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degree of summer heat is much greater there than in any
part of England; and perhaps the peculiar nature of the soil
renders peach-trees much more yielding to art there than
in this country.

For these reasons, however useful the plan of disbudding
in autumn or winter may be in the gardens of France, it
would be improper to practise it to any extent in those of
England. The system has been fairly tried in the garden
of the London Horticultural Society, but has long since
been discontinued.

sulting fractions of a foot. Hence the difficulties of INCOMMENSURABLE magnitudes, which arise from the failure of the attempt to represent flowing or continuous changes by the means of changes which always suppose finite intervals. as in passing from number to number."

But the arithmetical difficulty, being mtroduced antecedently to the express consideration of discontinuity, is rarely treated as belonging to this subject. In the higher parts of mathematics the necessity for the consideration of discontinuous expressions began with the investigation of partial differential equations. In the introduction of the arbitrary functions which those equations require, discontinuous funetions were thought to be admissible by Euler, an opinion which was controverted by D'Alembert, and supported, conclusively, it has always been thought, by Lagrange. It is our own opinion that not only the arbitrary function of a partial equation, but even the arbitrary constant of a common equation, may be allowed to be discontinuous, unless the contrary be a condition of the problem, expressed or implied. By a discontinuous constant, we mean one which preserves one value between certain limits of the value of the variable, which then suddenly changes its value, preserving the new value till the variable attains another limit, and so on.

It has however been proved, both there and from the long experience of men in private situations, that a judicious thinning of the buds after they have been unfolded in spring (when an experienced individual can foresee the strength of those which he is about to leave, and to which he looks for his fruit in the following year), is of great utility. DISC (discus, díocog), is used for the face of a circular plate, and frequently for a thin plate of any substance. Thus we speak of the sun's disc (referring to the appearance of the sun), and also of a disc of metal.

DISCIPLINE, MILITARY, the series of duties which are to be performed by military men. It also signifies a conformity to the regulations by which those who serve in the army are governed in all matters relating to the practice of their profession.

The subject has begun to force itself on the attention of DISCLAIMER (Law), a plea by a tenant in any Court mathematicians, and several remarkable cases have been of Record in which he disclaims to hold of his lord. This pointed out in which erroneous conclusions have been ardisclaimer of tenure is a forfeiture of the lands to the lord rived at for want of considerations connected with disconupon reasons most apparently feudal. And so likewise if tinuity. There is a full account of the state of this question in in any Court of Record the particular tenant does any act Mr. Peacock's Report on Analysis.' (Rep. Brit. Ass., 1834. which amounts to a virtual disclaimer, as if he claims a DISCORD, in inusic, a sound which, when heard with greater estate than was granted to him, or takes upon another, is disagreeable to the ear, unless treated according himself those rights which belong only to tenants of a su- to the rules of art. Discords are the 2nd, sharp 4th (Triperior class, or if he affirms the reversion to be in a stranger, tonus), flat 5th (Semidiapente), minor or flat 7th, and by attorning as his tenant, collusive pleading, and the like, major or sharp 7th. The ratios of these are 9: 8, 45: 32, such behaviour amounts to a forfeiture of his particular 64: 45, 9.5, and 15: 8. The 9th (9: 4) is also a discord, estate. The writ of right sur disclaimer was the old form and though only the octave to the 2nd, is considered in in which the lord took advantage of the forfeiture; but as harmony as a very different interval, and treated in a difit was decided that the tenant might be treated as a tres-ferent manner. The 4th (4:3) is either discord or concord, passer, and that notice to quit was not necessary, the more according to the manner in which it is accompanied, convenient action of ejectment was generally used, and [CONCORD.] Discords commonly, but not always, are prenow, since the stat. 3 & 4 W. IV., c. 27, the proceeding pared; i. e., the note which is to become the discord, is by writ of right sur disclaimer is abolished. first heard as a concord: and their resolution is absolutely necessary; i. e., the discord must pass into a concord, though the resolution is occasionally retarded. Examples :

Where a person by his plea denied that he was of the blood of another, he was also said to disclaim; and there is a disclaimer of goods as well as lands, as where on an arraignment of felony a man disclaims the goods, in which case, though he should be acquitted, he loses the goods.

