The Quarterly Journal of Science, Literature and the Arts, July-December, 1827

by Various

ASTRONOMICAL AND NAUTICAL COLLECTIONS. For Jan. 1828.

i. EPHEMERIS of the periodical COMET for its Return in 1828, computed with the consideration of a RESISTING MEDIUM. By Professor ENCKE. [◊]

Elements.

Mean anomaly 1829 Jan.9.72, mean time at Paris, = 0° 0′ 2″.83
Mean daily sidereal motion = 1069″.87572.

Longitude of the perihelion	=	157°	17′	26″.2		Mean Equinox 1829 Jan.9.72.
Ascending node	=	334	28	47.1
Inclination	=	13	20	47.9
Angle of the eccentricity	=	57	38	25.2

Ephemeris.

Mean Parisian time, 1829.		A. R.			Decl. N.			Log. Dist.
		°	′	″	°	′	″
Aug.	23.3	26	50	+..	22	42	+..	.34603	.19571
	24.3		50	...		52		.34411	.18983
	25.3		49	...	23	1		.34217	.18390
	26.3		48	...		10		.34022	.17791
	27.3		46	...		19		.33825	.17187
	28.3		44			29		.33626	.16577
	29.3		41			38		.33425	.15962
	30.3		37			47		.33222	.15341
	31.3		33			56		.33017	.14714
Sept.	1.3		28		24	6		.32810	.14082
	2.3		22			15		.32602	.13444
	3.3		16			24		.32392	.12801
	4.3		9			34		.32180	.12153
	5.3		1			43		.31966	.11499
	6.3	25	53			52		.31749	.10839
	7.3		44		25	2		.31531	.10174
	8.3		34			11		.31310	.09504
	9.3		23			20		.31087	.08829
	10.3		11			29		.30862	.08148
	11.3	24	58			38		.30635	.07462
	12.3		45			47		.30406	.06771
	13.3		30			56		.30174	.06075
	14.3		15		26	5		.29940	.05375
	15.3	23	58			14		.29704	.04670
	16.3		41			23		.29465	.03961
Sept.	17.3	23	22	+..	26	32	+..	.29224	.03247
	18.3		2			41		.28980	.02529
	19.3	22	41			49		.28733	.01806
	20.3		19			58		.28484	.01080
	21.3	21	56		27	6		.28232	.00350
	22.3		32			14		.27978	.99617
	23.3		6			22		.27721	.98881
	24.3	20	39			30		.27461	.98142
	25.3		10			37		.27198	.97400
	26.3	19	40			45		.26933	.96656
	27.3		9			52		.26665	.95910
	28.3	18	37			58		.26393	.95162
	29.3		3		28	5		.26118	.94413
	30.3	17	27			11		.25840	.93663
October	1.3	16	50			17		.25559	.92913
	2.3	16	11			22		.25275	.92164
	3.3	15	31			27		.24987	.91415
	4.3	14	49			32		.24696	.90668
	5.3		5			36		.24402	.89923
	6.3	13	20			39		.24104	.89181
	7.3	12	34			42		.23803	.88442
	8.3	11	45			44		.23498	.87707
	9.3	10	55			46		.23189	.86976
	10.3		4			47		.22876	.86251
	11.3	9	10			47		.22559	.85532
	12.3	8	15			46		.22238	.84820
	13.3	7	19			45		.21913	.84116
	14.3	6	21			43		.21584	.83421
	15.3	5	21			39		.21251	.82735
	16.3	4	20			35		.20913	.82059
	17.3	3	18			30		.20570	.81394
	18.3	2	14			24		.20223	.80741
	19.3	1	9			17		.19871	.80101
	20.3	0	3			9		.19515	.79474
	21.3	358	56			0		.19154	.78861
	22.3	357	47		27	49		.18787	.78263
