See Gabriel Hanotaux,Origines de l’institution des intendants des provinces(1884); D’Arbois de Jubainville,L’Administration des intendants d’après les archives de l’Aube(1880); P. Ardascheff,Provintzalnaya administratsiya vo Frantsii ve poshednoyo porou starago poryadka: provintsialny Intendanty(St Petersburg, 1900-1906).
See Gabriel Hanotaux,Origines de l’institution des intendants des provinces(1884); D’Arbois de Jubainville,L’Administration des intendants d’après les archives de l’Aube(1880); P. Ardascheff,Provintzalnaya administratsiya vo Frantsii ve poshednoyo porou starago poryadka: provintsialny Intendanty(St Petersburg, 1900-1906).
(J. P. E.)
1In Germany the titleIntendantis applied to the head of public institutions, more particularly to the high officials in charge of court theatres, royal gardens, palaces and the like. The director of certain civic theatres is now also sometimes styled Intendant. The titleGeneralintendantimplies the same official duties, but higher rank. In the German army theIntendanturcorresponds to the British quartermaster-general’s and financial departments of the War Office, the Frenchintendance militaire. Subordinate to these are theintendances(Intendanturen) under general officers commanding, the heads of which are in Germany calledKorpsintendanten, and in Franceintendants-généraux,intendants militaires, &c. (seeArmy, § 58).
1In Germany the titleIntendantis applied to the head of public institutions, more particularly to the high officials in charge of court theatres, royal gardens, palaces and the like. The director of certain civic theatres is now also sometimes styled Intendant. The titleGeneralintendantimplies the same official duties, but higher rank. In the German army theIntendanturcorresponds to the British quartermaster-general’s and financial departments of the War Office, the Frenchintendance militaire. Subordinate to these are theintendances(Intendanturen) under general officers commanding, the heads of which are in Germany calledKorpsintendanten, and in Franceintendants-généraux,intendants militaires, &c. (seeArmy, § 58).
INTENT(from Lat.intendere, to stretch out, extend, particularly in the phraseintendere animum, to turn one’s mind to, purpose), in law, the purpose or object with which an act is done. The question of intent is important with reference both to civil and criminal responsibility. Briefly, it may be said that in criminal law the constituent element of an offence is themens reaor the guilty intent. The commission of an act without the intent is not, as a general rule, sufficient to constitute a crime, nor, on the other hand, does the existence of a guilty intent without commission of the act amount to the legal conception of a crime (seeCriminal Law). In the case of civil wrongs, in general, the opposite holds good. A wrongful act done to the person or property of another carries with it legal liability, irrespective of the motive with which the act was done (seeTort). In reference to the construction of contracts, wills and other documents, the question of intention is material as showing the sense and meaning of the words used, and what they were intended to effect.
INTERAMNA LIRENAS,an ancient town of Italy in the Volscian territory near the modern Pignataro Interamna, 5 m. S.E. of Aquinum; the additional name distinguishes it from Interamna Praetuttianorum (mod. Teramo) and Interamna Nahartium (mod. Terni). It was founded by the Romans as a Latin colony in 312B.C.as a military base in the war against Samnium, no fewer than 4000 colonists being sent thither. It was among the Latin colonies which in 209B.C.refused to supply further contingents or money for the Hannibalic war. It became amunicipiumwith the other Latin colonies, but we hear no more of it—mainly, no doubt, because it lay off the Via Latina. Livy’s description of it as on the Via Latina is not strictly accurate, and cannot be used as an indication that the former course of the Via Latina was through Interamna. The city lay on a hill on the N. bank of the Liris, between two of its tributaries, thus lacking natural defences on the N. side alone. Many inscriptions have been found, and there are considerable remains of antiquity. One inscription bears the dateA.D.408, and the site was occupied in the middle ages by a castle called Terame or Termine.
(T. As.)
INTERCALARY(from Lat.intercalare, to proclaim,calare, the insertion of a day in the calendar), a term applied to a month, day or days inserted between other months or days in order to adjust the reckoning of time, based on the revolution of the earth round the sun, the day, and of the moon round the earth, the lunar month, to the revolution of the earth round the sun, the solar year (seeCalendar). From the meaning of something inserted or placed between, intercalary is used for something which interrupts a series, or comes between two types. In botany, the term is used of growth which is not apical but somewhere between the apex and base of an organ, such as the growth in length of an Iris leaf, or of the internode of a grass-haulm.
INTERCOLUMNIATION,in architecture, the distance between the columns of a peristyle, generally referred to in terms of the lower diameter of the column. They are thus set forth by Vitruvius (iii. 2): (a) Pycnostyle, equal to 1½ diameters; (b) Systyle, 2 diameters; (c) Eustyle, 2¼ diameters (which was the proportion preferred by him); (d) Diastyle, 3 diameters; and (e) Araeostyle or wide spaced, 4 diameters, a span only possible when the architrave was in wood. Vitruvius’s definition would seem to apply only to examples with which he was acquainted in Rome, or to Greek temples described by authors he had studied. In the earlier Doric temples the intercolumniation is sometimes less than one diameter, and it increases gradually as the style developed; thus in the Parthenon it is 1¼, in the Temple of Diana Propylaea at Eleusis, 1¼; and in the portico at Delos, 2½. The intercolumniations of the columns of the Ionic Order are greater, averaging 2 diameters, but then the relative proportion of height to diameter in the column has to be taken into account, as also the width of the peristyle. Thusin the temple of Apollo Branchidae, where the columns are slender and over 10 diameters in height, the intercolumniation is 1¾, notwithstanding its late date, and in the Temple of Apollo Smintheus in Asia Minor, in which the peristyle is pseudodipteral, or double width, the intercolumniation is just over 1½. Temples of the Corinthian Order follow the proportions of those of the Ionic Order.