One of the pleadings in a suit in Chancery is also called a disclaimer, as where a defendant, in his answer to the complainant's bill, disclaims all interest in the matter in question. [EQUITY.]

And where an estate is given either by deed or will to a person, he may by deed (which need not be enrolled, or, as it is called, made matter of record) disclaim all interest thereunder; but it seems that for this purpose a deed is necessary, and that a parol disclaimer would not be sufficient.

An executor is said to disclaim when he renounces probate of the will of his testator; and this is generally effected by verbal renunciation before some judge spiritual, or by simple writing under his hand, in either case the disclaimer being recorded in the spiritual court; but where the will contains a devise of lands to the executor, the disclaimer is generally made by deed, for although a disclaimer by the before-mentioned means would, it seems, be operative, yet the deed is preferred as affording evidence, in deducing a title to the lands, of the fact of disclaimer.

DISCONTINUITY (Algebra, &c.). Continuous changes are those which are so made that no two states exist without every possible intermediate state having been in existence between them. Thus the square on a line of 4 inches contains 16 square inches, while that on a line of 5 inches con tains 25 square inches; and there is no possible area between 16 and 25 square inches which is not equal to the square described on some line between 4 and 5 inches. That is,

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The perfect 5th in the chord of, and the 3rd in the chord of, are treated, so far as regards resolution, as discords. Examples

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DISCOUNT, a sum of money deducted from a debt in

if a straight line increase continuously, the square described consideration of its being paid before the usual or stipulated

on it increases continuously.

time. The circumstance on which its fairness is founded

The first introduction of discontinuity arises from the is, that the creditor, by receiving his money before it be attempt to represent all magnitudes by numbers. Arith- comes due, has the interest of the money during the inter nitude. If a foot be divided into 2, 3, 4, &c. equal parts, out to interest during the period in question, will realize inetical symbols cannot represent continuous change of mag-val. Consequently, he shouid only receive so much as, pue and so on ad infinitum, there exist infinite numbers of lengths | the amount of his debt at the time when it would have be

which will not be represented by any whatsoever of the re

P. C., No. 533

come due. For instance, 1007. is to be paid at the end of

VOL. IX.--D

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three years, what should be paid now, interest being 4 per cent.? Here it is evident that if we divide the whole debt into 112 (or 100 + 3 x 4) parts, 100 of these parts will make the other 12 in three years (at simple interest), whence the payment now due is the 112th part of 10,0001. or 891. 5s. 9d. The rule is, n being the number of years (a fraction or number and fraction), r the rate per cent., and D the sum due, 100 D Dnr Present value = ; discount = 100+nr 100+ nr In practice, it is usual not to find the real discount, but to allow interest on the whole debt in the shape of abatement. Thus it would be considered that, in the preceding example, three year discount upon 1007. at 4 per cent. is 121., or 881. would be considered as the present value. In transactions which usually proceed on compound interest, as in valuing leases, annuities, &c., the principle of discount is strictly preserved. The present value in the preceding case is, in its most usual form,

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where is the rate per pound (not per cent.: thus it is 04 p for 4 per cent.). But recourse is usually had to the tables of present values which accompany all works on annuities or compound interest. [INTEREST.]

The name of discount is also applied to certain trade allowances upon the nominal prices of goods. In some branches of trade these allowances vary according to the circumstances which affect the markets, and what is called discount is in fact occasioned by fluctuations in prices which it is thought convenient to maintain nominally at unvarying rates. This system is practised in some branches of wholesale haberdashery business, and we have now before us a list of prices furnished to his customers by a manufacturer of tools at Sheffield, in which the nominal price of each article is continued the same at which it has stood for many years, while to every different species of tool there is applied a different and a fluctuating rate of discount, this fluctuation constituting in fact a difference of price between one period and another: the rates of discount in this list vary from 5 to 40 per cent. upon the nominal prices of the different articles.

