a coincidence of maximum and minimum. (To avoid circumlocutions a harmony of two odd numbers will be called an odd binary harmony; a harmony of even and odd numbers will be called an even binary harmony.) Thus we see that in an odd binary harmony there is a phase-relation of coincident maximums and coincident minimums, and again an opposite phase-relation of coincident maximum minimum and minimum maximum. The former will be called the phase-relation of coincidences, the latter the phase-relation of oppositions. In an even binary harmony there is a phase-relation of coincident maxi |