Chapter XXII: Letter XIX: gives a definition of the ellipsis, which would be a (16)

3. That, in the Northern Hemisphere, on the shores of the Icy Sea, or at the furthest distance north, its situation is still observed to be the northward.

4. That, in the Southern Hemisphere, it has been observed to the east of the South Pole, and in the Northern, to the east and west of the North Pole.

5. That, upon the late occasion, the place of its columns, during the exhibition, was observed to change from the west of north to the east of north; but, so as always to have the north for the apparent centre of its strength.

6. That, in the Arctic regions, the appearance of the Aurora is said to be usually followed by clear and calm weather.

7. That the appearance of the Aurora Borealis is no wise peculiar to the winter solstice, but has been observed in each of the eight months of September, October, November, December, January, February, March, and April, and may be regarded, therefore, as coincident with the Arctic winter; and that the appearance of an Aurora in the Southern Hemisphere, in the month of February, or beginning of the Antarctic winter, as observed during the voyage of Captain Cook, in the year 1773, is [p423] consistent with the persuasion, that the Aurora _Australis_, in its turn, is a phenomenon of the Austral or Antarctic winter.

8. That _considerable_ or powerful displays of the Aurora are infrequent, even in the extreme Polar regions; and that it is _very considerable_ or powerful displays alone, which make themselves visible in the lower latitudes, north or south of the equator.

9. That no appearance belongs to the Aurora itself, but that of its coruscations, columns, spears, or streamers; and that all colours, therefore, or coloured figures, not belonging to the coruscations, but coincident in their appearance, are to be regarded only as reflections or refractions of light, derived from the coruscations by the clouds which happen to cover the sky.

10. That the colours, or coloured light, proper to the Aurora, or seen in the columns or coruscations themselves, are varied from column or coruscation to column or coruscation, and severally continued in the direction, and throughout the length or height, of each.

11. That, in the late example, the columns or coruscations situate in the due north, or apparent centre or focus of the phenomenon, exhibited a light at least comparatively white; and that the variation, from white to colour, had an apparent relation to the comparative remoteness of each column or coruscation from the column or coruscation in the central north.

12. That the direction or position of the columns or coruscations of the Aurora, are so far from being uniformly convergent toward the zenith, or uniformly vertical, or from the horizon to the zenith, that, in the late example, they _did not converge_ toward the zenith, but, contrariwise, _diverged_ from it; spreading themselves like the sticks of a fan, or like stalks in a flower-basket.

13. That the columns or coruscations of the Aurora are not uniformly rectilinear in their figure; but that, in the late example, those on the north-eastward were curved outwardly, or “bent sideways,” as described in the appearance of the columns or coruscations of an Aurora seen in the Southern Atlantic, during the voyage of Captain Cook, in the year 1773. [p424]

FOOTNOTES:

[120] Literary Gazette, Sept. 29, 1827.

[121] “PENCELLS.—Pencills, or flagges for horsemen, must be a yard and a halfe long.” Harleian MSS., cited in an interesting and valuable essay on the “Banners used in the English army, from the Conquest to the reign of Henry VIII.” By N. H. NICOLAS, Esq., F.S.A—_Retrospective Review_, Oct. 1, 1827.

“The Pensell, or Pennoncelle, was the diminutive of the Pennon, being a long narrow flag.”—MEYRICK’S ANCIENT ARMOUR.

“STREAMER.—A Streamer shall stand in the toppe of a shippe, or in the forecastle, and therein be put no armes, but a man’s conceit or device, and may be of the lengthe of twenty, forty, or sixty yards; and it is slit, as well as a guydhomme or standarde.”—HARLEIAN MSS.

An item, in a bill of parcels, charged to the Earl of Warwick, in 1437, consists of “a great Stremour for the Ship, of xl yerdis lenghth, and viij yerdis in brede.”—BANNERS USED IN THE ENGLISH ARMY, &c.

[122] Meteorological Journal, Literary Gazette, Sept. 29th.

[123] Literary Gazette, as above.

[124] _Encyclopædia Britannica_. Art. Aurora Borealis.

[125] Encyclopædia Britannica.

