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Fox teaching was not thrown away, and he early acquired extravagant and dissolute habits. In his younger days he was an outrageous fop, and led the fashion among the ‘macaronis.’ After his visit to Italy he and his cousin posted from Paris to Lyons simply in order to choose patterns for their waistcoats (ib. p. 74); he appeared in London in red-heeled shoes and blue hair-powder, and up to the age of twenty-five, sometimes at least, wore a hat and feather in the House of Commons. In later life he became careless both as to dress and cleanliness. He drank, though perhaps not so hard as many men in his position, and was much addicted to gambling. When a mere boy he became a member of Almack's [see ] gaming club, which was the scene of the most reckless play, and night after night lost sums that soon reached a ruinous amount.

In March 1768, when Fox was in his twentieth year, he was returned for the borough of Midhurst in Sussex, which his father and uncle, Lord Ilchester, had bought for their sons. He took his seat in the following November, and, influenced by the wishes and resentments of his father, joined the supporters of the Duke of Grafton's administration. His first speech was probably made on 9 March 1769, on a point of order. He took an active part in promoting the candidature of Colonel Luttrell for Middlesex, in opposition to Wilkes. On 14 April he spoke with some insolence in support of the motion that Luttrell ought to have been returned, and in the debate on the Middlesex petition on 8 May answered Wedderburn and Burke in a speech which, in spite of some boyishness, delighted his friends, and was praised even by the opposition (ib. p.53;, Debates, i. 406). This speech won him a place among the foremost members of the house. On 9 Oct. he went to Paris with his father and mother, and while there lost heavily at play (Lettres de la Marquise du Deffand, i. 355, 356). He returned to England early in January 1770, and won great applause by two speeches on the Middlesex election. On 24 Feb., when just past twenty-one, he entered Lord North's administration as one of the lords of the admiralty. Fox delivered his speeches without previous preparation, and their power lay not in rhetorical adornments, but in the vigour of the speaker's thoughts, the extent of his knowledge, the quickness with which he grasped the significance of each point in debate, the clearness of his conceptions, and the remarkable plainness with which he laid them before his audience. Even in his longest speeches he never strayed from the matter in hand; he never rose above the level of his hearers' understanding, was never obscure, and never bored the house. Every position that he took up he defended with a large number of shrewd arguments, plainly stated and well ordered. The training in elocution that he had received at Eton and his practice as an amateur actor gave him confidence and ease, while the accuracy and readiness of his memory supplied him with a store of quotations, and rendered him never at a loss for a word. At the same time he does not appear to have been particularly fluent until he became warmed with his subject; then he spoke with a stormy eloquence which carried his hearers with him. His voice was naturally poor, and though he generally modulated it skilfully, he was apt when excited to speak with shrillness. His action was ungraceful. His attempts at pathos generally failed; he was prone to invective, and is said to have been the wittiest speaker of his time. Although some of his speeches introducing subjects to the house are magnificent, he especially excelled in reply; for great as he was as an orator, he was certainly greater in debate. During the first period of his political career, when he was generally contemptuous of popular rights, he spoke with too much flippancy; but ‘in his best days,’ when he was attacking North's administration during the American war, he was in Grattan's opinion the best speaker he had ever heard (Last Journals, i. 85, with a comparison between Fox, Burke, and Townshend;, Preface to Speeches; , Statesmen, i. 236; Quarterly Review, art. by Frere, October 1810; Early Life, p. 331).

In June Fox was in Paris with his father; in November he was supping with Lauzun at the Clob à l'Anglaise, and he returned to England about the middle of January 1771. Much as he loved Paris, he was no favourite with Mme. du Deffand, who described him as ‘hard, bold, and ready;’ he did not, she complained, put his mind to hers, and cared only for play and politics (Lettre, 13 Jan. 1771, ii. 139. See also a somewhat similar character of him by Mme. Neckar, who in 1777 spoke of him as knowing everything, and as cold and cynical,, Miscell. Works, ii. 194). Of the two she preferred Richard Fitzpatrick [q. v.], Fox's connection by marriage, and his constant companion, who at this time shared the lodgings in Piccadilly where Fox lived when his father was absent from Holland House. After joining the administration Fox took a prominent part in several unpopular measures, and especially in the attempt to restrain the press. When on 6 Dec. a committee on the press laws was moved for, he opposed the motion, and jeered the opposition for their declaration that they wished to 'satisfy the


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Rh he will participate in a greatness that is greater even than his own—the supreme greatness of an impersonal work of art. Now, it is a chief glory of the Russian Ballet that it has not only afforded a perfect medium of expression to one of the most outstanding geniuses of the modern stage, but that it has also found in that genius an aptitude for subordination which is among the rarest and finest virtues an artist of the theatre can possess. It is for this reason that our treatment of the art of Nijinsky must needs go further than a consideration of its purely individual aspect. To do less would be unjust to various factors which contribute to the central effect. It would also be unjust to Nijinsky himself, whose method shows such a perfect sense of the action and interaction of elements as complex as they are mutually dependent.

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Rh Indeed, he had too much society, for Amans, a mouton, as prison spies were called, reported his real or alleged conversations to Robespierre. He is said to have drunk too freely, and to have amused himself with backgammon. When Danton, Desmoulins, and others, were arrested, a prison plot was invented as a pretext for cutting short their sham trial. The sole foundation for this was that prisoners, Dillon among them, had resolved, in the event of another September massacre, to sell their lives dearly. Dillon had also written to Desmoulins' wife, enclosing 3000 francs, as was alleged, to be used in exertions for her husband's release, but the turnkey, on second thoughts, detained the letter. (He was guillotined for not having divulged the fact.) After eight months' confinement, Dillon, with Lucile Desmoulins, and nineteen other persons, was tried for this prison plot, his name heading the list. They were all guillotined on the 14th April, 1794.

Dillon was twice married, and had two daughters. One of them, Fanny, married General Bertrand, and was with Napoleon at Elba and St. Helena. He was not, as is sometimes stated, the brother of General Theobald Dillon, who was murdered by his soldiers at Lille in 1792. Theobald was born at Dublin in 1745, entered Dillon's regiment in 1761, was at the attack on Grenada and the siege of


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176 Savannah in 1779, won the cross of St. Louis, and was a member of the Order of Cincinnatus. Brigadier-General under Dumouriez in Flanders, he was ordered to make a feigned attack on Tournay, to prevent it from assisting Mons, which was to be simultaneously attacked by Biron. According to Lord Gower's information, Dillon ventured too near the enemy, and was attacked while his horses were grazing. He ordered a retreat as prearranged, but the retreat became a headlong flight, and the soldiers, thinking Dillon had betrayed them, cut him to pieces in a barn where he had taken refuge, threw the body on a fire in Lille marketplace, and danced savagely round it. The Convention conferred a pension on Josephine Viefville, with whom Dillon had cohabited for nine years, but, as he said in a will made the previous day, had not had time to marry, as also on their three children.

Brigadier-General Thomas Ward, born at Dublin in 1748, had a similar fate to Arthur Dillon. He lodged at the same hotel as Paine, and related to Paine a conversation he had had with Marat. The latter said, Frenchmen were mad to allow foreigners to live among them. They should cut off their ears, let them bleed a few days, and then cut off their heads. "But you yourself are a foreigner," remarked Ward. Paine, in May 1793, informed the Convention committee of this episode, and suggested that they should question Ward upon it. Though


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The Government of the United States, by certain Joint Resolutions, bearing date on the first day of March, in the year A. D., 1845, proposed to the Republic of Texas, then a free, sovereign and independent nation, the annexation of the latter to the former, as one of the co-equal States thereof.

The people of Texas, by Deputies in Convention assembled, on the fourth day of July of the same year, assented to and accepted said proposals, and formed a constitution for the proposed State, upon which, on the twenty-ninth day of December, of the same year, said State was formally received into the confederated Union.