	23.3	356	38			38		.18415	.77681
	24.3	355	28			25		.18038	.77116
	25.3	354	17			11		.17656	.76568
	26.3	353	5		26	56		.17268	.76037
	27.3	351	53			40		.16874	.75525
	28.3	350	40			22		.16475	.75031
	29.3	349	27			4		.16069	.74557
	30.3	348	14		25	44		.15657	.74102
	31.3	347	1			23		.15239	.73668
Nov.	1.3	345	47			1		.14814	.73253
	2.3	344	34		24	38		.14382	.72858
	3.3	343	21	+..	24	14	+..	.13943	.72483
	4.3	342	9		23	49		.13497	.72128
	5.3	340	56			23		.13044	.71793
	6.3	339	44		22	56		.12583	.71477
	7.3	338	33			29		.12114	.71180
	8.3	337	23			0		.11638	.70902
	9.3	336	13		21	31		.11153	.70642
	10.3	335	4			1		.10660	.70399
	11.3	333	56		20	31		.10158	.70172
	12.3	332	48			0		.09647	.69961
	13.3	331	41		19	29		.09126	.69765
	14.3	330	36		18	57		.08595	.69583
	15.3	329	31			25		.08055	.69415
	16.3	328	27		17	52		.07505	.69258
	17.3	327	23			19		.06944	.69113
	18.3	326	21		16	46		.06372	.68979
	19.3	325	19			12		.05788	.68854
	20.3	324	18		15	39		.05193	.68738
	21.3	323	18			5		.04585	.68630
	22.3	322	19		14	31		.03965	.68529
	23.3	321	20		13	56		.03331	.68434
	24.3	320	21			22		.02684	.68345
	25.3	319	23		12	47		.02022	.68261
	26.3	318	26			12		.01346	.68182
	27.3	317	28		11	37		.00655	.68106
	28.3	316	31			2		.99948	.68034
	29.3	315	34		10	26		.99225	.67964
	30.3	314	37		9	50		.99484	.67897
December	1.3	313	40		9	14	N.	.97726	.67833
	2.3	312	43		8	37		.96949	.67772
	3.3	311	45		8	0		.96153	.67713
	4.3	310	48		7	23		.95336	.67657
	5.3	309	49		6	45		.94499	.67605
	6.3	308	50		6	6		.93640	67556
	7.3	307	50		5	27		.92759	.67513
	8.3	306	49		4	47		.91854	.67476
	9.3	305	47		4	6		.90924	.67446
	10.3	304	44		3	24		.89969	.67425
	11.3	303	40		2	42		.88987	.67415
	12.3	302	34		1	58		.87978	.67418
	13.3	301	27		1	13		.86939	.67437
	14.3	300	18		0	27	N.	.85870	.67475
	15.3	299	7		0	19	S.	.84770	.67536
	16.3	297	54		1	8		.83638	.67623
	17.3	296	40		1	58		.82474	.67742
	18.3	295	23		2	49		.81276	.67896
	19.3	294	5		3	42		.80042	.68091
Dec.	20.3	292	45	+..	4	36	+..	.78771	.68333
	21.3	291	23		5	32		.77465	.68627
	22.3	289	59		6	29		.76123	.68980
	23.3	288	34		7	27		.74746	.69399
	24.3	287	8		8	27		.73334	.69889
	25.3	285	42		9	27		.71890	.70456
	26.3	284	15		10	28		.70416	.71107
	27.3	282	48		11	30		.68917	.71845
	28.3	281	23		12	32		.67399	.72677
	29.3	280	0		13	34		.65871	.73604
	30.3	278	39		14	36		.64342	.74629
	31.3	277	21		15	37		.62830	.75750
1830 Jan.	1.3	276	9		16	37		.61348	.76969
	2.3	275	2		17	36		.59920	.78278
	3.3	274	1		18	33		.58572	.79670
	4.3	273	9		19	29		.57332	.81139
	5.3	272	25		20	22		.56231	.82673
	6.3	271	50		21	12		.55304	.84259