INTERDICT(Lat.interdictum, frominterdicere, to forbid by decree, lit., interpose by speech), in its full technical sense as an ecclesiastical term, a sentence by a competent ecclesiastical authority forbidding all celebration of public worship, the administration of some sacraments (baptism, confirmation and penance are permitted) and ecclesiastical burial. From general interdicts, however, are excepted the feast days of Christmas, Easter, Whitsunday, the Assumption and Corpus Christi. An interdict may be either local, personal or mixed, according as it applies to a locality, to a particular person or class of persons, or to a particular locality as long as it shall be the residence of a particular person or class of persons. Local interdicts again may be either general or particular; in the latter instance they refer only to particular buildings set apart for religious services. An interdict is a measure which seeks to punish a population or a religious body (e.g.a chapter) for the fault of some only of its members, who cannot be reached separately. It is a penalty directed against society rather than against individuals. In 869 Hincmar of Laon laid his entire diocese under an interdict, a proceeding for which he was severely censured by Hincmar of Reims. In theChronicleof Ademar of Limoges (ad ann.994) it is stated that Bishop Alduin introduced there “a new plan for punishing the wickedness of his people; he ordered the churches and monasteries to cease from divine worship and the people to abstain from divine praise, and this he called excommunication” (see Gieseler,Kirchengesch. iii. 342, where also the text is given of a proposal to a similar effect made by Odolric, abbot of St Martial, at the council of Limoges in 1031). It was not until the 11th century that the use of the interdict obtained a recognized place among the means of discipline at the disposal of the Roman hierarchy, which used it, without great success, to bring back the secular authorities to obedience. Important historical instances of the use of the interdict occur in the cases of Scotland under Pope Alexander III. in 1181, of France under Innocent III. in 1200, and of England under the same pope in 1209. So far as the interdict is “personal,” that is to say, applied to a particular individual, it may be regarded as a kind of partial excommunication; for instance, a bishop may, for certain faults, be interdicted from entering the church (ab ingressu ecclesiae), that is, without being excommunicated, he must not celebrate or assist at the celebration of divine offices. Interdicts cease at the expiration of the term, or by removal (relaxatio). General and local interdicts are no longer in use.
See the canonists in tit. 39lib.v.,De sententia excommun., &c.; L. Ferraris,Prompta bibliotheca canonica, &c., s.v. “Interdictum.”
See the canonists in tit. 39lib.v.,De sententia excommun., &c.; L. Ferraris,Prompta bibliotheca canonica, &c., s.v. “Interdictum.”
Interdict, in Scots law, is an order of court pronounced on cause shown for stopping any proceedings complained of as illegal or wrongful. It may be resorted to as a remedy against all encroachments either on property or possession. For the analogous English practice seeInjunction.
INTERDICTION,in Scots law, a process of restraint applied to prodigals and others who, “from weakness, facility or profusion, are liable to imposition.” It is either voluntary or judicial. Voluntary interdiction is effected by the prodigal himself, who executes a bond obliging himself to do no deed which may affect his estate without the assent of certain persons called the “interdictors.” This may be removed by the court of session, by the joint act of the interdictors and the interdicted, and by the number of interdictors being reduced below the number constituting a quorum. Judicial interdiction is imposed by order of the court, either moved by an interested party or acting in the exercise of itsnobile officium, and can only be removed by a similar order. Deeds done by the interdicted person, so far as they affect or purport to affect his heritable estate, are reducible, unless they have been done with the consent of the interdictors. Interdiction has no effect, however, on movable property.
INTERESSE TERMINI(Lat. for “interest in a term”), in law, an executory interest, being the right of entry which the grant of a lease confers upon a lessee. Actual entry on the lands by the lessor converts the right into an estate. If the lease, however, has been created by a bargain and sale or by any other conveyance under the Statute of Uses, which does not require an entry, the term vests in the lessee at once. Aninteresse terminigives a cause of action against any person through whose action entry by the lessee or delivery of possession to him may have been prevented. Aninteresse terminiis a rightin rem, alienable at common law, and transmissible to the executors of the lessee.
INTEREST,etymologically a state or condition of being concerned in or having a share in anything, hence a legal or other claim to or share in property, benefits or advantages. Further developments of meaning are found in the application of the word to the benefits, advantages, matters of importance, &c., in which “interest” or concern can be felt, and to the feeling of concern so excited; hence also the word is used of the persons who have a concern in some common “interest,”e.g.the trading or commercial interest, and of the personal or other influence due to a connexion with specific “interests.” The word is derived from the Latininteresse(literally “to be between”), to make a difference, to concern, be of importance. The form which the word takes in English is a substantival use of the 3rd person singular of the present indicative of the Latin verb, and is due to a similar use in French of the olderinterest, modernintérêt. The earlier English word wasinteress, which survived till the end of the 17th century; the earliest example of “interest” in theNew English Dictionaryis from theRolls of Parliamentof 1450.
