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THE THEORY OF LIGHT 
 
 PART I 
 
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THE THEORY OF LIGHT 
 
 A TREATISE ON PHYSICAL OPTICS 
 
 by 
 RICHARD C. MACLAURIN, M.A., LL.D. 
 
 \ \ 
 
 Formerly Fellow of St John's College, Cambridge ; 
 
 Professor of Mathematics and Mathematical Physics, Wellington, 
 
 New Zealand 
 
 IN THREE PARTS 
 PART I 
 
 7 
 
 OF THE 
 UNIVERSITY 
 
 CAMBRIDGE : 
 
 at the University Press 
 
 1908 
 
(STatnimfcge : 
 
 PRINTED BY JOHN CLAY, M.A. 
 AT THE UNIVERSITY PRESS. 
 
PREFACE. 
 
 THE aim of the treatise, of which this volume forms the first 
 part, is to give an account of the theory of physical optics 
 that is systematic, and as complete as possible within the somewhat 
 narrow limits within which I have thought it expedient to confine 
 myself. I hope, of course, that the work will be found useful to 
 the student, and it is mainly in his interests that I have begun 
 with a brief discussion of the aim and the method of science. 
 I place no special value on my statement of this matter ; but I do 
 not think that it is easy to exaggerate the importance of under- 
 standing clearly what we are really aiming at. Without an 
 agreement on this point we can have no criterion as to the 
 propriety or otherwise of our methods; and no final concord as to 
 our success or failure. And if I have marked out the limits of 
 science in a form that frees it as much as possible from meta- 
 physical difficulties, it is certainly not because I regard the 
 " intrusion " of metaphysics into science as necessarily baneful. 
 As a matter of fact every problem of science, in so far as it 
 involves thinking at all, abuts on a problem of metaphysics, and 
 we can scarcely be surprised if the workers in adjoining fields 
 occasionally cross the boundary. There might be advantages in 
 each keeping to his own domain; but if the physicist will be 
 a trespasser, what he must be careful to avoid is not so much 
 metaphysics as bad metaphysics. We can scarcely acquiesce in the 
 simple faith of some men of science that their metaphysics is to 
 be accepted merely because it is unconscious or naive. 
 
 A serious difficulty in discussing the aims and methods of 
 physics in a work that is mainly concerned with a special depart- 
 ment of that science, is to keep the discussion within reasonable 
 
 181410 
 
vi PREFACE 
 
 limits. In a few pages one has to touch on questions that might 
 easily occupy a book, and it may be that in my endeavour to 
 avoid obscurity on the one hand and diffuseness on the other, 
 I have "moved as in a strange diagonal" and satisfied no one. 
 As everything is made to rest on a dynamical basis perhaps I should 
 have subjected the fundamental concepts of that science to 
 a careful scrutiny. This, however, would be more appropriate to 
 a work on the Principles of Mechanics, and after some trial I have 
 concluded that it could not be undertaken here without taking up 
 more space than could be given to it. Perhaps, too, the student 
 may be content to work with the tools that he has inherited, 
 without denying that they would be the better for sharpening or 
 that they might even be replaced by more modern articles. In 
 any case, the writer has taken warning from his experiences in 
 practical mechanics that an unskilled attempt to sharpen a tool 
 is apt to remove whatever edge it has. 
 
 I hope that the foundation has been laid broadly enough to 
 support the whole superstructure and that the mode of treatment 
 will make clear the relation between light and electricity, when 
 these two sciences come into contact. This will appear more fully 
 in the second volume, which deals with those branches of our 
 subject not discussed in this, viz. such matters as dispersion, the 
 rotations structural and magnetic aberration, diffraction, etc. 
 The connection between optical and electrical phenomena is often 
 imperfectly presented. In some of the text-books it is scarcely 
 referred to, except in a sketchy way towards the close. In others 
 the science of optics is based on the electromagnetic theory in such 
 a manner that students who have not made a careful study of the 
 bases of electrical theory are ignorant of the real foundations. 
 The natural method seems to be to lay such a dynamical foundation 
 that all phenomena, electrical and optical, may be colligated with 
 the aid of the same ethereal medium. 
 
 The third, and concluding, volume of this work will be devoted 
 to a history of optical theories. Most modern works on light give 
 at least a passing reference to the older theories, such as the 
 elastic solid theory, sometimes even in such a way as to suggest 
 that these theories are still tenable. It seems preferable to state 
 the modern theory and, after working it out in detail, in order to 
 
PREFACE vii 
 
 test its usefulness, to supplement this with a history of the past. 
 We do this, not with the object of making dead bones live, but to 
 get an insight into the manner in which a great scientific theory is 
 actually built up, and to give a human interest to our study by 
 learning something of the lives and modes of thought of the great 
 men who have raised the structure. 
 
 A special feature of the work is the careful comparison between 
 theory and experiment at every stage. From our standpoint as to 
 the aim of science, such a comparison appears all-important. The 
 one difficulty in this matter has been to get access to the most 
 accurate and recent experimental results; a difficulty arising 
 entirely from the present isolation of the writer. At the ends of 
 the earth there are no great libraries to which to refer, and the 
 procuring of material from ' home ' is tedious and uncertain. In 
 dealing with experimental results I have deliberately refrained 
 from entering upon descriptions of the methods of experiment as, 
 in my judgment, quite out of place in such a work as this. 
 
 A large part of this volume embodies in a modified form the 
 substance of a series of papers of mine published within recent 
 years by the Royal Society. I have rarely, however, made any 
 direct reference to these or any other papers in the course of the 
 volume. Such references, to be complete, would occupy a great 
 deal of space, and the student who wishes to consult the original 
 ' authorities,' as every serious student must, will obtain practically 
 all the directions he can want in such a work of reference as 
 Winkelmann's Handbuch der Physik*. 
 
 My special thanks are due to Mr W. J. Harrison, of Clare, who 
 has undertaken the revision of the proof-sheets, a task rendered 
 impracticable by my residence at the antipodes. 
 
 R. C. M. 
 
 WELLINGTON. 
 June, 1907. 
 
 * However, at the suggestion of the Syndics of the Cambridge University Press, 
 a few references have been supplied in footnotes by Mr Harrison. 
 
CONTENTS. 
 
 CHAP. 
 
 I. THE SCOPE AND METHOD OF THE INQUIRY .... 
 
 II. PRELIMINARY IDEAS - 
 
 III. PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC MEDIA. 
 
 REFLECTION AND REFRACTION 
 
 IV. INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION 
 
 AND REFRACTION 
 
 V. ON TRANSPARENT ISOTROPIC PLATES AND ON NEWTON'S 
 RINGS 
 
 VJ. PROPAGATION OF LIGHT IN TRANSPARENT CRYSTALLINE 
 MEDIA 
 
 VII. REFLECTION AND REFRACTION WITH TRANSPARENT CRYSTALS . 
 
 VIII. TRANSPARENT CRYSTALLINE PLATES 
 
 IX. PROPAGATION OF LIGHT IN ABSORBING ISOTROPIC MEDIA . 
 
 X. ON NEWTON'S RINGS FORMED BY METALLIC REFLECTION 
 
 AND ON SOME OPTICAL PROPERTIES OF THIN METALLIC 
 PLATES 
 
 XI. PROPAGATION OF LIGHT IN ABSORBING CRYSTALLINE MEDIA . 
 INDEX 
 
CHAPTER I. 
 
 THE SCOPE AND METHOD OF THE INQUIRY. 
 
 " If, Theaetetus, you have a wish to have any more embryo thoughts, you 
 will be all the better for the present investigation and, if not, you will be 
 soberer and humbler and gentler to other men, not fancying that you know 
 what you do not know." PLATO. 
 
 " WHAT is the chief end of science ? " should be the first 
 question in the catechism of every physicist. We raise it here 
 not so much for the reasons suggested by the words that Plato 
 makes Socrates address to his pupil, although it is certainly well 
 to know our limitations and to shake assurance from our creeds, 
 as from the fact that in any inquiry the concepts that we employ 
 and the methods that we pursue must be determined, wholly or 
 in part, by the end we have in view. 
 
 What, then, is that end for physical science ? Of the various 
 answers that might be suggested we shall consider briefly the 
 most important. Probably the most popular is that the end of 
 science is to ' explain ' the world, and our agreement or dissent 
 must depend entirely on the meaning we attach to the word 
 1 explanation.' It is a word used in many senses, but the one 
 that will naturally occur to the mind of a person who has often 
 been called upon to explain anything and has attempted to do so 
 with any hope of success is the reference of the unknown to the 
 known. An explanation seeks to point out resemblances between 
 facts and to lay bare the connecting links between the unfamiliar 
 and the familiar. It is the desire to achieve this end that makes 
 the physicist so anxious to point out the traces that enable the 
 Sun to drag a planet round its orbit. It is this, too, that some- 
 times urges him to declare that all physical action is by impact, 
 
 M. L. 1 
 
2 THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 the idea of a collision and its consequences being so familiar. 
 And the historical fact that the science of mechanics was developed 
 before other branches of science has doubtless contributed to that 
 purely mechanical view of the universe, whose general acceptance 
 is, in the opinion of some, " that far-off divine event towards which 
 the whole creation moves." 
 
 However, although science may often strive, and sometimes 
 with success, to refer the unknown to the known, there can be no 
 doubt that it much more frequently reverses the process and 
 refers the known to the unknown. Innumerable instances will 
 occur to the student, and of these a few will suffice. In dynamics 
 we ' explain ' most things by means of Newton's Laws of Motion, 
 or by the conservation of energy, or the Principle of Action, or 
 the law of gravitation. It is clear that the ' explanation ' of so 
 well known and familiar a phenomenon as the fall of an apple 
 by the statement that "every particle in the universe attracts 
 every other particle with a force that varies directly as the 
 product of the masses of the particles and inversely as the square 
 of the distance between them " cannot be properly described as a 
 reference of the unknown to the known. Again in the theory of 
 sound we ' explain ' the music of our pianos by referring it to 
 waves in the air, with the attendant obscurities of elasticity, of 
 Boyle's law, Charles' law, the adiabatic law, and so forth. In heat, 
 we ' explain ' the heating of a poker by the vibrations of particles 
 that no one knows by experience, or we account for the failure of 
 an engine by the statement that "it is impossible for a self- 
 acting machine, unaided by any external agency, to convey heat 
 from one body to another at a higher temperature." Lastly in 
 electricity and optics we ' explain ' everything by reference to the 
 ether, which, for aught anyone can say, may be a mere figment of 
 the imagination. 
 
 It may, perhaps, be urged, that many of these are real 
 ' explanations,' for they refer things that we do not understand to 
 laws that are known. But how are these laws known ? The best 
 that can be said for them is that they are obtained by Induction 
 from the facts of experience. What, then, is Induction ? It was 
 described by Bacon as a process by means of which we can obtain 
 universal propositions from particular perceptions ; but neither 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 3 
 
 Bacon nor anyone since him has succeeded in making it clear how 
 this can be done by the method, or how, under any circumstances, 
 we can get more out of a thing than we put into it. The actual 
 process of building up a scientific law may be roughly described 
 thus. The phenomena under consideration are regarded from 
 particular aspects, as in all thinking, scientific or otherwise. In 
 this way conceptions are formed which correspond in a partial and 
 a symbolical manner to the phenomena. An endeavour is then 
 made to connect these concepts by a law. There is no logical 
 necessity for this, for it is conceivable that knowledge of universals 
 might be denied us. At the same time we realise that, if there is 
 to be any knowledge, there must be some law, so we assume that 
 it exists and that it may be found. The fact that we are 
 constrained to make this initial assumption is an interesting 
 psychological fact, and it is a striking result that the consequences 
 seem to justify the assumption. As to what the law is the 
 process of induction can give us only hints. What is really done 
 is to set out with some hypothesis and to compare its consequences 
 with the observed phenomena; but the primary hypothesis 
 cannot be found by rule and we can never through induction get 
 the idea of necessity into the sequence. Agreement between the 
 phenomena and the consequences of the hypothesis may make the 
 hypothesis tenable, it can never prove it. Hence all reference to 
 scientific 'laws' involve something that cannot be explained, so 
 that, except by imputing a special meaning to the word, we cannot 
 regard ' explanation ' as the real end of science. 
 
 Another answer to our fundamental question is that the aim 
 of science is to get rid of the arbitrary and the apparent and to 
 reveal the real. The aim is certainly a lofty one and we have no 
 a priori objections to the pursuit ; for to say at the outset that 
 the real is unknowable is to claim to know reality. Our objection 
 is that if science is to attempt so difficult a task, it must devote 
 itself to a much closer scrutiny of its methods and its concepts 
 than is its wont, and it must enter the lists with the champions of 
 different philosophical systems reasonably equipped for the fray. 
 So far, however, those who have made the most serious contribu- 
 tions to science have seldom had either the time, the inclination, 
 or the aptitude necessary for such work. Of recent years, at any 
 
4 THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 rate, the exhortation to " beware of metaphysics " seems to have 
 had its effect, and on the rare occasions when leading men of 
 science have disregarded it, the results have not been reassuring. 
 The claim to find reality, if made at all, is usually put forward by 
 the rank and file, which is less conscious of its limitations. These 
 men find certain concepts, such as matter and energy, useful for 
 the purposes of science. They spend their lives over problems in 
 which these are the only concepts needed. They assume that no 
 others can ever be needed. Next they take the extraordinary 
 step of declaring dogmatically that these concepts are real things, 
 "The only real things in the- physical universe are matter and 
 energy," say the authors of the Unseen Universe. This is 
 worse than the conduct of the artist who, after devoting his life to 
 portraiture, declared that the only real thing was paint. It takes 
 certain qualities of things, arbitrarily abstracts something, and 
 then declares the abstraction to be real and all that is real. This, 
 of course, is to take a ghost for reality, and there is actually a 
 school so frightened by the ghost as to worship it. 
 
 Some physicists who probably mean to convey the idea that 
 the end of science is to grasp the real, express themselves in 
 rather unintelligible language by assuring us that science reveals 
 the nature of " things in themselves." However, the idea of a 
 " thing in itself" proves, on examination, to be worthless, even if 
 it be not meaningless. Some of the difficulties are suggested in 
 the following words of von Helmholtz. " We speak of the solu- 
 bility of a substance, meaning its behaviour towards water; we 
 speak of its weight meaning its attraction to the earth ; we may 
 justly call a substance blue under the tacit assumption that we 
 are speaking of its action upon a normal eye. But, if what we 
 call a property always implies a relation between two things, 
 then a property or quality can never depend upon the nature of 
 one agent alone, but exists only in relation to and dependence on 
 the nature of some second subject acted upon. Hence, there is 
 really no sense in talking of properties of light that belong to it 
 absolutely, independently of all other objects, and that are 
 supposed to be representable in the sensations of the human eye. 
 The notion of such properties is a contradiction in itself." If you 
 agree with this, then when you have set aside all the qualities of a 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 5 
 
 thing, as necessarily involving a relation to something else, you 
 have not much left to be worthy of a name. If, however, you 
 insist on retaining the idea and the name of a " thing in itself," it 
 is worthless for scientific purposes. For either it is related to 
 phenomena or it is not. If it is not so related, it must be useless 
 to the man of science who is interested in phenomena, i.e. in things 
 as they appear to him. If, on the other hand, it is related to 
 phenomena, you have merely doubled his difficulties. All the old 
 problems of the phenomenal remain, with the added perplexity of 
 their relation to this empty " thing in itself." 
 
 And if science has no interest in " things in themselves " and 
 does not seek to reveal the real, it is equally true that it does not 
 get rid of the arbitrary and the apparent. As a matter of fact 
 the methods of science are often arbitrary in the extreme. Ordin- 
 ary knowledge involves a certain measure of arbitrariness, as it 
 selects what is of interest and dwells on certain aspects of things. 
 Scientific knowledge carries the process a stage further, and this, 
 not of course for any love of the arbitrary, but because it realises 
 that to succeed it must limit itself. And so far from getting rid 
 of appearances, these are what it specially strives to know and to 
 master. The mysteries of any hidden world " behind phenomena " 
 it is content to leave to others. 
 
 Let us turn to a third answer to the question with which we 
 began. " The final aim of physical science," says von Helmholtz, 
 " is to find the ultimate unchangeable causes of the processes in 
 nature." Here, just as in the case of a search for reality, we have 
 no a priori objections to raise against the quest. We do not, for 
 example, refuse to look for causes because, as has sometimes been 
 objected, the conception is ' anthropomorphic.' It is affirmed that 
 the conception of cause arises from the thought of one being 
 doing something to another and that this is 'anthropomorphic.' 
 Our answer is that all concepts are necessarily so, for it is a 
 truism, often strangely overlooked, that whether man be the 
 measure of all things or not, he is certainly the measure of his 
 own concepts. Our objection is that the conception of cause is a 
 popular and loose one, and must be carefully scrutinised before it 
 is admitted into the serious domain of science. Jones has met 
 his death at the hands of Sikes. What is the cause of the murder? 
 
6 THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 " The depravity of the human heart " says the moralist, " the 
 malice aforethought of Sikes " says the Crown Prosecutor, " the 
 conversion of the potential energy of gunpowder into the kinetic 
 energy of the bullet " says the physicist, " the rupture by the 
 bullet of some vital organ " says the doctor. Each of these 
 arbitrarily dwells on the special aspect that is interesting from his- 
 point of view and pronounces it the cause. The one point on 
 which they all agree is that cause is not an " unchangeable thing,'* 
 but is a condition. But which condition is to be described as the 
 true cause ? Here the man of science, seeking to avoid all 
 arbitrariness, is placed in a dilemma. The concept includes too 
 much or too little. The cause may mean all the conditions, in 
 which case it embraces the universe and is the cause of everything 
 that is, and not merely of the special event under consideration. 
 If on the other hand, he attempts to narrow the idea and always 
 to avoid the arbitrary, he has no alternative but to make the cause 
 and the effect identical, and so practically to get rid of causation 
 as a fruitful concept. It is this that has led him sometimes to 
 drive out ' force ' from mechanics, or to retain it only as a formal 
 concept as an aid in the statement of a law. If, however, he 
 wishes to retain causation, as he may well do, he must recognise 
 that there is something arbitrary about it. It singles out some 
 one of all the conditions of an event and speaks of this as the 
 cause. The principle on which this selection is made must depend 
 entirely on the end in view, so that cause is essentially an idea 
 that he brings to the phenomena and not one that the phenomena 
 impose on him. 
 
 And not only must it be admitted that there is something 
 arbitrary in the relation between cause and effect, but the most 
 important element in the relation as it is usually thought of and 
 described must present great difficulties to the man of science. 
 This is the element of necessity in the sequence. Whence comes 
 this idea of necessity, does it exist in the objects or only in our 
 minds ? To say that it exists in the objects and to ascribe it, as 
 is often done, to some " inner necessity of their development " 
 leaves this inner necessity the greatest of unsolved problems ; 
 while to account for it by means of ' laws,' such as the uniformity 
 of nature, requires the use of hypotheses that pass beyond 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 7 
 
 experience. If, on the other hand, we say that the idea of 
 necessity exists only in the mind, we must be prepared to enter 
 into investigations as to the nature and mode of working of the 
 mind and so raise almost innumerable questions that do not belong 
 to the sphere of physics as that science is ordinarily understood. 
 
 Thus far we have considered three different answers to the 
 question with which we began. To explain the world, to reveal 
 the real, or to show the causes of nature's processes, have each in 
 turn been put forward as the end of science. Our chief objections 
 are that the actual methods of science do not suggest any such 
 end and that it is inexpedient to involve science in metaphysical 
 and philosophical difficulties about which there is no immediate 
 prospect of agreement. Science is interested only in experience ; 
 its end is to know and to communicate. At the outset experience 
 is but an indefinite manifold, or a confused blur. Science, like 
 all thought, seeks to introduce definiteness and order. It soon 
 accumulates a mass of facts that no mind could grasp and no 
 memory retain. Its aim is to find a principle or ' law ' that will 
 bind all harmoniously together. As soon as it realises this, it 
 becomes a premeditated art, not only esthetic in its intensity, but 
 consciously esthetic in its aim. 
 
 If we recognise this as a true description of the purpose of 
 science, we shall admit that science is not much concerned with 
 reality and that the solution of the various philosophical difficulties 
 that have been raised is, for it, almost a matter of indifference. 
 The practical end it has in view is served equally well whether 
 its hypotheses correspond to reality or not. It may echo the 
 words of Descartes " Meme je crois qu'il est aussi utile pour la 
 vie de connaitre les causes ainsi imaginees que si on avait la 
 connaissance des vraies," and perhaps even assent to the remark 
 of Osiander in his preface to Kepler's books De revolutionibus 
 ccelestibus " It is not necessary that scientific hypotheses be true 
 or even probable ; the one thing required of them is to reconcile 
 calculation and observation." Of course no one reviewing the 
 work of the great men of science can fail to be impressed with 
 " the structure brave, the manifold music they build " and to ask 
 in what sense it is real. No answer worth listening to can, however, 
 be given without due consideration of the meaning of reality. 
 
THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 This is the province of the philosopher, and the man of science 
 who invades it rashly is in danger of making an exhibition of his 
 ignorance or of his incompetency for philosophic discussion. He 
 is probably wiser to confine himself to phenomena and their laws, 
 and to make no claim to account for everything. It is not for 
 him to say that there is no pathway to reality, or that there is a 
 fixed gulf between appearance and reality, or that it is not worth 
 the attempt to bridge it. All he need say is that the search for 
 reality and the ascertaining of ultimate truth are beyond his 
 province. For such an one it is a great relief to be freed as much 
 as possible from the incubus of metaphysics and to feel that he 
 need not stay his progress to await the issue of the conflict 
 between different schools of philosophy. Certain hypotheses for 
 him are not so much false as useless. 
 
 According to the view here presented the aim of science is 
 practical and esthetic. In its practical aspect it seeks to colligate 
 experiences with the object of enabling us to know as much as 
 possible and to communicate our knowledge. To achieve this 
 great end it seeks a harmony, the contemplation of which gives it 
 its esthetic interest. The harmony it expresses by a ' principle ' 
 or ' law.' If this be its aim, we should expect science to advance 
 by trying hypothesis after hypothesis until it reaches one that fits 
 in with all the facts. And whether this has been the conscious 
 aim of science or riot, there can be no doubt that this has been its 
 method. This might be exemplified from every department of 
 science and is strikingly illustrated in the field of optics as the 
 consideration of the history of our subject will show. A single 
 instance from the more widely known field of mechanics may be 
 taken. Let us recall the process by which Kepler obtained his 
 laws of planetary motion. He began with the hypothesis that 
 the distances of the planets from the Sun are determined by the 
 six regular solids of geometry. " The Earth's orbit is the sphere, 
 the measurer of all. Round it describe a dodecahedron ; the circle 
 including this will be (the orbit of) Mars. Round Mars describe 
 a tetrahedron ; the circle including this will be Jupiter. Describe 
 a cube round Jupiter; the circle including this will be Saturn. 
 Then describe in (the orbit of the) Earth an icosahedron; the 
 circle described in it will be Venus. Describe an octahedron 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 9 
 
 round Venus ; the circle inscribed in it will be Mercury." This 
 law seems fanciful enough ; but the only sound objection that 
 could have been advanced against it at the time was that it did 
 not harmonise well with the results of experiment. However, 
 Kepler was at first well satisfied. He declared that " he would 
 not barter the glory of the invention for the whole Electorate of 
 Saxony." His ardour was somewhat cooled by Tycho's advice 
 " first to lay a solid foundation for his views by actual observation." 
 So he turned aside from these speculations, and next gave his 
 mind to pondering over the forms of the planetary orbits. Natu- 
 rally, he tried circles first, but Tycho's observations convinced 
 him that, at least in the case of Mars, there was a considerable 
 departure from the circular form. He therefore tried curves of 
 various sorts, testing them by Tycho's observations, until he hit 
 upon the ellipse. Thus he discovered his second law. His next 
 step was to generalise this law by extending it to all the planets 
 and he found it still true. This great success revived his interest 
 in the problem that had first aroused his enthusiasm, that of the 
 planetary distances. He tried hypotheses of all kinds. He com- 
 pared the planetary distances with the intervals of the notes on 
 the musical scale, an idea suggested by the venerable notion of 
 the music of the spheres. At last he conceived the idea of 
 comparing the powers of the different numbers that represent the 
 distances of the planets with the powers of the numbers repre- 
 senting their periodic times. Thus he hit upon his third great 
 law. " The die is cast the book is written, to be read either now 
 or by posterity, I care not which. It may well wait a century for 
 a reader, as God has waited 6000 years for an interpreter of his 
 works." 
 