[126] The individual, social, and political importance of making the _art of drawing_ a branch of general education, is a subject which the author can never cease to urge upon the attention of his fellow-countrymen, and of all the civilised world. It is more than ten years since he first endeavoured to lead the public eye to its regard. In England, and with a view to the subsistence of a large and always increasing population, it is an EDUCATION IN THE ARTS which is the great want; and the _art of drawing_, besides being the assistant of all _knowledge_ whatever, is peculiarly so of all other _arts_ than itself, or of all other works of the _hand_. A recent Sermon, by the Lord Bishop of Bath and Wells, preached at Wells, for the benefit of the Diocesan National Schools, bears ample testimony to the deficiency, and even the dangers, to the poor not less than to others, in all the present popular education; and, so far, therefore, to the soundness of the author’s principles, and to the fitness of his remedy. His own design, however, is not only to remedy an evil arising from the present practice, but also to produce an independent good; and, not merely to aid the poor, nor merely to promote the political welfare of this kingdom, but to increase the resources, physical and intellectual, of all classes, and to promote the welfare of the whole world.

[127] The author has an opinion, that among the “agents of nature,” for equalising the temperature of the surface of the globe, is to be reckoned, not only the Northern and Southern Lights, but the entire Ocean; and that this agency is the immediate object aimed at in the existence of this last, as one body of water surrounding the entire globe. His evidence consists in geographical, hydrographical, meteorological, and physiological facts, as also in the apparent reason of the case. He supposes, in consequence, a perpetual circulation of the waters of the sea, longitudinally round the globe, or from North to South, and from South to North again; and the result of Captain Parry’s late attempt to reach the Arctic Pole, as also some of the facts which have transpired respecting Captain Franklin’s late land expedition, appear to confirm his theory, according to which the physical use, or final cause, of the existence of the Ocean has never previously been understood. His theory affects the question of the North-west Passage, which latter object he suspects to have never yet been pursued in the true direction; even the discoveries of Captain Parry appearing to him to have fallen short of ascertaining the communication with the Polar Sea by the channel of Davis’s Strait.—Some introductory observations upon this subject have been already made in an article in the New Monthly Magazine for October, 1826, (vol. xvii. p. 371.)

_Proceedings of the Royal Society_.

The anniversary meeting of the Royal Society for the election of a president, and other officers, was held as usual at Somerset House, on Thursday, the 30th of November, being St. Andrew’s day.

Till within a few days of the election, it was generally understood that the Rt. Hon. Robert Peel was a candidate for the chair; in consequence, however, of that gentleman having declined, Davies Gilbert, Esq., M.P., was put in nomination, and was almost unanimously elected the President of the Royal Society.

The late secretaries, Messrs. Herschel and Children, having resigned their respective offices, Dr. Roget and Captain Sabine were nominated in their places, and were duly elected.

The accession of Mr. Gilbert to the chair having rendered vacant the office of Treasurer, Major Kater was elected in his place.

The following council was elected, to continue in office until St. Andrew’s day, 1828.

Davies Gilbert, Esq. Major Kater. Dr. Roget, M.D. Captain Sabine, R.A. Dr. Wollaston, M.D. Dr. Fitton, M.D. Dr. Young, M.D. Dr. Paris, M.D. Dr. Prout, M.D. Dr. Goodenough, D.D. Dr. Buckland, D.D. J. W. Croker, Esq. Lord Colchester. Sir E. Home, Bart. Sir H. Davy, Bart. John Pond, Esq. Capt. F. Beaufort, R.N. Francis Baily, Esq. John Guillemard, Esq.

[p425]

In consequence of having been elected President _pro tempore_ by the council, (the chair having been vacated some weeks previous to the general election,) the duties of the office were performed by Mr. Gilbert, on occasion of the present anniversary. After having read over the list of members admitted, and of those deceased during the last year, he proceeded to announce the disposal of the Royal and Copley Medals, as awarded by the votes of the council.

Of the Royal Medals, one was awarded to Sir H. Davy, and the other to Professor Struve. Of the Copley Medals, one was given to Dr. Prout, and another to Lieutenant Forster. On this occasion Mr. Gilbert pronounced an eulogium upon the respective receivers of the medals; and, in adverting to the labours of the several individuals, he justified the decision of the council, in bestowing upon them these marks of distinction, in a learned and eloquent discourse.

_Proceedings of the Horticultural Society_.

_September_ 4th.