These meanings of “interest” are plainly derived from the ordinary uses of the Latininteresse. The origin of the application of the word to the compensation paid for the use of money or for the forbearance of a debt, with which, as far as present English law is concerned, this article deals, forms part of the history ofUsuryandMoney-Lending(q.v.). By Roman law, where one party to a contract made default, the other could enforce, over and above the fulfilment of the agreement, compensation based on the difference (id quod interest) to the creditor’s position caused by the default of the debtor, which was technically known asmora, delay. This difference could be reckoned according as actual loss had accrued, and also on a calculation of the profit that might have been made had performance been carried out. Now this developed the canonist doctrine ofdamnum emergensandlucrum cessansrespectively, which played a considerable part in the breaking down of the ecclesiastical prohibition of the taking of usury. The medieval lawyers used the phrasedamna et interesse(in Frenchdommages et intérêts) for such compensation by way of damages for the non-fulfilment of a contract, and for damages and indemnity generally. Thusinteresseandintérêtcame to be particularly applied to the charge for the use of money disguised by a legal fiction under the form of an indemnity for the failure to perform a contract.
At English common law an agreement to pay interest is not implied unless in the case of negotiable instruments, when it is supported by mercantile usage. As a general rule therefore debts certain, payable at a specified time, do not carry interest from that time unless there has been an express agreement that they should do so. But when it has been the constant practice of a trade or business to charge interest, or where as between the parties interest has been always charged and paid, a contract to pay interest is implied. It is now provided by the Civil Procedure Act 1833 that, “upon all debts or sums certain payable at a certain time or otherwise, the jury on the trial of any issue or in any inquisition of damagesmayif they shall think fit allow interest to the creditor at a rate not exceeding the current rate of interest, from the time when such debts or sums certain were payable, if such debts or sums be payable by virtue of some written instrument at a certain time; or if payableotherwise, then from the time when demand of payment shall have been made in writing, so as such demand shall give notice to the debtor that interest will be claimed from the date of such demand until the term of payment: provided that interest shall be payable in all cases in which it is now payable by law.” Compound interest requires to be supported by positive proof that it was agreed to by the parties; an established practice to account in this manner will be evidence of such an agreement. When interest is awarded by a court it is generally at the rate of 4%; under special circumstances 5% has been allowed.
INTERFERENCE OF LIGHT.§ 1. This term1and the ideas underlying it were introduced into optics by Thomas Young. His Bakerian lecture on “The Theory of Light and Colours” (Phil. Trans., 1801) formulated the following hypotheses and propositions, and thereby laid the foundations of the wave theory:—
Hypotheses.(i.) A luminiferous aether pervades the universe, rare and elastic in a high degree.(ii.) Undulations are excited in this aether whenever a body becomes luminous.(iii.) The sensation of different colours depends on the different frequency of vibrations excited by the light in the retina.(iv.) All material bodies have an attraction for the aethereal medium, by means of which it is accumulated in their substance, and for a small distance around them, in a state of greater density but not of greater elasticity.Propositions.(i.) All impulses are propagated in a homogeneous elastic medium with an equable velocity.(ii.) An undulation conceived to originate from the vibration of a single particle must expand through a homogeneous medium in a spherical form, but with different quantities of motion in different parts.(iii.) A portion of a spherical undulation, admitted through an aperture into a quiescent medium, will proceed to be further propagated rectilinearly in concentric superfices, terminated laterally by weak and irregular portions of newly diverging undulations.(iv.) When an undulation arrives at a surface which is the limit of mediums of different densities, a partial reflection takes place, proportionate in force to the difference of the densities.(v.) When an undulation is transmitted through a surface terminating different mediums, it proceeds in such a direction that the sines of the angles of incidence and refraction are in the constant ratio of the velocity of propagation in the two mediums.(vi.) When an undulation falls on the surface of a rarer medium, so obliquely that it cannot be regularly refracted, it is totally reflected at an angle equal to that of its incidence.(vii.) If equidistant undulations be supposed to pass through a medium, of which the parts are susceptible of permanent vibrations somewhat slower than the undulations, their velocity will be somewhat lessened by this vibratory tendency; and, in the same medium, the more, as the undulations are more frequent.(viii.) When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each.(ix.) Radiant light consists in undulations of the luminiferous aether.
Hypotheses.
(i.) A luminiferous aether pervades the universe, rare and elastic in a high degree.
(ii.) Undulations are excited in this aether whenever a body becomes luminous.
(iii.) The sensation of different colours depends on the different frequency of vibrations excited by the light in the retina.
(iv.) All material bodies have an attraction for the aethereal medium, by means of which it is accumulated in their substance, and for a small distance around them, in a state of greater density but not of greater elasticity.
Propositions.
(i.) All impulses are propagated in a homogeneous elastic medium with an equable velocity.
(ii.) An undulation conceived to originate from the vibration of a single particle must expand through a homogeneous medium in a spherical form, but with different quantities of motion in different parts.