Texas abandoned her separate national existence and Consented to become one of the confederated States, to promote her welfare, insure domestic tranquility and secure more substantially the blessings of liberty and peace to her people. She was received into the confederacy, with her own constitution, under the guarantees of the Federal Constitution and the compact of annexation, that she should enjoy these blessings. She was received as a commonwealth holding, maintaining and protecting the institution known as negro slavery—the servitude of the African to the white race within her limits—a relation that had existed from the first settlement of her wilderness by the white race, and which her people intended should continue to exist in all future time. Her institutions and geographical position


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Rh a general in the French army, Ward was arrested in October 1793, with all the British subjects, as a hostage for Toulon. Robert Plumer Ward, the author of "Tremaine," relates a singular story of his being arrested by mistake for General Ward. Plumer Ward had gone to Barèges for rheumatism, and had been visiting at various country houses. He was arrested, he says, "for having the same name and the same coloured waistcoat as another Ward, guilty of treason, was ordered without trial to Paris to be guillotined, and only escaped by their catching the real traitor. I was, however, banished the Republic merely for my name's sake." It is not easy to reconcile this story with the fact that Thomas was at least twenty years older than Robert, and that Englishmen were not banished but detained. Ward was condemned for the pretended prison plot at the Carmelites, and was executed on the 23rd July 1794. His servant, John Malone, born at Limerick in 1765, shared his fate. Five days more, and Robespierre's fall would have saved them. General James Ferdinand O'Moran had been guillotined four months previously. He was a native of Elphin, Ireland, and was fifty-six years of age. He was arrested in his camp at Cassel in Flanders, on a charge of designing to betray the army to the enemy. It was alleged that he had refused to advance on Furnes, pleading that the Austrians were in superior force, whereas he knew


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 increases. When the interval is very small the discrepancy, though mathematically existent, produces no practical effect; and the illumination at B due to P is as important as that due to A, the intensities of the two luminous sources being supposed equal. Under these conditions it is clear that A and P are not separated in the image. The question is to what amount must the distance AP be increased in order that the difference of situation may make itself felt in the image. This is necessarily a question of degree; but it does not require detailed calculations in order to show that the discrepancy first becomes conspicuous when the phases corresponding to the various secondary waves which travel from P to B range over a complete period. The illumination at B due to P then becomes comparatively small, indeed for some forms of aperture evanescent. The extreme discrepancy is that between the waves which travel through the outermost parts of the object-glass at L and L′; so that if we adopt the above standard of resolution, the question is where must P be situated in order that the relative retardation of the rays PL and PL’ may on their arrival at B amount to a wave-length (λ). In virtue of the general law that the reduced optical path is stationary in value, this retardation may be calculated without allowance for the different paths pursued on the farther side of L, L′, so that the value is simply PL − PL′. Now since AP is very small, AL′ − PL′ = AP sin α, where α is the angular semi-aperture L′AB. In like manner PL − AL has the same value, so that PL − PL′＝2AP sin α.

According to the standard adopted, the condition of resolution is therefore that AP, or ε, should exceed λ/sin α. If ε be less than this, the images overlap too much; while if ε greatly exceed the above value the images become unnecessarily separated.

In the above argument the whole space between the object and the lens is supposed to be occupied by matter of one refractive index, and λ represents the wave-length in this medium of the kind of light employed. If the restriction as to uniformity be violated, what we have ultimately to deal with is the wave-length in the medium immediately surrounding the object.

Calling the refractive index μ, we have as the critical value of ε,

λ0 being the wave-length in vacuo. The denominator μ sin α is the quantity well known (after Abbe) as the “numerical aperture.”

The extreme value possible for α is a right angle, so that for the microscopic limit we have

The limit can be depressed only by a diminution in λ0, such as photography makes possible, or by an increase in μ, the refractive index of the medium in which the object is situated.

The statement of the law of resolving power has been made in a form appropriate to the microscope, but it admits also of immediate application to the telescope. If 2R be the diameter of the object-glass and D the distance of the object, the angle subtended by AP is ε/D, and the angular resolving power is given by

This method of derivation (substantially due to Helmholtz) makes it obvious that there is no essential difference of principle between the two cases, although the results are conveniently stated in different forms. In the case of the telescope we have to deal with a linear measure of aperture and an angular limit of resolution, whereas in the case of the microscope the limit of resolution is linear, and it is expressed in terms of angular aperture.

It must be understood that the above argument distinctly assumes that the different parts of the object are self-luminous, or at least that the light proceeding from the various points is without phase relations. As has been emphasized by G. J. Stoney, the restriction is often, perhaps usually, violated in the microscope. A different treatment is then necessary, and for some of the problems which arise under this head the method of Abbe is convenient.

The importance of the general conclusions above formulated, as imposing a limit upon our powers of direct observation, can hardly be overestimated; but there has been in some quarters a tendency to ascribe to it a more precise character than it can bear, or even to mistake its meaning altogether. A few words of further explanation may therefore be desirable. The first point to be emphasized is that nothing whatever is said as to the smallness of a single object that may be made visible. The eye, unaided or armed with a telescope, is able to see, as points of light, stars subtending no sensible angle. The visibility of a star is a question of brightness simply, and has nothing to do with resolving power. The latter element enters only when it is a question of recognizing the duplicity of a double star, or of distinguishing detail upon the surface of a planet. So in the microscope there is nothing except lack of light to hinder the visibility of an object however small. But if its dimensions be much less than the half wave-length, it can only be seen as a whole, and its parts cannot be distinctly separated, although in cases near the border line some inference may be possible, founded upon experience of what appearances are presented in various cases. Interesting observations upon particles, ultra-microscopic in the above sense, have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy (Drude’s Ann., 1903, 10, p. 1).

In a somewhat similar way a dark linear interruption in a bright ground may be visible, although its actual width is much inferior to the half wave-length. In illustration of this fact a simple experiment may be mentioned. In front of the naked eye was held a piece of copper foil perforated by a fine needle hole. Observed through this the structure of some wire gauze just disappeared at a distance from the eye equal to 17 in., the gauze containing 46 meshes to the inch. On the other hand, a single wire 0·034 in. in diameter remained fairly visible up to a distance of 20 ft. The ratio between the limiting angles subtended by the periodic structure of the gauze and the diameter of the wire was (·022/·034) × (240/17) = 9·1. For further information upon this subject reference may be made to ''Phil. Mag., 1896, 42, p. 167; Journ. R. Micr. Soc.'', 1903, p. 447.

 6. Coronas or Glories.—The results of the theory of the diffraction patterns due to circular apertures admit of an interesting application to coronas, such as are often seen encircling the sun and moon. They are due to the interposition of small spherules of water, which act the part of diffracting obstacles. In order to the formation of a well-defined corona it is essential that the particles be exclusively, or preponderatingly, of one size.

If the origin of light be treated as infinitely small, and be seen in focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the absence of obstacles would be concentrated in the immediate neighbourhood of the focus. At other parts of the field the effect is the same, in accordance with the principle known as Babinet’s, whether the imaginary screen in front of the object-glass is generally transparent but studded with a number of opaque circular disks, or is generally opaque but perforated with corresponding apertures. Since at these points the resultant due to the whole aperture is zero, any two portions into which the whole may be divided must give equal and opposite resultants. Consider now the light diffracted in a direction many times more oblique than any with which we should be concerned, were the whole aperture uninterrupted, and take first the effect of a single small aperture. The light in the proposed direction is that determined by the size of the small aperture in accordance with the laws already investigated, and its phase depends upon the position of the aperture. If we take a direction such that the light (of given wave-length) from a single aperture vanishes, the evanescence continues even when the whole series of apertures is brought into contemplation. Hence, whatever else may happen, there must be a system of dark rings formed, the same as from a single small aperture. In directions other than these it is a more delicate question how the partial effects should be compounded. If we make the extreme suppositions of an infinitely small source and absolutely homogeneous light, there is no escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the chance distribution of apertures. If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in motion, and the discrimination of different directions imperfect, we are concerned merely with the mean brightness found by varying the arbitrary phase-relations, and this is obtained by simply multiplying the brightness due to a single aperture by the number of apertures (n) (see, § 4). The diffraction pattern is therefore that due to a single aperture, merely brightened n times.

