Page images

ment of Young's hypothesis of three distinct sets of retinal nerve-terminations? The more we study the minute structure of the retinal rods and cones, the further appears to remove an understanding of the mode of operation of the sensory apparatus of the eye. May not research in this direction be guided by the hypothesis that the molecular constitution of the retinal rods and cones is such that their molecules are severally tuned to the vibrations corresponding to the colours red, green, and violet ? This would lead us to look for effects of actinism on the retina as showing the link existing between the transmitting and sensory functions of the eye. Do not the facts of the known persistence of chemical action, after it has been once initiated, and the time which would be required for the retinal molecules to recombine, or rearrange themselves, after the ætherial vibrations had ceased, comport with the known durations of the residual visual sensations, and with the main facts of physiological optics, better than the hypothesis that masses of the retinal elements are set in vibration rather than their molecules?

5. Quantitative Applications of the Laws to the fundamental facts of Musical Harmony.

To show the full value of these laws in introducing quantitative precision in the explanations of consonance and dissonance would require an extended space; we here present only such application as will serve to show their importance in giving clear and simple guides in reasonings in the physiological theory of musical harmony.

We have seen that 26 beats of the simple sound C1, of 64 vibrations per second, give a continuous sensation; therefore, to determine the nearest consonant interval of this note, we have to obtain a sound which will make with C, the vibration-ratio of 64: 64+26. This would show that the nearest consonant interval of C1, on the natural scale, is its fourth plus of a semitone. The duration of the residual sensation of Ut, is of a second; hence, to determine by our law the nearest consonant interval of Ut, we must combine with it a note which will give with Ut, the vibration-ratio of 512: 512 +130. This note is the E above C-that is, its major third. In the following Table we give the determinations of the nearest consonant intervals of the Cs throughout five octaves*:


*We have, for simplicity of illustration, determined the above intervals on the basis of the pitch of the lower note; but as the beats are produced by the conjoined action of the two sounds, it would have been more accurate to have taken, as a second approximation, the mean pitch of the two sounds. Thereby the above determinations would be somewhat changed for lower, but not perceptibly for higher notes.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][ocr errors]

C1 512
C5 1024

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]



major third + = major third +


[ocr errors]
[ocr errors]
[ocr errors]

[ocr errors]

= minor third 3
= one tone ++'s

[ocr errors]

C6" We thus see that while in the neighbourhood of C, the nearest consonant interval is over a fourth, in the octave of C the nearest consonant interval has contracted to a tone. This result seems to show why it is that the middle portion of the musical scale is best adapted for expression, and is most used in musical composition; for while in the lowest octaves the available consonant intervals are few on account of the extended spaces separating them, in the highest octaves the consonances are so contracted that their highest consonant intervals lose the sharpness of definition given when these are bounded by distinctly marked dissonances.

It is here to be remarked that in our experiments we have obtained continuous and discontinuous sensation from beats produced by one sound of a constant pitch; but with musical intervals we obtain beats from two sounds differing in pitch. In the latter case De Morgan, Guéroult, Helmholtz, and Mr. Sedley Taylor have shown that there exists a variation, or oscillation, in pitch whenever the two sounds are not of the same intensity. Mr. Taylor*, from this fact, advances the idea that these oscillations in pitch cause a noise in place of a sound, and to this result is due, in great part, the dissonance produced by beats of two different sounds. That oscillations of pitch occur when the two sounds are of unequal intensity is a fact of which there can be no doubt; but that this oscillation of pitch is the principal cause of the dissonant sensations which are perceived when beats occur, my own experiments do not verify; yet I admit that the phenomenon has its effect in increasing slightly the dissonant character of the beats. But even assuming that Mr. Taylor's explanation of the dissonance of beats is correct, yet our views hold good when we regard the intervals as formed of sounds which are equal in intensity.

In concluding this paper, I should call attention to the evident difference existing between the dynamic constitution of the sonorous waves belonging to beating pulses produced by the action of a perforated rotating disk on a continuous stream of sonorous vibrations, and those waves which cause beats and which are formed by the joint action of sonorous vibrations differing in pitch. That these two kinds of beats are alike in their effects when following in the same rapidity I have assumed to be the fact in this paper.

* "On the Variations of Pitch in Beats," by Sedley Taylor, Esq., Phil. Mag. July 1872.

Phil. Mag. S. 4. Vol. 49. No. 326. May 1875.

2 C

XLI. On Extraordinary Reflection. By ARTHUR HILL CURTIS, LL.D., Professor of Natural Philosophy in the Queen's University*.