(iii.) A portion of a spherical undulation, admitted through an aperture into a quiescent medium, will proceed to be further propagated rectilinearly in concentric superfices, terminated laterally by weak and irregular portions of newly diverging undulations.
(iv.) When an undulation arrives at a surface which is the limit of mediums of different densities, a partial reflection takes place, proportionate in force to the difference of the densities.
(v.) When an undulation is transmitted through a surface terminating different mediums, it proceeds in such a direction that the sines of the angles of incidence and refraction are in the constant ratio of the velocity of propagation in the two mediums.
(vi.) When an undulation falls on the surface of a rarer medium, so obliquely that it cannot be regularly refracted, it is totally reflected at an angle equal to that of its incidence.
(vii.) If equidistant undulations be supposed to pass through a medium, of which the parts are susceptible of permanent vibrations somewhat slower than the undulations, their velocity will be somewhat lessened by this vibratory tendency; and, in the same medium, the more, as the undulations are more frequent.
(viii.) When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each.
(ix.) Radiant light consists in undulations of the luminiferous aether.
In thePhilosophical Transactionsfor 1802, Young refers to his discovery of “a simple and general law.” The law is that “wherever two portions of the same light arrive at the eye by different routes, either exactly or very nearly in the same direction, the light becomes most intense where the difference of the routes is a multiple of a certain length, and least intense in the intermediate state of the interfering portions; and this length is different for light of different colours.”
This appears to be the first use of the wordinterferingorinterferenceas applied to light. When two portions of light by their co-operation cause darkness, there is certainly “interference” in the popular sense; but from a mechanical or mathematical point of view, the superposition contemplated in proposition viii. would more naturally be regarded as taking place without interference. Young applied his principle to the explanation of colours of striated surfaces (gratings), to the colours of thin plates, and to an experiment which we shall discuss later in the improved form given to it by Fresnel, where a screen is illuminated simultaneously by light proceeding from two similar sources. As a preliminary to these explanations we require an analytical expression for waves of simple type, and an examination of the effects of compounding them.
§ 2.Plane Waves of Simple Type.—Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves isA cos{2π(Vt − x) + α}λ(1),in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and may here be left an open question. The phase of the wave at a given time and place is represented by α. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period isτ = λ/V(2).In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wave-length, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wave-lengths in harmonical progression.A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of thesame wave-length, are equivalent to a single train of the same type. ThusΣA cos{2π(Vt − x) + α}= ΣA cos α·cos2π(Vt − x) − ΣA sin α·sin2π(Vt − x)λλλ= P cos{2π(Vt − x) + φ}λ(3),whereP2= (ΣA cos α)2= Σ(A sin α)2(4),tan φ =Σ(A sin α)Σ(A cos α)(5).An important particular case is that of two component trains only.A cos{2π(Vt − x) + α}+ A′ cos{2π(Vt − x) + α′}λλ= P cos{2π(Vt − x) + φ},λwhereP2= A2+ A′2+ 2AA′ cos (α − α′)(6).The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.According to the phase-relation, determined by (α − α′), the amplitude of the resultant may vary from (A − A′) to (A + A′). If A′ and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is that described by Young as “interference.”§ 3.Intensity.—The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to thesquareof the amplitude.§ 4.Resultant of a Large Number of Vibrations of Arbitrary Phase.—We have seen that the resultant of two vibrations of equal amplitudeis wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distributed, and may vary from n2to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit?The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to twooppositephases. We may then conveniently discard the idea of phase, and regard the amplitudes as at randompositive or negative. If all the signs are the same, the intensity is n2; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from 0 to n2, the smaller values are much more probable than the greater.The simplest part of the problem relates to what is called in the theory of probabilities the “expectation” of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations arc positive is 2−n, and thus the expectation of intensity corresponding to this contingency is 2−n·n2. In like manner the expectation corresponding to the number of positive vibrations being (n − 1) is2−n·n·(n − 2)2,and so on. The whole expectation of intensity is thus1{1·n2+ n·(n − 2)2+n(n − 1)(n − 4)22n1·2+n (n − 1) (n − 2)(n − 6)2+ ...}1·2·3(1).Now the sum of the (n + 1) terms of this series is simply n, as may be proved by comparison of coefficients of x2in the equivalent forms(ex+ e−x)n= 2n(1 + ½ x2+ ... )n= enx+ ne(n−2)x+n (n − 1)e(n−4)x+ ...1·2The expectation of intensity is therefore n, and this whether n be great or small.The same conclusion holds good when the phases are unrestricted. From (4), § 2, if A = 1,P2= n + 2Σ cos (α2− α1)(2),where under the sign of summation are to be included the cosines of the ½ n(n − 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P2is n.The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, themeanintensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order n2, we can infer nothing as to the value of the sum of the series in comparison with n.Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved (Phil. Mag., 1880, 10, p. 73; 1899, 47. p. 246) that the probability of a resultant intermediate in amplitude between r and r + dr is2e−r2/nr drn(3).The probability of an amplitude less than r is thus2∫r0e−r2/nr dr = 1 − e−r2/nn(4),or, which is the same thing, the probability of an amplitude greater than r ise−r2/n(5).The accompanying table gives the probabilities of intensities less than the fractions of n named in the first column. For example, the probability of intensity less than n is .6321..05.0488.80.5506.10.09521.00.6321.20.18131.50.7768.40.32962.00.8647.60.45123.00.9502It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations.Themeanintensity, expressed by2∫∞0e−r2/n· r2· r dr,nis, as we have already seen, equal to n.It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two candles are twice as bright as one.