In his experiments upon this subject Fraunhofer employed plates of glass dusted over with lycopodium, or studded with small metallic disks of uniform size; and he found that the diameters of the rings were proportional to the length of the waves and inversely as the diameter of the disks.

In another respect the observations of Fraunhofer appear at first sight to be in disaccord with theory; for his measures of the diameters of the red rings, visible when white light was employed, correspond with the law applicable to dark rings, and not to the different law applicable to the luminous maxima. Verdet has, however, pointed out that the observation in this form is essentially different from that in which homogeneous red light is employed, and that the position of the red rings would correspond to the absence of blue-green light rather than to the greatest abundance of red light. Verdet’s own observations, conducted with great care, fully confirm this view, and exhibit a complete agreement with theory.

By measurements of coronas it is possible to infer the size of the particles to which they are due, an application of considerable interest in the case of natural coronas—the general rule being the larger the corona the smaller the water spherules. Young employed this method not only to determine the diameters of cloud particles (e.g. 1⁄1000 in.), but also those of fibrous material, for which the theory is analogous. His instrument was called the eriometer (see “Chromatics,” vol. iii. of supp. to Ency. Brit., 1817).

 7. ''Influence of Aberration. Optical Power of Instruments.''—Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In

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 Rh is enough. And if Nijinsky never danced again we should know that his fame would be safe—the fame of one who, more perhaps than any man living, has made beauty for his generation.

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The praise which the Russian Ballet has almost universally received in England seems to offer one of the rare instances of our ability to appreciate a good thing when we see it. On that we may congratulate ourselves. But I am reminded that this appreciation is by no means universal, and that in Russia itself there is a section of opinion to whom the later developments of the Diaghilew ballet are frankly abhorrent.

Hitherto, we must remember, the ballet in Russia has been an exclusively aristocratic form of art. It has relied 101

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Fox people.’ Where, he asked, was he to look for the complaints of the people? he refused to recognise the people apart from the majority of the house, their legal representatives (Speeches, i. 5). He took the same line on 25 March 1771, when urging the committal of Alderman Oliver for discharging the printers apprehended by the officers of the house. His action in this affair rendered him exceedingly unpopular, and on the 27th he and his brother were attacked by a mob as they drove down to the house, and he was rolled in the mud. Zealous for privilege of every kind, he gave much satisfaction to his party ‘by the great talents he exerted’ in opposing the Nullum Tempos Bill. Junius had hitherto virtually left him alone, but his opposition to the popular cause of the Duke of Portland called forth a sharp rebuke in the ‘Public Advertiser’ of 4 March, signed ‘Ulysses.’ Fox wished to challenge the writer, but was unable to identify him (Life of Sir P. Francis, i. 255). A letter of Junius in October provoked an answer signed ‘An Old Correspondent,’ which was attributed to Fox. A reply appeared signed ‘Anti-Fox,’ in which the writer warns ‘my pretty black boy’ that if provoked Junius might cease to spare Lord Holland and his family (Letters of Junius, ii. 384). His contempt for the wishes of the people provoked a caricature entitled ‘The Death of the Foxes’ in the ‘Oxford Magazine’ of February 1770. In this he appears with his father and brother, and his corpulence is ridiculed. Another caricature in the same magazine in December 1773 represents him as picking his father's pocket, in reference to his gambling debts.

On 6 Feb. 1772 Fox spoke against the clerical petition for relief from subscription to the articles, though he condemned the custom of requiring subscription from lads at the universities. He prepared himself for his defence of the church ‘by passing twenty-two hours in the pious exercise of hazard,’ losing during that time 11,000l. (, Miscellaneous Works, ii. 74). A twelvemonth later he supported a motion for a committee on the subject of subscription, and further showed that, in spite of his zeal for privilege, he was not to be reckoned among those who were content to forward the king's wishes on all points, for he acted as teller for a bill for the relief of protestant dissenters; the king declared that ‘his conduct could not be attributed to conscience, but to his aversion to all restraints’ (Speeches, i. 17; George III, Letters to Lord North, i. 89: this letter, dated 1772, seems to belong to 1773; comp. Parl. Hist. xvii. 758). On 20 Feb. 1772 he resigned office. Although he had some private grounds of dissatisfaction with North (Memorials, i. 73; Last Journals, i. 23), the chief cause of his resignation was that he intended to oppose the Royal Marriage Bill. The circumstances of his parents' marriage rendered him jealous of all needless restrictions on marriage; he had already obtained leave to bring in a bill to amend the marriage act, and he chose to sacrifice office rather than assent to the restrictions that the king was bent on placing on the marriages of his house. North was terrified by the report of his intended resignation, and withdrew one of the most objectionable clauses of the bill. Fox joined Conway and Burke in opposing the bill, and was ‘universally allowed to have seized the just point of argument throughout with amazing rapidity and clearness’ (ib. p. 59). At least as early as 1766 he had become acquainted with Burke, and had learnt to respect his opinion (Memorials, i. 26), and this temporary co-operation with him can scarcely have been without some effect on his later career. Fox introduced his own marriage bill on 7 April, having that morning, after a night spent in drinking, returned from Newmarket, where he had lost heavily; he spoke with effect, but took no more trouble about the bill, which was thrown out at a later stage. In December he re-entered the administration as a junior lord of the treasury. Although Olive had been absolved by parliament, Fox took the opportunity of a debate on the affairs of India in June 1773 to attack him with unsparing vehemence. He recommenced his assaults on the press. In a debate he had raised on this subject on 16 Feb. 1774 he rebuked T. Townshend for coupling the name of Johnson with that of Shebbeare (Speeches, i. 25). Johnson never forgot his warm defence (, Life, iv. 315). Fox had lately been elected a member of the club; he was generally silent when Johnson was present (ib. 179). He was naturally shy, but when in the society of those with whom he felt at ease would ‘talk on for ever with all the openness and simplicity of a child’ (, Table-talk, p. 75); his conversation was always easy and full of anecdote. Office exercised no restraint upon him. He forced North against his will to persist in a proposal that the printer Woodfall should be committed to the Gatehouse for printing a letter containing charges against the speaker. The minister was defeated, and the king, who already disliked Fox for the part he had taken against the Royal Marriage Bill, and in support of the relief bill of the year before, was furious at his presumption. ‘That young man,’ he wrote, ‘has so thoroughly cast off every principle of common honour and honesty that he must soon become as  Rh


text: Dictionary of National Biography, 1885-1900/Trotter, Thomas


text: Dictionary of National Biography, 1885-1900/Troubridge, Edward Thomas


text: Dictionary of National Biography, 1885-1900/Troubridge, Thomas (1758?-1807)


text: Dictionary of National Biography, 1885-1900/Troubridge, Thomas St. Vincent Hope Cochrane


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 practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wave-surface from its intended position. In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wave-length. Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding can have but little effect.

The only case in which the influence of small aberration upon the entire image has been calculated (Phil. Mag., 1879) is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx³. The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half. No terms in x or x2 need be considered. The first would correspond to a general turning of the beam; and the second would imply imperfect focusing of the central parts. The effect of aberration may be considered in two ways. We may suppose the aperture (a) constant, and inquire into the operation of an increasing aberration; or we may take a given value of c (i.e. a given wave-surface) and examine the effect of a varying aperture. The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 3), but it increases unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should accompany the increase of aperture.

If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e. when the wave-surface deviates at each end by a quarter wave-length from the true plane.

As an application of this result, let us investigate what amount of temperature disturbance in the tube of a telescope may be expected to impair definition. According to J. B. Biot and F. J. D. Arago, the index for air at t° C. and at atmospheric pressure is given by $\mu - 1 = \tfrac{\cdot00029}{1 + \cdot0037 t}.$ If we take 0° C. as standard temperature, ＝−1·1 × 10−6. Thus, on the supposition that the irregularity of temperature t extends through a length l, and produces an acceleration of a quarter of a wave-length, ＝1·1 lt × 10−6; or, if we take = 5·3 × 10−5, lt＝12, the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a stratum of air heated 1° C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would produce a not insensible effect. If the change of temperature progressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less difference of temperature. S. P. Langley has proposed to obviate such ill-effects by stirring the air included within a telescope tube. It has long been known that the definition of a carbon bisulphide prism may be much improved by a vigorous shaking.

