XXVII. On the Transformation of the Vibroscope into a Tonometer, and its employment for the Determination of the Absolute "Number of Vibrations. By A. TERQUEM*.

Y the optical study of vibrations and the construction of the vibroscope, M. Lissajous has endowed acoustics with means of investigation much more accurate than those based upon hearing only; nevertheless this optical method does not appear hitherto to have been conveniently applied to the determination of the absolute number of vibrations. I have thought that it could be utilized for this purpose, and that a tonometer might be more easily constructed with its aid than by the operation devised by Scheibler. The new tonometer, at least as accurate, would be much less costly; and it could be employed to give, by simple reading, the number of vibrations of any sonorous body whatever, and that through a very great extent of the musical scale.

I have had made by M. König four diapasons furnished with cursors, and each carrying at the extremity of one of its branches, like the diapason of the vibroscope, a small biconvex lens to serve as an objective. These diapasons can be successively fixed upon the same support, on which an ocular is placed. By shifting the cursors, we can obtain all the sounds comprised between UT2 (128 double vibrations) and UT,; moreover some sounds are common to two diapasons which immediately follow one another. The diapasons were divided by M. König after his tonometer, in such sort that, by shifting the cursors the space which separates two strokes, the sound is changed two double vibrations; but I consider this only a completely arbitrary division which might be replaced by any other made by means of a dividing machine. I shall give to these diapasons the name of standards.

I have, besides, other diapasons, not graduated, but likewise furnished with cursors and extending from UT, to UT. Two are sufficient, because spare cursors can be adapted to them. I shall give to these the name of auxiliary diapasons.

The first standard diapason and the first auxiliary are fixed the one in front of the other, on suitable supports, in one and the same horizontal plane, and at a right angle; moreover the vibrations of the standard diapason (which serves as a vibroscope) being made as usual, in a vertical plane, those of the auxiliary take place in a horizontal plane.

That the curves resulting from the coexistence of the two vibratory movements at a right angle might be readily perceived, particles of finely powered antimony have been fixed with gum

* Translated from a separate impression, communicated by the Author, from the Comptes Rendus de l'Académie des Sciences, January 12, 1874.

upon the terminal section of one of the branches of the auxiliary diapason. The facets of this crystalline powder, obliquely illuminated by means of a lamp and lenses, form luminous points of great brilliancy and extreme delicacy. The cursor of the standard having been fixed upon the first stroke of the division, the auxiliary diapason is put into exact unison with the former by shifting the cursors, finally putting on them small pieces of The operator is guided in this operation evidently by the transformations of the elliptic curve which is due to the coexistence of the vibrations; and he stops when it diminishes gradually and very slowly without changing its form.

wax.

This done, he moves the cursor of the standard diapason so as to obtain about one beat per second, and determines the exact duration of at least fifty beats by aid of a pointing telltale, observing the oscillations of the elliptic curve. If the conditions be good, the total duration of fifty beats scarcely varies a half second in several successive determinations, which gives an approximation within Too of a second for one beat, and consequently permits the number of vibrations to be determined within less than 0.01. The eursor of the auxiliary diapason is then shifted till unison is reestablished, which is ascertained by the fixity of the curve produced. The cursor of the standard diapason is then placed on the second stroke of the division, and the duration of the beats produced determined again.

The operation is continued in the same manner by shifting successively the cursors of the two diapasons until a sound is obtained sufficiently distant from UT, such as MI2, capable of giving with the sound UT, a sufficiently simple acoustic curve (4:5). When, by successive stops, the auxiliary diapason is put in unison with the sound MI2, the cursor of the first standard diapason must be carried back to the first stroke, and it must be ascertained if the sound MI, is perfectly true; if there is any difference, it is determined by the duration of the oscillations of the acoustic curve. The sum total of the numbers of beats per second (the inverse of their duration) which have been observed on the shifting of the cursors in succession, gives the difference of the numbers of vibrations of the sounds UT, and M12, of which we have the ratio which agrees with the principle given by Scheibler for the construction of the tonometer, and permits the absolute number of the vibrations of UT2 to be calculated. Continuing in the same manner, we arrive at UT. On the way numerous datum- and verification-points are met with in the simple intervals, such as the fourth, the fifth, the sixth, &c. At length we know in this manner the absolute number of vibrations corresponding to each stroke of the division of the standard diapason.

The only practical difficulty which I have encountered (and this I think I am on the point of surmounting) is in the mode of fixing the cursors on the diapasons. In fact they should be fixed there in a quite invariable manner, so that the sound may not suffer any variation of pitch when the diapason is set in vibration; this does not always happen, though the differences observed are very slight. In the second place, the cursors must be restored to the same position with mathematical accuracy.

I have already verified a certain number of the divisions traced by M. König on my diapasons from his tonometer, and have found only differences amounting to not more than a few hundredths of a vibration per second, which shows the degree of confidence that may be placed in the determinations made from the tonometer constructed by him.

By this method, then, it will be possible, with more facility than by the old process, though not without sustained attention and numerous determinations, to divide the interval from UT to UT, into sounds differing one from another by two vibrations, or even by one only, if it be desired. At the least, by the employment of auxiliary diapasons with cursors, the same procedure will permit the verification of the accuracy of the divisions of the standard diapasons mounted in the form of a vibroscope, and to construct a table of corrections if the divisions are not perfectly exact.