§ 2.Plane Waves of Simple Type.—Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is
(1),
in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and may here be left an open question. The phase of the wave at a given time and place is represented by α. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period is
τ = λ/V
(2).
In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).
There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wave-length, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wave-lengths in harmonical progression.
A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of thesame wave-length, are equivalent to a single train of the same type. Thus
(3),
where
P2= (ΣA cos α)2= Σ(A sin α)2
(4),
(5).
An important particular case is that of two component trains only.
where
P2= A2+ A′2+ 2AA′ cos (α − α′)
(6).
The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.
According to the phase-relation, determined by (α − α′), the amplitude of the resultant may vary from (A − A′) to (A + A′). If A′ and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is that described by Young as “interference.”
§ 3.Intensity.—The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to thesquareof the amplitude.
§ 4.Resultant of a Large Number of Vibrations of Arbitrary Phase.—We have seen that the resultant of two vibrations of equal amplitudeis wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distributed, and may vary from n2to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit?
The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to twooppositephases. We may then conveniently discard the idea of phase, and regard the amplitudes as at randompositive or negative. If all the signs are the same, the intensity is n2; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from 0 to n2, the smaller values are much more probable than the greater.
The simplest part of the problem relates to what is called in the theory of probabilities the “expectation” of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations arc positive is 2−n, and thus the expectation of intensity corresponding to this contingency is 2−n·n2. In like manner the expectation corresponding to the number of positive vibrations being (n − 1) is
2−n·n·(n − 2)2,
and so on. The whole expectation of intensity is thus
(1).
Now the sum of the (n + 1) terms of this series is simply n, as may be proved by comparison of coefficients of x2in the equivalent forms
(ex+ e−x)n= 2n(1 + ½ x2+ ... )n
The expectation of intensity is therefore n, and this whether n be great or small.
The same conclusion holds good when the phases are unrestricted. From (4), § 2, if A = 1,
P2= n + 2Σ cos (α2− α1)
(2),
where under the sign of summation are to be included the cosines of the ½ n(n − 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P2is n.
The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, themeanintensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order n2, we can infer nothing as to the value of the sum of the series in comparison with n.
Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved (Phil. Mag., 1880, 10, p. 73; 1899, 47. p. 246) that the probability of a resultant intermediate in amplitude between r and r + dr is
(3).
The probability of an amplitude less than r is thus
(4),
or, which is the same thing, the probability of an amplitude greater than r is
e−r2/n
(5).
The accompanying table gives the probabilities of intensities less than the fractions of n named in the first column. For example, the probability of intensity less than n is .6321.
It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations.
Themeanintensity, expressed by
is, as we have already seen, equal to n.
It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two candles are twice as bright as one.
§ 5.Interference Fringes.—In Fresnel’s fundamental experiment light from a point O (fig. 1) falls upon an isosceles prism of glass BCD, with the angle at C very little less than two right angles. The source of light may be a pin-hole through which sunlight enters a dark room, or, more conveniently, the image of the sun formed by a lens of short focus (1 or 2 in.). For actual experiment when, as usually happens, it is desirable to economize light, thepointmay be replaced by alineof light perpendicular to the plane of the diagram, obtained either from a linear source, such as the filament of an incandescent electric lamp, or by admitting light through a narrow vertical slit.
If homogeneous light be used, the light which passes through the prism will consist of two parts, diverging as if from points O1and O2symmetrically situated on opposite sides of the line CO. Suppose a sheet of paper to be placed at A with its plane perpendicular to the line OCA, and let us consider what illumination will be produced at different parts of this paper. As O1and O2are images of O, crests of waves must be supposed to start from them simultaneously. Hence they will arrive simultaneously at A, which is equidistant from them, and there they will reinforce one another. Thus there will be a bright band on the paper parallel to the edges of the prism. If P1be chosen so that the difference between P1O2and P1O1is half a wave-length (i.e.half the distance between two successive crests), the two streams of light will constantly meet in such relative conditions as to destroy one another. Hence there will be a line of darkness on the paper, through P1, parallel to the edges of the prism. At P2, where O2P2exceeds O1P2by a whole wave-length, we have another bright band; and at P3, where O2P3exceeds O1P3by a wave-length and a half, another dark band; and so on. Hence, as everything is symmetrical about the bright band through A, the screen will be illuminated by a series of bright and dark bands, gradually shading into one another. If the paper screen be moved parallel to itself to or from the prism, the locus of all the successive positions of any one band will (by the nature of the curve) obviously be an hyperbola whose foci are O1and O2. Thus the interval between any two bands will increase in a more rapid ratio than does the distance of the screen from the source of light. But the intensity of the bright bands diminishes rapidly as the screen moves farther off; so that, in order to measure their distance from A, it is better to substitute the eye (furnished with a convex lens) for the screen. If we thus measure the distance AP1between A and the nearest bright band, measure also AO, and calculate (from the known material and form of the prism, and the distance CO) the distance O1O2, it is obvious that we can deduce from them the lengths of O1P2and O2P2. Their difference is thelength of a waveof the homogeneous light experimented with. Though this is not the method actually employed for the purpose (as it admits of little precision), it