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Eic Aiorov Exovra Appočítηv. See likewise Montfaucon, Supplem. de l'Ant. Expliq, liv. iii., p. 64. DISDIAPA'SON, the name given by the Greeks to a scale of two octaves. [DIAPASON.]

DISK, a term in botany signifying any ring or who 1 of glands, scales, or other bodies that surround the base of an ovary, intervening between it and the stamens. In its most common state it is a fleshy wax-like ring as in the orange; it frequently forms a yellowish lining to the calyx, as in the plum and cherry, and not unfrequently rises up like a cup around the ovary as in the tree pæony. The latter renders it probable that the disk is nothing but an inner whorl of rudimentary stamens. Previously to the expansion of the flower the disk contains fæcula, and is dry and brittle; but after the blossom unfolds it perspires a sweet honey-like fluid, and becomes tough, absorbing oxygen and parting with carbonic acid. This phenomenon is similar to what occurs in the germination of seeds, and has led M. Dunal to the opinion that the conversion of the fæcula of the disk into sugar is for the purpose of forming a store of nutritive matter for the stamens and ovary at the time of fertilization, just as the same phenomenon in the germination of seed is for the purpose of supplying food to the young embryo.

DISLOCATION. Various parts of the body are liable to be displaced by the direct application of violence or by more gradual causes. But the term dislocation is commonly appropriated to displacements occurring about the joints. In this sense it is nearly synonymous with luxation, but not entirely; for the latter term carries with it more of the idea of external force, and is not quite so generally applied. It is usual, for instance, to speak of the dislocation, not the luxation, of the internal cartilage of the knee; and the latter term is seldom if ever used in describing the displacements of the small bones of the wrist or instep, or of single vertebræ.

The injuries classed under this title may be effected by external violence, or by the undue contraction of muscles, or by both of these causes combined; and they result in some instances from disease within the joints themselves, by which their ligaments are weakened or destroyed, and their sockets rendered insecure by ulceration and other gradual changes.

When, by the protrusion of the bone through the skin The term discount is also employed to signify other mer- or otherwise, the dislocation is complicated with an external cantile allowances, such for example, as the abatement of wound exposing the cavity of the joint, it is said to be com12 per cent. made upon the balances which underwriters, pound: and, as in the parallel case of fracture, this aggraor insurers of sea risks, receive at the end of the year fromvation of the injury is very serious, and the most skilful the brokers by whom the insurances have been effected. management is required to save the life or limb, where the The word discount is further used, in contradistinction to injury happens to one of the larger joints. premium, to denote the diminution in value of securities which are sold according to a fixed nominal value, or according to the price they may have originally cost. If, for example, a share in a canal company upon which 1007. has been paid is sold in the market for 987., the value of he share is stated to be at 2 per cent. discount.

DISCOVERY, in Law. [EQUITY.]

DISCUS (diokoç, discos), a quoit of stone, brass, or iron, with which the Greeks and Romans diverted themselves in the public games. The word is Greek. The discus, when perforated like our modern quoit, was thrown by the help of a thong, put through the middle of it. It was at other times of a solid piece, and was then hurled directly from the hand. This last method is illustrated by the celebrated statue of the Discobolus, or quoit thrower, attributed to Myro, an antient copy of which is among the marbles of the Townley Gallery. The figure is represented in action at the precise moment of delivering the discus. Ovid (Metam., li. x., v. 175) and Statius (Theb., vi., v. 646) have both described the diversion of the discus; see also Petri Fabri Agonisticon, sive de Re Athletica, Ludisque Veterum, 4to., Lugd. 1595, li. ii., c. i.