The opposition to the sun will be 1828, Oct. 12.34: while its light is weak, it may be observed on or near the meridian.

On the 10th of Nov. 1828, its distance from the sun will be the same as at the time of its discovery in 1818, and it will be considerably nearer to the earth; and on the 21st of December, its position with respect to the sun will be the same as at its last observation in 1819; and with respect to earth, its situation will be more advantageous. The 1st of January, 1829, it will set with the sun.

It follows, that the most advantageous time for seeing it will be during the whole of November, and the first 25 days of December. It will scarcely be seen before the end of September, as it has heretofore never been observed more than two months before the time of its perihelion, and even in the dark winter nights will scarcely be visible more than 14 or 15 weeks before that period. After the perihelion it will not be visible in these parts of the world.

ii. Elementary View of the UNDULATORY Theory of Light. By Mr. FRESNEL. [◊] [Continued from the last Number.]

IN order to complete the explanation of the conditions necessary for the formation of the fringes, it remains to show why a small luminous point must be employed in experiments on diffraction, and not an object of any considerable dimensions. If we resume the case of the interior fringes of the shadow of [p432] a narrow body, it will be easy to apply similar arguments to other cases of diffraction.

The middle of the central band, which is always formed by the simultaneous arrival of rays, which depart at the same instant from the luminous point, must be found in the plane drawn through this point, and the line bisecting the narrow body: because, since every thing is symmetrical on each side of this plane, the rays which unite in it must have passed through equal routes on each side, and must consequently arrive at the same instant, unless they have passed through different media, which is not the case to be considered at present. The situation of the middle stripe being determined, that of every other stripe must also be determined accordingly. Now it is evident that if the luminous point should change its situation a little, and be moved to the right, for example, the plane, which has been supposed, would incline to the left, and would carry with it all the fringes which accompany the middle stripe. And if, instead of supposing motion, we suppose the luminous point to become of sensible dimensions; the integral points of which it is composed will each produce a group of fringes, and their situations will be so much the more remote as the luminous object is larger; and ultimately, if its size is sufficiently increased, they will extinguish each other and disappear. This is the reason that, when the rays cross each other at sensible angles, as in all the phenomena of diffraction, it becomes necessary to employ a very fine luminous point, in order to discover their mutual influence: and the point must be so much the finer as the angle formed by the rays is greater.

However minute the luminous point may be, it is always composed, in reality, of an infinite number of centres of oscillations, and it is of each of these centres that we must understand what has been said of a luminous point. But as long as they are very near to each other in comparison with the breadth of the fringes, it is obvious that the different groups of fringes which they produce, instead of mixing with each other in a confused manner, will be superposed almost exactly, and instead of extinguishing, will co-operate with each other. [p433]

When the two systems of waves which interfere are parallel, the interval which separates their corresponding points must remain the same for a great portion of the surface of the waves, that is to say, in other words, the fringes will become almost infinite in breadth, so that a very considerable displacement of the centre of undulation will cause very little difference in the agreement or disagreement of their vibrations. And in this case it is no longer necessary to employ so small an object in order to perceive the effects of their mutual influence.

If the coloured rings, which are produced by the interference of two systems of undulations nearly parallel, exhibit, like the fringes, and often within a very short distance, alternations of dark and bright stripes; this circumstance depends entirely on the want of uniformity in the thickness of the plate of air interposed between the glasses, which causes a variation of the difference of the routes of the rays reflected at the first and at the second surface of this plate, of which the mutual interference produces the bright and dark rings.

We shall readily be able, to understand why the luminous rays, although they always exert a certain influence on each other, exhibit it to the eye so seldom, and in cases so much limited, if we consider that it is necessary, for such an exhibition, first, that the rays concerned shall have been derived from a common source; secondly, that the difference between their paths shall amount to a limited number of undulations only, even when the light is as homogeneous as possible; thirdly, that they shall not intersect each other at too great an angle, because the fringes would become so small as to be invisible even with the assistance of a strong magnifier; and fourthly, unless the rays are nearly parallel, that the luminous object should be of very small dimensions, and the smaller in proportion as the inclination of the rays is greater.

It has been thought necessary to insist so much at length on the theory of interferences, because of its numerous applications to the calculation of the most interesting of the laws of optical phenomena. These considerations may perhaps appear at first somewhat delicate and difficult of comprehension, notwithstanding the minuteness of the [p434] explanation; but with some reflection it will be found that nothing can be simpler than the principles on which they are founded, and their application will soon become familiar to the imagination.

In order to complete the bases of the general theory of diffraction, it remains for us to consider the principle of Huygens, which appears to be a rigorous consequence of the system of undulations.

The principle may be thus expressed: The vibrations of a luminous undulation, in each of its points, may be regarded as the result of the elementary motions which would be transmitted to that point, at the same instant, from all the points of the undulation, considered separately, as they existed in any one of its earlier situations.

It is a consequence of the principle of the co-existence of small motions, that the vibrations, produced at any point of an elastic fluid, by several agitations, are represented by the result of all the velocities belonging to that point at the same instant, as derived from the different centres of the undulations, combined according to the laws of motion, whatever may be the number and situation of the centres, and whatever the periods and nature of the undulations. This general principle is applicable to every particular case. We may suppose the agitations infinite in number, of the same kind, simultaneous, and taking place in contiguous points of a plane or a spherical surface: it will also be convenient to suppose the motions of the particles to take place in the same direction, perpendicular to the surface, their velocity being proportional to the condensation of the medium, and none of them retrograde in their direction. In this manner a derivative undulation will be produced by the union of these agitations, and the principle of Huygens may be truly applied to such a propagation. [This may be called a rigorous consequence of the system, but it can scarcely be considered as a proposition mathematically demonstrated: and the fundamental law of Huygens must perhaps be assumed as an axiom or a phenomenon. TR.]