has been thus fully explained here because it shows in a very simple way the possibility of measuring a wave-length.The difference between O1P1and O2P1becomes greater as AP1is greater. Thus it is clear that the bands aremore widely separated the longer the wave-length of the homogeneous light employed. Hencewhen we use white light, and thus have systems of bands of every visible wave-length superposed, the band A will be red at its edges, the next bright bands will be blue at their inner edges and red at their outer edges. But, after a few bands are passed, the bright bands due to one kind of light will gradually fill up the dark bands due to another; so that, while we may count hundreds of successive bright and dark bars when homogeneous light is used, with white light the bars become gradually less and less defined as they are farther from A, and finally merge into an almost uniform white illumination of the screen.If D be the distance from O to A, and P be a point on the screen in the neighbourhood of A, then approximatelyO1P − O2P = √{ D2+ (u + ½b)2} − √{ D2+ (u − ½b)2} = ub / D,where O1O2= b, AP = u.Thus, if λ be the wave-length, the places where the phases are accordant are given byu = nλD / b(1),n being an integer.If the light were really homogeneous, the successive fringes would be similar to one another and unlimited in number; moreover there would be no place that could be picked out by inspection as the centre of the system. In practice λ varies, and (as we have seen) the only place of complete accordance for all kinds of light is at A, where u = 0. Theoretically, there is no place of complete discordance for all kinds of light, and consequently no complete blackness. In consequence, however, of the fact that the range of sensitiveness of the eye is limited to less than an “octave,” the centre of the first dark band (on either side) is sensibly black, even when white light is employed; but it should be carefully remarked that the existence of even one band is due to selection, and that the formation of several visible bands is favoured by the capability of the retina to make chromatic distinctions within the visible range.The number of perceptible bands increasespari passuwith the approach of the light to homogeneity. For this purpose there are two methods that may be used.We may employ light, such as that from the soda flame, which possessesab initioa rather high degree of homogeneity. If the range of wave-length included be1⁄50000, a corresponding number of interference fringes may be made visible. The above was the number obtained by A. H. L. Fizeau. Using vacuum tubes containing, for example, mercury or cadmium vapour, A. A. Michelson has been able to go much farther. The narrowness of the bright line of light seen in the spectroscope, and the possibility of a large number of Fresnel’s bands, depend upon precisely the same conditions; the one is in truth as much an interference phenomenon as the other.In the second method the original light may be highly composite, and homogeneity is brought about with the aid of a spectroscope. The analogy with the first method is closest if we use the spectroscope to give us a line of homogeneous light in simple substitution for the artificial flame. Or, following J. B. L. Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu/(nD). In either case the number of bands observable is limited solely by the resolving power of the spectroscope, and proves nothing with respect to the regularity, or otherwise, of the vibrations of the original light.
If homogeneous light be used, the light which passes through the prism will consist of two parts, diverging as if from points O1and O2symmetrically situated on opposite sides of the line CO. Suppose a sheet of paper to be placed at A with its plane perpendicular to the line OCA, and let us consider what illumination will be produced at different parts of this paper. As O1and O2are images of O, crests of waves must be supposed to start from them simultaneously. Hence they will arrive simultaneously at A, which is equidistant from them, and there they will reinforce one another. Thus there will be a bright band on the paper parallel to the edges of the prism. If P1be chosen so that the difference between P1O2and P1O1is half a wave-length (i.e.half the distance between two successive crests), the two streams of light will constantly meet in such relative conditions as to destroy one another. Hence there will be a line of darkness on the paper, through P1, parallel to the edges of the prism. At P2, where O2P2exceeds O1P2by a whole wave-length, we have another bright band; and at P3, where O2P3exceeds O1P3by a wave-length and a half, another dark band; and so on. Hence, as everything is symmetrical about the bright band through A, the screen will be illuminated by a series of bright and dark bands, gradually shading into one another. If the paper screen be moved parallel to itself to or from the prism, the locus of all the successive positions of any one band will (by the nature of the curve) obviously be an hyperbola whose foci are O1and O2. Thus the interval between any two bands will increase in a more rapid ratio than does the distance of the screen from the source of light. But the intensity of the bright bands diminishes rapidly as the screen moves farther off; so that, in order to measure their distance from A, it is better to substitute the eye (furnished with a convex lens) for the screen. If we thus measure the distance AP1between A and the nearest bright band, measure also AO, and calculate (from the known material and form of the prism, and the distance CO) the distance O1O2, it is obvious that we can deduce from them the lengths of O1P2and O2P2. Their difference is thelength of a waveof the homogeneous light experimented with. Though this is not the method actually employed for the purpose (as it admits of little precision), it has been thus fully explained here because it shows in a very simple way the possibility of measuring a wave-length.
The difference between O1P1and O2P1becomes greater as AP1is greater. Thus it is clear that the bands aremore widely separated the longer the wave-length of the homogeneous light employed. Hencewhen we use white light, and thus have systems of bands of every visible wave-length superposed, the band A will be red at its edges, the next bright bands will be blue at their inner edges and red at their outer edges. But, after a few bands are passed, the bright bands due to one kind of light will gradually fill up the dark bands due to another; so that, while we may count hundreds of successive bright and dark bars when homogeneous light is used, with white light the bars become gradually less and less defined as they are farther from A, and finally merge into an almost uniform white illumination of the screen.