 If we are agreed as to the end of science, we may then profit- 
 ably discuss its method. In the field of optics we shall take it 
 as our aim to bind together harmoniously the enormous mass 
 of facts obtained by careful experiments on light, putting aside 
 those whose discussion involves anything but the simplest 
 physiological facts and theories. In any thinking about phe- 
 nomena we must consider them under particular aspects, and 
 speak of them in terms of certain concepts. It is all important 
 that the concepts be clearly formed and that they be conceived in 
 
10 THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 the same way by those with whom we wish to communicate. 
 Our choice of concepts is to a certain extent arbitrary ; but it 
 gives colour to the whole superstructure and determines the 
 language in which our facts are to be described. The concepts 
 that we shall employ most frequently are number, space, time, 
 motion, substance, and energy. We shall assume that these have 
 been cleared of contradictions as far as the nature of the mind 
 will permit. To make any progress we must make certain 
 postulates. We cannot begin with axioms and we certainly 
 cannot echo Newton's statement " hypotheses non fingo," nor do 
 otherwise than deny or explain away the remark with which he 
 closes the Principia " Hypotheses seu metaphysicae, seu physicae, 
 seu qualitatum occultarum, seu mechanicae, in philosophia 
 experimentali locum non habent." 
 
 In the first place we postulate the existence of the ether, an 
 all pervading medium through which what we call waves of light 
 may be propagated. This medium is literally metaphysical; but 
 it is a substance that has some of the qualities of ordinary matter. 
 Indeed it could not be so much as thought of, if it had not, since 
 our conception of substance is necessarily derived from our 
 sensations. It is an abstraction obtained by selecting some of the 
 qualities from media, such as water or jelly, with which we are 
 familiar, and rejecting other of the qualities. Our excuse for 
 postulating such a medium is that we see no prospect of attaining 
 our end without it, while with it great things can be accomplished, 
 not only in the domain of optics, but in the fields of electricity 
 and magnetism as well. At the same time we need not concern 
 ourselves with the 'reality' of the ether, nor take the unphilosophic 
 step of pronouncing it to be real merely because it is a conception 
 that is convenient for our special purpose. 
 
 Our next postulate, or set of postulates, comes in when 
 establishing a correspondence between our concepts and what is 
 given us in experience. We want, for example, to think of a 
 relation between the frequency of waves and our sensation of 
 colour, or between the energy associated with a certain portion of 
 the ether and our sense of the brightness of a light. To attempt 
 to establish such a relation in a thorough fashion would take us 
 into the depths of philosophical difficulties. We must be content 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 11 
 
 to postulate the correspondence, to say, for example, let it be 
 granted that a certain frequency corresponds to the sensation of 
 ' blue ' in a normal eye. This is no doubt arbitrary ; but it is 
 part of our art to suggest the correspondence that makes our 
 theory fit in with the facts. 
 
 The last postulate is made in laying down the fundamental 
 ' law ' or dynamical principle that is to bind all our concepts 
 together. What should be the characteristics of this law ? We 
 need scarcely say that it should fit in with the facts, for this is the 
 whole end of it. It should suggest all the experiences that we 
 have and none that we have not. W T e want it to be logical, that 
 is not self-contradictory, and it is desirable that it should not 
 contradict " well established " principles in other departments of 
 thought. Then it should be as simple as possible. This we want 
 for convenience and lucidity, if not from a conviction of the truth 
 of Newton's comment on the first of his Regulae Philosophandi : 
 " Natura enim simplex est et rerum causis superfluis non 
 luxuriet." We want, then, few concepts and the simplest relation 
 between them. Finally, we want a principle that will present the 
 facts as vividly as possible. 
 
 We should scarcely be surprised if we find it difficult to satisfy 
 all these demands, and if men differ as to which is the most 
 important. Some would sacrifice simplicity to vividness, others 
 will have simplicity at all costs. Thus some, in their anxiety to 
 work with the fewest concepts, drive force and energy out of 
 mechanics except as formal concepts to be defined in terms of 
 mass and motion, concealed or otherwise. Others think that even 
 if this does not render the definitions of mass and force ' circular,' 
 it abstracts so much as to leave mechanics almost ghostly. It 
 would be out of place to enter into the controversy here. We 
 shall make constant use of energy as a concept derived from our 
 experience of being able to exert ourselves and do work ; whether 
 this concept can be resolved into " simpler " ones or not will not 
 in any way affect the validity of what we do. Nor need we delay 
 our progress to decide the question whether it is possible or 
 expedient to regard all potential energy as the kinetic energy of 
 concealed masses,' for whether this be so or not will merely affect 
 the interpretation of certain terms in our formulae. 
 
12 THE SCOPE AND METHOD OF THE INQUIRY [CH. 
 
 The conflict between vividness and simplicity is also raised by 
 the question as to the part that the use of models should play in 
 the development of a scientific theory. Different views on this 
 question will be taken by men of different types of mind. One 
 can seize a large number of facts vividly, the other needs a general 
 idea to coordinate them all. Of these types of mind, the one 
 makes the engineer, the 'practical' man, the man of business or of 
 affairs, the other makes the philosopher. The first will delight in 
 models to ' explain ' his theory, the second will rest more content 
 in contemplating an all embracing law. Thus we find von 
 Helmholtz confessing that he 'is more at home with a differential 
 equation than with a model, while Lord Kelvin almost goes the 
 length of saying that we know only what we can construct. " I 
 never satisfy myself," he says, " until I can make a mechanical 
 model of a thing. If I can make a mechanical model I can under- 
 stand it. As long as I cannot make a mechanical model all the 
 way through I cannot understand." This is one of the many 
 interesting personal touches in the famous Baltimore Lectures, and 
 for these we are grateful as giving us an insight into the workings 
 of a mind that has done so much to enrich our science. We are, 
 however, less grateful when the lecturer appears to suggest that 
 his mode of thinking is the only possible one, and sets up the 
 question " Can we make a mechanical model of it ? " as the test 
 whether we do or do not understand any subject in physics. The 
 usefulness of models no one seriously questions. They are often a 
 great aid to the scientific imagination in suggesting analogies 
 between the unfamiliar and the familiar, and they may add 
 greatly to the vividness of our conceptions. In this way they are 
 particularly useful for purposes of exposition, when we are trying 
 to convey to others a conception that they have not yet clearly 
 formed. But their limitations and dangers are almost too obvious 
 to mention. At the best they present partial analogies, and 
 illuminate only special portions of the theory. No one would 
 suggest the possibility of constructing a model that would 
 represent all the phenomena of light and electricity, whereas it is 
 not only possible but probable that a single dynamical principle 
 will be found to comprehend them all. The dangers of a model 
 are that it is apt to suggest false analogies as well as true ones, 
 
l] THE SCOPE AND METHOD OF THE INQUIRY 13 
 
 that it is difficult to separate the truth from the falsehood, and 
 that it may easily give a groundless sense of security by suggesting 
 ' common sense ' notions, that is inductions from experience that 
 have not been rigorously scrutinised. 
 
 It must be clear, then, that if we are to fulfil the purpose of 
 science as we have described it, we cannot rely entirely on models, 
 but must use them only as convenient illustrations of special 
 aspects of some general dynamical principle. But how is this 
 principle to be obtained ? There is no royal road to it ; it can be 
 reached only by the patient and laborious process of trial and error, 
 of setting out with hypotheses and seeing how their consequences 
 fit in with the facts. Newton insisted that the only hypotheses to 
 be admitted are those suggested by induction from experience : 
 " quicquid enim ex phaenomenis non deducitur, hypothesis vocanda 
 est ; et hypotheses in philosophia experimentali locum non habent." 
 To this, however, we must demur. The hypothesis is the same 
 however it is reached, and it may be welcomed from any source. 
 No doubt we should expect greater things from hypotheses that 
 have been obtained by generalisations from experience, for we 
 know at the outset that they accord with some of the facts. But 
 the golden fruits of knowledge often ripen where they are least 
 expected, and a study of the history of science will soon convince 
 us that the process of induction as described in the text-books of 
 logic, has done far less to suggest the leading principles than 
 might be supposed. 
 
 Fortunately at this stage in the development of mechanics we 
 have not to go far in search of general principles. At first sight 
 our difficulty seems rather an embarras des richesses', we must make 
 a selection from different principles each of which may claim some 
 special advantage. In making our choice, we must, of course, be 
 guided by the special end we have in view, and the selection then 
 proves a simple matter. Three different courses seem open to us. 
 We might take as our fundamental principle or principles, the 
 Laws of Newton combined with d'Alembert's Principle. These 
 have the advantage of having been long in use, and so of being 
 thoroughly familiar to the student. However, even if the doubts 
 and difficulties with which they are involved be removed, they are 
 not convenient for the special problems of optics. We have to 
 
14 THE SCOPE AND METHOD OF THE INQUIRY [OH. 
 
 deal with a machine the inner workings of which are hidden and 
 do not concern us. To discuss its action by means of Newton's 
 Laws we should have to make various hypotheses as to the nature 
 of the mechanism. This would involve us in further doubts and 
 difficulties, and even if we had overcome them to our satisfaction, 
 we should not have obtained a formula more convenient for 
 purposes of mathematical development, than that derived from the 
 Principle of Least Action, which in this book will be taken as the 
 fundamental dynamical principle. 
 
 Another method of procedure would be to combine the Principle 
 of Inertia with Gauss's Principle of Least Constraint, and, follow- 
 ing Hertz in his Principles of Mechanics, formulate the " Law of 
 the Straightest Path " as the basis of dynamics. One objection to 
 this is that we should have to dress up our theory in a somewhat 
 unfamiliar garb. Another is that Hertz commits us to the dogma 
 that all potential energy is really the kinetic energy of concealed 
 masses in motion. This is an idea made familiar by the specula- 
 tions of physicists such as Lord Kelvin and von Helmholtz ; but it 
 seems inexpedient to make this hypothesis the very basis of all our 
 reasoning. At the same time it should be noted that the Principle 
 of Action that we are to adopt can be deduced, with proper 
 assumptions, from Hertz's Principle, and also from the more 
 classical Laws of Newton, so that anyone who prefers to build on 
 these foundations can soon reach a stage at w^hich our analysis 
 becomes intelligible. 
 
 Our procedure will be to postulate the Principle of Least or 
 Stationary Action. It is a principle that seems wide enough to 
 comprehend the whole of mechanical science, including all physics 
 in that term *. The only serious doubt as to this is the criticism 
 of Hertz f, that the principle cannot be applied to the motion of 
 rigid bodies rolling on one another. Larmor has indicated! how 
 the objection may be removed ; but even if we regard Hertz's 
 criticism as fatal to the Law of Action as the basis of all mechanics, 
 we may still admit the principle within the field of optics and 
 
 * See von Helmholtz : Ueber die physikalische Bedeutung des Princips der 
 kleinsten Wirkung. Wiss. Abh. in. 203; and Larmor: Aether and Matter. 
 Appendix B. 
 
 f The Principles of Mechanics (trans. Jones and Walley), p. 19. 
 
 J Aether and Matter, p. 277. 
 
I] THE SCOPE AND METHOD OF THE INQUIRY 15 
 
 electricity, with which alone we are at present concerned. In that 
 field we have nothing to do with the rolling of rigid bodies and the 
 course of our investigation will determine whether the principle is 
 wide enough for all the facts or not. 
 
 The Principle of Action has been presented in various forms 
 since its first hazy enunciation, a century and a half ago, by 
 Maupertius*. The form that we shall find most convenient is 
 that which makes everything depend on the variation of a 
 Principal Function, involving the difference between the kinetic 
 energy and the potential energy of the system in any configuration. 
 The principle states that the natural course of the motion from 
 one configuration A to another B is one that makes the time 
 integral of T W (the difference between the kinetic energy and 
 the potential energy) stationary as regards such slight variations 
 of the path from A to B as keep the time of passage unvaried. 
 Of course if the term 'potential energy' is to have any definite 
 meaning the work that the system can do must depend only on 
 the actual configuration and not on the path by which it has been 
 reached, so that the principle, as thus stated, is applicable only to 
 conservative systems. It may, however, be extended to non- 
 conservative systems by including in the variation the work of any 
 forces that are not involved in the potential energy. The virtual 
 work of these forces added to the variation of the Principal 
 Function must then be made to vanish. The only limitation on 
 the application of the principle now is that the coordinates must 
 be really independent, and that the configuration of the system 
 must be completely specified in terms of them without including 
 their differential coefficients with respect to the time. 
 
 Postulating this dynamical principle our aim is to show by 
 means of it that all the varied and complex phenomena of physical 
 optics may be woven together harmoniously by regarding them as 
 due to periodic disturbances in a medium that we call the ether. 
 
 * See von Helmholtz : Zur Geschichte des Princips der Ueinsten Achin. 
 Wiss. Abh. in. 249. 
 
CHAPTER II. 
 
 PRELIMINARY IDEAS. 
 
 IT has been said that we are to regard the phenomena of light as 
 due to periodic disturbances propagated in a medium that we call 
 the ether. The essential feature of a periodic disturbance is that 
 the characteristics of the motion are all repeated after a definite 
 interval of time which is called the period. One of the best 
 known examples of such a motion is that afforded by the pro- 
 gressive waves set up in the surface of a sheet of water by some 
 disturbance, and it is due to the analogy with such waves that the 
 movements of the ether discussed in optics are called waves and 
 the modern theory of light is spoken of as the wave theory. 
 While the wave presents the most familiar example of periodic 
 motion to the ordinary man, the type of this motion that the 
 student of science may be expected to know best is Simple 
 Harmonic Motion. In this case the displacement of a moving 
 point about a fixed centre is represented by an expression of the 
 form 77 = a cos p(t + e), where a is the amplitude, 2?r/p the period, 
 p/2-TT the frequency, and e the phase of the vibration expressed 
 in time. A displacement closely related to this and one that 
 is more important for our purposes is that represented by 
 77 = a cos p (t 4- e oc/v). 
 
 For a given value of x the motion is Simple Harmonic, so 
 that if 77 be at right angles to x, the point moves up and down at 
 right angles to the axis of x. Let the positions of P 1; P 2 , ... P n in 
 Fig. 1 indicate the displacements at any time at different points 
 on the axis of x, the curve drawn through P l , P 2 ... being the 
 well-known sine or cosine curve. At a later time the displace- 
 ments will be indicated similarly by P/, P 2 ', . . . P n ', lying on another 
 
CH. II] 
 
 PRELIMINARY IDEAS 
 
 17 
 
 sine or cosine curve. It thus appears that 77 = a cos p (t -f e xjv) 
 represents a wave form with a sine or cosine curve for its outline 
 moving with velocity v along the positive direction of the axis of x. 
 
 At any given moment 77 and ~ have equal values at points for 
 
 which px/v differs by 2?r, i.e. when x differs by Zjrv/p. This 
 quantity is consequently called the wave length and is usually 
 denoted by \, so that we may write the displacement in the form 
 77 = a cos 27T/A, . [vt x + ej, where 6j is the phase expressed as a 
 length. 
 
 Fig. 1. 
 
 More generally if y =f(x) is the equation of any curve, it is 
 obvious that y =f(x vt), where v is a constant, represents a wave 
 having a curve of this form for its outline, advancing with velocity 
 v along the positive direction of the axis of x. The consideration of 
 this more general type is readily reduced to that of the simpler 
 one dealt with above, by means of Fourier's theorem. According 
 to this theorem any function of x and t of the kind that presents 
 itself when we are dealing with optical problems connected 
 with plane waves can be expressed as a series of the form 
 
 oo 
 
 2a n cos n . 2-7T/X .(vt x + e n ), where v is a constant. It follows 
 i 
 
 from this that the consideration of the simpler type of wave 
 represented by a circular function will enable us to deal with 
 more complicated cases when they arise. As any periodic function 
 may be resolved into constituents of the simpler type, we may 
 regard a source of light as a centre from which waves of this type 
 emanate. If /be the frequency, v the velocity of propagation of 
 the wave, and X the wave length, we have \ = v/f, so that if v 
 is a constant the wave length is inversely proportional to the 
 frequency. We shall take it that the colour depends on the 
 M. L. 2 
 
18 PRELIMINARY IDEAS [CH. 
 
 frequency and therefore, in a given medium, on the wave length, 
 so that if all the waves are of the same length the light will be 
 monochromatic. Although it may be difficult, if not impossible, 
 to produce such light in actual practice, it will be none the less 
 convenient to begin by investigating the characteristics of its pro- 
 pagation and consider when necessary the modifications that must 
 be introduced when the light is no longer of this simple character. 
 
 The waves that we have been considering are plane waves, the 
 front of the advancing waves being always a plane perpendicular 
 to the axis of x. In attempting to solve the optical problems 
 actually presented in nature, it will not always be possible to 
 confine ourselves to waves of this very simple type. Thus if we 
 are dealing with a disturbance spreading out from a centre it may 
 be necessary to take account of the different directions in which 
 the waves are propagated. However, the same general principles 
 as before will enable us to deal with such a case. We shall 
 merely have to suitably modify the system of coordinates em- 
 ployed, and although this may complicate the analysis, it will 
 not otherwise modify the procedure. 
 
 Returning then to the case of a wave of simple type in which the 
 displacement is given by 77 = a cos 2?r/X . [vt x + e] we see that 
 there are four constants required to specify such a wave. These 
 four are the wave length X = v/f, the phase e, the velocity of pro- 
 pagation v, and the amplitude a. Our dynamical principles will, 
 as we shall see later, enable us to determine the magnitude of v 
 in any given case, but the other three constants may have 
 arbitrary values. The phase will depend on the choice of origins 
 for x and t, the frequency / on the colour of the light employed, 
 and we shall find that the amplitude is intimately related to the 
 intensity. The intensity of a beam of light might be measured 
 experimentally in various ways, e.g. by its physiological action, 
 by its chemical or photographic action, or by its heating effect. 
 These three measures will not usually be the same, so that in 
 speaking of intensity we must understand with which of these 
 intensities we are dealing. The photographic action is in some 
 circumstances the one most capable of exact measurement, and if 
 we are dealing with this we may say that two beams of light are 
 of equal intensity when they affect similar photographic plates 
 
II] PRELIMINARY IDEAS 19 
 
 similarly. This gives us an experimental means of comparing the 
 intensities of two beams and so of measuring any one in terms of 
 some unit. When, however, we come to apply our theory to the 
 results of experiment we are at once faced with the question 
 what quantity or quantities connected with a wave in the ether 
 may be taken as measuring the intensity ? As all the actions 
 referred to above, photographic or other, involve the expenditure 
 of energy it is clear that the intensity must be connected in some 
 way with the energy in the neighbourhood of the point con- 
 sidered ; but the exact relation between the two cannot be settled 
 a priori. It is plain, however, that in estimating the intensity 
 we are not concerned with the energy at any definite moment of 
 time, but with the mean energy over a large mimber of periods. 
 The period ^varies with the colour, but the variation is not great 
 within the limits of the visible spectrum where the period is 
 always of the order 10~ 15 of a second. The impression of light on 
 the retina persists for about one-tenth of a second, so that if we 
 measure the intensity of the beam by its physiological action an 
 enormous number of periods must elapse before there is any 
 measurable eifect. This will be true also for the most rapid 
 photographic plate, so that the intensity must depend on the 
 energy transmitted across a surface, or the energy per unit volume, 
 averaged over a large number of periods or on what may be called 
 the mean energy. We have still to determine whether the in- 
 tensity is measured by the mean energy, or the mean potential 
 energy, or the mean kinetic energy. Which, if any, of these is 
 to be taken as corresponding to the facts can be settled only by 
 comparing the results of the optical theory with those of experi- 
 ment. This will be undertaken in due course ; but it may be 
 noted here that very few of the numerous estimates of intensity 
 that have been made throw any light upon the question. These 
 experiments nearly all deal with sensibly plane waves moving in 
 the same direction in an isotropic medium and the beams com- 
 pared are in the same medium such as air. Under these 
 circumstances we shall prove later that the potential energy is 
 proportional to the kinetic, and so also to the total energy, and 
 hence all the measures of intensity suggested above will lead to 
 the same result. 
 
 22 
 
20 PRELIMINARY IDEAS [CH. 
 
 We have been concerned so far with a single disturbance 
 setting out from some source, but we may have to deal with 
 different disturbances proceeding from the same or from different 
 sources. As the actual displacements in the ether postulated by 
 our theory are extremely small according to any ordinary standard, 
 it will be natural to apply the general principle of the super- 
 position of small displacements to a problem such as that just 
 suggested. This principle directs us to consider the displacement 
 due to each disturbance separately and obtain the total displace- 
 ment by algebraic addition of the several parts. It is, of course, 
 conceivable that one disturbance might so alter the character of 
 another that this simple method of superposition would not be 
 legitimate, but it is reasonable to adopt the principle as a working 
 hypothesis which will be amply justified by the agreement of the 
 results with the facts of experience. The principle is usually 
 spoken of in optics as the Principle of Interference, an ill-chosen 
 term that has, however, become stereotyped in works on the 
 theory of light. 
 
 One of the first applications of the principle of superposition 
 in the domain of optics is the derivation of Huyghens' Principle, 
 which plays a leading part in most expositions of the theory of 
 light. If light be radiating from a source the medium around 
 this point will be disturbed and the discussion of the charac- 
 teristics of the disturbance is facilitated by considering the form 
 and position of a certain surface called the Wave Surface. This 
 surface we may define most conveniently for present purposes as 
 the locus of points at which the disturbances are in the same 
 phase. Its form will depend on the nature of the medium. In 
 the simple case of an isotropic medium, i.e. a medium whose 
 features are the same in all directions, it is obvious from symmetry 
 that the wave surfaces are spheres with as centre. In other 
 media, however, the form of the wave surface is less obvious, and 
 it will be the subject of investigation later. Let us consider a 
 disturbance spreading out from a source and so giving rise to 
 a wave surface S. Huyghens' principle asserts that in considering 
 the further propagation of this disturbance we may disregard the 
 origin of the disturbance and fix our attention on what is 
 taking place at S, that we may look upon each point of S as a new 
 
II] 
 
 PRELIMINARY IDEAS 
 
 21 
 
 A- 
 
 centre of disturbance provided that we arrange that the various 
 secondary waves proceeding from all the points of S, when com- 
 bined in accordance with the Principle of Superposition, produce 
 the same displacement at any subsequent time as that which 
 would have been produced by the original disturbance emanating 
 from 0. The analytical expression and critical discussion of this 
 principle will be more appropriately undertaken at a later stage, 
 if not as a matter of logical arrangement, at any rate on the 
 grounds of simplicity of treatment at the outset. Meanwhile it 
 will be assumed that the reader is familiar with some of the 
 simpler deductions from the principle to be 
 found in the most elementary works on the // / 
 
 theory of light from the descriptive side. Thus, 
 e.g. if S be the wave surface at time t and 
 S' at time t', then S' is the envelope of the 
 secondary waves setting out from the various 
 points of 8. If, moreover, the secondary wave 
 from A touches S' at A', then the disturbance 
 at A' due to all the secondary waves from the 
 various points of 8, except those in the immediate 
 neighbourhood of A, is null, the different dis- 
 placements neutralizing one another when 
 superposed. Thus the points A and A' may 
 be spoken of as corresponding points, for as far 
 as the effective disturbance is concerned the Fig. 2. 
 
 displacement at A' depends only on that at A. 
 
