When we calculate, by the same formulas, the intensity of the light in the centre of the projection of a small circular aperture, made in a large screen, we find that this centre will exhibit alternately a bright and a dark appearance, according to the distance at which the shadow is viewed; and that in homogeneous light this darkness must be perfect. This new inference from the general formulas may be deduced from the theory by very simple geometrical considerations. Thus we find that the values of the successive distances, at which the centre of the shadow becomes completly dark, areb=ar22ad−r2,b=ar24ad−r2,b=ar28ad−r2;and so forth;rbeing the semidiameter of the aperture,aandbits respective distances from the luminous point and from the micrometer, anddthe length of the undulation of the light employed. Now, if we place the micrometer at the distances indicated by these formulas, we observe, in fact, that the centre of the projection of the opening is so completely deprived of light, that it appears like a spot of ink in the middle of the illuminated part, at least with respect to the minimums of the first three orders, as indicated by the formulas here inserted: those of the subsequent orders, which are nearer to the screen, exhibiting no longer the same degree of darkness, on account of the want of homogeneity of the light employed.
When we calculate, by the same formulas, the intensity of the light in the centre of the projection of a small circular aperture, made in a large screen, we find that this centre will exhibit alternately a bright and a dark appearance, according to the distance at which the shadow is viewed; and that in homogeneous light this darkness must be perfect. This new inference from the general formulas may be deduced from the theory by very simple geometrical considerations. Thus we find that the values of the successive distances, at which the centre of the shadow becomes completly dark, areb=ar22ad−r2,b=ar24ad−r2,b=ar28ad−r2;and so forth;rbeing the semidiameter of the aperture,aandbits respective distances from the luminous point and from the micrometer, anddthe length of the undulation of the light employed. Now, if we place the micrometer at the distances indicated by these formulas, we observe, in fact, that the centre of the projection of the opening is so completely deprived of light, that it appears like a spot of ink in the middle of the illuminated part, at least with respect to the minimums of the first three orders, as indicated by the formulas here inserted: those of the subsequent orders, which are nearer to the screen, exhibiting no longer the same degree of darkness, on account of the want of homogeneity of the light employed.
There is still a multitude of other phenomena of diffraction, such as those of multiplied and coloured images, reflected by striated surfaces, as seen through a texture of fine fibres, as well as the coloured rings, produced by an irregular collection of such fibres, or of light powders, consisting of particles nearly equal, placed between the eye of the spectator and a luminous object; all of which may be explained and rigorously computed by means of the theory which has been laid down. It would, however, occupy too much of our time to describe them here, and to[p444]show how exactly they concur in confirming the theory; which indeed appears to be abundantly demonstrated by the numerous and diversified facts which have been already adduced in support of it. It will be sufficient to conclude this extract of the Memoir on Diffraction with a detailed description of an important experiment of Mr.ARAGO, which furnishes us with a method of determining the slightest differences of the refractive powers of bodies, with a degree of accuracy almost unlimited.
We have seen that the fringes, produced by two very narrow slits, are always placed symmetrically with regard to a plane passing through the luminous point and the middle of the interval between the slits, as long as the two pencils of light which interfere have passed through the same medium, for instance, the air, as happens in the ordinary arrangement of the apparatus. But the result is different when one of the pencils continues to pass through the air, and the other has to be transmitted by a more refractive body, a thin plate of mica, for example, or a piece of glass blown very thin: the fringes are then displaced, and carried towards the side on which the transparent substance is placed: and if its thickness becomes at all considerable, they are removed out of the enlightened space, and disappear altogether. This important experiment, which was first made by Mr. Arago, may also be performed with the apparatus of the two mirrors, if the plate be placed in the way of one of the pencils, either before or after its reflection.
Let us now see what inference may be drawn from this remarkable fact, by the assistance of the principle of interferences. The light stripe in the middle is always derived, as we have already seen, from the simultaneous arrival of rays which have issued at the same moment from the luminous point; consequently, in the common circumstances of the experiment, they must have described paths exactly equal, in order to arrive in the same time at the place of meeting: but it is obvious that if they pass through mediums in which light is not propagated with the same velocity, that pencil, which has travelled the more slowly, will arrive at the given point later than the other, and the point will[p445]therefore no longer be in the bright stripe. The stripe must therefore necessarily change its place towards the pencil which travels the more slowly, in order that the shortness of its path may compensate for the delay during its transmission through the solid: and the converse of the proposition enables us to conclude, that where the stripes are displaced, the pencil towards which they move has been retarded in its passage. The natural inference, therefore, “from Mr.ARAGO’s experiment,” is, that light is propagated more rapidly in the air than in mica or glass, and generally in all bodies more refractive than the air; a result directly opposite to the Newtonian theory of refraction, which, supposes the particles of light to be strongly attracted by dense substances, which would cause the velocity of light to be greater in these bodies than in rarer mediums.