The term discus was likewise applied to circular shields or bucklers, of a large size, placed in the temples, on which great actions were represented, or the names of those who had devoted themselves to the service of their country inscribed. One of the former of these is in the Townley Gallery at the British Museum, Room iii., No. 36, containing the names of the ephebi of Athens under Alcamenes, when he held the office of cosmetes. Such too was the shield of Scipio Africanus, found in the Rhone in 1656, engraved in Spon's Miscellanea Eruditæ Antiquitatis,' edit. 1685, p. 152. Anacreon has an ode on a disk of silver, representing Venus rising from the sea: Od. 51,

The particular dislocation takes its name either from the joint itself or from the furthest bone; and various terms are added to indicate the direction of the displacement, or the new situation of the head of the bone. Thus the most common form of the accident at the hip is called a dislocation of the head of the femur' (thigh-bone) backwards upon the dorsum ilii' (flat part of the haunch-bone).

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Any bone may be displaced in any direction, but the accident happens most frequently in those joints and directions in which the extent of motion is the greatest. Thus the most common dislocation is that of the shoulder, which is the most movable joint; and its most frequent variety is that in which the head of the humerus (or bone of the upper arm) is drawn downwards into the axilla (or arm-pit) by the sudden contraction of certain strong muscles. This happens when the arm is raised to the utmost, as in reaching to close a window; that is when it has moved through an angle of 180° degrees from its natural position. The most usual dislocation of the hip is that, already mentioned, on the dorsum ilii for the same reason. It is generally produced by sudden pressure or a blow on the knee when the thigh is bent upon the abdomen; the head of the femur is thus driven backwards from the socket, and is then drawn farther back and upwards by the powerful muscles of the buttock.

The jaw is sometimes thrown out of joint by the mere act of yawning; and that accident happened to a gentleman known to the writer in opening his mouth to make the usual response at church. The word was cut short at the first syllable; for in such cases the chin suddenly drops and is thrown forward, and it is impossible by any effort to shut the mouth. This distressing but irresisti bly ludicrous accident may be relieved immediately by any bystander wrapping a napkin round his thumbs and placing

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them firmly against the back teeth, so as to press them
downwards, while with the fingers and palms the chin is
steadily raised and pushed backwards. But the operator
should be on the alert to withdraw his hands the moment
the jaw snaps back into its place, or he may receive a very
unpleasant intimation of the success of his efforts.

It will be easily seen from these instances how important
a part is played by the muscles in determining both the
occurrence and direction of these accidents. Hence arises
in part their infrequency, often wondered at, during infancy
and childhood; for though the flexible joints of the young
have a greater extent of motion than those of the adult,
their muscular power is not only weaker as compared with
the strength of their ligaments, but is much more tardily
thrown into action, as may be observed in their tottering
gait. The fragility of their bones is another cause of this
infrequency, by rendering them more liable to be broken
than displaced by external violence. The only dislocation
that is at all common in children is that of the hip, which
is the consequence of scrofulous ulceration of the liga-
ments and the socket, and of the ball-shaped head of the
femur within it.

The reader will be prepared by what has been said to learn that the spasmodic and violent contraction of the muscles consequent upon these displacements is the chief or only obstacle to their reduction.

DISMAL SWAMP. [CAROLINA, NORTH; VIRGINIA.] DISPART, the difference between the semidiameter of the base ring, at the breech of a gun, and that of the ring at the swell of the muzzle.

On account of the dispart, the line of aim, which is in a plane passing through the axis of the gun, always makes a small angle with the axis; so that the elevation of the latter above the horizon is greater than that of the line of aim: an allowance for the dispart is consequently necessary in determining the commencement of the graduations on the tangent scale, by which the required elevation is given to the gun.