The intensity of the primitive undulation being uniform throughout the surface, it results from this “theoretical” [p435] consideration, as well as from other reasoning, that the uniformity will be preserved throughout the progress of the undulation, unless any part of it be intercepted or retarded; because the result of the elementary motions, which have been mentioned, will be the same for all the points. But if a portion of the undulation be intercepted by the interposition of an opaque body, then the intensity of each part will vary according to the distance from the margin of the shadow, and these variations will be particularly sensible in the neighbourhood of the tangent rays.

Let C be the luminous point, AG the screen, and AME the wave, arrived at A, and partly intercepted by the opaque body. We may suppose it to be divided into an infinite number of small arcs, Am′, m′m, mM, Mn″, n″n′, n′n, and so forth. In order to find its intensity at the point P, belonging to any subsequent situation of the undulation, BPD, we must find the result of all the elementary agitations which each of these portions of the primitive undulation would produce there if they acted separately.

The impulse, which has been given to every part of the [p436] primitive undulation, being perpendicular to its surface, the motions of the particles of ether in this direction must be more considerable than in any other; and the rays depending on these motions, if separately considered, would be so much the weaker as they deviated the more from this direction.

The investigation of the law by which their intensity would be governed, according to their direction, as derived from any separate centre of agitation, would certainly be of very difficult investigation: but happily we are not obliged to determine this law, for it is easy to see that when the inclination to the perpendicular is considerable, the effects of the different rays must very nearly destroy each other: so that these rays, which sensibly affect the quantity of light received at each point P, may safely be regarded as being equal in intensity.

When the centre of agitation has undergone a condensation, the expansive force tends to urge the molecules in every direction; and if they do not perform a retrograde motion, it is only because their initial velocities forwards destroy those which the expansion of the condensed fluid would otherwise generate backwards: but it does not follow from this that the agitation can only be propagated in the direction of the initial velocities; for the expansive force in a perpendicular direction, for example, will combine with the primitive impulse without any diminution of its effects. It is obvious that the intensity of the undulation thus produced may vary much at the different points of its circumference, not only from the nature of the initial impulse, but also because the condensations are not subject to the same law on every side of the centre of the agitated part[?]. But the variations of the intensity of the derivative undulation must necessarily be subjected to a law of continuity, and may consequently be considered as insensible in a very small angular interval, especially in the neighbourhood of the perpendicular to the surface of the primitive undulation; for the initial velocities of the molecules, referred to any given direction, being proportional to the cosines of the angles made by that direction with the perpendicular, these results vary much [p437] more slowly than the angles themselves, while they remain inconsiderable.

If, in fact, we consider rays sensibly inclined to each other, such as EP, FP, IP, meeting in the point P, which we may suppose at the distance of a great number of breadths from the undulation EA: and if we take two arcs, EF and FI, of such a length that the differences EP−FP and FP−IP may be equal to half an undulation: on account of the marked obliquity of the rays, and of the smallness of a semiundulation, in proportion to their length, these two arcs will be almost equal, and the rays which come from them to the point P will be nearly parallel; so that on account of the difference of a semiundulation between the corresponding rays of the two arcs, their effects will mutually destroy each other.

We may therefore suppose all the rays sent by the different parts of the undulation at AE, to the point P, to be of equal intensity, since the only rays, with respect to which this hypothesis would be incorrect, are such as have no sensible influence on the quantity of light which it receives. For the same reason, in order to simplify the calculation of the result of all these elementary undulations, we may consider their constituent motions as performed in the same direction, the angles which they form with each other being inconsiderable. The problem is thus reduced to that which has been solved in the Memoir on Diffraction, already quoted: To find the result of any number of systems of parallel undulations of light, of the same frequency, when their intensities and relative situations are given.—The intensities are here proportional to the length of the small illuminating arcs, and the relative situations are given from the differences of the paths described.