We will now consider the application of the principle to the formation of images, unassisted by reflection or refraction (Phil. Mag., 1881). The function of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture. If we suppose the diameter of the lens to be given (2R), and its focal length gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When ƒ attains a certain value, say ƒ1, the extreme error of phase to be compensated falls to. But, as we have seen, such an error of phase causes no sensible deterioration in the definition; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an object-glass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of in., about that of the pupil of the eye. The distance ƒ1, which the actual focal length must exceed, is given by √ (ƒ12 + R2) − ƒ1 ＝; so that

Thus, if =, R = , we find ƒ1＝800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66 ft., through a hole in. in diameter, is therefore at least as well defined as that seen direct.

As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 in., ƒ1 would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be in., an achromatic lens has no sensible advantage if the focal length be greater than about 11 in. If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1·7 in.

Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration. An admissible error of phase of will correspond to an error of λ in a reflecting and  in a (glass) refracting surface, the incidence in both cases being perpendicular. If we inquire what is the greatest admissible longitudinal aberration (ƒ) in an object-glass according to the above rule, we find

α being the angular semi-aperture.

In the case of a single lens of glass with the most favourable curvatures, ƒ is about equal to 2ƒ, so that 4 must not exceed /ƒ. For a lens of 3 ft. focus this condition is satisfied if the aperture does not exceed 2 in.

When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3 ft. focus might have an aperture of 2 in., and the image would not suffer materially from aberration.

On the same principle we may estimate the least visible displacement of the eye-piece of a telescope focused upon a distant object, a question of interest in connexion with range-finders. It appears (Phil. Mag., 1885, 20, p. 354) that a displacement ƒ from the true focus will not sensibly impair definition, provided

2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the ultimate optical efficiency in the method of “mirror-reading,” as commonly practised in various physical observations. A rotation by which one edge of the mirror advances (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, and of the employment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized.

A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest. The limiting efficiency of the microscope is attained when the angular aperture amounts to 180°; and it is evident that a lateral displacement of the point under observation through entails (at the old image) a phase-discrepancy of a whole period, one extreme ray being accelerated and the other retarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror.

We have seen that in perpendicular reflection a surface error not exceeding may be admissible. In the case of oblique reflection at an angle φ, the error of retardation due to an elevation BD (fig. 5) is QQ′ − QS＝BD sec φ(1 − cos SQQ′)＝BD sec (1 + cos 2)＝2BD cos ;

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 from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflection. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory. Let A0B0 be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A0B0 to A or B is determined by the condition that the optical distance, ∫ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus

We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A0B0 to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A0B0 to A is represented by ∫( + )ds, the integration being along the original path A0 A; and similarly the optical distance between A0B0 and B is represented by ∫( + )ds, the integration being along B0  B. In virtue of (4) the difference of the optical distances to A and B is

The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, ∫ ds＝.s, where s is simply the thickness traversed by the ray. If t2 and t1 be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion is given by ＝(t2 − t1)/a, or, if t1 be negligible,

The condition of resolution of a double line whose components subtend an angle is that  must exceed /a. Hence, in order that a double line may be resolved whose components have indices and  +, it is necessary that t should exceed the value given by the following equation:—

 8. Diffraction Gratings.—Under the heading “Colours of Striated Surfaces,” Thomas Young (Phil. Trans., 1802) in his usual summary fashion gave a general explanation of these colours, including the law of sines, the striations being supposed to be straight, parallel and equidistant. Later, in his article “Chromatics” in the supplement to the 5th edition of this encyclopaedia, he shows that the colours “lose the mixed character of periodical colours, and resemble much more the ordinary prismatic spectrum, with intervals completely dark interposed,” and explains it by the consideration that any phase-difference which may arise at neighbouring striae is multiplied in proportion to the total number of striae.

The theory was further developed by A. J. Fresnel (1815), who gave a formula equivalent to (5) below. But it is to J. von Fraunhofer that we owe most of our knowledge upon this subject. His recent discovery of the “fixed lines” allowed a precision of observation previously impossible. He constructed gratings up to 340 periods to the inch by straining fine wire over screws. Subsequently he ruled gratings on a layer of gold-leaf attached to glass, or on a layer of grease similarly supported, and again by attacking the glass itself with a diamond point. The best gratings were obtained by the last method, but a suitable diamond point was hard to find, and to preserve. Observing through a telescope with light perpendicularly incident, he showed that the position of any ray was dependent only upon the grating interval, viz. the distance from the centre of one wire or line to the centre of the next, and not otherwise upon the thickness of the wire and the magnitude of the interspace. In different gratings the lengths of the spectra and their distances from the axis were inversely proportional to the grating interval, while with a given grating the distances of the various spectra from the axis were as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of wave-lengths, and the method of separating the overlapping spectra by a prism dispersing in the perpendicular direction. He described also the complicated patterns seen when a point of light is viewed through two superposed gratings, whose lines cross one another perpendicularly or obliquely. The above observations relate to transmitted light, but Fraunhofer extended his inquiry to the light reflected. To eliminate the light returned from the hinder surface of an engraved grating, he covered it with a black varnish. It then appeared that under certain angles of incidence parts of the resulting spectra were completely polarized. These remarkable researches of Fraunhofer, carried out in the years 1817–1823, are republished in his Collected Writings (Munich, 1888).

The principle underlying the action of gratings is identical with that discussed in § 2, and exemplified in J. L. Soret’s “zone plates.” The alternate Fresnel’s zones are blocked out or otherwise modified; in this way the original compensation is upset and a revival of light occurs in unusual directions. If the source be a point or a line, and a collimating lens be used, the incident waves may be regarded as plane. If, further, on leaving the grating the light be received by a focusing lens, e.g. the object-glass of a telescope, the Fresnel’s zones are reduced to parallel and equidistant straight strips, which at certain angles coincide with the ruling. The directions of the lateral spectra are such that the passage from one element of the grating to the corresponding point of the next implies a retardation of an integral number of wave-lengths. If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture. For an investigation upon these lines the reader is referred to Airy’s Tracts, to Verdet’s Leçons, or to R. W. Wood’s Physical Optics. If, however, we assume the theory of a simple rectangular aperture (§ 3); the results of the ruling can be inferred by elementary methods, which are perhaps more instructive.

Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest. At the middle of this band there is complete agreement of phase among the secondary waves. The dark lines which separate the bands are the places at which the phases of the secondary wave range over an integral number of periods. If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width d, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed. At the central point there is still complete agreement of phase; but the amplitude is diminished in the ratio of a : a + d. In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period. The only effect of the ruling is to diminish the amplitude in the ratio a : a + d; and, except for the difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free.

The lateral (spectral) images occur in such directions that the projection of the element (a + d) of the grating upon them is an exact multiple of. The effect of each of the n elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a : d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mn, may be called the principal directions. On either side of any one of them the illumination is distributed according to the same law as for the central image (m＝0), vanishing, for example, when the retardation amounts to (mn ± 1). In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width d. The phase of the resultant effect is by symmetry that of the component which comes from the middle of a. The fact that the other components have phases differing from this by amounts ranging between ± am/(a + d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement).