DISPENSARY, an institution supported by voluntary contributions for the supply of the poor with medical and surgical advice, and with medicines gratuitously. Institutions of this kind are of very recent origin. They differ from hospitals in this, that the sick, when too ill to attend personally at the institution, are visited at their own homes by the medical officers of the charity. Each dispensary indeed is restricted to a certain district, beyond the limits of which the patients are not visited at their own houses. To every dispensary there are always attached one, and sometimes two physicians; one surgeon, and often a consulting surgeon, and a resident medical officer who dispenses the medicines prescribed by the physicians and surgeons. Every subscriber to the institution who pays annually a certain sum is called a governor, who is entitled to have at least one patient always on the books; a person who subscribes a larger amount in one sum is called a life-governor, who may have two or more patients on the list. The medicines, which are commonly purchased in considerable quantities at a time and at wholesale prices, are dispensed in unexpensive forms, and in this manner the extent of the relief afforded is great, while the cost is trifling. No other kind of charity affords so much real assistance at so small an expense, and perhaps fewer objections apply to this than to any other mode of giving eleemosynary aid to the poor. Its peculiar excellence is that it enables the sick poor to obtain advice on the very first day of their illness. Even the great metropolitan hospitals are often so full that urgent cases are constantly obliged to wait days and even weeks before admission can be obtained; but by means of the dispensary poor families, and even the heads of such families in regular employment, may procure medical and surgical assistance without leaving their occupation even for a day. It would be a great improvement in the principle of these institutions if some contribution towards their support on the part of the poor themselves were required to entitle them to avail themselves of the advantages which they afford. This would remove the only objection that can be urged against such establishments, and would enable the independent labourer, without asking charity, to procure the best advice for his sick family at a much cheaper rate than he can possibly do at present.

This object is effected by a process technically called
extension, consisting in the application of force in a proper
direction, and steadily kept up till the muscles are fatigued.
The head of the bone is thus drawn down a little below the
level of the joint; and being lifted over the edge of the
socket, slips easily into its place upon slightly relaxing the
extending force. This force is often required to be very
considerable, and in such cases it is customary to make use
of a block of pullies, the bone which contains the socket
having been first securely fixed to a staple in the wall by
proper bandages. Luxation of the hip is here supposed;
for the other joints are so inferior to that in strength that
their displacements may generally be reduced by less im-
posing means. It is sometimes necessary to favour the
relaxation of the muscles by emetics, warm baths, and
bleeding, and it is reckoned a point of good management to
call off the attention of the patient during the extension by
annoying him with questions and even exciting him to anger.
Almost all dislocations arising from accident may be
reduced in this way, and the joint rendered nearly or quite
as perfect as before: but this can only be done on condition
of perfect rest during a period sufficient for the firm union
of the ruptured ligaments; for if this precaution be not
strictly observed, and the ligaments are suffered to be
stretched by motion while the uniting substance is soft and
extensible, the accident is ever afterwards liable to re-
cur. No time should be lost in seeking assistance, for the
swelling that comes on soon renders the nature of the
accident obscure, and the reduction extremely difficult and
painful. When a joint has been unreduced for a certain
time, which varies with the particular joint, and with the
bodily strength of the individual-the weaker having the
advantage in this respect-it is unwise to make any attempt
at reduction. The parts have now become consolidated and
adapted to their new situation, and either the limb is per-
manently fixed or a new joint is under process of formation.
In the latter case the substitute is often better than might
be expected; and as this curious provision of nature cannot
be improved upon by art, it is better to leave it alone.

The most dangerous dislocations are those of the ver-
tebræ or bones of the spine, because in that case all the
parts of the body below the injury are paralyzed. But the
vertebræ are so curiously locked together, and have singly
so little motion, and are at the same time so well supported
by ligaments and muscles, that they are seldom dislocated
unless by a force sufficient to break as well as to displace
them. Such an injury is almost always fatal, and instantly
so in general when it takes place above the origin of the
nerves of respiration, that is, above the fourth vertebrae of
the neck. The object of the executioner in hanging a cri-
minal is to produce this effect, but he more often fails than

succeeds.

It would be out of place in this work to describe the various dislocations more particularly. The reader will find some additional information on the subject under the head of JOINTS. The best English treatise upon it is the large work of Sir Astley Cooper.

DISPENSATION (Law). The only kind of dispensation now used is that by which the bishop of a diocese licenses a clergyman within his jurisdiction to hold two or more benefices according to their value, or to reside out of the bounds of his parish, or dispenses with some other particular of his strict duty.