We have considered, correctly speaking, only the section of the undulation made by a plane perpendicular to the margin of the screen represented by A. We may now take into account the whole extent of the undulation, and suppose it to be divided, by equidistant meridians perpendicular to the plane of the figure, into infinitely thin wedges or strata; and we may apply to all of these the reasoning which has [p438] been employed for one section, and thus demonstrate that the rays which have a marked obliquity must destroy each other.

These strata, in the case here considered, being all parallel to the edge of the screen, and infinitely extended, while the undulation is intercepted but on one side; the intensity of the result of all the impressions, which they transmit to P, will be the same for each of them: for the rays emanating from them must be considered as of equal intensity, at least for the very small extent of the generating undulation, which has a sensible influence on the light received at P. Besides, each elementary result will evidently be retarded by the same quantity, with respect to the ray derived from the point of the stratum nearest to P, that is to say, to the point in which it cuts the plane of the figure: consequently the intervals between these elementary results will be equal to the differences of the paths described by the rays AP, m′P, mP, and so forth, which are in the plane of the figure, and their intensities will be proportional to the arcs Am′, m′, m, mM, and so forth. We may therefore consider the intensity of the general result as determined by the calculation already mentioned, as belonging to the section of the undulation made by a plane perpendicular to the margin of the screen.

While the outline of the screen remains rectilinear, it is sufficient, in order to determine the situations of the dark and light stripes, and their relative intensities, to consider the section of the undulation made by a plane perpendicular to that outline: but when it is curved, or composed of lines meeting at any angles, it becomes necessary to obtain the integral effect for two directions at right angles to each other, or for a circle surrounding the point considered. This last method is the most simple in some particular cases, as when we have to calculate, for example, the intensity of the light in the projection of the centre of a circular screen or opening:

It will now be easy to form a distinct idea of the method [p439] which must be followed, in order to calculate the situation and the intensity of the dark and bright stripes, in the different circumstances under which it is proposed to compare the theory with experiment. When the screen is infinitely extended on one side, or is broad enough to allow us to neglect the rays which pass beyond it, we are to determine, for any point P at the distance of the place at which the fringes are to be observed, the result of all the elementary undulations coming from the part AMF only of the incident wave; and comparing the intensities at different collateral points, P, P′, P″, we are to find the situation of the darkest and the brightest points. In this manner we find, for a screen closed on one side, 1st, that the intensity of the light decreases rapidly within the [shadow] beginning from the tangent CAB, and so much the more rapidly as the undulation is smaller; and this in a continuous manner, without any alternations of maxima and minima; 2ndly, that out of the shadow, the intensity of the light, after augmenting considerably to a certain point, which may be called a maximum of the first order, decreases to another point, which is the minimum of the first order: that it increases again to a second maximum, to which succeeds a second minimum, and so forth; 3rdly, that none of these minimums completely vanish, as in the case of fringes produced by the concourse of two luminous pencils of equal intensity, and that the difference between the maxima and minima diminishes in proportion as we go further from the shadow; whence we may understand why the fringes which surround shadows in a homogeneous light, are less marked and less numerous, than those which are obtained by a combination of two mirrors, and those in white light much less brilliant; 4thly, that the intervals been the maxima and minima are unequal, and diminish, as we depart from the shadow, in proportions which remain unaltered, whatever may be the distance from the screen at which we measure them; and 5thly, that the same maxima and minima, calculated for different distances from the screen, are situated in hyperbolas of a sensible curvature, of which the foci are the edge of the screen, and the luminous point. All these consequences of the theory are precisely confirmed by experiment. [p440]

The general formula gives the position of the maxima and minima for any distances whatever of the luminous point from the screen, and from the screen to the micrometer, when the length of the undulation of the light employed is known. In order to submit the theory to a decisive test, instead of determining the length of the undulation by measures of the external fringes, and then employing it in calculations of the same kind, I deduced it from an experiment on diffraction of a very different kind; and after having first verified it by the fringes obtained from two mirrors, of which it represented the breadth within a hundredth part of the truth, I introduced it into the formula which I afterwards compared with 125 measurements of exterior fringes, made under very different circumstances; for the distance of the radiant point from the screen was varied from four inches to six or seven yards, and the distance between the screen and the micrometer was varied from 113th of an inch to more than four yards: and the results of all these comparisons were perfectly satisfactory, as maybe seen in the comparative table published in the XIth volume of the Annales de Chimie et de Physique, p. 339, 343.