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text: Dictionary of National Biography, 1885-1900/Troy, John Thomas


text: Dictionary of National Biography, 1885-1900/Trübner, Nicholas


text: Dictionary of National Biography, 1885-1900/Trubshaw, James


text: Dictionary of National Biography, 1885-1900/Truman, Joseph


text: Dictionary of National Biography, 1885-1900/Trumbull, William (d.1635)


text: Dictionary of National Biography, 1885-1900/Trumbull, William (1639-1716)


text: Dictionary of National Biography, 1885-1900/Trusler, John


text: 1911 Encyclopædia Britannica/Diffraction of Light/8


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is a native of Warsaw. He was born some twenty-three years ago, the son of parents who were both closely connected with the branch of the Russian Imperial Ballet established in that city. His mother was a dancer. Likewise his father, who was early promoted to the rank of ballet-master, and took an honourable part in the development of a national school of dancing as opposed to the Italian school whose influence in Warsaw had previously been supreme. The young Nijinsky soon began to manifest the characteristics of genius, and at nine years of age he was transferred from

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 CHAPTER III

THE ART OF NIJINSKY

As we saw in the last chapter, Nijinsky's technique is the result of a system of training which, however superior, is similar in kind to that found in several other European schools of ballet dancing. We know how high a standard of skill may be attained by almost anyone with an aptitude for dancing if trained on these lines. Indeed there are several dancers in M. Diaghilew's company, and we have seen others at various times and places, whom one would not hesitate to put in the very foremost rank. But about Nijinsky's dancing there is something altogether unique—an exotic quality 24

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 If Bm denote the brightness of the mth lateral image, and B0 that of the central image, we have

If B denotes the brightness of the central image when the whole of the space occupied by the grating is transparent, we have B0 : B＝a2 : (a + d)2, and thus

The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only 1/m22 of the original light can be obtained in the mth spectrum. We conclude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to 1/2, or about, and that for this purpose a and d must be equal. When d = a the general formula becomes

showing that, when m is even, Bm vanishes, and that, when m is odd, Bm : B＝1/m22. The third spectrum has thus only of the brilliancy of the first.

Another particular case of interest is obtained by supposing a small relatively to (a + d). Unless the spectrum be of very high order, we have simply

so that the brightnesses of all the spectra are the same.

The light stopped by the opaque parts of the grating, together with that distributed in the central image and lateral spectra, ought to make up the brightness that would be found in the central image, were all the apertures transparent. Thus, if a = d, we should have $1 = \frac{1}{2} + \frac{1}{4} + \frac{2}{\pi^2} \left(1 + \frac{1}{9} + \frac{1}{25} + \ldots\right),$ which is true by a known theorem. In the general case $\frac{a}{a + d} = \left(\frac{a}{a + d}\right)^2 + \frac{2}{\pi^2}\sum^{m=\infty}_{m=1}\frac{1}{m^2}\sin^2\left(\frac{m\pi a}{a + d}\right),$ a formula which may be verified by Fourier’s theorem.

According to a general principle formulated by J. Babinet, the brightness of a lateral spectrum is not affected by an interchange of the transparent and opaque parts of the grating. The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero. So far as the application to gratings is concerned, the same conclusion may be derived from (2).

From the value of Bm : B0 we see that no lateral spectrum can surpass the central image in brightness; but this result depends upon the hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grating there is no opaque material present by which light could be absorbed, and the effect depends upon a difference of retardation in passing the alternate parts. It is possible to prepare gratings which give a lateral spectrum brighter than the central image, and the explanation is easy. For if the alternate parts were equal and alike transparent, but so constituted as to give a relative retardation of, it is evident that the central image would be entirely extinguished, while the first spectrum would be four times as bright as if the alternate parts were opaque. If it were possible to introduce at every part of the aperture of the grating an arbitrary retardation, all the light might be concentrated in any desired spectrum. By supposing the retardation to vary uniformly and continuously we fall upon the case of an ordinary prism: but there is then no diffraction spectrum in the usual sense. To obtain such it would be necessary that the retardation should gradually alter by a wave-length in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length (Phil. Mag., 1874, 47, p. 193). It is not likely that such a result will ever be fully attained in practice; but the case is worth stating, in order to show that there is no theoretical limit to the concentration of light of assigned wave-length in one spectrum, and as illustrating the frequently observed unsymmetrical character of the spectra on the two sides of the central image.

We have hitherto supposed that the light is incident perpendicularly upon the grating; but the theory is easily extended. If the incident rays make an angle with the normal (fig. 6), and the diffracted rays make an angle  (upon the same side), the relative retardation from each element of width (a + d) to the next is (a + d) (sin + sin); and this is the quantity which is to be equated to m. Thus

The “deviation” is ( + ), and is therefore a minimum when =, i.e. when the grating is so situated that the angles of incidence and diffraction are equal.

In the case of a reflection grating the same method applies. If and  denote the angles with the normal made by the incident and diffracted rays, the formula (5) still holds, and, if the deviation be reckoned from the direction of the regularly reflected rays, it is expressed as before by ( + ), and is a minimum when  =, that is, when the diffracted rays return upon the course of the incident rays.

In either case (as also with a prism) the position of minimum deviation leaves the width of the beam unaltered, i.e. neither magnifies nor diminishes the angular width of the object under view.

From (5) we see that, when the light falls perpendicularly upon a grating ( = 0), there is no spectrum formed (the image corresponding to m = 0 not being counted as a spectrum), if the grating interval or (a + d) is less than. Under these circumstances, if the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Fraunhofer argued that there must be a microscopic limit represented by ; and the inference is plausible, to say the least (Phil. Mag., 1886). Fraunhofer should, however, have fixed the microscopic limit at, as appears from (5), when we suppose = ,  =.

We will now consider the important subject of the resolving power of gratings, as dependent upon the number of lines (n) and the order of the spectrum observed (m). Let BP (fig. 8) be the direction of the principal maximum (middle of central band) for the wave-length in the mth spectrum. Then the relative retardation of the extreme rays (corresponding to the edges A, B of the grating) is mn. If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn + 1). Suppose now that +  is the wave-length for which BQ gives the principal maximum, then (mn + 1)＝mn( + ); whence

According to our former standard, this gives the smallest difference of wave-lengths in a double line which can be just resolved; and we conclude that the resolving power of a grating depends only upon the total number of lines, and upon the order of the spectrum, without regard to any other considerations. It is here of course assumed that the n lines are really utilized.

In the case of the D lines the value of is about 1/1000; so that to resolve this double line in the first spectrum requires 1000 lines, in the second spectrum 500, and so on.

It is especially to be noticed that the resolving power does not depend directly upon the closeness of the ruling. Let us take the case of a grating 1 in. broad, and containing 1000 lines, and consider the effect of interpolating an additional 1000 lines, so as to bisect the former intervals. There will be destruction by interference of the first, third and odd spectra generally; while the advantage gained in the spectra of even order is not in dispersion, nor in resolving power, but simply in brilliancy, which is increased four times. If we now suppose half the grating cut away, so as to leave 1000 lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved.

There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by Rowland, the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. However this may be, it would always be possible, with the aid of a grating of given resolving power, to construct artificially from white light mixtures of slightly different wave-length whose resolution or otherwise would discriminate between powers inferior and superior to the given one.

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 CHAPTER IV

THE BALLETS

the end of the book will be found a complete list of the ballets performed in England in which Nijinsky himself has taken part. Here we deal, rather more fully, with nine typical ballets, being those which Miss Dorothy Mullock has chosen as subjects for her pictures.

This ballet may be fitly taken first and by way of introduction to those which follow, since it stands rather by itself—neither definitely dramatic nor merely illustrative of its musical accompaniment. In spirit it belongs to that older kind of 34

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 If we define as the “dispersion” in a particular part of the spectrum the ratio of the angular interval d to the corresponding increment of wave-length d, we may express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to mnd. Hence, if a be the width of the diffracted beam, and d the angle through which the wave-front is turned, ad＝mn d, or

The resolving power and the width of the emergent beam fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power.

In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of m a high resolving power is attainable with a moderate number of lines. But this procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. According to the principle already laid down it can make but little difference in the principal direction corresponding to the first spectrum, provided each line lie within a quarter of an interval (a＋d) from its theoretical position. But, to obtain an equally good result in the mth spectrum, the error must be less than 1/m of the above amount.

There are certain errors of a systematic character which demand special consideration. The spacing is usually effected by means of a screw, to each revolution of which corresponds a large number (e.g. one hundred) of lines. In this way it may happen that although there is almost perfect periodicity with each revolution of the screw after (say) 100 lines, yet the 100 lines themselves are not equally spaced. The “ghosts” thus arising were first described by G. H. Quincke (Pogg. Ann., 1872, 146, p. 1), and have been elaborately investigated by C. S. Peirce (Ann. Journ. Math., 1879, 2, p. 330), both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously existing spectra of odd order vanish. But a very slight relative displacement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but complete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra; but in any case the compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate.