Formerly, not only in ecclesiastical jurisprudence, but also in the civil and criminal codes, the subject of dispensations occupied a large space. They formed a great source of the revenue of the court of Rome; for the pope's dis pensations prevailed against the law of the country in many if not most instances, indeed in all of an ecclesias tical nature; this abuse was however abolished by the statute 25 Henry VIII., c. 21; and the power of the | ope to grant dispensations not contrary to the law of God, but only to the law of the land, was granted to the archbishop of Canterbury under certain restrictions. It is hardly necessary to state that from the spirit of the times this power is never exercised in civil cases, and but in a few ca es of purely ecclesiastical cognizance, and in those the usage has become the law rather than the exception. This right of the archbishop in some cases, as to grant special licences of marriage, &c., has been expressly recognized by the legis

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by the Bill of Rights on the accession of William and DISPERSION. Light, as we receive it from the sun or from other original sources as a star, a fire, a candle, &c., appears to the senses as a simple undecomposable element by the instrumentality of which objects are perceived; and as for the peculiar colours of bodies, we naturally consider them, according to our early impressions, as belonging to the bodies themselves, or inherent in them. We are partly undeceived in this view by the changing colours of birds' feathers, soap-bubbles, compound silk textures, &c., but we are enabled to trace the immediate cause of the colours of bodies, whether permanent or transient, by the analysis of light furnished by the well-known experiments of the glass-prism. The triangular prism used for this purpose is a solid, terminated by two equal and exactly similar triangles, and having besides three plane faces of a rectangular form, contained by the sides of the triangles and by right lines or edges joining corresponding angles of the two triangular bases above-mentioned; and any imaginary right line within it parallel to these edges around which the prism is capable of revolving is called the axis of the prism.

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In the annexed figure the triangle B A C represents a section of the prism parallel to its basis or perpendicular to its axis. DE we shall suppose to be a ray or exceedingly narrow beam of solar light incident from vacuo or air on the prism at E; this ray of white light enters the prism at that point, and having undergone refraction by the dense medium of the glass, no longer proceeds as a simple ray E F, but is dispersed or divided into various rays of different colours over the space represented in the figure by ƒ E F, and emerging at ƒ F from the prism, undergoes another refraction, such that the portion f g of the ray proceeding from fis still more refracted than the portion FG from F, since the sines of the angles of incidence and refraction being in a constant ratio, that portion will be most refracted which has the greatest incidence: let now this dispersed beam gf F G be intercepted by a screen or wall P K, and from which extraneous light is as much as possible excluded, we shall then find the elongated space FG brilliantly painted over with tints passing gradually and insensibly from deep red to an attenuated violet, in the following order, as described by Newton, and since very generally concurred in,-red, orange, yellow, green, blue, indigo, violet. This experiment, which first opens the analysis of light, is easily made by letting a beam of light pass through a small circular hole in a shutter, in a darkened room, on a glass prism such as above described, the refracted and dispersed ray being received on the opposite wall, ceiling, or floor, according to the position of the prism. It would be still more effective by concentrating the light incident on a double convex lens in its focus, so that the beam EF may emanate more nearly from a point than it can when received through the hole of a shutter; for in the latter case rays are admitted which are inclined to each other at the angle subtended by the sun's disc to the eye. This primary experiment is, however, so familiar to almost all amateurs of science, that it will not be here necessary to enter into details respecting its most successful application. When the image of the sun or a star, candle, &c., is thus formed by admission through a small hole or narrow line, and the refraction of the prism, the coloured space G g, which has the same angular breadth as the object in a direction parallel to the axis of the prism (the screen being sup

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posed also parallel to it), but which is considerably elongated in the perpendicular direction, is called the spectrum; and that angle of the prism BAC the sides containing which, BA, AC, have been traversed by the ray D E F G is called the refracting angle of the prism.

Suppose, now, that a small orifice O is made in the screen at some point of the spectrum, so that rays of any particular colour, green for example, may be transmitted through it; and let the transmitted portion be again subjected to refraction through another prism, this beam being supposed very small, to ensure its purity or near uniformity of colour. It will not, after refraction, be again decomposed, or undergo any alteration of colour, except in respect to brilliancy, arising from absorption by the second prism: thus showing that light incident on the first prism, when once decomposed into homogeneous elements by refraction, is then, at least by refraction, not further decomposable.