 Thus far we have spoken of a wave, and this is the funda- 
 mental idea in the theory of light from the modern point of view. 
 There is, however, another idea that has played a very important 
 part in the development of the theory and is still convenient as a 
 subsidiary idea. This is what is called a ray. The idea of a ray 
 may be conceived in two different ways. From the first point of 
 view let AB be a wave front moving parallel to itself so as 
 afterwards to take up the position AB'. The disturbance due to 
 an element near A affects an element near A, its effect at other 
 points of A'B' being neutralized by the influence of the secondary 
 waves due to the other elements. Thus the effective disturbance 
 travels along A A' and this direction of propagation of a labelled 
 
22 PRELIMINARY IDEAS [OH. 
 
 disturbance is called a ray. The velocity with which the disturb- 
 ance travels along the ray is called the ray velocity, and it is 
 obvious that the velocity with which the wave 
 is propagated is equal to the component of 
 the ray velocity along the wave normal. 
 
 The second outlook is somewhat different. 
 From this we have regard to the energy flowing 
 from A as a source of light. If a screen S' were / 
 interposed between A and A'E' it would prevent 
 the energy that flows out of A from reaching 
 every part of the surface A'E' . There would 
 thus be a part of this surface that no energy B / 
 would reach. This part would not be illumi- 
 nated, nor if sensitive to light, like a photo- Fig 3 
 graphic plate, would it be affected. We could 
 therefore regard the energy as flowing out of A along certain 
 lines, such as AA, and these lines of flow of energy we may 
 call rays. Thus a ray from A to A' might be defined as the 
 locus of the points at which an opaque body must be placed to cut 
 off the influence of the light at A'. Of the two points of view 
 that we have indicated the second is the one more commonly 
 taken by the physicist; but it will appear later that the ray is 
 the same from whichever of these points of view it is regarded. 
 
 The problem of determining the directions of the reflected and 
 refracted rays corresponding to a given incident ray will be dealt 
 with later; but a theorem, due to Huyghens*, that is intimately 
 connected with this problem will be most conveniently given here. 
 In stating this theorem it will make for brevity if we represent 
 planes and lines by their intersections with the plane of the 
 paper. Let T be the interface between two media with different 
 optical properties and OR a wave front at any time t. As this 
 wave advances it will give rise to disturbances along OT, from 
 each point of which secondary waves will emanate in accordance 
 with the principles explained above. After a unit interval, i.e. at 
 time + 1, the secondary waves will all touch a plane TQ where T 
 is the line of intersection of the interface and the wave front in 
 the first medium at time + 1. This plane TQ is the refracted 
 
 * Huyghens : (Euvres ix. p. 382. 
 
Il] PRELIMINARY IDEAS 23 
 
 wave and the corresponding ray is the line OQ where Q is the 
 point of contact of the plane TQ with the wave surface whose 
 centre is 0, for and Q are corresponding points in the sense 
 mentioned above, since Q is the point where the secondary wave 
 from touches the wave surface at time + 1. If the lower 
 medium had been the same as the upper we could, by Huyghens' 
 
 Fig. 4. 
 
 principle, have found the disturbance at time t + 1 by considering 
 the envelope of the secondary waves emanating from the various 
 points of OT. This envelope would be a plane TA through the 
 line T, a plane representing the position of the incident wave at 
 time t + 1 had there been no change of medium. Hence if we 
 draw the wave surface characteristic of the upper medium and 
 having as centre, the incident ray is OA where A is the point 
 of contact of the plane TA with this wave surface. Thus we 
 have the following construction for the refracted ray corresponding 
 to a given incident ray. Let OA be the direction of the incident 
 ray, a point in the interface OT. With centre draw the wave 
 surface for unit interval of time characteristic of the upper 
 medium and let OA cut this in A. Draw the tangent plane to 
 the wave surface at A and let this cut the interface in the line T. 
 Through this line draw a tangent plane to the wave surface for 
 unit interval of time characteristic of the lower medium and let 
 the point of contact be Q. Then OQ is the refracted ray. Of 
 course if the wave surface consists of more than one sheet this 
 
24 PRELIMINARY IDEAS [CH. 
 
 construction will apply to each of the sheets separately and we 
 shall have more than one refracted ray. The construction thus 
 described is called Huyghens' construction. It may, if desired, be 
 replaced by a modification due to Hamilton whose construction is 
 derived from Huyghens' by reciprocating with respect to a unit 
 sphere of which is the centre. 
 
 So far we have said nothing of the displacement at any point 
 except that it is a periodic function of the time which, by means 
 of Fourier's theorem, may be resolved into a series of circular 
 functions. In the most general case the displacement at any 
 point (x t y, 2} may be resolved into its components (, 77, f). Each 
 of these components may then be expressed as Fourier series, so 
 that we shall have f =/j (x, y, z, t), rj =/ 2 (x, y, z, t), f =/ 3 (x, y, z } t). 
 If these functions are quite independent, there will be no relation 
 between f, 77, so that we cannot find a definite orbit for any 
 point of the medium by eliminating x, y, z and t from the equa- 
 tions for f, 77, f. However in many cases that actually present 
 themselves the relations are such that this elimination can be 
 performed. Under such circumstances each point of the medium 
 on a given wave surface is describing a definite orbit, and the 
 light is then said to be polarised. If the orbit is a straight line, 
 a circle, or an ellipse, the light is said to be rectilinearly, circularly, 
 or elliptically polarised as the case may be, although rectilinear 
 polarisation is more commonly spoken of as plane polarisation. 
 
 We have seen that a wave of simple type where the displace- 
 ment is represented by a circular function practically includes all 
 cases that arise, and as any wave surface may be regarded as the 
 envelope of its tangent planes we may substitute a system of 
 plane waves for a wave of any form. Thus the consideration of a 
 displacement whose components are given by 
 
 = u cos 2-7T/X . [vt + 1 (Ix + my + nz)~\ ; 
 77 = v cos 2?r/X . [vt 4- e 2 (Ix + my 4- nz)~\ ; 
 = w cos 2?r/X . [vt + e 3 (Ix + my + nz)~\ ', 
 which for brevity we may denote by 
 
 ( 7 ;, f) = (u, v, w) cos 2?r/X [vt + e (Ix + my + nz)] , 
 
 will lead us to results of a far reaching character, although such a 
 displacement is primarily characteristic of a plane wave of simple 
 
II] PRELIMINARY IDEAS 25 
 
 type, moving with velocity v along a line whose direction cosines 
 are I, m, n. The labour of discussing such a displacement is 
 greatly lightened by replacing the circular functions by their 
 exponential equivalents and employing a kind of shorthand in 
 which (f, ?;, f) is written in the form 
 
 (U, V, W) e i2nl * W - (lx + my + n*)] _ fa ^ w )00 
 
 say, it being understood that as , 77, f are necessarily real, only 
 the real parts of the complex quantities are to be retained. With 
 this convention we see that if u, v, w are real and equal to 
 (\,fji,v)A we have f = XJ.cos#, 77 = p, A cos 6, %=vAcos0 so 
 
 o o 
 
 that | - = - = -. Hence the displacement is always directed along 
 
 a fixed straight line and the light is polarised rectilinearly. If, 
 however, it, v, w are not real we may put 
 
 (u, v, w) = (\ + i\' y fji + i/jf, v + iv ) A 
 
 and we then have % = A (A, cos 6 \' sin 6), y = A (/m cos 6 // sin 6), 
 f = A (v cos 6 v sin 0). Solving any two of these for sin 6 and 
 cos 6 and eliminating we get an equation representing an 
 elliptic cylinder. If, further, as is always the case in optical 
 problems, we have l\ + m/j, + nv and l\' + ra/// + nv = 0, it 
 follows that /f+ mrj + n = 0. Hence the point (f , 17, f) moves on 
 a plane section of an elliptic cylinder, so that its orbit is an ellipse. 
 In this case the light is said to be elliptically polarised, and of this 
 type circular polarisation is only a special case. 
 
 We have now to discuss some special examples of these different 
 kinds of polarisations and to consider more particularly the effect 
 of superposing two differently polarised beams. The simplest case 
 is that of two rectilinear vibrations in the same direction. If this 
 direction be taken as the axis of x we may represent the two dis- 
 placements by expressions of the form g 1 = r 1 e i91 and % 2 = r 2 e i0 *. 
 Any complex such as f = re* 9 may be represented in the well- 
 known manner by a straight line of length r making an angle 
 with some fixed line. With this geometrical symbolism the 
 length of a line corresponds to the amplitude of the vibration 
 and the angle between two lines to the difference of phase 
 between the corresponding vibrations. If, then, we wish to 
 superpose two different vibrations of this simple type, we have 
 
26 PRELIMINARY IDEAS [CH 
 
 merely to compound two vectors, such as OPj and OP 2 , by means 
 of the ordinary law for compounding vectors, i.e. by the parallelo- 
 gram law. A formula for the resultant amplitude and phase is 
 readily obtained from this geometrical construction. Thus if the 
 components be represented by f 1 = a 1 e i( ^ +e ' ) and f 2 = a 2 e i(pt+e * ] , 
 and the resultant by f = ae i{pt+e ^ we have 
 
 cti sin e, 4- a z sin e 2 
 tan e = 
 
 tti cos e l + a 2 cos e 2 
 
 and a 2 = a^ + a 2 2 4- Za^ cos (e l e 2 ) = a^ 4- a 2 2 -I- 2a : a 2 cos A, where 
 A is the difference of phase between the two components. Three 
 special cases of this formula for the amplitude of the resultant 
 vibration are of frequent occurrence. When the difference of 
 phase is zero or an even multiple of IT we have a = a a + a 2 ; when 
 this difference is an odd multiple of TT we have a = a^ a 2 ; and 
 when A = ?r/2, or any odd multiple thereof, we have a 2 = of + a 2 2 . 
 We shall prove later that the intensity of the light is proportional 
 to the square of the amplitude of the vibration, so that if A = ?r/2 
 the intensity of the resultant beam is equal to the sum of the 
 intensities of the components. Vibrations for which this is true 
 are called conjugates. 
 
 If we wish to compound several vibrations we have merely to 
 make repeated applications of the parallelogram law. Thus, if 
 there are n components whose amplitudes are represented by 
 OP 15 OP,,... OP n and phases by the angles P, OX, P 2 OX, ...P n OX 
 where P n OX = e n ; then the amplitude of the resultant vibration 
 is n.OG, where G is the centre of mean position of the points 
 P lt P 2 ,...P n , and the phase of the 
 resultant is measured by the angle 
 GOX. The same result may be 
 expressed in another geometrical 
 form, which is more convenient for 
 some purposes. To obtain this we 
 draw OP, equal to a, and making 
 an angle e x with a fixed line OX, P X P 2 
 equal to 2 and making an angle * 2 
 with OX and so on. Then OP W 
 represents the magnitude of the resultant vibration and the angle 
 
II] PRELIMINARY IDEAS 27 
 
 P n OX its phase. In the special case when the component vibrations 
 form a series in which the amplitudes vary continuously and the 
 phase difference between consecutive members is infinitesimal, the 
 part of the polygon OP l P 2 ...P n becomes a continuous curve. 
 The element of length at any point P of this curve represents 
 the amplitude of the vibration in the corresponding constituent, 
 while the angle that the tangent at P makes with OX represents 
 the phase of this constituent. 
 
 Turning from the case of displacements all in the same 
 direction, we shall next consider the effect of compounding two 
 vibrations that are at right angles to one another e.g. along the 
 axes of x and y respectively. The displacements in such a case 
 can be represented by f = ae ipt and rj = be i(pt+ ^ ) ) where a and b are 
 the amplitudes of the vibrations and A is the difference of phase 
 between them. Confining ourselves to real quantities we have 
 f = a cos pt and 77 = 6 cos (p + A), and on eliminating t between 
 these equations we get 
 
 2 7f 2 77 cos A 
 
 + Ti- -^ -- =sm 2 A. 
 a 2 6 2 ab 
 
 The orbit in the resultant vibration is therefore an ellipse and the 
 light is elliptically polarised. The elements of the elliptic orbit 
 are obtained from the above equation by means of the formulae 
 investigated in the discussion of Conic Sections. In particular 
 the azimuth (ty) of the major axis of the ellipse is given by 
 the equation 
 
 2h' 2ab cos A 
 tan 2\/r = - r/ = , . 
 a - b' a 2 - b' 2 
 
 In the special case when A = Tr/2, we have -^ = 0, and the axes of 
 the elliptic vibration coincide with the directions of the component 
 vibrations. For some purposes it is important to decide the sense 
 of the oscillation, i.e. to know whether the elliptic orbit is described 
 clockwise or counter-clockwise as viewed from the positive side 
 of the plane z = 0. This is easily determined by considering 
 
 the signs of f, 77, -y- and ~ at any time. Thus, if a, 6, p are all 
 d/t (tt 
 
 positive and a > b, the motion is clockwise if A lies between and 
 TT, and counter-clockwise if A lies between TT and 2?r. A change of 
 the sign of p necessitates the reversal of this statement. 
 
28 PRELIMINARY IDEAS [CH. 
 
 The elliptic orbit may degenerate into a straight line or it may 
 become a circle. The degeneration occurs if sin A = 0, or if 
 A = mr. If A is an even multiple of TT the ellipse becomes the 
 
 straight line - ^ = 0, while if A is an odd multiple of TT the straight 
 
 v 
 
 line is - + 7- = 0. The conditions for circular polarisation are 
 a b 
 
 A = (2n -f 1) ^ an d a = b. 
 
 The components of the vibration are then %=acospt and 77 = a siupt, 
 and the motion is clockwise or counter-clockwise according as p is 
 negative or positive. It is convenient occasionally to replace a 
 rectilinear or elliptic vibration by a pair of circular vibrations. 
 
 Thus we may replace f = a cos pt by f 1 = ^ cos pt, % = - sin pt and 
 
 = - cos pt, rj 2 = - - sin pt = H sin ( p) t. The first corresponds to 
 _ Zi & 
 
 a counter-clockwise and the second to a clockwise circular motion. 
 Similarly we may replace the elliptic vibration ' f = a cos pt, 
 
 a+6 a+6 . 
 
 77 = 6 smj# by the circular pair = ~ cospt, rj l = sm.pt 
 
 a 5 , x a 6 . , v . 
 
 and 2 = o cos ( P) ^ ^2 = ^ sin ( p) t. 
 2 
 
 The principles already enumerated enable us to combine two 
 elliptical vibrations into a single one. Thus, if the component vibra- 
 tions be represented by ^ = a^ cos (pt + 1), % = 61 cos (pt 4- l Aj) 
 and 2 = a 2 cos (p* + e 2 ), ^ 2 = 6 2 cos (pt + e 2 - A 2 ), the resultant 
 vibration is f = a cos (p< + e), T; =6 cos(p^+ e A), where 
 
 a 2 = a! 2 + a 2 2 + Sa^a cos (e l - e 2 ) ; 
 
 6 2 = 6^ + 6 2 2 + 2M 2 cos (e! - e 2 - A! + A 2 ) ; 
 
 a T sin 61 + a 2 sin e 2 
 tan e = - ; 
 
 a x cos e x + a 2 cos e 2 
 
 6j sin (e l - Aj) + 6 2 sin (e 2 - A 2 ) 
 tan i ^A ) , T r , T r . 
 
 6i cos (ej - Aj) + 6 2 cos (c, - A 2 ) 
 
 If the component vibrations are similar in form and sense, and 
 have their corresponding axes parallel we have A! = A 2 and = , 
 
II] PRELIMINARY IDEAS 29 
 
 and in this case A = Aj = A 2 and - = = so that the resultant 
 
 a a x a 2 
 
 vibration is similar in form and sense to the components. 
 
 Denoting the amplitudes of the component and resultant 
 vibrations by r l} r 2 and r we have 
 
 r 2 = r x 2 + r 2 2 + 2 [a^ cos (ej - e 2 ) + M 2 cos fo - e 2 - A x + A 2 )]. 
 The condition that the vibrations should be conjugate is 
 r a _ rj 2 + ^ i e a^s cos(e 1 -6 2 ) + 6 1 6 2 cos(6 1 - 6 2 - Aj + A 2 )=0. 
 In order that this condition may be satisfied for all values of 
 e l e 2 we must have 
 
 i a 2 + &j 6 2 cos (Aj A 2 ) = and &A sin ( A x A 2 ) = , 
 
 whence, excluding the case of rectilinear vibrations already dis- 
 cussed, we must have sin (Aj A 2 ) = and c^a* b^ = 0, the 
 upper sign being taken if A] A 2 is an even multiple of TT and 
 the lower if it is an odd multiple. Putting bi/a^ = K we get 
 a 2 = + K& 2 , and the component vibrations are represented by 
 
 f ! = ttj COS (pt + j), rj 1 = ! COS (pt + j Aj), 
 
 and f 2 = + tfk> cos (^)^ + 2 ), r) 2 = 6 2 cos ( jo + 6 2 A^. 
 
 From this it appears that the two ellipses are similar, have the 
 same axes, the major axis of one being parallel to the minor axis 
 of the other, and that the motion in one is clockwise and in the 
 other counter-clockwise. 
 
 We shall conclude this chapter by a brief discussion of some of 
 the most important features of natural or white light. The first 
 point to notice is that this light is not homogeneous, but is 
 proved by Newton's experiments with prisms to consist of various 
 differently coloured beams superposed. The displacement must 
 therefore be represented by the addition of a large number of 
 terms corresponding to displacements in waves of different 
 lengths. Here, however, the question naturally arises what is 
 the character of the vibration in each of the component waves, 
 has it a definite orbit such as an ellipse ? The answer forced 
 upon us by the experimental evidence is that we must regard 
 the light as polarised elliptically (to take the most general case 
 possible) for an interval of time which is long compared with 
 the period of a vibration, but very short compared with the 
 
SO PRELIMINARY IDEAS [CH. 
 
 time required to make any impression on the retina or on a 
 photographic plate. The form and orientation of the elliptic 
 orbit is constantly changing so that in an appreciable interval 
 of time the mean effect is that of a displacement that is perfectly 
 symmetrical about the direction of propagation. 
 
 As regards polarisation, the distinguishing features of a stream 
 of natural light revealed by experiment are : (1) it can be resolved 
 into two streams rectilinear ly polarised at right angles to one 
 another, (2) the intensities of these streams are independent of 
 the azimuths of the planes of polarisation, and (3) if the streams 
 be superposed once more the resultant is the same whatever be 
 their relative retardation. These facts enable us to determine the 
 conditions that must be satisfied by the displacements in a 
 train of waves representing natural light. Thus suppose we are 
 considering light propagated along the axis of z. It will appear 
 later that the vibrations are always in the wave front so that the 
 component displacements may be taken parallel to the axes of 
 x and y. If the elliptic orbit in the nth constituent of the stream 
 be such that its semi-axes are a n cos a n and a n sin a n , and that its 
 major axis makes an angle ft n with the axis of x, then the dis- 
 placements parallel to the principal axes of the ellipse may be 
 represented by 
 
 i( e -- 
 n e l n = A n e n and a n sin a n e \ n 
 
 a n cos a 
 
 i6 n r> id n 
 
 = ia n sin a n e n = iB n e n . 
 
 Resolving along the axes of x and y and summing the results for 
 all the constituents we get as the components of the resultant 
 displacement 
 
 f = 2 (A n cos n + iB n sin /3 n ) e ie *> 
 
 and 77 = 2 (A n sin @ n - iBn cos /3 n ) e dn . 
 
 If now the component parallel to the axis of y receive a retardation 
 A and the light be then resolved into two components in directions 
 making angles </> and 90 + < with the axis of x t the displacement 
 in the direction ^ is represented by 
 
 2 [A n (cos @ n cos $ + sin @ n sin < e~* A ) 
 
 + iB n (sin fi n cos (/> cos /3 n sin $ e~* A )] e l n . 
 
II] PRELIMINARY IDEAS 31 
 
 The intensity / is thus given by 
 / - 2 (A n * cos 2 + B n 2 sin 2 ) cos 2 </> 
 + 2 (A n 2 sin 2 n + ' cos 2 fa) sin 2 < 
 + sin cf) cos < [22 (A n * B n 2 ) sin /3 n cos ft n cos A 2^A n B n sin A] 
 
 = (4 + B) cos 2 + (A - B) sin 2 < + sin 2< (Ocos A - D sin A) 
 where 2A = 2 (4 n 2 + 5 n 2 ) = Sa ?l 2 ; 
 
 2# = 2 (A n * - B n 2 ) cos 2/3 n = 2a n 2 cos 2a w cos 2/3 n \ 
 2(7 = 2 (A n 2 - B n 2 ) sin 2/3 n = 2a n 2 cos 2o sin 2/3 n ; 
 2D = 22^1 nB n = 2a 7l 2 sin 2a n . 
 
 The condition that the stream thus represented should be 
 equivalent to natural light is that / should be independent of 
 </> for all values of A. This requires B = C = D=Q. A case of 
 special interest is that in which we have two polarised streams. 
 The elliptic orbit is constant in form and orientation for all the 
 constituents of each stream, so that a n and j3 n are equal to a' and 
 ff for one stream and to a" and ft" for the other. We thus have 
 
 25 = cos 2a r cos 2/3' . 2a /a + cos 2a" cos 2/3"2a n " 2 , 
 2 C = cos 2a x sin 2/3'2a n ' 2 + cos 2a" sin 2/3"2a w " 2 , 
 and 2D = sin 2a / Sa n ' 2 + sin 2a"2a n " 2 . 
 
 The intensities, as we shall see, are proportional to 2a n /2 and 2a n " 2 , 
 
 so that if K 2 : 1 be the ratio of the intensities the conditions 
 
 B=C = D = Q give 
 
 cos 2a cos 2/3' + & cos 2a" cos 2/3" = 0, 
 cos 2a' sin 2/3' + *? cos 2a" sin 2" = 0, 
 
 and sin 2a' 4- /c 2 sin 2 a" = 0. 
 
 Hence we must have /c 2 = 1, so that the two streams are of equal 
 intensity. Putting K? = 1 and confining ourselves to values of 
 a and /3 which represent distinct ellipses we get a" = a.' and 
 ft" = ft' -f 90. Thus the two ellipses are similar, are described in 
 opposite senses and have their major axes at right angles. The 
 two vibrations are therefore conjugates. The fact that natural 
 light may be represented by two such streams of equal intensity 
 will be found of frequent use in the development of the theory. 
 
CHAPTER III. 
 
 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC 
 MEDIA. REFLECTION AND REFRACTION. 
 