When the screen, instead of extending infinitely on one side, is narrow enough to admit some light on that side, not too much weakened by the rapid decrease of intensity produced by obliquity, we must take into the calculation the light on both sides, and find, for each point of the shadow, the general result of all the elementary undulations derived from the points on the right and left. We thus demonstrate that the interior parts of the shadow must be divided by a series of dark and bright stripes, nearly equal in breadth, of which the situations differ very little from those which would be deduced from the approximative formula which has already been given for the same purpose, when they are still separated from the borders of the shadow by an interval of several of their breadths. But when the opaque body is narrow enough, and the micrometer far enough removed for the observed stripes to be very near the exterior stripes, then the results of this more exact calculation, as well as those of experiment, show that the approximation is no longer accurate. The [p441] calculation determines also, with remarkable precision, the singular alterations which the exterior fringes often undergo, when the other series extends beyond the shadow, and mixes its effects with those of the exterior.

I have also verified the theory by examining the fringes derived from a narrow slit of indefinite length; and determining, for the different points enlightened by the luminous pencil, the result of all the elementary undulations derived from the part of the primitive wave comprehended in the breadth of the slit; and I have found a satisfactory agreement between the calculation and the observations, even when the fringes thus obtained afforded the most capricious and apparently irregular appearances.

In this mode of considering the problems relating to diffraction, we have not taken into the calculation the greater or less thickness of the edges of the screen, but merely the extent of the primitive wave which is capable of sending elementary undulations to the points for which we are to find the intensity of illumination; and the opaque substance has no other effect than simply to intercept a part of the wave: for this reason the result is necessarily independent of the nature of the body, of its mass, and of the thickness of its edges. Nevertheless, if the surface of the edges were very extensive, it would be impossible to consider the portion of the wave as quitting the slit without having received some previous modification, and it would be necessary to take into the calculation the small fringes derived from the effect of the remoter parts of the slit. But while the thickness is moderate, or the edges rounded off into a well marked curve, the small fringes derived from this cause may be neglected, and the emerging wave may be considered as of equal intensity throughout, at the moment of its quitting the screen, especially if the intensity of the light is to be calculated for a pretty considerable distance from the screen. We must not, indeed, forget, that according to the reasoning which has been employed, the formulas for diffraction are only sufficiently exact when this distance is very considerable, in comparison with the breadth of an undulation, since it is in this case only that we can neglect the rays that are decidedly oblique, and [p442] can suppose all those, which are essentially concerned in the effect, to be nearly of equal intensity. It is not, however, surprising that the same formulas will give the position of the fringes with sufficient accuracy at small distances from the screen, when its edges are thin, since, the mean breadth of an undulation being but about one fifty thousandth of an inch, a tenth of an inch becomes comparatively a very considerable distance.

These are the three principal kinds of phenomena presented to us by diffraction, when the edges of the screen, or of the opening made in it, are sufficiently extensive to afford fringes independent of any effect from their terminations: and in such cases it is sufficient to make the integral calculation for the plane perpendicular to the edges of the screen only, in order to determine the position of the dark and bright stripes, and their comparative intensities. But when the screen or the opening are of small dimensions in every direction, it becomes necessary to extend the integration to the effects produced in two perpendicular planes: and the results of the calculation agree perfectly with observation, as will appear from two curious instances.

When the screen is circular, the calculation leads to this singular result, that the centre of the shadow projected by it must be as much enlightened as if the screen were not in existence. It was Mr. POISSON that first pointed out this consequence of my formulas, which I did not at first observe, though it is immediately deducible from the theory by very simple geometrical considerations. Mr. ARAGO made the experiment with the shadow of a screen 113th of an inch in diameter, perfectly round, and fixed on a plate of glass. The result confirmed the fact which had been announced by the theory. It is only the centre itself that possesses this property, and the same brightness is only extended to a sensible distance from this mathematical point when the screen is of very small diameter, and when its shadow is observed at a great distance: for the wider that the screen becomes, the more the little bright circle is contracted; and when the screen is four tenths of an inch in diameter, we only see a single point of light, at the distance of a yard, even with a powerful magnifier. It must be observed, that if the screen [p443] were too large, the reasoning, from which the formulas have been deduced, would no longer be rigorously applicable to the rays inflected into the shadow, because of their too great obliquity, which would render it impossible to consider their effects as equal in intensity to those of the direct rays.