The effect of a gradual increase in the interval (fig. 9) as we pass across the grating has been investigated by M. A. Cornu (C.R., 1875, 80, p. 655), who thus explains an anomaly observed by E. E. N. Mascart. The latter found that certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spectrum under observation, and decreases towards the hinder edge. It is evident that the waves from both halves of the grating are accelerated in an increasing degree, as we pass from the centre outwards, as compared with the phase they would possess were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a convex lens, which accelerates the marginal relatively to the central rays. On the other side the effect is reversed. This kind of irregularity may clearly be present in a degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect.

It may be worth while to examine further the other variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane. If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface

and, as we have just seen, the term in x2 corresponds to a linear error in the spacing. In like manner, the term in y2 corresponds to a general curvature of the lines (fig. 10), and does not influence the definition at the (primary) focus, although it may introduce astigmatism. If we suppose that everything is symmetrical on the two sides of the primary plane y = 0, the coefficients B, vanish. In spite of any inequality between and ′, the definition will be good to this order of approximation, provided  and  vanish. The former measures the thickness of the primary focal line, and the latter measures its curvature. The error of ruling giving rise to is one in which the intervals increase or decrease in both directions from the centre outwards (fig. 11), and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope. The term in corresponds to a variation of curvature in crossing the grating (fig. 12).

When the plane zx is not a plane of symmetry, we have to consider the terms in xy, x2y, and y3. The first of these corresponds to a deviation from parallelism, causing the interval to alter gradually as we pass along the lines (fig. 13). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y = ± x. The term in x2y corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 14); and that in y3 would be caused by a curvature such that there is a point of inflection at the middle of each line (fig. 15).

All the errors, except that depending on, and especially those depending on and , can be diminished, without loss of resolving power, by contracting the vertical aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating.

The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings purposely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra.

A similar idea appears to have guided H. A. Rowland to his brilliant invention of concave gratings, by which spectra can be photographed without any further optical appliance. In these instruments the lines are ruled upon a spherical surface of speculum metal, and mark the intersections of the surface by a system of parallel and equidistant planes, of which the middle member passes through the centre of the sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 16) represent the surface of the grating, O being the centre of the circle. Then, if Q be any radiant point and Q′ its image (primary focus) in the spherical mirror AP, we have $\frac{1}{v_{1}} + \frac{1}{u} = \frac{2\cos\phi}{a},$|undefined where v1 = AQ′, u = AQ, a = OA, = angle of incidence QAO, equal to the angle of reflection Q′AO. If Q be on the circle described upon OA as diameter, so that u = a cos, then Q′ lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon ) vanishes.

This disposition is adopted in Rowland′s instrument; only, in addition to the central image formed at the angle ′ =, there are a series of spectra with various values of ′, but all disposed upon the same circle. Rowland’s investigation is contained in the paper already referred to; but the following account of the theory is in the form adopted by R. T. Glazebrook (Phil. Mag., 1883).

In order to find the difference of optical distances between the courses QAQ′, QPQ′, we have to express QP − QA, PQ′ − AQ′. To find the former, we have, if OAQ =, AOP = , QP2＝u2＋4a2sin2 − 4au sin sin ( − ) ＝(u＋a sin sin )2 − a2 sin2 sin2＋4a sin2 (a − u cos).

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42 "As we were straightened to the very utmost, I undertook to draw portraits at the low price of five dollars per head, in black chalk. I drew a few gratis, and succeeded so well that ere many days had elapsed I had an abundance of work." His fame spread, his orders increased. A settler came for him in the middle of the night from a considerable distance to have the portrait of his mother taken while she was on the eve of death, and a clergyman had his child's body exhumed that the artist might restore to him the lost features. Money flowed in and he was soon again established with his family in a house in Louisville. His drawings of birds still continued and, he says, became at times almost a mania with him; he would frequently give up a head, the profits of which would have supplied the wants of his family a week or more, "to represent a little citizen of the feathered tribe."


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 Now as far as 4 4 sin2 ＝sin2 + sin4, and thus to the same order QP2＝(u + a sin sin )2 − a cos (u − a cos ) sin2 + a(a − u cos ) sin4. pose that Q lies on the circle u = a cos, the middle term vanishes, and we get, correct as far as 4, $\mathrm{QP} = (u + a \sin \phi \sin \omega ) \sqrt{\bigg.} \left\{1 + \frac{a^2 \sin^2 \phi \sin^{4}\omega}{4u}\right\};$ so that

in which it is to be noticed that the adjustment necessary to secure the disappearance of sin2 is sufficient also to destroy the term in sin3.

A similar expression can be found for Q′P − Q′A; and thus, if Q′A = v, Q′AO = ′, where v = a cos ′, we get

If ′ =, the term of the first order vanishes, and the reduction of the difference of path via P and via A to a term of the fourth order proves not only that Q and Q′ are conjugate foci, but also that the foci are exempt from the most important term in the aberration. In the present application ′ is not necessarily equal to ; but if P correspond to a line upon the grating, the difference of retardations for consecutive positions of P, so far as expressed by the term of the first order, will be equal to ± m (m integral), and therefore without influence, provided

where denotes the constant interval between the planes containing the lines. This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions. They are here focused (so far as the rays in the primary plane are concerned) upon the circle OQ′A, and the outstanding aberration is of the fourth order.

In order that a large part of the field of view may be in focus at once, it is desirable that the locus of the focused spectrum should be nearly perpendicular to the line of vision. For this purpose Rowland places the eye-piece at O, so that = 0, and then by (11) the value of ′ in the mth spectrum is

If now relate to the edge of the grating, on which there are altogether n lines, n＝2a sin , and the value of the last term in (10) becomes nsin3 sin ′ tan ′, or

This expresses the retardation of the extreme relatively to the central ray, and is to be reckoned positive, whatever may be the signs of, and ′. If the semi-angular aperture be, and tan ′ = 1, mn might be as great as four millions before the error of phase would reach. If it were desired to use an angular aperture so large that the aberration according to (13) would be injurious, Rowland points out that on his machine there would be no difficulty in applying a remedy by making slightly variable towards the edges. Or, retaining constant, we might attain compensation by so polishing the surface as to bring the circumference slightly forward in comparison with the position it would occupy upon a true sphere.

It may be remarked that these calculations apply to the rays in the primary plane only. The image is greatly affected with astigmatism; but this is of little consequence, if in (8) be small enough. Curvature of the primary focal line having a very injurious effect upon definition, it may be inferred from the excellent performance of these gratings that is in fact small. Its value does not appear to have been calculated. The other coefficients in (8) vanish in virtue of the symmetry.

The mechanical arrangements for maintaining the focus are of great simplicity. The grating at A and the eye-piece at O are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, AQ at right angles to each other. A tie between the middle point of the rod OA and Q can be used if thought desirable.

The absence of chromatic aberration gives a great advantage in the comparison of overlapping spectra, which Rowland has turned to excellent account in his determinations of the relative wave-lengths of lines in the solar spectrum (Phil. Mag., 1887).

For absolute determinations of wave-lengths plane gratings are used. It is found (Bell, Phil. Mag., 1887) that the angular measurements present less difficulty than the comparison of the grating interval with the standard metre. There is also some uncertainty as to the actual temperature of the grating when in use. In order to minimize the heating action of the light, it might be submitted to a preliminary prismatic analysis before it reaches the slit of the spectrometer, after the manner of Helmholtz.

In spite of the many improvements introduced by Rowland and of the care with which his observations were made, recent workers have come to the conclusion that errors of unexpected amount have crept into his measurements of wave-lengths, and there is even a disposition to discard the grating altogether for fundamental work in favour of the so-called “interference methods,” as developed by A. A. Michelson, and by C. Fabry and J. B. Pérot. The grating would in any case retain its utility for the reference of new lines to standards otherwise fixed. For such standards a relative accuracy of at least one part in a million seems now to be attainable.