The vibroscope tonometer once constructed and verified, in order to determine the pitch of any sound whatever, it will be sufficient to place one of the diapasons of which it consists by the side of the vibrating body in such a way that the vibrations of the one shall be perpendicular to those of the other. On this body (which may be a rod, a plate, a string, &c.) some particles of antimony powder are to be fixed; then the cursors will be shifted until an acoustic curve of well-recognizable form is obtained, such as that due to vibrations in the ratio of 1:2, 1:3, 1:4, 1:5,.... If the sound studied is higher or deeper than those comprised in the interval from UT, to UT, it is not even necessary to obtain the absolute fixity of the curve; in fact, by determining the duration of the period of return of the same figure, we can know the difference between the number of vibrations of the sound and that which it ought to have in order that that fixity may exist.

With this tonometer I purpose to resume, with more accuracy than has hitherto been attainable, the study of the vibrations of rigid bodies, particularly plates, in order to arrive, if possible, at elucidating the question, still so much controverted, of the propagation of sound in bodies which present at least two dimensions of the same order of magnitude.

XXVIII. Notices respecting New Books.

An Introduction to the Elements of Euclid, being a familiar explanation of the first twelve propositions of the first book. By the Rev. STEPHEN HAWTREY, A.M., late Assistant Master at Eton. London: Longmans, Green, and Co. 1874. (Pp. 105.)

ONE'S first impression on looking at this book might be to ask,

Is it possible that the first twelve propositions of Euclid's first book require 105 pages of explanation? But if the querist has ever taught the elements of geometry to a child and remembers in what his verbal instruction consisted, he will probably find on reading the book that his explanations were at least as voluminous as those here printed; and doubtless he will be both interested and instructed in this record of the method of oral teaching which was used for many years by Mr. Hawtrey "with some success."

We will notice very briefly the leading points of the book. In the first place the author insists on the learner being provided with "a pair of compasses having a pen- and pencil-leg, a small flat ruler, and a hard pencil;" besides these, he adds, it is well to have "a little box-wood triangle, called. . . . 'a set square.' He also insists on having the data of each problem drawn in ink before the construction is completed in pencil. In the next place he treats the definitions very fully, those at least which the learner needs to begin upon; the postulates and axioms are noticed, with a sufficiency of comment, the twelfth axiom being left out as not needed at present. The propositions are then taken in order, directions are given for drawing the figure line by line, and each step of the demonstration commented on. At the end of each exposition the proposition is given as it stands in Euclid-general enunciation, particular enunciation, construction, demonstration; and the learner is directed to write it out.

The fourth proposition is treated at great length, and its importance in regard to what follows insisted on more forcibly (not than it deserves to be, but) than we have ever seen it insisted on before. "It is a great and most important proposition, and has been called the key to Euclid." The exposition of the proposition is followed by seven exercises, all fully worked out; and as they cannot be done without assuming something more than the first four propositions, the assumptions are carefully explained; as a deduction from two of the exercises the fifth proposition is proved in fact though not in name. After this the learner is ready to encounter the fifth proposition in its proper form, which is, as Mr. Hawtrey says, nothing but an exercise on the fourth proposition, though too hard to be taken immediately after it.

We fear that this account of the contents is somewhat meagre ; but there is, in fact, not much to say about the book in the way of description. Its merit consists in the admirable clearness with which all the points of a well-trodden road are brought under notice and much insisted on, which, in these days of quick travelling, is

apt to be passed over. The book is interesting in another way; it is quaint without affectation, and produces a very distinct impression of being written by a man of a genial and original character. Here and there little pieces of classical lore are felicitously introduced, and the more appropriately as Eton was the scene of Mr. Hawtrey's labours. The dedication runs thus :-" Morais suis ayarηrois hoc opusculum dicat S. H." The "initiated" are those pupils who have mastered the fourth proposition--who can not only write it out, but understand and are able to apply it -who having got hold of the key, can go on to investigate the hidden treasures of Euclid. We cordially recommend the book to all who are engaged in teaching the elements of geometry to the young; they cannot fail to be interested, and will probably learn from it something as to the best way of performing a difficult and important task.

XXIX. Proceedings of Learned Societies.

ROYAL SOCIETY.

[Continued from p. 74.]

May 8, 1873.-Francis Sibson, M.D., Vice-President, in the Chair. THE following communication was read:

"The Action of Light on the Electrical Resistance of Selenium." By Lieut. Sale, R.E.

It having been recently brought to notice that selenium in the crystalline condition exhibits the remarkable property of having a conductivity varying with the degree of light to which it is exposed, the following experiments were undertaken with a view to the further elucidation of the matter:

Experiment 1.-A bar of crystalline selenium measuring approximately 1.5′′ × 5′′ x '05" was procured, and platinum wire terminals were fastened to the ends.

The bar itself was then enclosed in a box having a draw-lid, so as to admit or exclude the light at pleasure.

Then, the lid of the box being on, the resistance of the selenium was measured by means of a high-resistance galvanometer and a Wheatstone's bridge, with dial-coils capable of measuring up to 10,000,000 ohms. The battery-power was 2 cells Daniell.

The measurement was made on a dull cloudy day, and in a room of equable temperature.

The resistance having been carefully balanced, the lid of the box was withdrawn, when the resistance of the selenium fell instantaneously and considerably, as indicated by the rapid movement of the spot of light on the galvanometer-scale.

Experiment 2.-The transition from darkness to the light given by an ordinary gas-burner (conditions as before), caused a slight and barely perceptible fall in the resistance.

Experiment 3.-The bar of selenium was next tried in the solar