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, are b = ar²2ad−r², b = ar²4ad−r², b = ar²8ad−r²; and so forth; r being the semidiameter of the aperture, a and b its respective distances from the luminous point and from the micrometer, and d the 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. ARAGO and 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.]

iii. Remarks on the Action of CORPUSCULAR FORCES. In a Letter to Mr. POISSON. [◊]

My dear Sir,

I AM very 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 the Modulus of Elasticity of 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, is Q cos. (ω−θ) = (2ς−ς′) K sin. θ. Now this, in my opinion, is a perfect reductio ad absurdum: for Q must always be incomparably less than K; the attraction of the particles lying between a cylinder and its tangent plane being always infinitely 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, and Q vanishes 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, 18 Nov. 1827.

[p450]

iv. Calculations of LUNAR PHENOMENA. By THOMAS HENDERSON, Esq. [◊]

PrincipalLUNAROCCULTATIONS of the Fixed Stars in theMonths of January, February, March, and April, 1828; calculatedfor 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.	4	κ Cancri	5.6	Imm.	10	51	43	13	18	S.	172	R.
				Em.	11	44	48	7	19	S.	91	R.
	31	α1 Cancri	6	Imm.	11	14	52	7	45	S.	134	L.
				Em.	12	35	56	2	19	N.	88	R.
	"	κ Cancri	5.6	Imm.	18	38	0	0	58	N.	47	L.
				Em.	Under			Horizon.
Feb.	7	α2 Libræ	3	Imm.	20	29	54	1	35	S.	69	L.
				Em.	21	37	54	4	33	N.	107	R.
	22	δ3 Tauri	5	Imm.	7	0	9	3	47	S.	90	L.
				Em.	8	16	39	6	34	S.	146	R.
	28	ω Leonis	6.7	Imm.	11	24	25	14	57	S.	165	L.
				Em.	12	3	15	9	17	S.	145	R.
March	10	ρ1 Sagit- tarii	5	Imm.	16	14	54	4	24	N.	105	L.
				Em.	17	20	39	1	25	N.	62	R.
	23	u Gemin- orum	5.6	Imm.	8	4	36	2	48	N.	55	L.
				Em.	9	18	11	7	46	N.	94	R.
	24	k Gemin- orum	5	Imm.	9	12	5	3	53	S.	78	L.
				Em.	10	28	34	3	55	N.	111	R.
	26	κ Cancri	5.6	Imm.	7	41	25	7	6	S.	132	L.
				Em.	9	3	38	3	14	N.	83	R.
April	2	ν1 Libræ	6	Imm.	14	7	43	12	43	N.	38	L.
				Em.	14	34	56	15	49	N.	10	R.
		ν2 Libræ	6.7	Imm.	13	58	36	2	1	S.	99	L.
				Em.	15	13	58	6	19	N.	76	R.
	29	α1 Libræ	6	Imm.	16	15	38	14	48	S.	126	L.
				Em.	16	48	16	11	55	S.	174	R.
		α2 Libræ	3	Imm.	16	33	5	15	54	S.	145	L.
				Em.	16	43	5	15	3	S.	162	L.

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 α² Libræ, 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]

ELEMENTSfor computing theECLIPSESof theSUN andOCCULTATIONS of thePLANETS by theMOON, in the Year 1828.
Conjunction in A. R. Apparent Time.					Diff. Dec.			Relative H. M.		Relative Orb. Ang.				or Planet’s A. R. at			or Planet’s N. P. D. at
	D.	H.	M.	S.	′	″		′	″		°	′		H.	M.	S.	°	′	″
Jan.	11	10	47	11	21	1	S.	34	6	S.	76	58	E.	14	35	47	104	2	30
Jan.	11	16	40	36	4	29	N.	33	14	S.	78	4		14	49	29	105	12	34
Feb.	7	22	17	54	5	44	N.	33	26	S.	77	35		14	46	40	104	48	7
Mar.	6	4	48	7	16	46	N.	33	29	S.	77	47		14	49	7	104	53	57
April	2	8	6	49	9	6	N.	34	12	S.	77	26		14	42	44	104	21	7
April	13	21	23	53	8	50	N.	31	23	N.	74	7		1	30	20	80	32	14
April	29	10	49	53	9	3	S.	34	50	S.	76	40		14	30	21	103	22	5
May	12	8	58	41	7	44	N.	27	31	N.	78	28		2	30	50	76	35	6
May	26	14	58	46	20	58	S.	34	38	S.	75	54		14	17	53	102	23	57
June	22	21	27	33	14	27	S.	33	40	S.	75	26		14	11	5	101	55	51
July	13	12	0	30	68	43	S.	30	15	S.	76	50		8	55	0	76	17	27
July	20	6	18	33	10	12	N.	32	33	S.	75	28		14	12	35	102	11	28
Aug.	16	17	18	6	44	49	N.	31	50	S.	75	59		14	22	10	103	7	21
Sept.	5	3	7	12	4	10	S.	28	8	S.	78	3		8	13	4	75	7	21
Oct.	8	12	23	35	6	39	S.	29	6	S.	73	2		12	57	44	96	10	27
Dec.	3	13	30	46	39	10	S.	29	39	S.	75	23		14	3	58	100	19	38