 IN the discussion of optical problems two vectors will appear 
 throughout. These are the. displacement at any point of the 
 medium and the curl of the displacement. At any point (x, y, z) 
 the displacement may be represented by its components (, rj, f), 
 and the curl by its components (f, g, h), the relation between the 
 two vectors being a geometrical one indicated by the equations 
 
 /=<_?? qjl-^ & = ??_< 
 
 dy dz ' y dz dx' dx dy ' 
 
 Before we can apply the dynamical Principle of Action to the 
 discussion of the propagation of a disturbance in the ether, it will 
 be necessary to obtain expressions for the kinetic energy T and 
 the potential energy W in terms of the vectors (f, 77, f) and (/, g, h). 
 Taking the density of the ether as the unit density we have 
 
 where dr is an element of volume. The form of TTwill depend on 
 the assumptions made as to the nature of the elasticity of the 
 etherial medium. We shall suppose that the ether offers resist- 
 ance to any rotation, but not to any translation, of its elements*. 
 The work function for such a medium must, if the medium be 
 isotropic, be given by an expression of the form 
 
 Applying the Principle of Action to any portion of the ether 
 bounded by a surface S we have to make 
 
 ;f(T- 
 
 * This type of ether is analogous to the Elastic Solid Theory of Green and 
 MacCullagh. For a complete discussion see Stokes : Collected Works, iv. p. 177. 
 
CH. Ill] REFLECTION AND REFRACTION 33 
 
 Making use of the ordinary rules of variation we get 
 
 - W) dt = 
 two similar terms involving 77 and f. Also we have 
 
 while the term containing f in the variation of 8 W is 
 ^- h jjl\ dr = c 2 j(ng - mh) SdS 
 
 where (I, m, n) are the direction cosines of the outward normal to 
 the bounding surface 8. 
 
 Picking out the coefficients of Sf, 77, and S in the variation, 
 we obtain the three dynamical equations 
 
 while the surface conditions require ^(ng mh), c 2 (lh nf), and 
 c 2 (mf Ig) to be continuous. Of course the displacement (f , T/, ) 
 must also be continuous if there is to be no rupture of the 
 medium. 
 
 In the domain of optics we are interested only in periodic 
 functions of the time, so that we may put everything proportional 
 to e ipt , where Zir/p is the period. We then have f = p^ and so for 
 77 and . Thus the dynamical equations may be written in the 
 form 
 
 _ ___ r___ 
 
 ~p*(dy dz)' ~p*(dz fa)' p\&* 
 This shows that (f , 77, f) may be regarded as the curl of a vector 
 
 and that + + .O. 
 
 ox oy oz 
 
 Thus, whether the medium be compressible or not, there is no 
 compression involved in the motion. 
 
 M. L. 3 
 
34 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 Returning to the dynamical equations in their more general 
 form we have 
 
 t = ^ ( V _H _ ''1 = .2 J ( G _5_ _ ^ \ _ } _ u :n _ 
 
 ^ I ^ ^../ I g^ I g^. o^./ Ci.. I a o,. 
 
 with similar equations for rj and f. The solution of these equations 
 in various systems of coordinates is a familiar problem to the 
 student of mathematics, but for our present purposes we are 
 concerned only with some simple types of solution. If we are 
 dealing with plane waves moving in any direction we may take 
 this direction as that of the axis of z, and this without loss of 
 generality since the medium is isotropic, Under such circum- 
 stances we have to solve equations of the type 
 
 the most general solution of which is 
 
 where F l and F 2 are arbitrary functions. This represents waves of 
 arbitrary form moving with velocity c, in the positive and negative 
 directions of the axis of z. Hence the quantity c measures the 
 velocity of light in free ether. 
 
 As was pointed out in the last chapter we are specially con- 
 cerned with wave forms represented by a sine or cosine curve. 
 For such forms we have, for progressive waves moving in the 
 positive direction of the axis of z, 
 
 If LMN be real, the light is plane polarised, and LMN are the 
 direction cosines of the direction of displacement. If, on the other 
 hand, LMN be complex, the polarisation is elliptical, and the real 
 parts of LMN are the direction cosines of one axis and the 
 imaginary parts those of the other axis of the elliptic orbit. Since 
 
 +% + $.<) 
 
 doc dy dz 
 we have ^V=0, so that the displacement is in the wave front. 
 
Hi] MEDIA. REFLECTION AND REFRACTION 35 
 
 The axes of as and y have not yet been fixed, and by taking them 
 parallel to the principal axes of the elliptic orbit we may put 
 L = 1 and M= 0, we then have 77 = f = and % = Ae i ? (t ~ zlc) , from 
 which we find 
 
 /=0, g^-^A&t-, A = 0. 
 c 
 
 Hence the curl of the displacement is also in the wave front, and 
 its amplitude is proportional to that of the displacement, the 
 magnitude of the former being p/c times that of the latter. 
 The kinetic energy per unit volume is 
 
 T = $(!* + if 4- ? 2 ) = if 2 = 
 Similarly the potential energy per unit volume is 
 
 W = ic 2 (f- + g* + A 2 ) = - ^ ^ {t ~ zlc) . 
 
 2t 
 
 Hence the potential and kinetic energies per unit volume are the 
 same. The total energy E in a tube of unit cross section extending 
 along the axis of z from the origin to z = z l} is 
 
 **' fa 
 
 J 
 
 From this we obtain 
 
 [ 
 
 J 
 
 pi r 1*1 
 
 ~ e-'P t~ z!c] dz = - p 2 A 2 e^ (t - zle} .. 
 Jo L Jo 
 
 Hence the energy may be regarded as flowing across z = z^ at the 
 rate p*A 2 e^P (t ~ Zl ' c) , which is the same quantity as the total energy 
 per unit volume in the neighbourhood of z l . Thus in comparing 
 the intensities of two beams we shall get the same ratio whether 
 we measure the intensity by the mean potential energy, the mean 
 kinetic energy, the mean energy, or the rate at which energy is 
 flowing across a surface. 
 
 Since the velocity is a constant (c), we see that the wave 
 surface is a sphere and the radius vector drawn from any origin to 
 represent a ray is normal to the sphere. Hence the ray and the 
 wave normal coincide. As there is perfect symmetry round the 
 origin, the energy must flow along the radius vector and so along 
 the ray. Thus the two definitions of a ray suggested in Chap. II. 
 are identical. 
 
 32 
 
36 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 The formulae just obtained for the energy have reference to 
 progressive plane waves moving along some fixed direction. These 
 formulae must be modified if we have simultaneously two trains 
 of waves moving in opposite directions. A specially important 
 case of this occurs when there are two waves of the same 
 amplitude and frequency. If 6 be the difference of phase be- 
 tween the waves when z = 0, the displacement f is given by 
 the equation 
 
 f = A cosp (t z/c) + A cos [p (t 4- z/c) + 0] 
 
 = 2A cos (pt + 0/2) cos (pz/c + 6/2). 
 
 Under such circumstances the phase is constant everywhere, but 
 the amplitude is 
 
 + 2A cos (pz/c + (9/2), 
 
 which is a periodic function of z. Waves of this kind are called 
 stationary waves, and the points where the amplitude is a maximum 
 are loops, and those where it is zero are nodes in the waves. The 
 loops occur when 
 
 pZ/C + 6/2 = ftTT, 
 
 i.e. when z = n\/2 X0/47T, 
 
 where X is the wave length, and the nodes occur when 
 
 Thus the distance between successive loops or successive nodes 
 is X/2. Taking 
 
 f = A cosp (t z/c) + A cos {p (t + z/c) + 0} ; 77 = ; and f = 0, 
 we get 
 
 f = 0, h = 0, and g=pA/c . [smp(t - z/c) sin {p (t + z/c) + 0}]. 
 Hence 
 
 T = f /2 =p*A 2 /2 . [smp (t - z/c) + sin {p (t + z/c) + <9}] 2 , 
 and 
 
 W = cy/2 =p*A*/2 . [sin^ (t - z/c) - sin {p (t + z/c) + 6>}] 2 . 
 
 Thus since 
 
 /2r/JP C^/P 
 
 sitfp (t - z/c) dt = TT/P = sin 2 {p (t - z/c) + 6} dt, 
 Jo J o 
 
 ip 
 
 sin p (t - z/c) sin [p (t + z/c) + 6}dt= ir/p . cos (2pz/c + 6\ 
 . 
 
 o 
 and 
 
Ill] MEDIA. REFLECTION AND REFRACTION 37 
 
 we have T =27rpA*cos*(pz/c + 0/2), 
 
 and W= 2-TrpA* sin 2 (pz/c + 0/2), 
 
 where T and IT are the mean values of T and W throughout a 
 period. It is therefore no longer true that the mean value of the 
 kinetic energy is equal to that of the potential energy. T and W 
 are both periodic functions of z, T vanishing at the nodes and W at 
 the loops of the stationary waves. 
 
 So far in this chapter we have been considering the pro- 
 pagation of light in free ether. When we come to discuss the 
 corresponding problems for material media, and to deal with 
 reflection and refraction at the surface of a transparent isotropic 
 body, we are faced with the difficulties connected with the relations 
 between ether and matter. However, it will not be necessary to 
 delay our progress until these difficulties are overcome. A large 
 amount of experimental evidence goes to show that matter is 
 not a continuum, but is made up of discrete particles or atoms ; but 
 even if we were satisfied with our knowledge of the nature of these 
 atoms, we should not be concerned at this stage with the action of 
 the individual atoms on a wave of light. We are interested only 
 in their average effect, the influence of all the atoms in a finite 
 portion of matter. We assume as a working hypothesis that the 
 presence of matter affects the ether in such a way that ether and 
 matter may be replaced ideally by a continuum whose rotational 
 elasticity is different from that of the free ether, but whose density 
 is the same. The elastic constant of this medium obtained by 
 mentally smoothing out the atoms and spreading their influence 
 uniformly throughout the body is no longer c as for free ether, 
 but some other constant C//JL. With this slight change all the 
 results previously obtained for the propagation of light in the 
 ether apply to its propagation in any transparent isotropic body. 
 In particular, the dynamical equations become 
 
 the boundary conditions require the continuity of 
 
 /I 2 
 
38 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 and the velocity of propagation is c/fi. The quantity /x will be 
 identified with the refractive index of the medium. 
 
 In applying these formulae to the problem of reflection and 
 refraction we shall confine ourselves in this chapter to the ideal 
 case of an abrupt transition from one medium to the other. The 
 sequel will prove that in many cases actually presented experi- 
 mentally the transition is not of this abrupt character, and the 
 formulae will require modification before they can be applied to 
 such cases. The nature of this modification will be considered in 
 the next chapter. 
 
 In dealing with reflection and refraction it is convenient to 
 begin with a discussion of plane polarised light and to consider 
 separately the cases where the displacements are perpendicular 
 and parallel to the plane of incidence. The results for any other 
 case can be deduced from these by compounding vibrations on the 
 principles explained in the last chapter. We shall take the 
 surface of separation to be x = 0, the plane of xy to be the plane 
 of incidence, and then all the vectors are independent of z. 
 
 Let the displacement (f, 77, f) be perpendicular to the plane 
 of incidence and consequently the curl (f, g, h) parallel to this 
 
 plane. Let AO, OB and OC represent the incident, reflected and 
 refracted rays, (/>, (^ and (/>' the angles of incidence, reflection and 
 refraction. Let the arrows indicate the positive direction of the 
 curl (f, g, h\ these being chosen so as all to be parallel and in the 
 same sense in the case of normal incidence. Taking the refractive 
 index of the first medium to be unity, let /JL be that of the second. 
 
Ill] MEDIA. REFLECTION AND REFRACTION 39 
 
 We have f = = 77, and in the first medium 
 
 _ e ip [t - (x cos < + y sin <f>)/c] _ r& ip [t - (x cos ir - 0, + ?/ sin TT - </>,)/c] 
 
 the first term corresponding to the incident and the second to the 
 reflected wave. In the second medium we have 
 
 _ se ip It - (x cos ^' + y sin $')p-/e] 
 
 which represents the refracted wave. In writing down these 
 formulae we have made use of the theorems already proved, 
 that the ray coincides with the wave normal and that the velocity 
 of propagation of the wave is c/yu,. The boundary conditions 
 require f to be continuous at the interface x = Q for all values 
 of y and t. Hence we must have sin <f> = sin fa = //, sin </>'. These 
 are the fundamental laws of reflection and refraction, amply verified 
 by experiment, and familiar to everyone acquainted with the 
 elements of the science of optics. The law connecting the angles 
 of incidence and refraction is sometimes ascribed to Snell, and some- 
 times to Descartes, who first stated it in the form here presented. 
 The dynamical principles already explained give us much more 
 than Snell's law determining the directions of the reflected and 
 refracted rays. They enable us to determine the amplitudes of 
 the displacements and their curls in each of the waves, and thus 
 to estimate their relative intensities. We have seen that for 
 progressive waves, such as those now under discussion, it is 
 immaterial whether we measure the intensity by the mean 
 kinetic energy, the mean potential energy, or the rate at which 
 energy is flowing across a surface. The kinetic energy per unit 
 volume is \ (f 2 + rf -f j*) = 2 , in this case and this is proportional 
 to f 2 . Hence the intensities of the incident, reflected and refracted 
 rays are in the ratios 1 : r 2 : s 2 . It may be noted that while the 
 amplitudes of the displacements are in the ratios 1 : r : s, those of 
 the curls are in the ratios 1 : r : /JLS. These various ratios are 
 determined by means of the boundary conditions which require 
 
 1 r)f 
 
 the continuity of f and of g/fj,*, i.e. of f and of -. Whence we 
 
 o 
 
 get 1 r = s, and cos <p(I + r) = - cos <j>, so that 
 cos (' IJL cos <f> tan (^> <') 
 
 cos </>' + /JL cos <f) tan (</> + </>') ' 
 
40 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 since sin = /u, sin 0' ; and 
 
 2//, cos sin 20 
 
 cos 0' + jj, cos sin (0 + 0') cos (0 0') ' 
 
 s _ 2 sin 0' cos 
 
 //, sin (0 + 0') cos ((f> 0') ' 
 
 Before discussing these formulae we shall deal similarly with 
 the other case in which the displacement (f, rj, f) is parallel to the 
 plane of incidence, and its curl (/, g, h) perpendicular to this plane. 
 For this purpose it is convenient to introduce a new vector 
 (f '; v'> ") of which (f, fj, f ) is the curl, the possibility of this 
 having been pointed out before. We then have 
 
 In the first medium we may take 
 
 ' _ gp [ (xcos<f> +y sintfOA 1 ] _j_ ^'gzp [ + (x cos </> - 
 
 the first term representing the incident and the second the re- 
 flected wave. In the second medium we have similarly 
 
 f ' = - e*P \t - (x cos </>' 
 
 A* 
 
 The continuity of the displacement at all points of x = 0, leads as 
 before to Snail's law sin < = /j, sin / . The amplitudes of the dis- 
 placements in the different waves are in the ratios 1 :r'\s, and 
 those of the curls in the ratios 1 : r : ps. The ratios of the 
 intensities are 1 : r /2 : s' 2 . As before these ratios are determined 
 from the boundary conditions which require the continuity of 
 f, 77, and h/fj?. From these we get 
 
 1 + r = - and cos < (1 r') = s f cos $', 
 
 so that 
 
 cos <f) + fjb cos </> sin (</> + < ) 
 
 i , sin 20 , s 2 sin (/ cos 
 
 ~~ sin (0 + 0')' l " JL ~ sin (0 + r ) ' 
 
 These formulae giving the intensities in the different waves 
 
 are substantially the same as those first obtained by Fresnel, and 
 
 they will be referred to as Fresnel's formulae*. The formulae for 
 
 r and r f are the most interesting from the point of view of the 
 
 * Fresnel : (Euvres i. No. xxx. p. 774. 
 
Ill 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 41 
 
 experimenter. The march of these functions is represented in 
 Fig. 7, which gives the graphs of r 2 and r' 2 for ordinary glass 
 
 90' 
 
 whose refractive index is /JL = T52. From the formulae, or the 
 graph, it appears that r' 2 and r 2 are equal at normal incidence 
 (as they must be since there is then no distinction between 
 directions parallel and perpendicular to the plane of incidence) 
 and also at grazing incidence where their value is 1. As the 
 angle of incidence increases r' 2 constantly increases, slowly at first, 
 but afterwards somewhat rapidly. On the other hand r 2 begins by 
 diminishing and continues in this course until it vanishes, when 
 (f) + 0' = 90, i.e. when tan 4> = p. The angle at which r vanishes 
 is called the polarising angle for reasons that will appear presently. 
 After this angle is passed r 2 begins to increase and continues to do 
 so rapidly until it reaches the value unity at grazing incidence. 
 
 Let us consider a beam of natural light incident on any 
 reflecting surface of a transparent isotropic body. We have seen 
 that the light may be replaced analytically by two plane polarised 
 streams of equal intensity, the planes of polarisation being at right 
 angles. Thus we have two waves, the displacements in one being 
 
42 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH 
 
 parallel to the plane of incidence and those in the other at right 
 angles to this plane. The amplitudes of these displacements are 
 equal. After reflection, however, the amplitudes will no longer be 
 equal. Let e be the ratio of the amplitude of the displacement 
 perpendicular to the plane of incidence to that of the displacement 
 parallel thereto. Then by Fresnel's formulae we have 
 
 e = + r/r f = cos (</> + $')/cos (</> <'), 
 
 the sign being chosen so as to make e positive, for e being the ratio 
 of two amplitudes cannot be negative. When <f> + <' = 90, i.e. 
 when tan < = /JL, both r and e vanish. Thus the displacements 
 perpendicular to the plane of incidence are wanting from the 
 reflected beam, in other words, the reflected light is polarised 
 in the plane of incidence. The fact that natural light can thus 
 be polarised by reflection was first observed by Malus, and a few 
 years later Brewster* deduced from his experiments that the 
 polarising angle is given by the formula tan <j> = p. More modern 
 experiments indicate a very slight departure from Brewster's law, 
 but they show that the differences between observation and calcula- 
 tion are extremely small when the reflecting surfaces are thoroughly 
 clean and freshly polished. An explanation of the slight discrepancy 
 will appear in the next chapter, when the influence of the layer of 
 transition from one medium to the other is considered. The 
 following table gives the polarising angles of various substances 
 calculated from Brewster's law and compared with experiment. 
 
 Theory 
 
 Experi- 
 ment 
 
 53 Q 7' 
 53 l f 
 
 53 18' 
 53 18' 
 
 55 33' 
 
 55 33' 
 
 55 37' 
 55 37' 
 
 58 13' 
 58 12' 
 
 58 36' 
 
 5836' 
 
 59 41' 
 
 59 44' 
 
 60 30' 
 60 30' 
 
 63 33' 
 63 34' 
 
 67 7' 
 676' 
 
 67 32' 
 67 26' 
 
 67 4 
 67 c 
 
 The quantity denoted above by e can be deduced from experi- 
 ment and the results compared with those derived from theory. 
 This comparison is made for reflection from a diamond in the 
 following table and in the accompanying Fig. 8 which represents 
 the results graphically. 
 
 In this figure, as in all that follow, the curve corresponds to 
 the theory and the crosses to the experiments. The agreement is 
 as close as could be desired except in the neighbourhood of the 
 
 * Phil. Trans, p. 123. 1815. 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 e, 2 (Theory) 
 
 0421 
 
 0324 
 
 0234 
 
 0166 
 
 0104 
 
 005(5 
 
 e 2 (Experiment) 
 
 0420 
 
 0312 
 
 0213 
 
 0178 
 
 0102 
 
 0057 
 
 Difference 
 
 + 0001 
 
 + 0012 
 
 + -0021 
 
 - -0012 
 
 + : 0002 
 
 - -0001 
 
 
 
 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 e a 2 (Theory) 
 
 0049 
 
 0103 
 
 0177 
 
 0275 
 
 0399 
 
 0552 
 
 e 2 (Experiment) 
 
 0054 
 
 0106 
 
 0184 
 
 0296 
 
 0469 
 
 0576 
 
 Difference 
 
 - -0005 
 
 - -0003 
 
 - -0007 
 
 - -0021 
 
 - -0070 
 
 - -0024 
 
 05 
 04 
 03 
 -02 
 -01 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 , 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 K 
 
 
 
 
 
 
 ri 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 7 
 
 Fig. 8. 
 
 polarising angle. The explanation of the discrepancy in this 
 region will be discussed in the next chapter. 
 
 Fresnel's formulae give us the amplitudes of the displacements 
 and their curls for vibrations confined either to the plane of 
 incidence or to the plane perpendicular to this. It is, however, 
 
44 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 a simple matter to derive from them formulae for these amplitudes 
 when the incident vibrations are in any given azimuth. Thus if 
 the displacement in the incident wave be of amplitude A, and 
 make an angle 6 with the plane of incidence, we may resolve 
 this vector into its components A cos and A sin respectively 
 parallel and perpendicular to the plane of incidence. If then the 
 displacement in the reflected wave be of amplitude A and it 
 make an angle & with the plane of incidence, Fresnel's formulae 
 give us 
 
 A' cos 0' = r'A cose=-Acos0. S ^ "*2 ..... C0 
 
 sm (< + <') 
 
 and A' sin 6' = rA sin d = - A sin . tan ffi ~ f 2 ..... (2), 
 
 tan (< + </>') 
 
 which determine A and 6'. 
 
 If we eliminate 6 from (1) and (2) we get 
 
 41 - cos2 & . s " 12 ^ 
 
 Z 7 " 2 ~ ~^~ ~^~ ' 
 
 from which it follows that if we keep the amplitude (A) constant, 
 but vary the azimuth (6), the end of the line representing the 
 displacement in the reflected wave traces out an ellipse. Also 
 from (1) and (2) we obtain by division 
 
 n , sec (< 6'), 
 tan#=- - ^{tanfl. 
 sec (0 + </>') 
 
 Differentiating this and making use of (1) and (2) we find 
 
 so that the area swept out by a radius vector representing the 
 displacement in the reflected wave is proportional to that swept 
 out by the corresponding vector for the incident wave. 
 
 The relation tan & = sec (ft ~ < ) tan Q 
 
 sec (< + </> ) 
 
 can be put to the test of comparison with experiment by measuring 
 the azimuths of the planes of polarisation of the incident and reflected 
 waves for different angles of incidence. The two following tables 
 and figures exhibit the results of such a comparison, the experi- 
 ments being those of Brewster* made in 1830, and the refractive 
 index 1*4826. 
 
 * Phil. Trans, p. 74. 1830. 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 45 
 
 
 
 10 
 
 20 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 86 
 
 tf (Theory) 
 
 43 49' 
 
 40 4' 
 
 23 1' 
 
 9 
 
 6 16' 
 
 21 3' 
 
 33 46' 
 
 40 36' 
 
 0' (Experiment) 
 
 44 
 
 39 
 
 22 37' 
 
 9 
 
 6 10' 
 
 22^6' 
 
 33 13' 
 
 40 43' 
 
 Difference 
 
 -011' 
 
 + 14' 
 
 + 024' 
 
 
 
 + 06' 
 
 -13' 
 
 + 033' 
 
 -07' 
 
 This table shows how & varies with the incidence, being 
 constant and equal to 45 throughout. Fig. 9 below represents 
 the same results graphically. 
 
 1 pc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / rf 
 
 
 *- 
 
 \. 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f 
 
 
 
 
 
 "N 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 orf 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 N 
 
 v 
 
 
 
 
 
 / 
 
 / 
 
 
 
 9ft 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 ^ x 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 in r 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 V 
 
 > 
 
 
 
 
 
 
 
 * 
 
 10' 
 
 
 20 
 
 3C 
 
 1 
 
 4cr 
 K 
 
 5 
 
 ?. 9. 
 
 
 
 60 
 
 
 70 
 
 80 
 
 
 
 90 
 
 The next table shows how ff varies with 9 when the angle of 
 incidence is constant and equal to 75 throughout. Fig. 10 
 exhibits the same results graphically. 
 