If D be the distance from O to A, and P be a point on the screen in the neighbourhood of A, then approximately
O1P − O2P = √{ D2+ (u + ½b)2} − √{ D2+ (u − ½b)2} = ub / D,
where O1O2= b, AP = u.
Thus, if λ be the wave-length, the places where the phases are accordant are given by
u = nλD / b
(1),
n being an integer.
If the light were really homogeneous, the successive fringes would be similar to one another and unlimited in number; moreover there would be no place that could be picked out by inspection as the centre of the system. In practice λ varies, and (as we have seen) the only place of complete accordance for all kinds of light is at A, where u = 0. Theoretically, there is no place of complete discordance for all kinds of light, and consequently no complete blackness. In consequence, however, of the fact that the range of sensitiveness of the eye is limited to less than an “octave,” the centre of the first dark band (on either side) is sensibly black, even when white light is employed; but it should be carefully remarked that the existence of even one band is due to selection, and that the formation of several visible bands is favoured by the capability of the retina to make chromatic distinctions within the visible range.
The number of perceptible bands increasespari passuwith the approach of the light to homogeneity. For this purpose there are two methods that may be used.
We may employ light, such as that from the soda flame, which possessesab initioa rather high degree of homogeneity. If the range of wave-length included be1⁄50000, a corresponding number of interference fringes may be made visible. The above was the number obtained by A. H. L. Fizeau. Using vacuum tubes containing, for example, mercury or cadmium vapour, A. A. Michelson has been able to go much farther. The narrowness of the bright line of light seen in the spectroscope, and the possibility of a large number of Fresnel’s bands, depend upon precisely the same conditions; the one is in truth as much an interference phenomenon as the other.
In the second method the original light may be highly composite, and homogeneity is brought about with the aid of a spectroscope. The analogy with the first method is closest if we use the spectroscope to give us a line of homogeneous light in simple substitution for the artificial flame. Or, following J. B. L. Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu/(nD). In either case the number of bands observable is limited solely by the resolving power of the spectroscope, and proves nothing with respect to the regularity, or otherwise, of the vibrations of the original light.
In lieu of the biprism, reflectors may be invoked to double the original source of light. In one arrangement two reflected images are employed, obtained from two reflecting surfaces nearly parallel and in the same plane. Glass, preferably blackened behind, may be used, provided the incidence be made sufficiently oblique. In another arrangement, due to H. Lloyd, interference takes place between light proceeding directly from the original source, and from one reflected image. Lloyd’s experiment deserves to be better known, as it may be performed with great facility and without special apparatus. Sunlight is admitted horizontally into a darkened room through a slit situated in a window-shutter, and, at a distance of 15 to 20 ft., is received at nearly grazing incidence upon a vertical slab of plate glass. The length of the slab in the direction of the light should not be less than 2 or 3 in., and for some special observations may advantageously be much increased. The bands are observed on a plane through the hinder vertical edge of the slab by means of a hand-magnifying glass of from 1 to 2 in. focus. The obliquity of the reflector is, of course, to be adjusted according to the fineness of the bands required.
From the manner of their formation it might appear that under no circumstances could more than half the system be visible. But according to Sir G. B. Airy’s principle (see below) the bands may be displaced if examined through a prism. In practice all that is necessary is to hold the magnifier somewhat excentrically. The bands may then be observed gradually to detach themselves from the mirror, until at last the complete system is seen, as in Fresnel’s form of the experiment.