[ocr errors]

no elementary treatise on Experimental Physics with which I am acquainted is any reference made to extraordinary Reflection; and I have known several instances where, from this omission, students have been led, though somewhat illogically, to the inference that in crystals producing double Refraction the ordinary law of Reflection is satisfied. On the other hand, in memoirs on molecular mechanics attention has frequently been drawn to the fact that the construction of Huyghens is as applicable to reflection as to refraction. Professor Haughton draws attention to this fact in his memoir on the equilibrium and motion of solid and fluid bodies (Trans. of the Royal Irish Academy, vol. xxi. part 2). Mr. Stoney has applied Huyghens's construction to the doubly reflected rays producing rings in striated calc-spar (Trans. R. I. A. vol. xxiv.); and Professor MacCullagh has, in the case of light, by the aid of his theorem of the polar plane, determined, on his theory, the intensities of the two reflected rays (Trans. R. I. A. vol. xviii. part 1). Still, in an experimental point of view, the existence of double reflection seems to be generally overlooked. If we suppose a wave-plane to traverse a crystal surrounded by any medium, whether ordinary or extraordinary, when this wave-plane reaches the extreme surface of the crystal it will give rise to two reflected rays, and to one or two refracted rays according as the surrounding medium is ordinary or extraordinary. To take the case most easily subjected to experiment, let us suppose a crystal surrounded by air, and a ray of light to fall upon it; part of this light is reflected, and part is refracted, the latter portion being in general divided into two rays, whether the crystal be uniaxial or biaxial; each of these rays will suffer double reflection at the point where it reaches the bounding surface of the crystal; and in the case where the two surfaces of contact of the crystal with the surrounding medium are parallel planes, or, more generally, when they intersect in a line parallel to the incident wave-plane, it is easy to see that the planes of polarization of the pair of reflected rays corresponding to any one of the two refracted rays produced at the first incidence, are the same as those corresponding to the other; for it is obvious that, when the above condition is fulfilled, the rays and wave-normals of one pair are parallel to those of the other; and whatever theory of light be adopted, the plane of polarization is known when the plane of the ray and wavenormal is determined. This is not only true of the two pairs of reflected rays while within the crystal, but, for the same reason,

* Communicated by the Author.

continues true of the rays into which they may be divided subsequently by reflection and refraction at a common plane surface, and consequently is true, not only of the rays which reach the eye on emergence from the crystal, with which we are at present concerned, but also of the rays reflected back into the crystal at surface.

its upper

The intensities of the four rays, however, are, as might be expected, in general different; and in fact, varying the position of the crystal, any one may be made to vanish while the other three continue to exist. The phenomena of crystalline reflection may be exhibited by the apparatus represented in the annexed figure.

[blocks in formation]

It consists of a horizontal circular stage, B, movable round, and along, a vertical axis, C, passing through its centre. The crystal D (Iceland spar, well polished, for example) is placed on B. The light falls on it through a tube, A, covered with a cap, in which a small orifice is made. E is a tube situated similarly to A but at the opposite side of the crystal, while the height of the stage B is so adjusted that the light which passes through the tube A will, after reflection at the surface of the crystal which is in contact with B, and refraction at the opposite surface, pass through the tube E. The tubes A and E may be made movable round horizontal axes perpendicular to the plane of the figure, so as to allow of the angle of incidence being altered at will. The apparatus then being adjusted, if the eye be applied to the tube E, five images of the small orifice in the cap of the tube A will be seen

-one formed by reflection at the upper surface of the crystal, which requires no consideration, and the other four by the double reflection of each of the refracted rays. As the stage is turned round its vertical axis these images may be four, three, or twofour in general, three when the azimuth of the crystal is such as to cause the intensity of one of the reflected rays to vanish, and two when the incidence and azimuth are such as to give but one

refracted wave-plane within the crystal, viz. when the plane of incidence contains the optic axis or axes, and the wave-normal corresponding to the two coincident refracted wave-planes coincides with an optic axis. If the cap of the tube A be replaced by a Nicol's prism with a small orifice, the images may be four, three, two, or one as the stage B is rotated (the last corresponding to the same arrangement as that which gives two images in the previous case) when in addition the Nicol's prism is so adjusted as to polarize the incident light in such an azimuth as to cause the intensity of one of the reflected vibrations to vanish. The planes of polarization of the several refracted rays may be determined by examining them by means of a Nicol's prism introduced into the tube E. If the tubes A and E be made of the same diameter, one Nicol's prism will be sufficient for all purposes. Queen's College, Galway,

April 1875.

XLII. On Graphical Methods of solving certain simple Electrical Problems. By Prof. G. CAREY FOSTER, F.R.S.*

[ocr errors]

[With a Plate.]

thiet te

is probably very seldom that the geometrical representation of the mathematical relations between physical quantities is as convenient, for purposes of investigation, as the corresponding analytical expression; but for purposes of exposition, as distinguished from those of investigation, geometrical constructions often possess considerable value. In particular, whenever they can be put into a simple form, their physical significance is more easily understood, than that of an algebraical formula, by those who are but little accustomed to mathematical modes of expression; and they generally exhibit with greater clearness the effect of a variation of any one of the related magnitudes. I have therefore thought that it might be worth while, for the sake of trying to draw increased attention to the utility of geometrical methods in elementary physical teaching, to point out in detail their application to a few important electrical problems, such as some of the simplest cases which come under Ohm's law of the relations between electromotive force, strength of current, and resistance.

If the equation which expresses Ohm's law be written in the three following ways—


C = R,


[blocks in formation]



* Read before the Physical Society, November 7, 1874. Communicated by the Society.

« PreviousContinue »