 
 
 10 
 
 20 
 
 35 
 
 45 
 
 55 
 
 60 
 
 70 
 
 80 
 
 d' (Theory) 
 
 4 29' 
 
 10 16' 
 
 19 12' 
 
 26 27' 
 
 35 23' 
 
 40 45' 
 
 53 49' 
 
 70 29' 
 
 0' (Experiment) 
 
 4 54' 
 
 10 
 
 20 
 
 26 20' 
 
 35 30' 
 
 40 
 
 53 
 
 70 
 
 Difference 
 
 -025' 
 
 + 016' 
 
 -048' 
 
 + 07' 
 
 - 07' 
 
 + 045' 
 
 + 049' 
 
 + 029' 
 
46 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 The experiments were not very refined and the differences 
 between observation and calculation were nearly all within the 
 
 70 
 
 60 
 
 50 
 
 40= 
 
 30? 
 
 20 
 
 10 
 
 - 10 
 
 20 
 
 30 
 
 60 
 
 TOT 
 
 80 
 
 40 5 OP 
 
 Fig. 10. 
 
 limits of the errors of experiment. We shall see in the next 
 chapter that if the transition from one medium to the other is 
 gradual and not abrupt, the reflected beam is elliptically polarised 
 instead of being plane polarised. Under most circumstances, 
 however, the minor axis of the ellipse is extremely small, and the 
 azimuth of the major axis is very near that of the plane of polari- 
 sation calculated on the hypothesis of an abrupt transition. 
 
 The refracted wave may be dealt with on exactly similar lines. 
 Thus if the amplitude of the displacement be A and its azimuth 
 #! we have 
 
 A l cos#! = s'A cos and A l sin 6 l = s. A sin 9, 
 which determine A l and 6 1 when s and s are known from Fresnel's 
 formulae. The azimuth of the plane of polarisation of the 
 refracted wave is given by the equation 
 
 tan 1 - tan 6 = sec (< (/>') tan 6. 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 47 
 
 The two tables and figures that follow compare the results 
 obtained from this formula with those found experimentally by 
 Brewster. 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 45 
 
 55 
 
 65 
 
 80 
 
 6 l (Theory) 
 
 45 6' 
 
 45 25' 
 
 46 
 
 46 56' 
 
 47 34' 
 
 48 59' 
 
 52 7' 
 
 59 5' 
 
 d l (Experiment) 
 
 45 13' 
 
 45 27' 
 
 45 40' 
 
 46 30' 
 
 46 47' 
 
 48 54' 
 
 51 48' 
 
 58 40' 
 
 Difference 
 
 -07' 
 
 -02' 
 
 + 020' 
 
 + 026' 
 
 + 047' 
 
 + 05' 
 
 + 019' 
 
 + 025' 
 
 This with the corresponding Fig. 11 below shows how X varies 
 with the incidence when 6 = 45. 
 
 60' 1 
 
 55 
 
 5G 
 
 4Cf 
 
 Fig. 11. 
 
 The next table shows how O l varies with for a given angle 
 of incidence, < = 80. These results are shown graphically in 
 Fig. 12. 
 
 The formulae 
 
 tan 0j = tan . sec ((/> - <') = tan 0' sec (</> 4- </>') 
 
48 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 e 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 55 
 
 65 
 
 80 
 
 e l (Theory) 
 
 16 25' 
 
 31 19' 
 
 43 57' 
 
 54 31' 
 
 63 19' 
 
 67 15' 
 
 74 24' 
 
 83 58' 
 
 X (Experiment) 
 
 17 10' 
 
 32 30' 
 
 44 10' 
 
 54 36' 
 
 63 10' 
 
 66 58' 
 
 74 8' 
 
 83 23' 
 
 Difference 
 
 -045' 
 
 -111' 
 
 - 13' 
 
 -05' 
 
 + 09' 
 
 + 017' 
 
 + 016' 
 
 + 035' 
 
 80 
 
 60 
 
 4Cf 
 
 3C? 
 
 10 
 
 2QT 
 
 30? 
 
 4CP 
 
 BO 1 
 
 60" 
 
 70 
 
 80 
 
 90 
 
 Fig. 12. 
 
 enable us to give a very simple geometrical construction for the 
 direction of the displacement in the reflected and refracted waves 
 when that in the incident wave is given. However, instead of 
 deriving this construction from Fresnel's formulae we shall obtain 
 it by another method which is more direct and will be of use when 
 dealing with the corresponding problem of crystalline reflection 
 and refraction. The displacement (f, 77, ) is a vector whose square 
 is proportional to the energy per unit volume, or to the rate at 
 which energy is flowing across any surface. Hence, on the 
 doctrine of energy which is involved in our fundamental hypotheses, 
 the magnitude of this vector is not altered in going from 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 49 
 
 medium to another. Moreover to avoid a rupture of the medium 
 the components (f, 77, f) must be continuous at an interface. 
 Hence it follows that the refracted displacement is the resultant, in 
 the mechanical sense, of the incident and reflected displacements. 
 These three are therefore in one plane and are connected by the 
 parallelogram law. If Z = be the plane of incidence and the 
 origin, the three waves pass through the line OZ and are inclined 
 
 Fig. 13. 
 
 to one another at angles determined by Snell's law and the law of 
 reflection. Thus if the incident wave be given the other waves 
 are easily obtained, and we have now to investigate the directions 
 of the displacements in these waves. Let OT be the direction 
 of the incident displacement, OP the direction of its curl, so 
 that TOP is a right angle, T and P being points on the unit 
 sphere whose centre is 0. Through OP draw a plane perpen- 
 dicular to the incident wave to cut the refracted wave in OP 1? and 
 through OP l draw a plane perpendicular to the reflected wave to 
 cut it in OP'. Then OP' and OPj are the directions of the curls 
 in the reflected and refracted waves and OT' and OT 1} at right 
 angles to these lines, are the directions of the displacements. 
 Since TP and TPP l are right angles, P^T is a right angle. 
 Similarly P,T' is a right angle. Hence P l is the pole of TT.T', 
 
 M. L. 4 
 
50 
 
 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 so that the plane of the three displacements is the polar plane 
 of the curl of the refracted displacement. 
 
 The same general principles also enable us to prove, indepen- 
 dently of Fresnel's formulae, the important theorem that at the 
 polarising angle the angles of incidence and refraction are comple- 
 mentary. Let OT (Fig. 14) be the section of the interface by the 
 
 Fig. 14. 
 
 plane of incidence, ON the normal to the interface, OQ and OQ' 
 the directions of the incident and refracted rays. If OQ l be 
 parallel to the front of an incident wave, the direction of the wave 
 front is altered so that after refraction it becomes TQ'. If TQ be 
 parallel to OQi then TQ and TQ' are, by Huyghens' construction, 
 tangents to the circles whose radii are proportional to the velocities 
 of propagation of the waves in the two media. Hence the volumes 
 of disturbance in the two media are proportional to the areas of 
 the triangles OQT and OQ& i.e. OQT and OQT, since OQT = OQ 1 T. 
 We have seen that the refracted displacement is the resultant of 
 the incident and reflected displacements. Thus at the polarising 
 angle where there is no reflected displacement the incident and 
 refracted displacements are equal in magnitude and direction. 
 Hence the energy per unit volume is the same in the two waves, 
 and therefore, since there is no change of energy, the areas OQT 
 
III] MEDIA. REFLECTION AND REFRACTION 51 
 
 and OQ'T are equal and QQ' is parallel to OT. From this it 
 easily follows that the angles </> and <f> are complementary. For 
 OQ'QT can be inscribed in a circle of which ON is the tangent at 
 0. Hence 
 </>' = Q'ON= angle in alternate segment = OQQ' ' = QOT= 90 - </>. 
 
 Returning to the consideration of Fresnel's formulae we see 
 that at normal incidence 
 
 so that these terms are negative. The displacements in the incident 
 and reflected waves are represented by e ip (t ~ x/e) and re ip(t+xlc) . 
 Hence as r is negative, the displacements in the two waves 
 are in the same direction and there is no change of phase on 
 reflection. As the angle of incidence increases r and r' take 
 different values, but r' retains the same sign throughout. Thus 
 for light polarised parallel to the plane of incidence there is no 
 change of phase throughout. It is different, however, for light 
 polarised at right angles to the plane of incidence, for r vanishes at 
 the polarising angle and changes its sign there. Hence there is a 
 sudden change of phase of half a wave length on passing through 
 the polarising angle. If we are dealing with the curl of the 
 displacement it must be noted that in Fig. 6 the directions were 
 so chosen that the curls in the incident and reflected waves were 
 taken positively in the same direction for normal incidence. Hence 
 the negative value of r at normal incidence shows that the curls 
 are in opposite directions in the two waves, or that there is a 
 change of phase of half a wave length on reflection. 
 
 For purposes of comparison with experiment we are usually 
 more interested in the difference of phase after reflection between 
 the components parallel and perpendicular to the plane of incidence 
 than with the actual change of phase for either of these components. 
 We shall put r = Re ip and r' = R'e i ( > ', so that 
 
 where RjR' = e and A = p p. If the displacement in the 
 incident wave be resolved into components of unit amplitudes 
 parallel and perpendicular to the plane of incidence, then after 
 reflection the component displacements are represented by 
 
 42 
 
52 
 
 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC 
 
 CH. 
 
 fi = R' cos (pt + p\ and ^ = R cos (p -}- p). If then p be 
 greater than p the crest (or hollow) of the wave 77! is reached 
 sooner than the crest (or hollow) of the wave fj. Thus fj lags 
 behind ^ , the amount of the lag being measured by p - p' = A. 
 According to Fresnel's formulae A is zero for all angles of incidence 
 less than the polarising angle. At this angle A changes suddenly 
 to TT or TT, so that there is an acceleration or retardation of phase 
 of half a wave length. It is found, however, by experiment that 
 the change of phase does not take place suddenly but enters by 
 degrees. The change is very rapid in the neighbourhood of the 
 polarising angle and very slow outside of that neighbourhood. At 
 the polarising angle itself A is found to be very nearly ?r/2. 
 Fig. 15 represents the results obtained by reflection from diamond, 
 
 60 
 
 70 
 
 80 
 
 90 
 
 10 20f 3QT 4(T 50" 
 
 Fig. 15. 
 
 the dotted line corresponding to Fresnel's formulae and the crosses 
 along the continuous curve to the experimental results. A 
 increases from zero to TT throughout the range. Under such 
 circumstances the reflection is said to be negative, to distinguish 
 this from what is called positive reflection, when A diminishes 
 from to TT. 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 53 
 
 The intensity of the light reflected from a transparent isotropic 
 body is easily calculated from Fresnel's formulae. If natural 
 light of unit intensity be employed, we can replace it by two 
 beams whose intensity is -J, polarised in planes parallel and 
 perpendicular to the plane of incidence. The intensity of the 
 reflected light is thus (r 2 + r' 2 )/2, which at normal incidence 
 becomes [(/*- !)/(/* + 1)] 2 . In 1870 Rood* concluded from his 
 experiments that " the reflecting power of glass conforms in the 
 closest manner to the predictions of theory" or to Fresnel's 
 formulae. However, in 1886, this conclusion was shown to be 
 untenable by Lord Rayleigh f. The difficulties of measuring the 
 intensity of the reflected beam directly are very considerable, and 
 Rood had contented himself with measuring the transmitted 
 light and deducing the amount that was reflected. Rayleigh 
 showed that when this fact was considered the difference between 
 theory and Rood's experimental results might amount to 7/ of 
 the reflected light, a difference much too great to be regarded as 
 insignificant. Rayleigh found from his own experiments that 
 recently polished glass surfaces have a reflecting power differing 
 not more than 1 /o or 2 /o froni that calculated from Fresnel's 
 formulae ; but that after some months or years the reflection may 
 fall off 10 /o r more and that without any apparent tarnish. 
 About the same time Conroy carried out a careful series of 
 experiments on the same subject, his results confirming those of 
 Lord Rayleigh. The following table gives Conroy's estimates 
 of the percentage of light reflected from glass that had been 
 polished a considerable time before the experiments were made. 
 It also compares the results with the calculations from Fresnel's 
 formulae. 
 
 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 65 
 
 70 
 
 Reflecting power 
 (Experiment) 
 
 0378 
 
 0378 
 
 0377 
 
 0392 
 
 0437 
 
 0553 
 
 0854 
 
 1116 
 
 1604 
 
 (Theory) 
 
 0419 
 
 0419 
 
 0421 
 
 0434 
 
 0477 
 
 0598 
 
 0916 
 
 1231 
 
 1737 
 
 Difference 
 
 0041 
 
 0041 
 
 0044 
 
 0042 
 
 0040 
 
 0045 
 
 0062 
 
 0115 
 
 0133 
 
 * Amer. Journ. Sci. Vol. XLIX. 1870 (March). Vol. L. 1870 (July). 
 t Scientific Papers, Vol. n. p. 523. Proc. Roy. Soc. XLI. 
 
54 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 These results are exhibited graphically in Fig. 16. 
 
 16 
 
 14 
 
 12 
 
 10 
 
 08 
 
 06 
 
 04 
 
 02 
 
 RR 
 
 1 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 Fig. 16. 
 
 There can thus be no doubt of a decided departure from 
 Fresnel's formulae in certain circumstances. The difference 
 between observation and calculation is too great to be put down 
 to experimental errors, and there is no evidence of such errors 
 
Ill] MEDIA. REFLECTION AND REFRACTION 55 
 
 being considerable, seeing that the results of experiment are 
 fairly consistent. Nor can there be very much doubt as to the 
 direction in which to look for an explanation of the apparent 
 divergence between theory and observation. Everything points 
 to a changing condition of the reflecting surface, and this suggests 
 that a consideration of the layer of transition will show how 
 Fresnel's laws are violated in this as in some other directions. 
 This will be discussed further in the next chapter. Meanwhile it 
 should be remarked that Lord Rayleigh found when dealing with 
 reflection from liquids*, that the departure from Fresnel's formula 
 diminished almost to zero when care was taken to remove all 
 greasy contamination from the surface, and concluded that " there 
 is no experimental evidence against the rigorous applicability of 
 Fresnel's formulae to the ideal case of an abrupt transition between 
 two uniform transparent media." 
 
 Total Reflection. 
 
 Thus far we have supposed that when a ray of light is going 
 from one medium to another, the index of refraction of the second 
 medium is greater than that of the first. We have now to 
 consider the case when the media are interchanged, so that the 
 first medium is of higher refractive index than the second. The 
 formulae in this case can, in general, be deduced from those already 
 obtained by replacing //, by I///,. There is one matter, however, in 
 which the results are so completely modified as to deserve a special 
 investigation. The relation between c/> and <p', the angles of 
 incidence and refraction, is given by Snell's law 
 
 sin<f>=- sinrf/. 
 f* 
 
 Let (/h be the angle whose sine is I//*, then since 
 sin </>' = sin </sin fa 
 
 we see that <f>' is real or not according as (f> is less or greater 
 than fa. The angle fa is called the critical angle, and if the angle 
 of incidence be greater than this there is no refracted ray in the 
 ordinary sense. The reflection is then said to be total. 
 
 * Scientific Papers, Vol. in. p. 496. Phil. Mag. jcxxm. pp. 119. 
 
56 PKOPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 The investigation of the change of amplitude and phase in 
 the case of total reflection follows the same lines as in the case 
 of ordinary reflection considered above. Thus, if the displacement 
 (?> *!> ) be perpendicular to the plane of incidence, we have 
 f = = 77, and in the first medium 
 
 _ ip [t+ (x cos 4>- 
 
 the first term corresponding to the incident and the second to the 
 reflected wave. In the second medium we have 
 
 t = se ip ^ ~ ( nx +y sin *) pl e \ 
 
 which represents the refracted wave. In order to satisfy the 
 fundamental dynamical equation in the second medium we must 
 
 have 
 
 (ri 2 + sin 2 <) /r 5 = 1, 
 
 and so n = i* sin 2 < - l//x, 2 . 
 
 The refracted wave is thus represented by 
 
 .e 
 
 The boundary conditions require the continuity of f and 
 
 i p\ I* 
 i.e. of f and ^ . From these we get 
 
 . p?s Vsin 2 </> - l//x 2 . 
 and l+r= -^r = - i - -^ - -* f tan a, 
 
 cos <p cos 9 
 
 where tan a = c r 
 
 cos c/> 
 
 Whence s = 2cosa.e z ' a and r = - e zi = 
 
 Thus the reflected wave has the same amplitude (and therefore 
 the same intensity) as the incident wave, but there is a change of 
 phase represented by 2cc TT. The disturbance in the second 
 medium is represented by 
 
 = 2q cos a . ' 
 
 -^^ Vsin^-l/^ 
 
 where g = e 
 
 X being the wave length in the second medium. There is thus a 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 57 
 
 change of phase represented by a. The amplitude at the interface 
 is 2 cos a, but owing to the " factor q, it diminishes very rapidly 
 as x increases, so that the disturbance practically disappears at 
 a depth of a few wave lengths. The following table gives the 
 values of the factor q, for different incidences, and values of x, in 
 the case where /z = T596, the critical angle being 38 47'. 
 
 0... 
 
 39 
 
 40 
 
 45 
 
 50 
 
 60 
 
 75 
 
 85 
 
 xl\ = l 
 
 5433 
 2952 
 
 2353 
 0553 
 
 0372 
 0014 
 
 0120 
 0001 
 
 0025 
 10- 6 x 6 
 
 0005 
 10~ 7 x 3 
 
 0004 
 lO- 7 x 2 
 
 xl\--=3 
 
 1604 
 
 0130 
 
 10- 5 x 5 
 
 10~ 6 x 2 
 
 10- 8 xl-5 
 
 10- 10 xl-6 
 
 10- 11 x 7-6 
 
 ,/X = 4 
 
 0873 
 
 0031 
 
 10~ 6 x 2 
 
 10- 8 x 2 
 
 10- !1 x4 
 
 10~ 14 x9 
 
 10- 14 x 3 
 
 Similarly if the displacement (f, 77, f) be parallel to the plane 
 of incidence we have, with the notation employed in the 
 corresponding case of ordinary reflection, 
 
 ' e ip [t -(xcos$+y sin <J>) //c] _j_ r ' & ip [t+ (x cos - y sin <j>) n/c] 
 
 in the first medium, and 
 
 - (nx+y sin 0) /*/c] 
 
 in the second. In order to satisfy the dynamical equations, 
 we must give n the same value as before. The boundary 
 conditions require f, 7; and A///, 2 to be continuous, whence we 
 have 1 +r' = s, and cos < (1 r) = ns. These give s = 2 cos V V , 
 and r' = e zia> , where 
 
 , _ Vsin 2 </> 1//A 2 _ tan a 
 
 cos 
 
 The intensity of the reflected beam is equal to that of the 
 incident ; but there is a change of phase of 2a'. The disturbance 
 in the second medium disappears, as before, at a depth of a few 
 wave lengths. 
 
 From the formulae 
 
 , tan a Vsin 2 6 - l//u, 2 
 tan a = - = - 
 
 /, 2 cos <) 
 
 we see that a' is less than a, except at the critical angle where 
 
58 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 a and a.' are both zero, and at grazing incidence where they 
 are both Tr/2. Since 
 
 it follows that as the angle of incidence increases from the critical 
 angle to 90, p increases from TT to 0, and p' from to TT, and 
 their difference is always an obtuse angle. The component of the 
 displacement perpendicular to the plane of incidence lags behind 
 that parallel to this plane by an amount A' in phase, where 
 
 A' = p' p = TT 2 (a a'). 
 
 The component of the curl perpendicular to the plane of 
 incidence is in advance of that parallel to this plane by the same 
 amount A'. The above formulae for a and a give 
 
 cot A'/ 2 = Vsin 2 <j> - l//x' 2 /sin </> tan </>. 
 
 The following table gives the values of a, a' and A' for various 
 incidences beyond the critical angle, for the case mentioned above 
 where //, = T596 and the critical angle is 38 47'. 
 
 0- 
 
 39 
 
 40 
 
 45 
 
 50 
 
 60 
 
 75 
 
 85 
 
 a 
 
 11 17' 
 
 25 38' 
 
 49 46' 
 
 60 14' 
 
 71 50' 
 
 82 17' 
 
 87 28' 
 
 a' 
 
 4 29' 
 
 10 40' 
 
 24 53' 
 
 34 27' 
 
 50 6' 
 
 70 57' 
 
 83 35' 
 
 A' 
 
 166 24' 
 
 150 4' 
 
 130 14' 
 
 128 26' 
 
 136 32'. 
 
 157 20' 
 
 172 14' 
 
 These results are exhibited in Fig. 17 below, which represents 
 the march of A' as the angle of incidence increases from normal to 
 grazing incidence. A corresponds to the polarising angle and G to 
 the critical angle. 
 
 As A' is equal to TT at the critical angle and at grazing 
 incidence, it will have a minimum value somewhere within the 
 range of total reflection, as appears from the figure below. The 
 position of this minimum is found by making A' stationary, 
 so that 
 
 This gives 
 
 whence 
 
 tan 2 a 
 
 sn 2 e - 
 
 1 sin 2 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 59 
 
 At this angle we have 
 
 tan a = ft, and tan a = 1 //A, 
 so that a + a' = Tr/2, and A' = 4a'. 
 
 18Cf 
 
 I2rf 
 8Cf 
 4Cf 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 ^ 
 
 x^ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 s 
 
 
 
 <* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _^ KT 20 30 :40 50 60 70 80 90 
 Fig. 17. 
 
 Thus at the angle for which A' is a minimum, a is Brewster's 
 angle tan" 1 l//n. The following table gives the values of these 
 quantities for various refractive indices. 
 
 ft 
 
 a' (Brewster's 
 angle) 
 
 0j (critical 
 angle) 
 
 <j> (angle for 
 which A' is least) 
 
 Least value of 
 A'=4' 
 
 1-46 
 
 34 24' 
 
 43 14' 
 
 53 3' 
 
 137 38' 
 
 1-50 
 
 33 41' 
 
 41 49' 
 
 51 40' 
 
 134 45' 
 
 1-51 
 
 33 31' 
 
 41 28' 
 
 51 20' 
 
 134 3' 
 
 1-596 
 
 32 4' 
 
 38 49' 
 
 48 40' 
 
 128 16' 
 
 1-60 
 
 32 0' 
 
 38 41' 
 
 48 33' 
 
 1-28 V 
 
 2-371 22 52' 
 
 24 57' 
 
 33 20' 
 
 91 28' 
 
 2-414 
 
 2'2 30' 
 
 24 28' 
 
 32 46' 
 
 90 
 
 2-434 
 
 22 20' 
 
 24 15' 
 
 32 31' 
 
 89 20' 
 
 2-454 
 
 22 10' 
 
 24 3' 
 
 32 15' 
 
 88 40' 
 
60 PROPAGATION OF LIGHT IN TRANSPARENT ISOTROPIC [CH. 
 
 The minimum value of A' being 4a', we see that if this is Tr/2 
 we must have 
 
 a' = 7r/8, and p = cot a! = 2'414. 
 