If the original prism BAC be turned gradually round its axis, preserving always to the incident light the same refracting angle A, the spectrum Gg may be made to descend towards K, but after arriving at a certain point where the deviation, that is the inclination of D E produced to F G, is a minimum, it then reascends, and it is usual to make the chromatic experiments in this definite position of minimum deviation. This occurs when the position of the prism is such that the angles of incidence and emergence, or their complements DE B, G FC, are equal; for when the moving point G has reached its lowest place, it is for a moment in the condition of a fixed point like the point D, through which we may suppose the incident beam admitted; hence rays proceeding from D, notwithstanding a small variation of incidence arising from the rotation of the prism, reach G, as if it were a fixed point; and since in dioptrics it is of no consequence to the path in what direction we suppose the rays to move, it follows that rays proceeding from G, notwithstanding a small alteration of the angle C F G, would arrive at the fixed point or orifice D; and consequently the data for the determination of the angles DE B, G F C, in the position of minimum deviation, are precisely the same, and therefore these angles must then be equal.

This being premised, the following easy calculus will give the necessary angle of incidence to produce a minimum deviation

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Since the angles of incidence and emergence are equal, the angles formed by the interior ray E F with both sides of the prism are equal, or the triangle A E F is an isosceles; let 2 a be the refracting angle of the prism, then drawing AM perpendicular to E F, we have EAM = a, which. being the complement of A E M, is necessarily the angle of refraction; if therefore be the index of refraction for rays of any given colour, the angle of incidence P, corresponding to a minimum deviation, is given by the equation, Sin. (P) μ sin. (a)

=

For distinctness, suppose the preceding index of refraction to belong to the extreme red rays, and let u' be the index for the extreme violet rays of the spectrum; then, if P' denote the angle of incidence corresponding to the minimum deviation of the latter, we have

Sin. P'μ' sin. a;

and since a is always less than a right angle, and u' is greater than u, therefore P' is greater than P. In other words, when the red rays of the spectrum, having arrived at their lowest position on the screen, begin to reascend by the continued rotation of the prism, the violet rays will still descend a little before they arrive at their lowest position. Under these circumstances, the extent of the spectrum contracts from both ends, and an angle of incidence P intermediate to P and P', which do not greatly differ, corresponds to the minimum or brightest spectrum; and it

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We have seen that compound light, the sun's for example, may be decomposed into its homogeneous constituent rays by refraction through a transparent prism. Conversely it may be recompounded into light similar to the original, merely by making the rays, thus separated, by another refraction to occupy the same place. This may be effected by placing a prism of exactly similar material and form to that already used, with its refracting angle turned in a direction opposite to that of the former, so that the near faces of both prisms may be parallel; for the rays entering the second prism are in the same condition as if we supposed their direction inverted, that they may repass through the first; and therefore they emerge in a similar compound ray with the original, which may also be easily confirmed by experiment.

The rays issuing from the second face of the refracting prism may also be collected by means of a double convex lens, so as all to meet very nearly in its principal focus, where, if the image be received on a sheet of paper, the original compound light will be reproduced.

When the light of the sky, admitted through a small hole in a shutter in a dark room, is refracted by a prism, if an eye is placed behind the prism in the position which the spectrum would occupy on a screen, the hole will appear of the particular colour of the ray which reaches the eye, changing continually from one colour to another as the eye occupies different parts of the spectrum.

The prismatic analysis of light, together with the phenomeua relative to the transmission and absorption of light, enabled Newton to conclude that the colours of natural bodies are not inherent qualities of those bodies, but depend on their powers of reflecting, transmitting, or absorbing the rays of some colours more than others from the compound light incident on them; for all bodies placed in homogeneous light of any colour appear themselves to be of that colour, though the vividness of tint is greatest when placed in that coloured light which they reflect most copiously. Hence also arise the different colours of the liquids exhibited frequently in chemists' shops, according as they are viewed by transmitted or reflected light which would necessarily be complementary colours if no absorption or extinction of light occurred in its passage through the fluid. Many of the prismatic colours may be imitated by mixing colours taken as in the spectrum of greater and less refrangibility, as orange from red and yellow, &c., but such compound colours are not identical with the homogeneous light of the same colour, being immediately decomposable when viewed through a prism.

