d. P. The double vertical lines in the above diagrams show the period of recurrence in the unitates of the positive powers of a given number; this, in the function U,a", is 6 terms, in U,, 4 terms, and in U,,a", 10 terms. The Table in the margin gives the periods of recurrence for the values of & up to 10, P being the corresponding period. In the above " unitation squares," the series of unitates is continued towards the left hand to show the law of formation of U; a-", which in some cases is different from that of Usa+n. By means of these unitation squares, the unitate of any power of any number to the given base may be calculated by inspection, knowing the recurring figures and the period of recurrence; for instance, U143562-U,74=7; U,256647-U,43=2; 122496264 234567806 10 Practically there is not much difficulty in finding Us/q when it is finite or integral; for it may be ascertained from the root itself (by extraction), or from looking out Us q in the table of unitates belonging to the series Usam and ascertaining the corresponding value of Usa. It will, however, sometimes be found that there are several values of Us a that will fulfil the conditions; for instance, U,/729=U,/9; and, looking for 9 in the series 1, 8, 9, 1, 8, 9, 1, 8, 9,..., it is seen to belong to 3, 6, or 9*; a method of ascertaining the true unitate by means of a unitation square will be given further on. It will be discovered upon examination that in unitates, as in some other numerical results, there may be m mth roots, n nth roots, and so on. Considering unitates as remainders, and inspecting the series of unitates in unitation squares, it will be evident that the unitates of some surd roots have finite and integral values. These values may be determined in the same manner as the unitates of rational roots; for instance, from the series Uga2 (=1,4,9,7,7,9,4,1,9,...), Ug/7=4 or 5. Further, it will be found, from these reasons, that Us/q (m and q being whole numbers, and /q being irrational) is finite and integral when 8 is of the form nm-q; in this case n is any whole number excepting unity, taking care to choose n so large that the expression n-q may not be negative. δ Unitation squares, formed in geometrical progression with respect to exponents (in the horizontal series), may be used to obtain the unitates of certain roots, the base of the system being determined by the expression 8=n"-q, as above stated. The following are some of the series that may be used :— aï3, a3, a1, a3, a1, a2, a1, a3 ; ao1, a*, a‡, a1, a3, ao, ao, a81 ; aïs, a's, a3, a', a3, a25, a125, a625 ̧ Thus U, 2-3 or 4. Taking the leading principle of the function U1on as true when x/q (an irrational quantity)-an extension that must not be made without realizing the extent of the step involved in it (namely, that the extreme right-hand figure or figures is or are given by this class of unitates)-it would seem possible to assign a value to the last figures of some incommensurable quantities; it would appear, for instance, that U10/5 must be 5. If this be true, the real terra incognita of incommensurable quantities does not always lie at their extreme right-hand end, but in the middle region of their interminable decimal. This supposition would make good an analogy between the curves to certain equations (namely those that have a curve of finite perimeter at an infinite distance from the origin) and certain incommensurable quantities. Although the subject must be dealt with very cautiously, there does not seem any incongruity in the conception; and the range of thought thus opened to the mind is new. The points brought forward in this paper have been verified and tested by inductive reasoning, by the laws of their existence, and by examples, according to the methods more particularly set forth in the last paper-that upon negative and fractional unitates. The foregoing remarks and elucidation show that unitation is not identical with any known process. As an operation it is a direct process, dependent upon repetition for its completion. A unitate is a function which, in some of its results, gives a finite and integral value when applied to quantities that are neither finite nor integral. The further the investigation of unitation proceeds, the more unitates manifest themselves as functions that may be found throughout all the domain of quantity, and as, in some instances, interpretable when the quantity from which they are derived is but imperfectly known. 74 Brecknock Road, N., March 1875. XL. Researches in Acoustics. By ALFRED M. MAYER*.—No. VI. [Continued from vol. xlviii. p. 525.] CONTENTS. 1. The Determination of the Law connecting the Pitch of a Sound with the Duration of its Residual Sensation. 2. The Determination of the numbers of Beats, throughout the musical scale, which produce the greatest dissonances. 3. Application of these Laws (1 and 2) in a New Method of Sonorous Analysis, by means of a perforated rotating_disk. 4. Deductions from these Laws leading to new Facts in the Physiology of Audition. 5. Quantitative Applications of these Laws to the fundamental facts of Musical Harmony. "Consonanz ist eine continuirliche, Dissonanz eine intermittirende Tonempfindung."-Helmholtz. 1. The Determination of the Law connecting the Pitch of a Sound with the Duration of its Residual Sensation. WHILE the durations of the residual sensations on the eye, corresponding to lights of various colours and intensities, have been the subjects of many masterly memoirs, I know of no attempts to determine the durations of the residual sonorous sensations. Helmholtz founds indeed his Physiological Theory of Music on the facts that a certain number of beats per second produce in the ear a maximum dissonant sensation, while a greater number may blend into a smooth continuous sound; and in discussing the position in his scale of the "dampingpowers" of the covibrating parts of the organ of Corti, Helmholtz (Tonempf. p. 212 et seq.) infers, from the difficulty of trilling on the bass notes, that the covibrating parts of the ear set in motion by sounds of low pitch maintain their vibrations longer than those excited by sounds belonging to higher portions of the musical scale. He says:-"Trills of this kind, of ten notes per second, are of a sharp and clear execution in the greatest portion of the musical scale; below the la of 110 vibrations in the grand and contra octaves, however, they sound bad, harsh, and the sounds begin to blend." Yet it does not appear that Helmholtz ever attempted to determine that quantitative relation existing between the pitch of a sound and the duration of its residual sensation which I will now endeavour to establish. This law in its further applications will render quantitative many of the qualitative statements contained in Helmholtz's renowned work. The method of obtaining the facts (of which our law expresses the general relation) is similar to the method used in the study of the analogical phenomena of light. A simple sound was obtained by vibrating a fork before the mouth of its corresponding resonator; and this sound was broken up into flashes, or explosions by alternately screening and unscreening the mouth of the * Communicated by the Author. rtesonator, by means of a perforated disk which rotated between the resonator and the fork, as is shown in the accompanying figl. The mean diameter of the open sectors of the disk equalled the diameter of the mouth of the resonator, while the spaces of cardboard between the open sectors was twice the width of these openings. Thus the resonator's mouth was exposed to the vibrations during an interval which equalled that during which it was screened from them. A rubber tube led from the nipple of the resonator to one ear, while the other ear was tightly closed with a lump of bees-wax. In my first experiment I firmly clamped an Ut, resonator, and vibrated opposite its month an Ut, fork. I now placed the tube in the ear, and on slowly rotating the disk I perceived a series of sharply separated explosions or beats. On gradually increasing the velocity of the disk these explosions gradually approached each other; and on reaching a certain frequency in their succession they blended into a continuous smooth sensation, similar to that experienced when the disk was removed and the fork vibrated gently before the resonator. I now kept the disk at the velocity required just to blend the separate beats; and I found, on timing its rotations, that the resonator was sending into my ear about thirty explosions or beats per second. Hence sonorous waves of Ut, cut into thirty parts per second, or, in other words, divided into lengths of about four waves separated by the same lengths of quiescence, produce the same sensation as that caused by an uninterrupted flow of these sonorous waves into the |