This experiment furnishes a method of comparing the velocity of the propagation of light in different mediums, [or, in other words, the refractive density, which is always supposed in this theory, to be reciprocally proportional to it.] If, in fact, we measure very accurately, by means of a spherometer, the thickness of the thin plate of glass which has been placed in the way of one of the luminous pencils, and if the displacement of the fringes has been measured by the micrometer; since we know that, before the interposition of the glass, the paths described were equal for the middle of the central stripe, we may calculate how much difference is occasioned by the change of position, and this difference will give the retardation in the plate of glass, of which the thickness is known: so that, by adding this thickness to the difference calculated, we shall find the little path which the other pencil has described in the air, while the former was transmitted by the plate of glass; and this path, compared with the thickness of the plate of glass, will give the proportion of the velocity of the light in the air, to its velocity within the glass.
We may also consider this problem in another point of view, with which it is convenient to make ourselves familiar. The duration of each undulation, as we have seen, does not depend on the greater or less velocity with which the[p446]agitation is propagated along the fluid, but merely on the duration of the previous oscillation which gave it birth; consequently, when the luminous waves pass from one medium into another, in which they are propagated more slowly, each undulation is performed in the same interval of time as before, and the greater density of the medium has no other effect than that of diminishing the length of the undulation, in the same proportion as the velocity of light is diminished: for the length of the undulation is equal to the space that the first agitation describes during the time of a complete oscillation. We may therefore calculate the relative velocities of light in different mediums, by comparing the length of the undulations of the same kind of light in those mediums. Now, the middle of the central stripe is formed by the reunion of such rays of the two pencils as have performed the same number of undulations, in their way from the luminous point, whatever may be the nature of the mediums transmitting the light. If then the central stripe is brought towards the side of the pencil which has passed through the glass, it is because the undulations of light are shorter within the glass than in the air; and it is necessary, in consequence, that the path described on this side should be shorter than the other, in order that the number of undulations may remain the same. Let us suppose, then, that the central stripe has been displaced to the extent of twenty breadths of fringes, for example, or of twenty times the interval between the middle points of two consecutive dark stripes; we must necessarily conclude that the interposition of the plate of glass has retarded the progress of the pencil passing through it to the extent of twenty undulations; or that it has performed within the plate twenty undulations more than the same pencil would have performed in an equal thickness of air, since each breadth of a fringe answers to the difference of a single undulation. If then we know the thickness of the plate, and the length of an undulation of the light employed, which is easily deduced from the measurement of the fringes, by the formula that has been given, we can calculate the number of undulations comprehended in the same thickness of air, and by adding twenty to the number, we shall have that of the[p447]undulations performed in the thickness of the glass; and the proportion of these two numbers will be that of the velocities of light in the different mediums. Now this proportion is found by experiment the same with that of the sines of incidence and of refraction between air and glass; which agrees with the theory of the refraction of undulations, as will be seen hereafter.
The same experiment may be employed, on the other hand, for determining with extreme precision the thickness of a thin plate of a substance of known refractive density; placing it in the way of one of the two pencils of light, and measuring the displacement of the fringes which it occasions.
This method of determining refractive densities is however liable to some difficulties, when we wish to apply it to a body much more dense than air, such as water, or glass, for example; since it is necessary to employ a very thin plate only, in order that the fringes may not be too much displaced for observation; and then it becomes difficult to measure the thickness of such a plate with sufficient accuracy. We may, indeed, place in the way of the other pencil a thick plate of a transparent substance, of which the refractive density has been ascertained by the ordinary methods, and we can then employ as thick a plate of the new substance. But then it becomes simpler to measure its refractive density by the common method: [unless we choose to immerse the whole apparatus in a fluid very nearly approaching to it in refractive density, which may sometimes be done without inconvenience.TR.]
The case, in which Mr. Arago’s experiment has a decided advantage over the direct method, is when we desire to determine very slight differences of velocity in mediums of nearly equal refractive density: for by lengthening the passage of the light in the two mediums of which we wish to compare the refractive density, we can increase the accuracy of the results almost without limit. In order to form an idea of the extreme precision that may be attained by these measurements, it is sufficient to observe that the length of the yellow undulations in air being about .000021 E.I., there are two millions of them in the length of about 42 inches. Now[p448]it is very easy to observe the difference of one fifth of a fringe, which corresponds to a retardation of one fifth of an undulation in one of the pencils, that is, the ten millionth part of the whole length of 42 inches; we might therefore, by introducing any gas or vapour into a tube of this length, terminated by two plane glasses, estimate very accurately the variation of its refractive power.
I take the length of an undulation of the yellow rays, which are the most brilliant of the spectrum, and of which the dark and light stripes consequently coincide with the darkest and brightest stripes of the fringes produced by white light, which is commonly employed in these experiments, both because of its greater brightness, and because of the more marked character which it gives to the central stripe, so as to prevent any other from being mistaken for it.
It was an apparatus of this kind that Mr.ARAGOand myself employed for measuring the difference of the refractive powers of dry air, and of air saturated with moisture at 80° F., which is so small, that it would escape every other method of observation, because the greater refractive power of aqueous vapour is almost exactly compensated by the less specific gravity of moist air. But, in the generality of cases, the slightest mixture of one vapour or gas with another produces a considerable displacement in the fringes: and if we had a series of experiments of this kind, made with care, the apparatus might become a valuable instrument of chemical analysis.