 The angle of incidence for which the difference of phase is ?r/2 
 is called the Principal Incidence. Below the critical angle this 
 coincides with the polarising angle. Within the limits of total 
 reflection there will be no Principal Incidence if //, be less than 
 2*414 and two principal incidences for substances such as diamond 
 and realgar for which p is greater than 2'41A Thus we cannot 
 obtain a phase difference of ?r/2 by means of total reflection at a 
 single surface for substances whose refractive indices are less than 
 2'414. This, however, can be effected by arranging, as is done in 
 Fresnel's rhomb, for two reflections with a phase difference of 3-7T/4 
 at each reflection, since a retardation of 3?r/2 is virtually the same 
 as an acceleration of vr/2. Fig. 17 above shows that, as a rule, 
 there will be two angles of incidence for which the difference 
 of phase is 37T/4, and it is easy to obtain a formula for their 
 magnitudes. Since 
 
 we have 
 
 A' , 
 
 tan -= = cot ( 
 
 sin $ tan 6 
 
 - r 
 
 Putting A' = 3-7T/4, this gives 
 
 l# sin 4 $ (4 - 2^2-) - (p* + 1) sin 2 + 1 = 0, 
 
 a quadratic equation in sin 2 <f>, which has real roots provided /JL be 
 greater than 1*496. The following table gives the two values of </> 
 for different kinds of glass, the refractive index being /JL. 
 
 V 
 
 1-5 
 
 1-51 
 
 1-55 
 
 1-596 
 
 1-6 
 
 Smaller angle 
 of incidence 
 
 50 13' 46" 
 
 48 37' 27" 
 
 45 14' 29" 
 
 42 41' 15" 
 
 42 29' 45" 
 
 Larger angle 
 
 53 15' 31" 
 
 54 37' 23" 
 
 57 4' 42" 
 
 58 37' 35" 
 
 58 44' 2" 
 
 Fresnel's rhomb is a rhomb of glass, a principal section of which 
 is shown in Fig. 18. The angles of the rhomb are chosen so that 
 a ray incident at right angles to one face, suffers total reflection at 
 
Ill] 
 
 MEDIA. REFLECTION AND REFRACTION 
 
 61 
 
 two parallel faces, and emerges normally to the face opposite that 
 at which it enters. The difference of phase produced by passing 
 
 Fig. 18. 
 
 through the rhomb being Tr/2, it is evident that the acute angle of 
 the rhomb must be one or other of the two values of $ determined 
 above. Of these two possible values the larger is to be preferred, 
 because with it the errors due to slight changes in //, for different 
 colours are less than with the smaller value of <f>, and also because 
 a small deviation from < will introduce less error when <f> is large 
 than an equal deviation when $ is small. For this latter reason 
 also, the refractive index should exceed 1*496 by as small a quantity 
 as possible. If the light incident at A were plane polarised at an 
 angle of 45 with the plane of incidence, so that the amplitudes of 
 the displacements parallel and perpendicular to this plane were 
 equal (and we have seen that natural light could be so represented), 
 then the light on emerging from the rhomb would be circularly 
 polarised, for the amplitudes of the displacements parallel and 
 perpendicular to the plane of incidence would be equal, and their 
 difference of phase would be Tr/2. 
 
 The following table gives the values of A', expressed as a 
 fraction of the half wave length, for total reflection from a 
 substance whose refractive index is 1*619 and critical angle 38 9', 
 
 0... 
 
 38 13' 
 
 39 58' 
 
 41 59' 
 
 44 2' 
 
 46 4' 
 
 47 54' 
 
 49 58' 
 
 51 57' 
 
 53 58' 
 
 55 57' 
 
 A' (Theory) 
 
 960 
 
 799 
 
 744 
 
 718 
 
 707 
 
 705 
 
 707 
 
 712 
 
 721 
 
 731 
 
 A' (Exp.) 
 
 978 
 
 813 
 
 754 
 
 728 
 
 718 
 
 712 
 
 715 
 
 719 
 
 726 
 
 730 
 
 Difference 
 
 018 
 
 014 
 
 010 
 
 010 
 
 Oil 
 
 007 
 
 008 
 
 007 
 
 005 
 
 001 
 
62 
 
 KEFLECTION AND REFRACTION 
 
 [CH. Ill 
 
 and compares them with the results of experiment. Fig. 19 
 represents the same results graphically. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 >0 38 40 42 44" 46 48 5CT 52 54 56 
 
 Fig. 19. 
 
CHAPTER IV. 
 
 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION 
 AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA. 
 
 IT appears from the last chapter that the theory of an abrupt 
 transition from one medium to another leads to expressions for the 
 amplitude and phase of the displacement in the reflected and 
 refracted waves that cannot be regarded as better than a close 
 approximation to the facts as derived from careful experiments. 
 As a large number of such experiments indicate that the optical 
 effects of a reflecting or refracting body depend on the condition 
 of its surface, such as its freedom from tarnish or the state of^its 
 polish, it is clear that we must, in our analysis, take some account 
 of these surface conditions. We shall, therefore, in the present 
 chapter, investigate to what extent the previous formulae must be 
 modified when the refractive index, instead of changing abruptly, 
 changes continuously throughout a very thin layer of transition 
 between the two media. As the law under which the refractive 
 index changes within the layer will vary with the condition of the 
 surface, we shall not at the outset make any hypothesis as to the 
 nature of this law. 
 
 In dealing with the problem of reflection and refraction we 
 shall use the same notation as in the last chapter and shall 
 suppose that the layer of transition extends from x = to x = d, 
 and is continuous, as regards the refractive index JJL, with the 
 media bounding it. It will be convenient to put 
 
 sin cf> = /A sin $' v ; cos = tc ; /u- cos <' = K. 
 The dynamical equations and boundary conditions for the 
 
64 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 layer may be obtained as before from the Principle of Action, by 
 taking 
 
 but it must be remembered that //, is no longer a constant. The 
 term containing S% in the variation of W is 
 
 fcV 85f_ 
 
 , 
 or 
 
 Hence the dynamical equations consist of 
 
 and two similar equations. Also, since //, is continuous at the 
 boundaries of the layer, the boundary conditions are satisfied by 
 the continuity of ng mh, Ih nf, and mf Ig, i.e. of g and h for 
 surfaces parallel to the plane x = 0, coupled, of course, with the 
 continuity of the displacement (, ?;, f). 
 
 If the incident light be polarised at right angles to the plane 
 of incidence we have f = = 77, while f is given in the first and 
 second media by the formulae on p. 39. Within the layer itself 
 we may take ^=ue ip(t ~ vy l c \ where u is a function of x only. This 
 gives 
 
 /= = _ *(-,*>, an d g = - = _ 
 8y c a# ao; 
 
 Substituting in the dynamical equation 
 
 rI*fi//w 
 
 ' LayUv 
 we get 
 
 d / 1 dw\ , ^ 2 /- 
 
 S\x5/ + ^( 
 
 Putting x/d = x l and <h =pd/c = 27rd/X, where \ is the wave length 
 in the first medium, we get 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 65 
 
 This equation, of course, cannot be solved completely until we 
 know yu, 2 as a function of ac 1 , i.e. until we know the law of variation 
 of fi in the layer. However, in all cases to which we shall apply 
 the solution, ^ is a small quantity (as will appear later), and we 
 can thus solve the above equation by successive approximations. 
 
 A first approximation is obtained by neglecting d^, when the 
 equation becomes 
 
 d / 1 du\ 1 du Q 
 
 --j- 1 = 0, whence - T -=/3 
 dx l \/j? dxj ya 2 dx l 
 
 and u = A + B %&, = A+ Bu?M, 
 
 Jo 
 
 where M is a function of x lf As the next approximation we put 
 
 u = A + Bp*M 
 and the equation for v is 
 
 Integrating this, we get 
 
 ^- 4- Affac! + BtfMi - Av*Ms - Bv*i*M 4 = 0, 
 where 
 
 A second integration then gives 
 
 v + Ap 2 faM - M,) + B^M 5 - A v 2 M 6 - Bv*ti*M 7 = 0, 
 where 
 
 ft'M, = \ Xl ^M l dx l , 
 
 Jo 
 
 o o 
 
 When a?! = all the i/'s are zero, and when ^ = 1 we have 
 
 ^=#, Jf^ff, ^ 3 =,P, J/4=J r , ^5 = ^, ^6 = ^ ^ 7 = ^ 
 
 where E, ..., N are constants depending on the law of variation of 
 yu- within the layer. 
 
 Thus when x l = we have u = A, dufdx l = 5, and when x l = 1 
 we have similarly 
 
 w = 4 + B^E + d, 2 [ J. [i* (H-E) + v 2 L] + B 
 and dujdx, = B^ + rf x 2 [JL (i/lF - /i 2 ) + 5yLt 2 (v 2 
 
 M. L. 5 
 
66 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 The boundary conditions give 
 
 ^ [A (yu 2 (H-E) + v*L] 
 - i/c's = B^/d, + d 1 [A (v 2 F - ^) + B/J? (z/ 2 / - 
 
 On eliminating A and B from these equations, we get two 
 equations to determine r and s, which give the amplitude and 
 phase of the reflected and refracted beams. If r = Re ip and 
 s = Se i<r , then R and $ are the amplitudes, and p and cr the phases, 
 of the displacements in the reflected and refracted waves. 
 
 In dealing with the above equations we shall, as the first 
 approximation, neglect squares and higher powers of d l . The 
 equations then yield 
 
 s + r [1 + idiKEp*] = 1 - idiicEn* 
 
 and K'S - r [K/JI? - id, (v*F - yu 2 )] = KL? + id, (v 2 F - //, 2 ). 
 
 From these we obtain 
 
 ^^ 
 K [1 + id.fcE/j, 2 ] + Kp? - id, (v 2 F - ^) 
 
 ) + id l (lac'Efj? + p* - v*F)]~ l , to our order, 
 
 = &^, 
 
 where 
 
 2i6 2 cos <> sin2< 
 
 ~~ 
 
 K 4- KfJ? ft cos <' + /j? cos (/> sin(^) + <')cos (</> </>') ' 
 
 and tan o- = -, . d, . 
 
 K + /c/r 
 
 Hence, to this order, the amplitude of the refracted wave is the 
 same as in the ideal case of an abrupt transition ; but there is a 
 small change of phase. 
 
 Similarly for the reflected wave we get 
 
 9 f i ' 7 / ' TJ* 9 9 i 9 TJ?\ 
 
 "'I K + Id, ( KK J&LL LL + IT/f ) _. . 
 
 ' = jR^^, 
 
 *' + id, 
 
 D , tcp? K tan (& &' 
 where R = -^ = + 
 
 ~ 
 
 ~ tan 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 67 
 
 the sign being chosen to make R positive, and 
 
 2Kp*[EK'* + Fit-fjf] , 
 tan p = -- -S- -r A . d 1 
 
 2 2 
 
 K 
 
 2/x 2 cos cf> [(F - E) sin 2 <f> - (1 - E) ^] 
 
 (^ - 1) [sin 2 <f> - /x 2 cos 2 (/>] ~ ' 1J 
 
 except in the neighbourhood of Brewster's angle, where 
 
 KfJ? K 0, 
 
 and we have 
 
 where / denotes the integral 
 
 I /a dx lt 
 
 Jo A 6 
 
 /// being the refractive index in the layer. 
 
 In this neighbourhood also tan p is infinite, so that 
 
 From these formulae we see that with reflection as well as with 
 refraction the amplitude is, in general, given correctly, to our 
 order of approximation, by Fresnel's formulae, but that there is a 
 small change of phase depending on d l . The only region in which 
 there is a marked departure from Fresnel's formulae is in the 
 neighbourhood of Brewster's angle. Here R does not vanish, 
 although it is very small, and there is a change of phase of a 
 ^quarter wave length. 
 
 These results have been obtained by neglecting squares of d lt 
 For some purposes it is necessary to proceed to a higher order of 
 approximation and retain terms in df. The equations for r and 5 
 then become 
 
 s + r [1 + idiicEn* + d? {/z 2 (H - E) + v*L}] 
 
 = 1 - id lK Ep? + d* {p? (H-E) 
 
 K'S - r O// 2 - id l (v*F - /j?) + dficp* (v 2 J - f^ 
 
 52 
 
68 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 Whence we get 
 
 2/c// 2 [1 + d,* (i/ 2 (L + J 4- EH) - 
 
 ^/^^^ 
 = Be*, 
 
 where 8 = ? Kf * , \l + d^ \v z (L + J + EH) 
 
 K + KjJ? \_ [ 
 
 2(' 
 
 , 
 
 and tan cr = --- J - 7 - . d l . 
 
 K + /c/A 2 
 
 Similarly 
 /^-/g'+^K/c^Ey-// 2 ^^ 
 
 where 
 
 and 
 
 except at Brewster's angle, where 
 
 where tan 
 
 /c' + /CyLt 2 ) (/f/t^yLt 2 - yu, 2 + Z^ 2 
 
 . 
 
 We must next consider the case in which the incident light is 
 polarised parallel to the plane of incidence. We then have 
 ( ' = = 7)', while " in the first and second media is given by the 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 69 
 
 formulae on p. 40. Within the layer f ' = ue 1 ^*' 1 ^ 1 ^ y where u is a 
 function of x only. This gives 
 
 c 
 
 t=0, /=<> = </, 
 
 We have seen above that the dynamical equations are 
 
 =c2 L^(J)~sytvj' 
 
 and two similar equations. These are all satisfied if 
 
 ,3. r 2 
 
 1' = -^ = ^'- 
 Hence we require 
 
 or 
 
 Solving this by successive approximations, we get in the first 
 place u = A 4- Bx l , and as the second approximation 
 
 u = A + Bx l + c?j 2 v, 
 
 where d?v/dxi> + (/^ 2 - z/ 2 ) (A + B^) = 0. 
 
 Integrating this equation, and making use of the relation 
 
 we get 
 
 fl *?J 
 
 5^ = L 2 (^la?, + J^ a ) - /t 2 (^llf + BMx l - BMJ, 
 Q/XI 
 
 and v = v 2 (^Aa^ + $Bxf) - ^ (AM, 
 
 /%\ 
 
 
 When x l = 0, il/ 2 = 0, and when ^ = 1, Jf 2 = (7. Thus we have 
 u = A and dufdx^ = 5 when ^ = 0, and when x l = 1, 
 
 ii = J. + 5 + ^ 2 [i/ 2 (^1/2 + B/Q) - rf [H (A+B)- 2BG}] 
 and du/dic, = 5 + 4" [i/ 2 (4 + 5/2) - /* 2 [E (A+B)- BE}]. 
 
70 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 The boundary conditions give 
 
 A . = !+/, B = -id lK (l-r'), 
 
 (A/2 + 5/6) - ^ {H (A+B)- 2BG}] } 
 [> 2 (A + 5/2) - tf [E(A +B)- BE}], 
 from which r' and s' can be determined, by eliminating A and B. 
 
 As the first approximation we shall retain only the first power 
 of d l} and we then get s'/p r' (1 + id^) = 1 id,*, 
 and K'S'/IA + r' {K + id, (E^ 2 - v 2 )} = K- id 1 (E/j 2 - v 2 ). 
 Thus 
 
 / {1 - id,K\ {K + id, (Efji 2 - v 2 )} + (1 + id lf c] {K - i 
 
 + id : (KK 
 
 * + *'.+ id, (KK + Efj? - v 2 ) 
 where 
 
 a , ZfjiK sin 20 , KK + Eu 2 v 2 
 
 /o = / = - TT ~TT^ j anc i tan a- = . . cL . 
 
 /c + /c sin (0 + ) /c + K 
 
 a . ., , , K K +id, (KK E/J? 4- z/ 2 ) D , . , 
 
 Similarly r = ? ry-p -, ~ ( = R e lp , 
 
 K + K -f- id, (KK + M/n v ) 
 
 where 
 
 , /c' 8^(0 00 , 2K/ji 2 (lE) 
 
 K = -J+T- = . ;]7 , ;T>( , and tan p = ^ - ^ . d,. 
 
 K +K Sin (0 + ) y^ 2 1 
 
 Hence, both in the reflected and in the refracted wave, the ampli- 
 tude is the same as that given by Fresnel's formula, but there is 
 a small change of phase. 
 
 If we proceed to a higher order of approximation and retain 
 squares of d, we get similarly 
 
 2/JLK q 
 
 K + K+ id, (KK + Eft 2 - v 2 ) + d 2 [(K + K) v 2 /2 - ^{EK+H(K -K)}] = 
 where 
 
 and tan a' = 
 
 - 2^ 2 (K + K) (Etc + Etc' - HK)]\ , 
 
 KK + E/JL 2 V 2 , 
 
 / ~" A 7 
 
 K + K 
 
 .2/17 77V _i_ 77".M 
 
 = Re*, 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 71 
 
 where 
 
 and tan p = __ 
 
 It will be observed that the change of phase (p' and <r') is the 
 same as in the first approximation. 
 
 The quantities that can be most accurately measured experi- 
 mentally are e and A, the ratio of the amplitudes and the 
 difference of phase for the displacements perpendicular and parallel 
 respectively to the plane of incidence. These are given in terms 
 of the quantities dealt with above, by means of the relation 
 
 and from what we have proved, it appears that it is only in the 
 neighbourhood of Brewster's angle that there is any appreciable 
 + departure from the values given for e and A by Fresnel's 
 formulae. The value of e at Brewster's angle is called the 
 coefficient of ellipticity and will be denoted by e^ Neglecting d t 
 we have, at Brewster's angle, 
 
 2 Vl 
 
 Hence l = -^-^- 
 
 Since 
 
 . ( 
 
 J o 
 
 
 
 we see that if p lies between 1 and //,, I is negative, so that we 
 must take the lower sign with 6 1} which is essentially positive. If 
 we write e! in the form 
 
 we see that 6 X is a maximum along with yu/ 2 + yu- 2 /// 2 ? i.e. when 
 //- = fj,. If ^ had this value throughout the layer we should 
 have 
 
72 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 This enables us to obtain an upper limit for the thickness of the 
 layer necessary to produce the ellipticity observed at Brewster's 
 angle in any given case. Thus with Kurz's experiments on 
 reflection from glass for which //,= 1/5963 and j= 0*0344, we have 
 d/\ = 0*0025 and d^ = 0*0159 ; while with Jaurin's experiments on 
 reflection from diamond for which //,= 2*434 and e 1 = 0*0241, we 
 have d/\ = 0'0074 and d 1 = 0'0464. 
 
 Since e l can be readily determined from experiment, it is 
 convenient to express e and A in terms of this quantity. Thus 
 we have 
 
 sn 
 
 <//) V( 
 
 cos < - cos 
 
 I) 3 
 
 R r ~~ sin (</> </>') cos </> + cos </>'///, 
 
 which is practically equivalent to Fresnel's formula 
 
 e = + cos ((/> + <')/cos (0 </>') 
 
 outside the neighbourhood of Brewster's angle. The following 
 table gives the values of e 2 for different angles of incidence near 
 Brewster's angle and compares them with the results of experi- 
 ment. The reflection is from diamond for which //, = 2'434, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 04 
 03 
 02 
 01 
 
 1 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 ' 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 \-^ 
 
 
 _^ 
 
 / 
 
 
 
 
 
 
 Fig. 20. 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 73 
 
 Brewster's angle is 67 40', and the coefficient of ellipticity 
 observed was 0'0241. 
 
 d>.... 
 
 66 
 
 66 30' 
 
 67 
 
 67 30' 
 
 68 
 
 68 30' 
 
 69 
 
 
 
 
 
 
 
 
 
 e- (Theory) 
 
 00282 
 
 00168 
 
 00092 
 
 00061 
 
 00068 
 
 00126 
 
 00200 
 
 e 2 (Experiment) 
 
 00303 
 
 00173 
 
 00092 
 
 00057 
 
 00068 
 
 00128 
 
 00265 
 
 Difference 
 
 - -00021 
 
 - -00005 
 
 
 
 + -00004 
 
 
 
 - -00002 
 
 - -00065 
 
 We have already seen (p. 42) that outside of this region 
 Fresnel's formula gives an admirable representation of the facts. 
 These results are exhibited in Fig. 20. 
 
 For A we have approximately, 
 
 , 
 
 tan A = tan p - tan p = - 2*^ 
 
 L 2 * 
 
 sin $ tan 
 
 Vl + fjL 2 ' tan 2 fj? ' 
 
 The values of A deduced from this formula for the case of 
 reflection from diamond mentioned above are given in the next 
 table, and the results compared with those observed by Jaurin. 
 
 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 30' 
 
 67 
 
 67 30' 
 
 - A (Theory) 
 
 033 
 
 038 
 
 045 
 
 055 
 
 070 
 
 096 
 
 197 
 
 282 
 
 430 
 
 -A(Exp.) 
 
 033 
 
 042 
 
 047 
 
 063 
 
 073 
 
 105 
 
 202 
 
 288 
 
 437 
 
 Difference 
 
 
 
 -004 
 
 - ;002 
 
 -008 
 
 -003 
 
 -009 
 
 -005 
 
 -006 
 
 -007 
 
 
 
 68 
 
 68 30' 
 
 69 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 -A (Theory) 
 
 618 
 
 749 
 
 822 
 
 891 
 
 923 
 
 941 
 
 952 
 
 961 
 
 967 
 
 - A (Exp.) 
 
 640 
 
 769 
 
 826 
 
 897 
 
 928 
 
 940 
 
 948 
 
 955 
 
 962 
 
 Difference 
 
 -022 
 
 -020 
 
 -004 
 
 - -006 
 
 -005 
 
 + 001 
 
 + 004 
 
 + 006 
 
 + 005 
 
74 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 Fig. 21 shows the march of A graphically, A being expressed 
 throughout as a fraction of the half wave length. 
 