The fringes now under discussion are those which arise from the superposition of two simple and equal trains of waves whose directions are not quite parallel. If the two directions of propagation are inclined on opposite sides of the axis of x at small angles α, the expressions for two components of equal amplitude arecos2π{Vt − x cos α − y sin α},λandcos2π{Vt − x cos α + y sin α},λso that the resultant is expressed by2 cos2πy sin αcos2π{Vt − x cos α},λλfrom which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity V/cos α. Considered as depending on y, the vibration is a maximum when y sin α is equal to O, λ, 2λ, 3λ, &c., corresponding to the centres of the bright bands, while for intermediate values ½λ,3⁄2λ, &c., there is no vibration.From (1) we see that the linear width Λ of the bands, reckoned from bright to bright or dark to dark, isΛ = λD / b(2).The degree of homogeneity necessary for the approximate perfection of the nthFresnel’s band may be found at once from (1) and (2). For if du be the change in u corresponding to the change dλ, thendu / Λ = ndλ / λ(3).Now clearly du must be a small fraction of Λ, so that dλ/λ must be many times smaller than 1/n, if the darkest places are to be sensibly black. But the phenomenon will be tolerably well marked if the proportional range of wave-length do not exceed 1/2n, provided, that is, that the distribution of illumination over this range be not concentrated towards the extreme parts.So far we have supposed the sources at O1, O2to be mathematically small. In practice, the source is an elongated slit, whose direction requires to be carefully adjusted to parallelism with the reflecting surface or surfaces. By this means an important advantage is gained in respect of brightness without loss of definition, as the various parts of the aperture give rise to coincident systems of bands.The question of the admissiblewidthof the slit requires consideration. We will suppose that the light issuing from various parts of the aperture is without permanent phase-relations, as when the slit is backed immediately by a flame, or by an incandescent filament. Regular interference can then only take place between light coming fromcorrespondingparts of the two images, and a distinction must be drawn between the two ways in which the images may be situated relatively to one another. In Fresnel’s experiment, whether carried out with the mirrors or with the biprism, the corresponding parts of the images are on the same side; that is, the right of one corresponds to the right of the other, and the left of the one to the left of the other. On the other hand, in Lloyd’s arrangement the reflected image is reversed relatively to the original source; the two outer edges corresponding, as also the two inner. Thus in the first arrangement the bands due to various parts of the slit differ merely by a lateral shift, and the condition of distinctness is simply that the projection of the width of the slit be a small fraction of the width of the bands. From this it follows as a corollary that the limiting width is independent of the order of the bands under examination. It is otherwise in Lloyd’s method. In this case the centres of the systems of bands are the same, whatever part of the slit is supposed to be operative, and it is the distance apart of the images (b) that varies. The bands corresponding to the various parts of the slit are thus upon different scales, and the resulting confusion must increase with the order of the bands. From (1) the corresponding changes in u and b are given bydu = −nλD db/b2;so thatdu/Λ = −n db/b(4).If db represents twice the width of the slit, (4) gives a measure of the resulting confusion in the bands. The important point is that the slit must be made narrower as n increases if the bands are to retain the same degree of distinctness.
The fringes now under discussion are those which arise from the superposition of two simple and equal trains of waves whose directions are not quite parallel. If the two directions of propagation are inclined on opposite sides of the axis of x at small angles α, the expressions for two components of equal amplitude are
and
so that the resultant is expressed by
from which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity V/cos α. Considered as depending on y, the vibration is a maximum when y sin α is equal to O, λ, 2λ, 3λ, &c., corresponding to the centres of the bright bands, while for intermediate values ½λ,3⁄2λ, &c., there is no vibration.
From (1) we see that the linear width Λ of the bands, reckoned from bright to bright or dark to dark, is
Λ = λD / b
(2).
The degree of homogeneity necessary for the approximate perfection of the nthFresnel’s band may be found at once from (1) and (2). For if du be the change in u corresponding to the change dλ, then
du / Λ = ndλ / λ
(3).
Now clearly du must be a small fraction of Λ, so that dλ/λ must be many times smaller than 1/n, if the darkest places are to be sensibly black. But the phenomenon will be tolerably well marked if the proportional range of wave-length do not exceed 1/2n, provided, that is, that the distribution of illumination over this range be not concentrated towards the extreme parts.
So far we have supposed the sources at O1, O2to be mathematically small. In practice, the source is an elongated slit, whose direction requires to be carefully adjusted to parallelism with the reflecting surface or surfaces. By this means an important advantage is gained in respect of brightness without loss of definition, as the various parts of the aperture give rise to coincident systems of bands.
The question of the admissiblewidthof the slit requires consideration. We will suppose that the light issuing from various parts of the aperture is without permanent phase-relations, as when the slit is backed immediately by a flame, or by an incandescent filament. Regular interference can then only take place between light coming fromcorrespondingparts of the two images, and a distinction must be drawn between the two ways in which the images may be situated relatively to one another. In Fresnel’s experiment, whether carried out with the mirrors or with the biprism, the corresponding parts of the images are on the same side; that is, the right of one corresponds to the right of the other, and the left of the one to the left of the other. On the other hand, in Lloyd’s arrangement the reflected image is reversed relatively to the original source; the two outer edges corresponding, as also the two inner. Thus in the first arrangement the bands due to various parts of the slit differ merely by a lateral shift, and the condition of distinctness is simply that the projection of the width of the slit be a small fraction of the width of the bands. From this it follows as a corollary that the limiting width is independent of the order of the bands under examination. It is otherwise in Lloyd’s method. In this case the centres of the systems of bands are the same, whatever part of the slit is supposed to be operative, and it is the distance apart of the images (b) that varies. The bands corresponding to the various parts of the slit are thus upon different scales, and the resulting confusion must increase with the order of the bands. From (1) the corresponding changes in u and b are given by
du = −nλD db/b2;
so that
du/Λ = −n db/b
(4).
If db represents twice the width of the slit, (4) gives a measure of the resulting confusion in the bands. The important point is that the slit must be made narrower as n increases if the bands are to retain the same degree of distinctness.
§ 6.Achromatic Interference Bands.—We have already seen that in the ordinary arrangement, where the source is of white light entering through a narrow slit, the heterogeneity of the light forbids the visibility of more than a few bands. The scaleof the various band-systems is proportional to λ. But this condition of things, as we recognize from (2) (see § 5), depends upon the constancy of b,i.e.upon the supposition that the various kinds of light all come from the same place. Now there is no reason why such a limitation need be imposed. If we regard b as variable, we see that we have only to take b proportional to λ, in order to render the band-interval Λ independent of colour. In such a case the system of bands isachromatic, and the heterogeneity of the light is no obstacle to the formation of visible bands of high order.