[To be continued.]
My dear Sir,
IAMvery glad to see that you have been applying your analytical powers to the investigation of the acustical effects of corpuscular forces, and that, among many more refined determinations, you, have confirmed several of the results relating to sounding bodies, which were published twenty years ago in my Lectures on Natural Philosophy: though they were generally such as might have been derived from the calculations of Bernoulli and Euler; which I attempted in some[p449]measure to simplify by the introduction of the element which I called theModulus of Elasticityof each substance. You have very properly observed that it is often difficult to represent the combination of these corpuscular forces by an integral, since in many practical cases the integral must vanish, where it would naturally be applied to the phenomena: and, from similar considerations, I trust you will be prepared to admit the objections that I made long ago, to the reasoning of your great predecessor, Mr. Laplace, to whose station in the mathematical world you appear so eminently qualified to succeed.
The equation, which may be called final, in Mr. Laplace’s Supplement to the Xth Book, p. 47, isQcos. (ω−θ) =(2ς−ς′)Ksin. θ.Now this, in my opinion, is a perfectreductio ad absurdum: forQmustalwaysbeincomparablyless thanK; the attraction of the particles lying between a cylinder and its tangent plane beingalwaysinfinitely less than that of the particles in an angular or prismatic edge: or if this were denied in general, it would obviously become true when the cylinder itself becomes a plane, andQvanishes altogether; which will always be the state of the problem, when the surface of the solid is so inclined to the horizon, that the surface of the fluid may remain horizontal, the appropriate angle of contact being unaltered in these circumstances, as it is easy to show by making the experiment with mercury.
I entreat you to consider this objection with patient attention, and to tell me if you can find any arguments to supersede it. I would also presume to ask your opinion of my own method of deducing the force of capillarity from the elementary attractions and repulsions of bodies, at the end of my Illustrations of the Celestial Mechanics, Art. 382; Appendix A, p. 329 to 337. The volume is in the Library of the Academy; or I should have taken the liberty of sending you a copy, as an inadequate return for so many valuable communications with which you have had the kindness to favour me.
Believe me always, dear Sir,
Very truly yours,
* * * *
London, 18Nov.1827.
[p450]
PrincipalLUNAROCCULTATIONSof the Fixed Stars in the Months of January, February, March, and April, 1828; calculated for the Royal Observatory at Greenwich.Date.Names of Stars.Magn-itude.Immersion and Emersion. Mean Time.Apparent Difference of Declination.*Point of Moon’s Limb.H.M.S.′″°Jan.4κ Cancri5.6Imm.1051431318S.172R.Em.114448719S.91R.31α1Cancri6Imm.111452745S.134L.Em.123556219N.88R."κ Cancri5.6Imm.18380058N.47L.Em.UnderHorizon.Feb.7α2Libræ3Imm.202954135S.69L.Em.213754433N.107R.22δ3Tauri5Imm.709347S.90L.Em.81639634S.146R.28ω Leonis6.7Imm.1124251457S.165L.Em.12315917S.145R.March10ρ1Sagit-tarii5Imm.161454424N.105L.Em.172039125N.62R.23uGemin-orum5.6Imm.8436248N.55L.Em.91811746N.94R.24kGemin-orum5Imm.9125353S.78L.Em.102834355N.111R.26κ Cancri5.6Imm.7412576S.132L.Em.9338314N.83R.April2ν1Libræ6Imm.147431243N.38L.Em.1434561549N.10R.ν2Libræ6.7Imm.13583621S.99L.Em.151358619N.76R.29α1Libræ6Imm.1615381448S.126L.Em.1648161155S.174R.α2Libræ3Imm.163351554S.145L.Em.16435153S.162L.The fifth column shows the apparent difference of declination between the Star and Moon’s centre at the immersion and emersion; the letters N and S denoting the Star to be north or south from the Moon. The sixth or last column shows the point of the Moon’s limb where the immersion and emersion take place, reckoning from the vertex or highest point; the letters L and R signifying to the left hand or right hand of the observer.An error of 11 seconds in the computed difference of declination between the Moon and Star, will be sufficient to convert the expected Occultation of α2Libræ, on 29th April, into an Appulse; and a less error will considerably affect the times and places of immersion and emersion.
The fifth column shows the apparent difference of declination between the Star and Moon’s centre at the immersion and emersion; the letters N and S denoting the Star to be north or south from the Moon. The sixth or last column shows the point of the Moon’s limb where the immersion and emersion take place, reckoning from the vertex or highest point; the letters L and R signifying to the left hand or right hand of the observer.
An error of 11 seconds in the computed difference of declination between the Moon and Star, will be sufficient to convert the expected Occultation of α2Libræ, on 29th April, into an Appulse; and a less error will considerably affect the times and places of immersion and emersion.
[To be continued.][p451]