 1 
 
 8 
 
 fi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 f > 
 
 ! ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 4 
 2 
 
 A 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 </_ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L _^ - ^ J 
 
 ~~~^* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60 62 64 66 68 70 72 74 
 *<*> 
 
 Fig. 21. 
 
 If we are dealing with light polarised at right angles to the 
 plane of incidence we have r = Re if> = x + iy, say, and in expressing 
 the various terms of these formulae it will be convenient to 
 introduce the coefficient of ellipticity ej. Thus we have 
 
 t/an p 
 
 sec 
 
 (tan 2 (/> - /x 2 
 
 The values of p calculated for diamond from this formula are given 
 in the following table. 
 
 0... 
 
 60 
 
 61 
 
 62 
 
 63 
 
 64 
 
 65 
 
 66 30' 
 
 67 
 
 67 30' 
 
 -p 
 
 041 
 
 046 
 
 053 
 
 062 
 
 077 
 
 103 
 
 203 
 
 288 
 
 436 
 
 *... 
 
 68 
 
 68 30' 
 
 69 
 
 70 
 
 71 
 
 72 
 
 73 
 
 74 
 
 75 
 
 -p 
 
 624 
 
 754 
 
 828 
 
 897 
 
 928 
 
 946 
 
 957 ! -965 
 
 1 
 
 971 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 75 
 
 Also we have very approximately 
 tan (<- 
 
 X 
 
 tan 
 
 and 
 
 (the r of Fresnel's formula), 
 2 (^ _ i) 6l 
 
 + l) f [COS </> + COS 
 
 The denominator in the formula for y will vary very little 
 throughout a considerable range on each side of Brewster's angle, 
 so that throughout this range y will be very nearly a constant. 
 The following are the values of x and y for diamond. 
 
 <t> 
 
 X 
 
 y 
 
 
 
 X 
 
 y 
 
 60 
 
 - -1314 
 
 0168 
 
 68 
 
 0071 
 
 0173 
 
 61 
 
 - -1162 
 
 0169 
 
 69 
 
 0287 
 
 0173 
 
 62 
 
 - -1016 
 
 0170 
 
 70 
 
 0514 
 
 0173 
 
 63 
 
 - -0858 
 
 0170 
 
 71 
 
 0754 
 
 0172 
 
 64 
 
 - -0690 
 
 0171 
 
 72 
 
 1007 
 
 0172 
 
 65 
 
 - -0513 
 
 0172 
 
 73 
 
 1247 
 
 0170 
 
 66 
 
 - -0328 
 
 0173 
 
 74 
 
 1525 
 
 0170 
 
 67 
 
 - -0136 
 
 0173 
 
 
 
 
 Since y is practically constant throughout this range, if we 
 draw a graph to represent the amplitude and phase of the dis- 
 placement in the reflected light, we get very approximately a 
 straight line parallel to the axis of x, and all the points of departure 
 from Fresnel's formulae due to the layer are indicated, as regards 
 both amplitude and phase, by the slight shifting of a straight line 
 from the axis of x to a parallel position. This is illustrated in 
 Fig. 22, where OP represents the amplitude of the displacement 
 in the reflected wave, and the angle POx represents the change of 
 phase produced by reflection. 
 
 The formulae already obtained enable us to estimate the 
 influence of the surface layer of transition on the reflecting power 
 of the substance. This power is measured, as we have seen in the 
 
76 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 last chapter, by (R 2 + R 2 )/2, and as the formulae show that R 2 
 and R' 2 involve df, but not d 1} it follows that if the layer be so thin 
 
 01 
 
 12 10 --08 --06 --04 --02 Q -02 -04 -06 -08 -10 -12 
 
 ^0 61 63 65 67 69 71 
 
 Fig. 22. 
 
 that the square of d^ can be neglected, it has no influence on the 
 reflecting power. If, then, we are considering experiments such 
 as those referred to on p. 53, where there is a decided departure 
 from Fresnel's formulae, it will be necessary to include terms 
 containing d- 2 in our approximations. This has been done above, 
 and the formulae for R and R there obtained may be written 
 thus: 
 
 p fJL COS (f) COS <j)' f %fJ?d 2 COS <j) COS (f)' 
 
 ~~ /A COS <j) + COS <' [ (/-1 2 COS 2 < COS 2 </>') 2 
 
 i 2 cos 2 </> cos 2 (/>' + (1 Fjfj 2 . sin 2 
 
 - F/JJ? . sin 2 <) (/u, 2 cos 2 < + cos 2 c/>') 
 
 - 2 J 
 
 ( + (p? cos 2 $ - cos 2 <t>')(E-2H+(J- L)/fj? . sin 2 </>) 
 
 a cos 6 cos 6' f 2 cos 6 cos </>' 7 . . . 1 
 
 = -7; , 1 + , TT^ T\ /V) (a + 6 sm 2 (f> + c sin 4 *) , 
 
 /A cos </> + cos $ L (/^ 2 cos 2 ^ - cos 2 ^ r y J 
 
 where 
 
 -i)j|], 
 
 and 
 
 -i/M 2 )- 
 
 c = fjbcii' I ^" + F 2 //^ EF (1 + I//-*- 4 ) (/ Z) (1 l//^ 4 )], 
 
 COS (f> COS (f)' 
 
 u cos 6' cos 6 F 
 = ...T/ . r5 ** 
 
 D 
 
 yLt COS ^ + COS (/) I (ft. 2 COS 2 <f>' COS 2 <^)) 2 
 
 /Lt 2 cos 2 d> cos 2 d> -|- {Eu? sin 2 rf>) 2 
 
 j + a 2 (^ 2/Z) (/u, 2 cos 2 6' cos 2 6) 
 
 ( - 
 
 /u- cos <// cos <^> r 2a cos cos ^> r 
 
 p COS </)' + COS (f) [ ^ (/Lt 2 COS 2 ^-COS 2 </)) 2 J ' 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 77 
 
 Thus, as far as the influence of the layer is concerned, the intensity 
 of the reflected beam can be expressed in terms of three constants 
 a, b, c, whose values depend on the thickness of the layer and the 
 law of distribution of the refractive index within it. 
 
 Putting R=R l -\- p l and R = R{ -f p/, where R^ and R^ have 
 the values given by Fresnel's formulae, we see that the addition 
 to the reflecting power due to the layer is R l p l + R^p\. This 
 addition is obtained from the above formulae in the form 
 
 2 cos (f> cos </>' [a A 4- B sin 2 $ (6 + c sin 2 $)], 
 where A = l/(/-t cos < -f cos <') 4 + !/(//- cos $ + cos $) 4 , 
 and B = l/(/j, cos </> + cos </>') 4 . 
 
 At normal incidence this quantity is positive or negative, according 
 as a is positive or negative, so that the layer increases or diminishes 
 the reflecting power at normal incidence, according as H is less or 
 greater than 
 
 The values of the constants a, b, c could be determined from the 
 reflecting powers at three different angles of incidence. Their 
 values could also be calculated in terms of di by means of the 
 above formulae, for any given law representing the refractive index 
 at different points within the layer. Thus in considering Conroy's 
 experiments on reflection from glass we shall suppose that the 
 refractive index within the layer is given by the formula 
 
 and take p = l/fj, 4 I so as to preserve continuity when x l =l. 
 The value of fj, found by experiment was T5145, and taking 
 c?! = 0-481 we get 
 
 a = --041, b = '06642, c = - '08079. 
 
 The following table gives the intensity of the reflected light, for 
 different angles of incidence, calculated from the above formulae, 
 with these values of the constants, and compares them with the 
 results of Conroy's experiments. 
 
 Fig. 23 represents these results graphically. 
 
78 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH 
 
 0... 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 65 
 
 70 
 
 Intensity 
 (Theory) 
 
 (Exp.) 
 
 0378 
 0378 
 
 0378 
 0378 
 
 0380 
 0377 
 
 0392 
 0392 
 
 0439 
 0437 
 
 0537 
 0553 
 
 0831 
 0854 
 
 1128 
 1116 
 
 1612 
 1604 
 
 Difference 
 
 
 
 
 
 + -0003 
 
 
 
 + -0002 
 
 - -0016 
 
 - -0023 
 
 + -0012 
 
 + -0008 
 
 14 
 12 
 10 
 08 
 
 Hfi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 O4 
 
 
 
 
 
 
 
 
 , 
 
 7 
 
 
 
 
 
 
 
 
 
 
 >^ 
 
 ^ 
 
 
 
 
 
 02 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 20 30 40 50 60 70 
 
 Fig. 23. 
 
IV] AND REFKACTION IN TRANSPARENT ISOTROPIC MEDIA 79 
 
 We have seen that one effect of a surface layer of transition is 
 that there is no longer any angle of incidence, at which the com- 
 ponent of the displacement at right angles to the plane of incidence 
 is absent from the reflected beam. Thus the reflected light can 
 never be plane polarised. However, as the coefficient of ellipticity 
 is always very small, the departure from plane polarisation will be 
 slight, and it may be convenient for some purposes to speak of the 
 polarising angle as the angle of incidence, for which the component 
 of the displacement at right angles to the plane of incidence is 
 least in the reflected light. The position of this angle is easily 
 obtained from our formulae. We have 
 
 _ n cos <> cos < + pi LI cos cos < + /* . sn< 
 ~ (LI, cos $ + cos (/>') + ifjidi (ELI cos < cos <' + 1 F//J? . sin 2 <) ' 
 
 At Brewster's angle LI cos $ cos <' = 0, so that in its neighbour- 
 hood we have R 2 = a? + y 2 , where 
 
 _ /JL COS (f> COS (/>' 
 fJb COS (j> + COS </>' ' 
 
 7 (flu, cos d> cos <t> 1 + F/LL 2 . sin 2 d>) 
 and y = LL^ /, . 
 
 yLfc COS <f) + COS (f) 
 
 The polarising angle, as here defined, occurs when R is a 
 minimum, i.e. when 
 
 Since y, being proportional to di, is of the first order of small 
 quantities, ydyjdfy is of the second order. Hence if we neglect df 
 we have X = 0, i.e. JJL cos cos 0' = 0, which corresponds to 
 Brewster's angle a = tan" 1 LI. To this order, then, the layer does 
 not alter the position of the polarising angle. Proceeding to a 
 higher order of approximation R will be a minimum when < = a ty, 
 where -v/r is a small angle to be determined. Since 
 
 approximately, the equation to determine o/r may be written 
 (as - 
 
80 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 where (f> is to be put equal to a after differentiating. Since x 
 when < = a, this equation gives 
 
 
 [(1 + /i 2 ) (1 + yLt 4 ) - F (1 + V + At 4 ) + > 2 (1 + A* 4 )], 
 
 where ty is expressed in circular measure, and represents the 
 diminution of the polarising angle produced by the layer. With 
 the law giving the refractive index within the layer that was 
 discussed above, the diminution amounts to 10' 12", and the 
 polarising angle becomes 56 23' 38". The mean of the results 
 obtained experimentally by Conroy was 56 23' 30", so that here, 
 as elsewhere, there is a very close agreement between theory and 
 observation. 
 
 The methods, hitherto used in discussing the influence of the 
 layer of transition, may be applied equally well when dealing with 
 the problem of total reflection. Thus for light polarised at right 
 angles to the plane of incidence we have f = = 77. In the first 
 and second media f is given by the formulae of p. 56, and in the 
 layer by the formula %=m ip(t ~ vyic} ) when u has the same value as 
 on p. 65. Neglecting squares of d l the boundary conditions give 
 A = 1 r, Bfj? = id 1 /c(\ -f r), 
 
 s = A 4- BE, inps = B/d, + d l A (v 2 F I), 
 
 where K = /j, cos $. 
 
 Putting n = i Vsin 2 < 1/yu, 2 = ip, 
 
 and eliminating A and B, we get 
 
 s + r (1 + id^KEjfjf) = 1 id, /cE//j?, 
 and p^s - r [IK//JI? - d l (1 - Fv 2 )] = %K\^ + d : (1 - Fv*). 
 
 From these r and s can be determined. For reflection we are 
 interested only in r, which is given by the formula 
 
 (%* =z^ 
 
 where 
 
 tan a x lt 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 81 
 
 p^" fjb 2 v'sm^ - 
 where tan a = f^- = ~ 
 
 K cos (f> 
 
 so that a is the same angle as that used in dealing with the case 
 of an abrupt transition, and 
 
 x l = t j.d l {l-E-(F-E) ^ 2 }/cos (f>. 
 
 Similarly, for light polarised parallel to the plane of incidence, 
 we have f ' in the first and second media given by the formulae 
 of p. 57, and in the layer by f = ue ip(t ~ vy/c) , where u has the same 
 meaning as on p. 69. The boundary conditions give 
 
 A = I + r', B = - id lf c (1 - r), 
 
 s' = A 4- B, - in^s = B/d l + d,A (z/ 2 - E). 
 
 Eliminating A and B, we get 
 
 s r (1 + id l /c) = 1 idi K, 
 
 and p l fjis' + r {i/c - d l (E - v*) } = iic + d l (E - i/ 2 ), 
 
 whe / _ - 
 - 
 
 = (tan a + ^ 2 ), where ^ = ^(1 - E)/cos </>. 
 
 The difference of phase between the components of the dis- 
 placement parallel and perpendicular to the plane of incidence is 
 
 A 7 = TT - 2 (' - a"), 
 so that A x is given by the formula 
 
 A A' tan 
 
 cot = tan (a - a' ) = 
 
 1) ^ sec <j> - fa + a- 2 //i 2 ) 
 
 sec 
 
 sin tan $ (^ 1 ) sin </> tan <j> 
 
 sin <^> tan <^> ' 
 M. L. 
 
82 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 M 
 
 where .V 
 
 Here e : is the coefficient of ellipticity as formerly defined, and ^ is 
 positive if // is less than yu,. 
 
 The following table gives the values of A' calculated for 
 different angles of incidence from the above formula, and compares 
 them with the results of experiment. 
 
 The refractive index is /uu= 1'619, the critical angle 38 9', and 
 the coefficient of ellipticity e x = "0250. 
 
 0... 
 
 38 13' 
 
 39 58' 
 
 41 59' 
 
 44 2' 
 
 46 4' 
 
 47 54' 
 
 49 58' 
 
 51 57' 
 
 53 58' 
 
 5557' 
 
 A' (Theory) 
 
 975 
 
 813 
 
 758 
 
 728 
 
 716 
 
 712 
 
 712 
 
 719 
 
 727 
 
 736 
 
 A' (Exp.) 
 
 978 
 
 813 
 
 754 
 
 728 
 
 718 
 
 712 
 
 715 
 
 719 
 
 726 
 
 730 
 
 Difference 
 
 -003 
 
 
 
 + 004 
 
 
 
 -002 
 
 
 
 -003 
 
 
 
 + 001 
 
 + 006 
 
 These results are shown in Fig. 24. 
 
 38 s 4(T 42 44 46 s 4lT 50 52 54 56 
 
 Fig. 24. 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 83 
 
 Having seen that the deductions from theory agree thoroughly 
 with the results of careful experiment, wherever a comparison is 
 possible, we feel that we are on solid ground when we investigate 
 the details of the polarisation of the reflected light, even although 
 our results cannot be put directly to the test of experiment. Let 
 us suppose, then, that the incident light is polarised in a plane 
 making an angle 6 with the plane of incidence, and that the dis- 
 placement is of unit amplitude. This displacement may be 
 resolved into two components 
 
 ! = cos 6 cos pt, and ^ = sin 9 cos pt, 
 
 parallel and perpendicular respectively to the plane of incidence. 
 After reflection the components will be given by the formulae 
 
 f ! = R cos (pt 4- p), and ^ = R cos (pt + p), 
 
 where R, R', p, p have the values assigned to them earlier in this 
 chapter. We have seen (p. 52) that lags in phase behind ^ by 
 an amount A = p p', so that in cases of positive reflection for 
 which A is negative, the component of the displacement perpen- 
 dicular to the plane of incidence lags behind the other component. 
 On eliminating t from the equations for ^ and ^ , we get 
 
 tf/R'* cos 2 6 + r)i*/R 2 sin 2 - 2^, cos &/RR' sin cos 6 = sin 2 A. 
 
 Thus the reflected beam is, in general, elliptically polarised, the 
 elements of the elliptical orbit being determined by means of the 
 above equation. 
 
 Putting tan a. = e tan 0, where e = R/R, this equation becomes 
 
 f T 2 tan 2 a. + V - 2&i tan a . cos A = R 2 sin 2 A sin 2 6. 
 The sense in which this orbit is described is determined, by 
 
 considering the signs of fj, t] l , ~, and ~. We shall take the 
 
 ct vt 
 
 case of positive reflection and look along the ray, whether incident 
 or reflected, in the direction of the propagation of the light. If 
 be positive, we have to turn from the plane of incidence counter- 
 clockwise through an angle to get to the direction of the dis- 
 placement in the incident wave, and in this case the orbital motion 
 in the reflected wave is clockwise. 
 
 The orbit is represented by the ellipse in Fig. 25, in which OE 
 and OF represent the amplitudes of the displacements parallel and 
 
 62 
 
84 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 perpendicular to the plane of incidence, so that OE = R' cos 6 and 
 
 F, H 
 
 Fig. 25. 
 
 OF= R sin 0. Let OA and OB be the semi-axes (a and b) of the 
 ellipse, AOE = ^, HO A = %, HOE = a, so that 
 
 tan a = OF JOE = e tan 0. 
 
 The azimuth ^r of the major axis of the ellipse is given by the 
 equation 
 
 2h' 2 tan a . cos A 
 
 tan 2\1/ = r , = - = tan 2a . cos A, 
 
 a b 1 - tan 2 a 
 
 and the eccentricity e by the equation 
 
 e 4 /(2 - e 2 ) 2 = 1-4 (a'b' - h'*)/(a + bj = 1 - sin 2 2ct . sin 2 A. 
 Putting sin 27 = sin 2a . sin A, we have e = sec 7 Vcos 27, 
 
 a = R sin cos 7 cosec or, b = a tan 7. 
 
 We shall consider first the variation of ^ for different inci- 
 dences (0) and azimuths (6). At normal incidence 6 = 1 and 
 A = 0, so that ^r a = 6. As the angle of incidence increases 
 fy diminishes, and as a rule continues to do so until it vanishes at 
 the Principal Incidence, where cos A = 0. After this ^r becomes 
 negative, and continues to diminish algebraically until the incidence 
 is grazing, when ^ = 0. For some azimuths, however, ^ does 
 not vanish at the Principal Incidence, but is 90 there. At this 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 85 
 
 incidence we have e = e a = cot ft, so that if 6 be greater than ft, we 
 have tan a greater than unity, and therefore 2a greater than 90. 
 Hence, from the formula tan 2-v/r = tan 2a . cos A, we see that just 
 below the Principal Incidence tan 2-^r is negative, and -v/r is rather 
 less than 90, while just beyond the Principal Incidence tan 2>/r is 
 positive, and -v/r is rather greater than 90. It appears, then, that 
 if 6 be greater than ft the curve representing t|r in the neighbour- 
 hood of the Principal Incidence takes the form represented in 
 Fig. 26. 
 
 ,9Cf 
 
 w \ 
 
 t ! 
 
 B'l 
 
 -90 
 
 Fig. 26. 
 
 Thus at the Principal Incidence in passing through the 
 azimuth = ft, the angle i/r changes suddenly from to 90. 
 This means, of course, that the longer axis of the ellipse is the 
 one at right angles to the plane of incidence, instead of being in 
 that plane as before. There is, however, no real discontinuity 
 at 6 ft, for at that azimuth the ellipse is a circle, as we shall find 
 immediately. 
 
 If there were no layer the reflected light would be plane 
 polarised in an azimuth & given by tan 6 f e tan 6, the upper 
 or lower sign being taken, according as the angle of incidence is 
 less or greater than the Principal Incidence. Except near the 
 Principal Incidence the value of e is the same whether there is a 
 layer or not, so that tan & = + tan a, or & = ot. Hence OH in 
 Fig. 25 represents the direction of the displacement when the 
 transition is abrupt, and the angle %= ^ represents the turning 
 towards the plane of incidence in going from the azimuth of plane 
 polarisation, when there is no layer, to the major axis of the 
 elliptical orbit when the layer is present. It is obvious from the 
 figure, or the formulae, that % is positive if a < 45, and negative 
 if a > 45. We have a > 45 when e tan 6 > 1, and this will be the 
 case at low and high incidences, when e is nearly unity, for 
 
86 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 azimuths greater than 45, and for all incidences with azimuths 
 greater than H. In the neighbourhood of the Principal Incidence 
 e is not the same for an abrupt transition as when there is a 
 transition layer, so that 6' is no longer equal to + a. When there 
 is no layer we have e = at the Principal Incidence, and 6' = 0, so 
 that % = 6'~ty = if 6 < H, and ^ = 90 if 6 is equal to or greater 
 than H. 
 
 The table below gives the values of ^ for different incidences 
 and azimuths for reflection from diamond and realgar. 
 
 These substances have nearly equal refractive indices, but the 
 coefficients of ellipticity found experimentally by Jaurin were very 
 different, so that a comparison of the two cases indicates the effect 
 of varying the surface layer. For diamond we have //,= 2'434, 
 ej = '0241, so that Brewster's angle is- 67 40', and H = 88 37'. For 
 realgar we have //, = 2'420, e^'ll 20, so that Brewster's angle is 
 67 33', and 11 = 83 37'. In the table with the various azimuths 
 (0), the first row gives the values of % for diamond and the second 
 for realgar. 
 
 e 
 
 = 10 
 
 30 
 
 60 
 
 65 
 
 67 30' 
 
 67 40' 
 
 68 
 
 70 
 
 75 
 
 80 
 
 10 
 
 
 
 
 
 
 
 0!' 
 14 
 
 02' 
 00 
 
 00' 
 00 
 
 00' 
 00 
 
 00' 
 00 
 
 02' 
 00 
 
 or 
 
 012 
 
 00' 
 06 
 
 30 
 
 
 
 
 
 
 
 02 
 45 
 
 07 
 02 
 
 01 
 01 
 
 00 
 01 
 
 01 
 01 
 
 08 
 03 
 
 02 
 039 
 
 01 
 016 
 
 45 
 
 
 
 
 
 
 
 06 
 1 12 
 
 11 
 
 08 
 
 02 
 01 
 
 00 
 00 
 
 01 
 01 
 
 014 
 07 
 
 04 
 11 
 
 01 
 019 
 
 60 
 
 
 
 
 
 
 
 05 
 1 40 
 
 10 
 25 
 
 02 
 01 
 
 00 
 
 02 
 
 03 
 04 
 
 023 
 023 
 
 05 
 1 18 
 
 01 
 010 
 
 75 
 
 
 
 
 
 
 
 02 
 058 
 
 035 
 23 
 
 06 
 05 
 
 00 
 Oil 
 
 05 
 038 
 
 15 
 112 
 
 01 
 020 
 
 01 
 -0 13 
 
 85 
 
 
 
 
 
 
 
 05 
 -145 
 
 12 
 - 25 17 
 
 034 
 
 -8747 
 
 00 
 
 -8426 
 
 015 
 -6934 
 
 023 
 
 - 24 48 
 
 08 
 -116 
 
 01 
 -013 
 
 89 
 
 
 
 
 
 
 
 -0 1 
 -033 
 
 -031 
 -936 
 
 -5826 
 -8550 
 
 -900 
 -791 
 
 - 39 36 
 -5142 
 
 -042 
 -924 
 
 -01 
 -022 
 
 00 
 -03 
 
 The march of the quantity b/a is easily followed by means of 
 the relations b/a = tan 7, and sin 2y = sin A . sin 2 z. For a given 
 incidence sin 2y varies as sin 2a, so that 7 is greatest when a = 45, 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 87 
 
 and its greatest value is A/2 below the Principal Incidence, and 
 (TT - A)/2 beyond that. Thus as the azimuth (9) varies, b/a is a 
 maximum when tan 0= 1/e, and its greatest value is tan A/ 2 below 
 the Principal Incidence, and cot A/2 beyond that. At the Principal 
 Incidence itself A = Tr/2, so that the greatest value of b/a is 1, and 
 this occurs when tan = l/e a , i.e. when 0=H. Thus at the Principal 
 Incidence, when 6 = H the reflected light is circularly polarised. 
 The following table gives the maxima for different incidences and 
 their corresponding azimuths. 
 
 
 
 10 
 
 30 
 
 60 
 
 65 
 
 67 30' 
 
 67 40' 
 
 68 
 
 70 
 
 75 
 
 80 
 
 6/a (diamond) 
 
 0006 
 
 0057 
 
 0515 
 
 1521 
 
 8016 
 
 1 
 
 6822 
 
 1727 
 
 0515 
 
 0233 
 
 6 (diamond) 
 
 45 42' 
 
 5155' 
 
 78 24' 
 
 85 43' 
 
 88 35' 
 
 88 37' 
 
 88 30' 
 
 85 
 
 7647' 
 
 66 43' 
 
 bja (realgar) 
 
 0026 
 
 0262 
 
 2469 
 
 5441 
 
 9879 
 
 9702 
 
 8889 
 
 5403 
 
 2132 
 
 1051 
 
 6 (realgar) 
 
 45 44' 
 
 5156' 
 
 78 30' 
 
 82 21' 
 
 83 37' 
 
 83 35' 
 
 83 30' 
 
 82 18' 
 
 76 40' 
 
 6635' 
 
 Figs. 27 and 28 represent these results graphically, giving the 
 
 90 
 85 
 
 75 
 65 
 55 