Since the time of Fraunhofer many skilled mechanicians have given their attention to the ruling of gratings. Those of Nobert were employed by A. J. Ångström in his celebrated researches upon wave-lengths. L. M. Rutherfurd introduced into common use the reflection grating, finding that speculum metal was less trying than glass to the diamond point, upon the permanence of which so much depends. In Rowland’s dividing engine the screws were prepared by a special process devised by him, and the resulting gratings, plane and concave, have supplied the means for much of the best modern optical work. It would seem, however, that further improvements are not excluded.

There are various copying processes by which it is possible to reproduce an original ruling in more or less perfection. The earliest is that of Quincke, who coated a glass grating with a chemical silver deposit, subsequently thickened with copper in an electrolytic bath. The metallic plate thus produced formed, when stripped from its support, a reflection grating reproducing many of the characteristics of the original. It is best to commence the electrolytic thickening in a silver acetate bath. At the present time excellent reproductions of Rowland’s speculum gratings are on the market (Thorp, Ives, Wallace), prepared, after a suggestion of Sir David Brewster, by coating the original with a varnish, e.g. of celluloid. Much skill is required to secure that the film when stripped shall remain undeformed.

A much easier method, applicable to glass originals, is that of photographic reproduction by contact printing. In several papers dating from 1872, Lord Rayleigh (see Collected Papers, i. 157, 160, 199, 504; iv. 226) has shown that success may be attained by a variety of processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. For many purposes the copies, containing lines up to 10,000 to the inch, are not inferior. It is to be desired that transparent gratings should be obtained from first-class ruling machines. To save the diamond point it might be possible to use something softer than ordinary glass as the material of the plate.

 9. Talbot’s Bands.—These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g. of glass or mica. The view of the matter taken by the discoverer (Phil. Mag., 1837, 10, p. 364) was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago’s theory of stellar scintillation. It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of

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Rh In 1819 he was offered the position of taxidermist in the museum at Cincinnati, and soon moved there with his family. His pay not being forthcoming from the museum, he started a drawing school there, and again returned to his portraits. Without these resources, he says, he would have been upon the starving list. But food was plentiful and cheap. He writes in his journal: "Our living here is extremely moderate; the markets are well supplied and cheap, beef only two and one half cents a pound, and I am able to supply a good deal myself. Partridges are frequently in the streets, and I can shoot wild turkeys within a mile or so. Squirrels and Woodcock are very abundant in the season, and fish always easily caught."

In October, 1820, we again find him adrift, apparently with thought of having his bird drawings published, after he shall have further added to them by going through many of the southern and western states.


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44 Leaving his family behind him, he started for New Orleans on a flatboat. He tarried long at Natchez, and did not reach the Crescent City till midwinter. Again he found himself destitute of means, and compelled to resort to portrait painting. He went on with his bird collecting and bird painting; in the meantime penetrating the swamps and bayous around the city. At this time he seems to have heard of the publication of Wilson's " Ornithology," and tried in vain to get sight of a copy of it.

In the spring he made an attempt to get an appointment as draughtsman and naturalist to a government expedition that was to leave the next year to survey the new territory ceded to the United States by Spain. He wrote to President Monroe upon the subject, but the appointment never came to him. In March he called upon Vanderlyn, the historical painter, and took with him a portfolio


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Rh of his drawings in hopes of getting a recommendation. Yanderlyn at first treated him as a mendicant and ordered him to leave his portfolio in the entry. After some delay, in company with a government official, he consented to see the pictures. "The perspiration ran down my face," says Audubon, "as I showed him my drawings and laid them on the floor." He was thinking of the expedition to Mexico just referred to, and wanted to make a good impression upon Yanderlyn and the officer. This he succeeded in doing, and obtained from the artist a very complimentary note, as he did also from Governor Robertson of Louisiana. In June, Audubon left New Orleans for Kentucky, to rejoin his wife and boys, but somewhere on the journey engaged himself to a Mrs. Perrie who lived at Bayou Sara, Louisiana, to teach her daughter drawing during the summer, at sixty dollars per month, leaving him half


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46 of each day to follow his own pursuits. He continued in this position till October when he took steamer for New Orleans. "My long, flowing hair, and loose yellow nankeen dress, and the unfortunate cut of my features, attracted much attention, and made me desire to be dressed like other people as soon as possible."

He now rented a house in New Orleans on Dauphine street, and determined to send for his family. Since he had left Cincinnati the previous autumn, he had finished sixty-two drawings of birds and plants, three quadrupeds, two snakes, fifty portraits of all sorts, and had lived by his talents, not having had a dollar when he started. "I sent a draft to my wife, and began life in New Orleans with forty-two dollars, health, and much eagerness to pursue my plan of collecting all the birds of America." His family, after strong persuasion, joined him in December, 1821, and his former life of drawing portraits, giving


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Rh lessons, painting birds, and wandering about the country, began again. His earnings proving inadequate to support the family, his wife took a position as governess in the family of a Mr. Brand.

In the spring, acting upon the judgment of his wife, he concluded to leave New Orleans again, and to try his fortunes elsewhere. He paid all his bills and took steamer for Natchez, paying his passage by drawing a crayon portrait of the captain and his wife. On the trip up the Mississippi, two hundred of his bird portraits were sorely damaged by the breaking of a bottle of gunpowder in the chest in which they were being conveyed.

Three times in his career he met with disasters to his drawings. On the occasion of his leaving Hendersonville to go to Philadelphia, he had put two hundred of his original drawings in a wooden box and had left them in charge of a friend. On his return, several


text: Dictionary of National Biography, 1885-1900/Trussell, John


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48 months later, he pathetically recounts what befell them: "A pair of Norway rats had taken possession of the whole, and reared a young family among gnawed bits of paper, which but a month previous, represented nearly one thousand inhabitants of the air!"

This discovery resulted in insomnia, and a fearful heat in the head; for several days he seemed like one stunned, but his youth and health stood him in hand, he rallied, and, undaunted, again sallied forth to the woods with dog and gun. In three years' time his portfolio was again filled.

The third catastrophe to some of his drawings was caused by a fire in a New York building in which his treasures were kept during his sojourn in Europe.

Audubon had an eye for the picturesque in his fellow-men as well as for the picturesque in Nature. On the Levee


text: Dictionary of National Biography, 1885-1900/Trussell, William


text: Dictionary of National Biography, 1885-1900/Trye, Charles Brandon


text: Dictionary of National Biography, 1885-1900/Tryon, George


text: Dictionary of National Biography, 1885-1900/Tryon, Thomas


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Rh in New Orleans, he first met a painter whom he thus describes: "His head was covered by a straw hat, the brim of which might cope with those worn by the fair sex in 1830; his neck was exposed to the weather; the broad frill of a shirt, then fashionable, flopped about his breast, whilst an extraordinary collar, carefully arranged, fell over the top of his coat. The latter was of a light green colour, harmonising well with a pair of flowing yellow nankeen trousers, and a pink waistcoat, from the bosom of which, amidst a large bunch of the splendid flowers of the magnolia, protruded part of a young alligator, which seemed more anxious to glide through the muddy waters of a swamp than to spend its life swinging to and fro amongst folds of the finest lawn. The gentleman held in one hand a cage full of richly-plumed nonpareils, whilst in the other he sported a silk umbrella, on which I could plainly read 'Stolen from


text: Dictionary of National Biography, 1885-1900/Tryon, William


text: Dictionary of National Biography, 1885-1900/Tuathal


text: Dictionary of National Biography, 1885-1900/Tucker, Abraham


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 the strongest ties between her and the other slaveholding States of the Confederacy. Those ties have been strengthened by the association. But what has been the course of the government of the United States, and of the people and authorities of the non-slaveholding States, since our connection with them?

The controlling majority of the Federal Government, under various pretences and disguises, has so administered the same as to exclude the citizens of the Southern States, unless under odious and unconstitutional restrictions, from all the immense territory owned in common by all the States, on the Pacific ocean, for the avowed purpose of acquiring sufficient power in the common government, to use it as a means of destroying the institutions of Texas and her sister slaveholding States.