A paper by Mr. Lindley was read upon the new hardy plants which had flowered in the Society’s garden; among them a number of new shrubs were mentioned, which appeared likely to prove acquisitions to the public. A thermometer was exhibited by Mr. Bregazzi, of Derby, for ascertaining the temperature of bark-beds. It consisted of a thermometer enclosed in a shaft of copper with a wooden handle, and a door in its side, by which the temperature can be ascertained with precision. It is needless to point out the superiority of this plan, over the common mode of determining this very essential point, by feeling of a stick previously stuck in the bed; the sensation of heat when the stick is grasped in the hand will obviously depend in a great degree upon the temperature of the hand itself. As usual, there was an extensive display of all the choicest flowers and fruit of the season. One hundred and seventy-two subjects of this description were placed upon the table. Among the flowers, the most remarkable was a new hardy climber from Mexico, with deep purple blossoms studded [p426] with glittering green glands, called Maurandya Barclaiana; among the fruit was a fine specimen, from Lord Grantham’s garden, of the Papaw, a tropical fruit never ripened in England before.

_September_ 18th.

The exhibitions of this day were chiefly confined to a display of Dahlias, which for magnificence exceeded any thing of the kind we ever witnessed before. The large meeting-room was filled with masses of the richest and most lively colours. In the whole, eight hundred and fifty-one varieties were shown, among which the finest were from the garden of William Wells, Esq., of Redleaf; but where all are so excellent, it is almost invidious to particularize. The time will be remembered by many of our readers when gardens in the autumn contained little besides marigolds, sun-flowers, and sweet-peas; by the aid of dahlias and chrysanthemums the autumn has now become the liveliest season of the year, and the beauty of the flower-garden is only destroyed by the severest of the winter frosts. Among the grapes upon the table was a remarkably excellent yellow-berried kind, from Portugal, from the garden of Mr. Holford, of Hampstead, which was quite new to this country. Apples, nectarines, peaches, and pine-apples abounded.

_October_ 2nd.

Among the flowers was a fine bunch of ranunculuses, from Mr. Groom, of Walworth, a rare sight in October; they were obtained by having been planted in July and carefully protected by tulip-shades when coming into flower. The season for softer fruits being nearly over, pears and apples formed the chief display; of these a vast number, upwards of one hundred and eighty of the latter, were upon the table: the Blenheim orange, or Woodstock pippin, pomme gris, scarlet nonpareil, courtpendu plat, golden reinette, and packhorse apples; and Chapman’s, Marie Louise, and brown beurré pears, appeared to us to excel all their rivals. The famous gloux morceau and beurré d’Aremberg pears were also exhibited, but were not ripe. [p427]

_October_ 16th.

The first number of a new periodical work, called the _Pomological Magazine_, consisting of coloured figures of the fruits cultivated in Great Britain, was placed upon the table. Among the apples were specimens of a variety sent from England to Connecticut, in the year 1636, and reimported from America within a few years. It proved to be a kind not known at the present day in this country, but still cultivated in France. In the gardening books of the sixteenth and seventeenth centuries it is mentioned under the name of the haute bonté. The specimens exhibited served to disprove the opinion that many of the American apples are European kinds altered by climate; these, although the produce of trees which have been growing in America for nearly two hundred years, differed in no respect from French samples exhibited at a subsequent meeting of the Society.

_November_ 6th.

An excellent paper was read upon the method of cultivating horse radish, in Denmark. The roots are cut into slips, and planted _horizontally_, the lower end inclining a little upwards, and the crown of the plant hanging over the alleys, by which the beds are separated. From time to time the roots are uncovered, and all the _lateral_ fibres are carefully removed, by which the size and length of the roots are much increased. The place hitherto occupied by dahlias, was now taken by Chinese chrysanthemums, of which a large number, consisting of twenty-two different varieties, was exhibited at the bottom of the room.

_November_ 20th.

Cuttings of the fine new Portugal grape, of which fruit was exhibited on the 18th of September, were distributed to the members present. A few dahlia flowers still showed themselves, notwithstanding the unusual severity of some early frosts, and the room was crowded with chrysanthemums. The gloux morceau and beurré d’Aremberg pears were tasted, and found to retain the station which has been assigned to them at the head of all known varieties. [p428]

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 Ascending node = 334 28 47.1} 1829 Jan.9.72. 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, A_m′_, _m′__m_, _m_M, M_n″_, _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, _m_P, and so forth, which are in the plane of the figure, and their intensities will be proportional to the arcs A_m′_, _m′_, _m_, _m_M, 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: [a simplification which, though sufficiently obvious, had perhaps not occurred to Mr. FRESNEL, until it was pointed out to him by the Translator of this paper.]

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 1/13th 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 1/13th 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_ = _a__r_^2/(2_a__d_−_r_^2), _b_ = _a__r_^2/(4_a__d_−_r_^2), _b_ = _a__r_^2/(8_a__d_−_r_^2); 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.