It would be difficult, if not altogether impracticable, to judge of the dispersive powers of transparent media by measuring the length of the spectra which they produce in a prismatic form, in consequence of the indefiniteness of their termini. The light at the violet end is so feeble that it requires some continued application of the eye to perceive a colour where we had first imagined the spectrum terminated: and, on the other hand, the influence of imagination, after we have recognised it, is apt to extend it momentarily far beyond its limits. Fortunately Nature has herself fur ́nished a scale of definite limits in the beautiful discovery made by Wollaston and Fraunhofer of the existence of dark spaces, bands transverse to the length of the spectrum, and now generally designated Fraunhofer's lines.

&c., each essentially different source having a peculiar system of deficient rays.

Substances which have not a great difference of refractive powers possess frequently very different dispersive powers, and the angular dispersion by a medium is not proportional to the angular deviation, and therefore by a system of prisms, two or more, white light incident on the first may be reproduced from the last, though on the whole refracted from its original direction. Such a system is called achromatic.

Conversely, by forming an achromatic system experimentally, where the angles of the prisms are small, and in the position of minimum deviation, if the dispersive power of the material of one of them be taken as a standard, that of the other may be readily obtained, the dispersion being δμ measured by μ being the index of refraction, and dμ the difference of its extreme values for any class of rays. This method has been much used in practice, particularly by Dollond.

με

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The formula for achromaticity in systems of prisms or lenses, though not difficult of investigation, are in general too complicated and tedious for a popular work; (see Mémoires de l'Acad. de Sciences, 1765; Mém. par D'Alembert.)

The rainbow is a beautiful natural exhibition of the dispersion of light into the spectral colours. [RAINBOW.] To find the longitudinal chromatic aberration of a lens, or the interval of the axis between the foci of extreme red

and violet rays:

Let the red rays converge to the point R, and the violet to the point V in the axis.

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Let f F be respectively the focal distances for the given system of rays, and a parallel system; then the fundamental equations for lenses (neglecting their thickness), give FFconstant, since the rays of all colours in the compound incident beam have a common origin; now differentiate relative to μ, the variable index of refraction: hence, df ƒ dF αμ F2 αμ

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To find, in the same case, the radius of the circle of least chromatic dispersion

By referring to the same figure, we may observe that the foci RV are respectively the vertices of red and violet conical surfaces, having the lens as a common base. Let these surfaces intersect in a circle, of which the radius is DE; then it is plain that all the intermediate coloured rays pass through this circle. It is therefore that of least dispersion:

These bands are best observable by forming the spectrum
of a luminous line instead of a point, by means of a prism
of great purity, and viewing it through a telescope of good
magnifying power, though some of them may, when care-
fully pointed out, be recognised by the unassisted eye, and
after one recognition are in future easily found. They are
spaces totally deficient of light, of very unequal width, and
exceedingly numerous; the large bands near the extre-
mities of the spectrum serve, however, as definite limits, so
as to furnish the requisite criteria for the dispersive powers
of different substances; it is also very remarkable that these
bands, always the same in number and relative position for
the same light, are different when the source of light is
varied. Thus sun-light, moon-light, planet-light, sky-light,
derived from a common source, have the same lines, but they
are different from those of star-light, fire light, candle-light, ized bodies in such a manner as to display their structure.

The preceding figure, representing a plane section of the
whole system taken through the axis, it is obvious that,
from the smallness of RV relative to C R, the angles CV B,
CR A, are sensibly equal, or the triangle VRD is exceed-
ingly nearly isosceles, and therefore DE bisects VR, or
of
CA h f
ER = and DE ER.
2
CR 2 F
rallel incident rays DE

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