By the disloyalty of the Northern States and their citizens, and the imbecility of the Federal Government, infamous combinations of incendiaries and outlaws have been permitted in those States and the common territory of Kansas, to trample upon the Federal laws, to war upon the lives and property of Southern citizens in that territory, and, finally, by violence and mob law, to usurp the possession of the same, as exclusively the property of the Northern States.

The federal government, while but partially under the control of these our unnatural and sectional enemies, has, for years, almost entirely failed to protect the lives and property of the people of Texas against the Indian savages on our borders; and, more recently, against the murderous forays of banditti from the neighboring territory of Mexico; and when our State Government has expended large amounts for such purposes, the Federal government has refused re-imbursement therefor—thus rendering our condition more insecure and harrassing than it was during the existence of the Republic of Texas.

These and other wrongs we have patiently borne, in the vain hope that a returning sense of justice and humanity would induce a different course of administration.

When we advert to the course of individual non-slaveholding States and that of a majority of their citizens, our grievances assume far greater magnitude.

The States of Maine, Vermont, New Hampshire, Connecticut, Rhode Island, Massachusetts, New York, Pennsylvania, Ohio, Wisconsin, Michigan, and Iowa, by solemn Legislative enactments, have deliberately, directly or indirectly, violated the third clause of the second section of the fourth article of the Federal Constitution, and laws passed in pursuance thereof; thereby


text: Dictionary of National Biography, 1885-1900/Tucker, Benjamin


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50 I,' these words being painted in large white characters. He walked as if conscious of his own importance; that is, with a good deal of pomposity, singing, 'My love is but a lassie yet'; and that with such thorough imitation of the Scotch emphasis that had not his physiognomy suggested another parentage, I should have believed him to be a genuine Scot. A narrower acquaintance proved him to be a Yankee; and anxious to make his acquaintance, I desired to see his birds. He retorted, 'What the devil did I know about birds?' I explained to him that I was a naturalist, whereupon he requested me to examine his birds. I did so with much interest, and was preparing to leave, when he bade me come to his lodgings and see the remainder of his collection. This I willingly did, and was struck with amazement at the appearance of his studio. Several cages were hung about the walls, containing specimens of birds, all


text: Dictionary of National Biography, 1885-1900/Tucker, Charlotte Maria


text: Dictionary of National Biography, 1885-1900/Tucker, Henry St. George


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Rh of which I examined at my leisure. On a large easel before me stood an unfinished portrait, other pictures hung about, and in the room were two young pupils; and at a glance I discovered that the eccentric stranger was, like myself, a naturalist and an artist. The artist, as modest as he was odd, showed me how he laid on the paint on his pictures, asked after my own pursuits, and showed a friendly spirit which enchanted me. With a ramrod for a rest, he prosecuted his work vigorously, and afterwards asked me to examine a percussion lock on his gun, a novelty to me at the time. He snapped some caps, and on my remarking that he would frighten his birds, he exclaimed, 'Devil take the birds, there are more of them in the market.' He then loaded his gun, and wishing to show me that he was a marksman, fired at one of the pins on his easel. This he smashed to pieces, and afterward put a rifle bullet exactly


text: Dictionary of National Biography, 1885-1900/Tucker, Josiah


text: Dictionary of National Biography, 1885-1900/Tucker, Thomas Tudor


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52 through the hole into which the pin fitted." Audubon reached Natchez on March 24, 1822, and remained there and in the vicinity till the spring of 1823, teaching drawing and French to private pupils and in the college at Washington, nine miles distant, hunting, and painting the birds, and completing his collection. Among other things he painted the "Death of Montgomery" from a print. His friends persuaded him to raffle the picture off. This he did, and taking one number himself, won the picture, while his finances were improved by three hundred dollars received for the tickets. Early in the autumn his wife again joined him, and presently we find her acting as governess in the home of a clergyman named Davis. In December, there arrived in Natchez a wandering portrait painter named Stein, who gave Audubon his first lessons in the use of oil colours, and was


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Rh by Audubon in turn in chalk drawing.

There appear to have been no sacrifices that Mrs. Audubon was not willing and ready to make to forward the plans of her husband. "My best friends," he says at this time, "solemnly regarded me as a mad man, and my wife and family alone gave me encouragement. My wife determined that my genius should prevail, and that my final success as an ornithologist should be triumphant." She wanted him to go to Europe, and, to assist toward that end, she entered into an engagement with a Mrs. Percy of Bayou Sara, to instruct her children, together with her own, and a limited number of outside pupils.

Audubon, in the meantime, with his son Victor, and his new artist friend, Stein, started off in a wagon, seeking whom they might paint, on a journey through the southern states. They wandered as far as New Orleans, but


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54 appears to have returned to his wife again in May, and to have engaged in teaching her pupils music and drawing. But something went wrong, there was a misunderstanding with the Percys, and Audubon went back to Natchez, revolving various schemes in his head, even thinking of again entering upon mercantile pursuits in Louisville. He had no genius for accumulating money nor for keeping it after he had gotten it. One day when his affairs were at a very low ebb, he met a squatter with a tame black wolf which took Audubon's fancy. He says that he offered the owner a hundred dollar bill for it on the spot, but was refused. He probably means to say that he would have offered it had he had it. Hundred dollar bills, I fancy, were rarer than tame black wolves in that pioneer country in those days.

About this time he and his son Victor were taken with yellow fever, and Mrs.


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Rh Audubon was compelled to dismiss her school and go to nurse them. They both recovered, and, in October (1823), set out for Louisville, making part of the journey on foot. The following winter was passed at Shipping Port, near Louisville, where Audubon painted birds, landscapes, portraits and even signs. In March he left Shipping Port for Philadelphia, leaving his son Victor in the counting house of a Mr. Berthoud. He reached Philadelphia on April 5, and remained there till the following August, studying painting, exhibiting his birds, making many new acquaintances, among them Charles Lucien Bonaparte, giving lessons in drawing at thirty dollars per month, all the time casting wistful eyes toward Europe, whither he hoped soon to be able to go with his drawings. In July he made a pilgrimage to Mill Grove where he had passed so many happy years. The sight of the old familiar scenes filled him with the deepest emotions.


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178 the contrary, and that when forced into marching he laid a mine on the road, intending to blow up his own men. His real offence seems to have been a refusal to drink to Marat's health. Custine superseded him. O'Moran was at first screened by Carnot, who, however, said, "He has a prudence which makes me desperate, and which I should call pusillanimity if I did not respect his talents." Released from St. Lazare, he was rearrested, tried for "manœuvres tending to favour the enemy," and executed on the 6th March 1794. His aide-de-camp, Etienne de Jouy, charged with complicity, fled from Cassel and took refuge in the house of Robert Hamilton, a Jamaica planter claiming kinship with the Dukes of Hamilton. Hamilton had married a daughter of Lord Leven and Melville, widow of a Dr. James Walker. They had two daughters, one of whom fell in love with the refugee, a modern d'Artagnan, who had lost two fingers in battle, and who had probably many tales to tell of his adventures in America and the East. Jouy fled for greater security first to Paris, where he saw O'Moran pass on his way to execution, and then to Switzerland. Returning after the Terror, he married Miss Hamilton, but this alliance, soon followed by a separation, exposed him to imputations of intrigues with England. He threw up his commission in the army out of disgust with these repeated vexations, and became a prolific dramatist.


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56 In August he left Philadelphia for New York, hoping to improve his finances, and, may be, publish his drawings in that city. At this time he had two hundred sheets, and about one thousand birds. While there he again met Vanderlyn and examined his pictures, but says that he was not impressed with the idea that Vanderlyn was a great painter.

The birds that he saw in the museum in New York appeared to him to be set up in unnatural and constrained attitudes. With Dr. De Kay he visited the Lyceum, and his drawings were examined by members of the Institute. Among them he felt awkward and uncomfortable. "I feel that I am strange to all but the birds of America," he said. As most of the persons to whom he had letters of introduction were absent, and as his spirits soon grew low, he left on the fifteenth for Albany. Here he found his money low also. Abandoning the


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