ELEMENTSfor computing theECLIPSESof theSUN andOCCULTATIONS of thePLANETS by theMOON, in the Year 1828. [continuedfrom above]
Conjunction in A. R. Apparent Time.					Nearest Appr- oach.		Time of nearest Approach, Apparent Time.				or Planet’s
											Horary	Motion	Semi-diam-eter	Hor. Par.
											in A. R. in Time.	in N. P. D.
	D.	H.	M.	S.	′	″	D.	H.	M.	S.	SEC.	″	″	″
Jan.	11	10	47	11	20	28	11	10	38	51	+1·3	+6	17	2
Jan.	11	16	40	36	4	23	11	16	42	16	+5·9	+27	3	5
Feb.	7	22	17	54	5	36	7	22	20	7	+0·6	+2	18	2
Mar.	6	4	48	7	16	23	6	4	54	29	−0·2	−1	20	2
April	2	8	6	49	8	53	2	8	10	18	−0·9	−4	21	2
April	13	21	23	53	8	30	13	21	19	16	+9·2	−54	958	9
April	29	10	49	53	8	48	29	10	46	17	−1·2	−6	22	2
May	12	8	58	41	7	35	12	8	55	19	+19·8	−119	3	7
May	26	14	58	46	20	20	26	14	49	55	−0·9	−4	21	2
June	22	21	27	33	13	59	22	21	21	5	−0·3	−1	20	2
July	13	12	0	30	66	55	13	11	29	28	−3·4	+22	26	27
July	20	6	18	33	9	53	20	6	23	16	+0·5	+3	18	2
Aug.	16	17	18	6	43	29	16	17	38	31	+1·2	+6	17	2
Sept.	5	3	7	12	4	5	5	3	5	22	+5·8	−1	18	19
Oct.	8	12	23	35	6	21	8	12	19	35	+9·2	+57	963	9
Dec.	3	13	30	46	37	54	3	13	10	46	+11·5	+61	7	8

The places of the Sun and Moon have been taken from the Nautical Almanac, those of Mercury from Lindenau’s Tables, and those of the other Planets from Schumacher’s Ephemeris.——The sign + denotes the motion in A. R. to be direct; the sign −, retrograde. The sign + denotes the motion in N. P. D. to be towards the South; the sign − towards the North.——None of the preceding Conjunctions will prove to be an Eclipse or Occultation visible at Greenwich. [p452]

Apparent Distance of Jupiter’s Satellites from Jupiter’s Centre, at his Conjunctions in A. R. with the Moon.
Date.	Satellite.	Distance.
1828.		′  ″
January 11	I.	1 14	East
	II.	0 51	——
	III.	4 16	West
	IV.	2 21	——
February  7	I.	1  6	West
	II.	2 40	East
	III.	2 16	West
	IV.	4 27	——
March  6	I.	1 51	East
	II.	2 24	West
	III.	3 14	East
	IV.	8 44	——
April  2	I.	1 44	West
	II.	0 15	——A
	III.	4 58	East
	IV.	7 30	West
29	I.	0 14	WestB
	II.	2 51	East
	III.	0 32	West
	IV.	1  6	East
May 26	I.	1 44	East
	II.	3  8	West
	III.	5 15	——
	IV.	6 38	East
June 22	I.	1 59	West
	II.	0  4	EastC
	III.	3 10	West
	IV.	8 56	——
July 20	I.	1 51	East
	II.	2 50	——
	III.	1 55	——
	IV.	4 51	——
August 16	I.	1 43	West
	II.	1 12	——
	III.	4 21	East
	IV.	1 53	——
A: On Jupiter’s disk. B: Eclipsed. C: On Jupiters disk. These Configurations have been computed from De Lambre’s Tables.[p453]