 45 
 
 6 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 diamo 
 
 \ 
 
 id 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 'realgi 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 7 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 > 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 ^- 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 E 
 
 1(T 20 3tf 40 50 60 70 8CP 9CT 
 ~~V 
 Fig. 27. 
 
88 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 plane and elevation of the ridge of maxima. Fig. 27 shows the 
 values of where the maxima occur, and Fig. 28 the values of 
 these maxima. 
 
 Fig. 28. 
 
 In seeking for the azimuth (6) that makes b a maximum at 
 any given angle of incidence, we take the equation 
 
 b 2 = a 2 tan 2 7 = (E 2 sin A/2e) sin 20 tan 7 = K sin 20 tan 7. 
 
 This gives 
 
 2bdb/K = 2 cos 20 tan. 7^0 + sin 20 sec 2 7^7 
 
 = sin 27 sec 2 ydO [cos 20 + cos 2a sec 27]. 
 
 Hence b is a maximum when cos 20 = cos 2a sec 27. For 
 brevity we shall put tan 2 a = x, sin 2 A = s, e 2 = c, and we then have 
 to solve the equation 
 
 
 Squaring, and clearing of fractions, and noting that the process of 
 squaring may introduce an irrelevant root, we get 
 x> (1 - c - s) + 2scx - c + c 2 (1 - s) = 0, 
 so (l-c)Vc(l -s) 
 
 whence 
 
 I c s 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 89 
 
 To exclude the irrelevant root we must make the sign of 
 
 negative, and as (c + #)/(! + x) is necessarily positive, this requires 
 (1 x)l(c x) to be negative. Hence we get 
 
 (1 - x)j(c x) = + cos A/e, 
 
 taking the upper sign below the Principal Incidence and the 
 lower sign beyond it. Thus we have 
 
 tan 2 6 = (1 + e cos A)/e (e cos A), 
 
 which gives real values of 0, whether the incidence is below or 
 above the Principal Incidence. The maximum value of b corre- 
 sponding to this azimuth is 
 
 R sin A/Vl 26 cos A + e 2 . 
 
 The following table gives the magnitude and position of the 
 maximum value of b for various incidences. 
 
 <t> 
 
 
 
 10 
 
 30 
 
 60 
 
 65 
 
 67 30' 
 
 6 (diamond) 
 
 45 
 
 45 22' 
 
 48 29' 
 
 65 40' 
 
 75 
 
 85 33' 
 
 b (diamond) 
 
 
 
 0002 
 
 0023 
 
 0112 
 
 0143 
 
 0170 
 
 6 (realgar) 
 
 45 
 
 45 22' 
 
 48 30' 
 
 66 19' 
 
 73 46' 
 
 83 17' 
 
 b (realgar) 
 
 
 
 0011 
 
 0107 
 
 0521 
 
 0714 
 
 0786 
 
 <i> 
 
 67 40' 
 
 68 
 
 70 
 
 75 
 
 80 
 
 90 
 
 6 (diamond) 
 
 88 37' 
 
 84 14' 
 
 75 34' 
 
 64 11' 
 
 56 45' 
 
 45 
 
 b (diamond) 
 
 0172 
 
 0172 
 
 0161 
 
 0155 
 
 0119 
 
 
 
 6 (realgar) 
 
 .82 48' 
 
 80 51' 
 
 73 40' 
 
 64 40' 
 
 56 47' 
 
 45 
 
 b (realgar) 
 
 0785 
 
 0783 
 
 0769 
 
 0625 
 
 0539 
 
 
 
 These results are represented in the figures below. Fig. 29 
 gives the values of the azimuths, and Fig. 30 the corresponding 
 
 maxima. 
 
90 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 A simple geometrical construction for tan 2 6, and for the corre- 
 sponding maximum value of 6, is easily obtained, see Fig. 31. Take 
 
 I, AC = e, 4(7 = 7r-A or A, 
 
 90 
 85 
 
 75 
 65 
 
 55 
 45 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 dinmo 
 
 nd 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 yrealff 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 /\ 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 S 
 
 
 
 
 
 
 \ 
 
 
 - 
 
 . 
 
 . 
 
 
 
 ^-^ 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 ^ 1(T 2CT 3CT 40 50 60 70 80 90 
 
 Fig. 29. 
 
 according as A is less or greater than ir/2. Draw AD perpen- 
 dicular to EG, and take P on AD such that AP = R f x AD. Then 
 b = AP, and tan 2 d = BD/CD. For 
 
 sin A/Vl 2e cos A + e 2 = BE/BC = sin C, 
 and R = eR' = R f . AC. Hence 
 
 6 = Esin A/Vl 2ecos & + e 2 = R . AC sinC= R' . AD. 
 Also tan 2 = (1 e cos A)/e (e cos A) = BF/AC . CE, 
 and BF = BF.AB = BC.BD, 
 
 while AC.CE=CD.CB- 
 
 hence tatf6 
 
 The changes in the magnitudes of 6 and 6 are easily discussed by 
 means of this construction. As A increases from to Tr/2, BC/CD 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 91 
 
 increases from unity. It afterwards diminishes to unity. Thus 
 increases from 45 to H, and then diminishes to 45. Moreover, as 
 
 08 
 
 07 
 
 06 
 
 
 05 
 
 04 
 
 03 
 
 02 
 
 01 
 
 40 
 
 50 
 
 60 
 
 Fig. 30. 
 
 70* 
 
 80' 
 
 9(f 
 
 A increases from to ?r/2, AD increases from to e^x/1 +e-f, and 
 R increases, so that b increases in this range from to Rj/^l + e^, 
 where R t is the value of R at the Principal Incidence. After this 
 
92 INFLUENCE OF A LAYER OF TRANSITION ON REFLECTION [CH. 
 
 AD diminishes to zero, while R' increases to unity, so that b 
 diminishes to zero. In the immediate neighbourhood of the 
 
 Principal Incidence we have R sin A = jR 1} so that in this region 
 b = RijBC. As the incidence increases beyond the Principal 
 Incidence, BC increases so that b diminishes. Thus the maximum 
 value of b is found at the Principal Incidence, and its value there 
 
 s 
 
 10 
 
 
 
 45 60 
 
 Fig. 32. 
 
 75 
 
 87 4Q 
 
IV] AND REFRACTION IN TRANSPARENT ISOTROPIC MEDIA 93 
 
 We have already observed that at the Principal Incidence the 
 major axis of the elliptical orbit is in the plane of incidence for 
 azimuths less than XI, and at right angles to this for greater 
 azimuths. The lengths of the axes at this incidence are most 
 simply calculated by remembering that the amplitudes of the 
 displacements parallel and perpendicular to the plane of incidence 
 are always R cos 6 and R sin 0. Hence the lengths of the semi- 
 axes in these two directions are RJ cos and R l sin 6 respectively. 
 Fig. 32 above and Fig. 33 below represent half the elliptical orbit 
 for different azimuths at the Principal Incidence, Fig. 32 referring 
 to diamond, and Fig. 33 to realgar. 
 
 \ 
 
 30 
 
 45 
 
 60 
 Fig. 33. 
 
 75 
 
 873Q 
 
 90 
 
CHAPTER V. 
 
 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS. 
 
 THE results obtained in the last two chapters enable us to 
 investigate the nature of the reflected and transmitted beam, when 
 light is incident on a transparent isotropic film or plate, and so to 
 explain, among other things, the colours of soap bubbles, of thin 
 films of glass, or of a thin layer of oil on water. The phenomenon 
 of this character that has been most carefully investigated experi- 
 mentally is that of Newton's rings, and with this we shall deal 
 more particularly; but we shall begin by obtaining a formula 
 applicable to all such problems. 
 
 Suppose that we have a plate of thickness c lf and of different 
 refractive index from the media on each side of it. These media 
 need not be of the same refractive index, e.g. we may have a film 
 of soap between air and glass. A wave of unit amplitude repre- 
 
 
 <U fc 
 
 soap 
 
 qs,r a qV 2 r 3 cfs.rjr q%,^r s 
 
 _J t . J t. 
 
 Fig. 34. 
 
 sented by & (pt-to-^y-w t will, at the passage from air to soap, give 
 rise to a reflected wave r^ (pt ~ lx ~ my ~ nz) in the air, and a refracted 
 wave s^ ^-ix-my-nz) [ n ^ e soa p. Let the quantities corresponding 
 
CH. V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 95 
 
 to T! and s l be r 2 and s 2 at the passage from soap to glass, and 
 r 3 and s s at the passage from soap to air. Suppose, further, that 
 a wave of unit amplitude, represented by e 1 (pt ~ lx ~ m y~ nz} , becomes 
 q e i( P t-ix-my-nz) a f ter traversing a distance c, of the film or plate 
 (of soap). Waves will be reflected to and fro between the air and 
 glass, and those that return into the air will constitute a reflected 
 beam represented (omitting the factor e l (pt-te-my-vw ) by 
 
 n + q 2 Sir 2 s s (1 + gV 2 r 3 + # 4 r 2 2 r 3 2 + ... ad inf.) 
 - , provided | q 2 r 2 r 3 < 1. 
 
 Similarly the beam transmitted through the plate is repre- 
 sented by 
 
 qs,s 2 (1 + 2 r 2 r 3 + 4 r 2 2 r 3 2 +...)= g ^f 2 . 
 
 1 - q*r z r s 
 
 The values of the quantities r l} r 2 etc., are given in Chap, in., 
 for the ideal case of an abrupt transition, to which for the present 
 we shall confine ourselves. From the formulae there obtained it 
 appears that r^ + r s = 0, and s^ = 1 - r^ = 1 - r 3 2 , results that can 
 also be obtained by noting that when the first and third media are 
 the same, and the thickness of the film is indefinitely diminished, 
 r 3 = r 2) s s = s 2 , and q = 1, and that in this case all the light is trans- 
 mitted, and none reflected. Making use of these relations we obtain 
 for the reflected beam 
 
 .. g 2 , and for the transmitted beam ^ \/ 2 . 
 
 From these formulae the details both of the reflected and the 
 transmitted system can be readily discussed. As, however, the 
 media are transparent, there is no loss of light, so that the trans- 
 mitted system is complementary to the reflected. Hence the 
 investigation of the properties of one system involves those of the 
 other, brightness in one case being replaced by darkness in the 
 other. 
 
 If the incident light is of unit intensity, and is plane polarised 
 in an azimuth 6 with the plane of incidence, the intensity of the 
 reflected beam is 
 
 |2 
 
 7=sin 2 
 
 + COS 2 d 
 
96 
 
 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 where r and r' are given by Fresnel's formulae, and <f by the 
 relation cf e~^ = e~ ilfS , where K = 2ir/\ J and S = jjujfi . c^ cos </>'. 
 Here //, is the refractive index of the first medium, /// that of the 
 second, and X the wave length in the first. 
 Thus we have 
 
 cos 
 
 1 + (? Vjr* 1 + rjVa 2 + 2rjr 2 cos i/r ' 
 
 This is simply represented by the following geometrical construc- 
 tion. Take OA = r lt OB = r Z) AOB = 7r-^, OA' = 1, and draw 
 AH parallel to A'B. Then OB' = r l r^ and 
 
 B' 
 
 A O 
 
 Fig. 35. 
 
 From this construction, or by means of the calculus, we see that 
 the intensity / is stationary when ty = mr, there being alternations 
 of maxima and minima for different integral values of ??. When 
 ir = 2?tt7r we have 
 
 and when o/r = (2m + 1) TT we have 
 
 I = sin 2 (9 
 
 /j. T \ 2 
 
 M 1 2 \ 
 
 + cos 2 e 
 
 r, n 
 
 .i-nV 2 V ' 
 
 The difference between these two intensities is 
 
 Which of these stationary values of / is the bigger depends on the 
 signs of /*! and r a . If rfa be positive, so that r x and r 2 have the 
 same sign, then the maxima occur when ^r=2w7r, while if rjr. 2 
 be negative the maxima occur when i|r = (2m + I)TT. 
 
V] TRANSPARENT IStJTftSW&^PffATES. NEWTON'S RINGS 97 
 
 For a beam of parallel light for which <, and therefore </>', is 
 constant, ^ is constant, if the thickness of the film is everywhere 
 the same. Such a film would therefore appear uniformly bright 
 throughout its surface, the brightness depending on the thickness. 
 If, however, as is nearly always the case in practice, the film is not 
 uniformly thick, there will be alternations of brightness and dark- 
 ness over the surface depending on the differences of thickness at 
 various points. In the case of Newton's rings these alternations 
 are distributed round a centre, where the thickness of the film is 
 negligible. At the centre, where ^ = 0, we have -v/r = 0, so that 
 the centre corresponds to n = in the series of stationary values of 
 /. Hence the central spot corresponds to the lowest of this series. 
 and it is bright or dark, according as D is positive or negative. 
 
 If we are dealing with light polarised in the plane of incidence, 
 then 6 0, and the sign of D is the same as that of r^i\, i.e. 
 
 f sin ($ - $') sin ($' - </>") rr ,. . ... ., ,, . . 
 
 of - T-J- -^ . ^^7 JT~. This is positive, if <f> is mter- 
 sm (< + ) sm (0 4- $ ) 
 
 mediate between </> arid <", and negative otherwise. Hence the 
 central spot is bright, if the refractive index of the film lies between 
 those of the other media, but otherwise dark. In particular the 
 central spot is dark, if the first and third media are the same. 
 
 For light polarised at right angles to the plane of incidence we 
 have 6 90, and the sign of D is the same as that of r^, i.e. 
 
 tan (4> - <') tan (<f>' - <") T T , , . . . 
 
 of , ;^ . \,( . - ;^, , [.' . Let < = /! be the polarising angle 
 tan (c/> + </)) tan ((/> + (f> ) 
 
 at the first reflection, and < = / 2 that at the second. The first is 
 always real ; but the second may not be so. The greatest possible 
 angle of incidence at the second surface is <', where sin <' = //-/yu/, 
 so that tan <' = /A/ V// 2 - ft 2 . For the polarising angle at the second 
 surface we have tan <f>' = JJL" '///, so that for / 2 to be real we must 
 
 Let us suppose that yu/ is intermediate between //, and //'. 
 Then, if <f> lies between zero and /,, r^ is positive, and the rings 
 are bright centred; if </> lies between I I and / 2 , r^ is negative^ 
 and the rings are dark centred, while if <f> lies between I 2 and 90% 
 7\r 2 is positive, and the rings are once more bright centred. If, on 
 the other hand, // is not intermediate between jj, and /n", we must 
 interchange the words bright and dark in this description of the 
 M. L. 7 
 
98 
 
 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 centres of the rings. The transition from bright to dark centred 
 rings occurs when D 0, in which case there are no maxima and 
 minima, for the rings disappear. The equation D = gives us 
 
 belli " "~ /-m~~ IT\ 7^ ^T\ s^ / / ,\ % 
 
 which determines the azimuth at which the rings disappear. From 
 this we get 6 = 90 when r l = or r 2 = 0. The first corresponds to 
 < = /!, the polarising angle at the first surface, and the second to 
 (f) = J 2 which, as has been seen, may not be real. In general, for 6 to 
 be real, we must have r/r,/ of opposite sign to r^, and as these are 
 always of the same sign if < be less than l l or greater than / 2 , the 
 rings cannot disappear for incidences outside these limits. 
 
 The following table gives the values of 6 calculated from the 
 above formula for different incidences, and compares the results 
 with the observations of Brewster. The media considered are air, 
 soap, and diamond ; p = T475, / a = 55 52', //," = 2'44. 
 
 
 
 55 52' 
 
 60 
 
 65 
 
 67 43' 
 
 70 
 
 75 
 
 6 (theory) 
 
 90 
 
 74 10' 
 
 67 43' 
 
 64 51' 
 
 62 52' 
 
 59 13' 
 
 (experiment) 
 
 90 
 
 73 
 
 68 30' 
 
 66 20' 
 
 63 30' 
 
 59 15' 
 
 Difference 
 
 
 
 1 10' 
 
 47' 
 
 - 1 29' 
 
 - 38' 
 
 -02' 
 
 The differences are well within the limits of error of Brewster's 
 experiments. The rings disappear gradually, and not suddenly, so 
 that it is difficult to determine with great precision when the 
 disappearance takes place. 
 
 Whether the rings have bright or dark centres for any given 
 incidence and azimuth is made evident by representing 90 6 
 graphically for different incidences. 
 
 We have 
 
 tan 2 (90 - 0) = - 
 
 so that there are two equal and opposite values of 90 6 for each 
 value of c/>, and 90 # vanishes with r a and ?* 2 , i.e. when <f) = I l 
 or / 2 . The figure will take different forms, according as I 2 is real 
 
TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 
 
 99 
 
 or not. Fig. 36 corresponds to the case where 7 2 is real, and 
 Fig. 36 a to that in which 7 2 is not real. 
 
 Fig. 36. 
 
 Fig. 36 a. 
 
 The regions are marked ' Bright centres ' or ' Dark centres ' for 
 the case, where // is intermediate between /JL and // '. For any 
 other case we must interchange the words Bright and Dark. For 
 azimuths and incidences corresponding to points on the curves in 
 these figures, the rings disappear. 
 
 The apparent brightness of the rings is proportional to the 
 difference of the maxima and minima intensities, i.e. to D. The 
 following table gives the values of D for different incidences and 
 azimuths, for the case of air-soap-diamond already mentioned. 
 
 
 
 8 = 
 
 30 
 
 45 
 
 60 
 
 75 
 
 90 
 
 
 
 172 
 
 172 
 
 172 
 
 172 
 
 172 
 
 172 
 
 30 
 
 218 
 
 195 
 
 173 
 
 151 
 
 134 
 
 128 
 
 55 52' 
 
 364 
 
 273 
 
 182 
 
 091 
 
 024 
 
 
 
 60 
 
 399 
 
 291 
 
 183 
 
 076 
 
 -003 
 
 -032 
 
 65 
 
 441 
 
 312 
 
 182 
 
 053 
 
 -041 
 
 -076 
 
 70 
 
 475 
 
 325 
 
 175 
 
 025 
 
 -085 
 
 -125 
 
 75 
 
 496 
 
 328 
 
 160 
 
 -008 
 
 -131 
 
 -176 
 
 90 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 37 represents these results graphically. 
 
 The rings are bright centred or dark centred, according as D is 
 positive or negative. The table, or the figure, shows clearly how 
 the intensity varies, and how the change from bright to dark 
 centres takes place. The shaded portion of the figure indicates 
 the region in which D is negative, and the rings dark centred. 
 
 72 
 
100 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 For azimuths less than 45 the distinctness of the rings increases 
 with the angle of incidence as far as /^but it diminishes for greater 
 
 90" 
 
 azimuths. When the angle of incidence lies between / : and 90, 
 the brightness increases for azimuths less than 45, passes through 
 a maximum, and then diminishes rapidly to zero ; but for greater 
 azimuths the brightness diminishes until the centres become dark, 
 then passes to a minimum, and increases to zero at grazing inci- 
 dence. Putting D in the form 
 
 D = D l sin 2 + A cos 2 0, 
 
 we see that for a given incidence the brightness continually 
 diminishes as the azimuth increases. The brightest centred ring 
 is found when 6 = 0, and the darkest centred one when 6 = 90. 
 From Fig. 37 we find the magnitude and position of the maxima 
 and minima of brightness for the different azimuths represented 
 are given as follows : 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 101 
 
 e o 
 
 30 
 
 4o 
 
 60 
 
 75 90 
 
 Maximum : D 
 
 497 
 
 328 
 
 183 
 
 -045 
 
 -154 
 
 -190 
 
 <t> 
 
 76 
 
 75 
 
 66 
 
 82 
 
 79 
 
 78 
 
 As an example of the same problem when / 2 is real, we shall 
 take the case when the three media are air, water, and fluor-spar. 
 For this we have 
 
 / = T336, /*" = 1-437, I^SFU', I 2 = 784'. 
 
 The azimuths at which the rings disappear for different 
 incidences are obtained from the formula of p. 98, which yields 
 the following results. 
 
 
 
 e 
 
 ii 
 
 90 
 
 55 
 89 2' 
 
 60 
 
 87 18' 
 
 65 
 86 51' 
 
 70 
 87 29' 
 
 75 
 88 52' 
 
 J 2 
 90 
 
 The brightness for different incidences and azimuths is set out 
 in the following table of D, and in Fig. 38 below. 
 
 
 
 = 
 
 30 
 
 45 
 
 60 
 
 75 
 
 90 
 
 
 
 018 
 
 018 
 
 018 
 
 018 
 
 018 
 
 018 
 
 30 
 
 029 
 
 025 
 
 021 
 
 018 
 
 015 
 
 014 
 
 55 
 
 060 
 
 045 
 
 031 
 
 014 
 
 003 
 
 -001 
 
 60 
 
 074 
 
 055 
 
 039 
 
 016 
 
 002 
 
 -004 
 
 65 
 
 085 
 
 063 
 
 045 
 
 018 
 
 001 
 
 -005 
 
 70 
 
 098 
 
 073 
 
 051 
 
 021 
 
 003 
 
 -004 
 
 75 
 
 109 
 
 081 
 
 055 
 
 026 
 
 005 
 
 -002 
 
 80 
 
 111 
 
 083 
 
 056 
 
 029 
 
 009 
 
 + '002 
 
 85 
 
 080 
 
 061 
 
 042 
 
 023 
 
 009 
 
 + 004 
 
 90 
 
 
 
 
 
 
 
 
 
 
 
 
 
 All the noteworthy features of the phenomena are made 
 evident by the figure. The rings pass from bright centred to 
 
102 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 dark centred at / 1? and change again at / 2 . Between J^and / 2 the 
 intensity is always small, reaching a maximum about half-way 
 
 f 
 
 -01 
 
 y 
 
 10" 
 
 20 
 
 30 
 
 40 50 I, 
 
 Fig. 38. 
 
 60 
 
 1,80 
 
 90 
 
 between J x and 7 2 . The azimuth required to produce these dark 
 centred rings is very nearly 90. When the white centred rings 
 reappear, their intensity is nowhere large, reaching a maximum 
 about midway between / 2 and 90, though rather nearer the latter. 
 The magnitudes and positions of the various maxima and minima 
 obtained from the above figure are given in the following table. 
 For values of 6 greater than 45, D has both a maximum and a 
 minimum ; the first entry in the table refers to the maxima and 
 the second to the minima. 
 
 e 
 
 
 
 30 
 
 45 
 
 60 
 
 75 
 
 90 
 
 Max. and 
 min. : D 
 
 112 
 
 083 
 
 057 
 
 030 and -014 
 
 009 and -001 
 
 004 and - -005 
 
 <t> 
 
 79 
 
 80 
 
 80 15' 
 
 81 and 55 
 
 83 and 65 
 
 85 and 67 
 
 These various deductions from theory relative to the behaviour 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 103 
 
 of the rings for different incidences and azimuths are in complete 
 harmony with the experimental results. 
 
 A specially simple and important application of the principles 
 employed in this chapter is met with, when we are dealing with 
 the case where the first and third media are the same. This 
 happens in the ordinary arrangement for the production of 
 Newton's rings, where a thin film of air separates two pieces of 
 glass of the same kind. In this case we have to put 
 
 in the formulae of p. 96, whence we find that when ^ = 2m7T, 
 / = 0, so that the central spot and all the minima are not only 
 dark, but absolutely black. The intensity of the bright rings, 
 which occur when ty (2m + I)TT, is 
 
 4r 2 4r' 2 
 
 - 
 
 
 cos' 
 
 from which it appears that the rings cannot disappear, except 
 when 6 = 90 and r = 0, i.e. when the incident light is polarised at 
 right angles to the plane of incidence, and the angle of incidence 
 is the polarising angle. The following table gives the brightness 
 of the rings for incidences less than the critical angle in the case 
 of glass, for which ^ = 1'596, the polarising angle being 32 1', 
 and the critical angle 38 47'. Fig. 39 represents these results 
 graphically. The rings are black centred throughout. 
 
 
 
 = 
 
 30 
 
 45 
 
 60 
 
 75 
 
 90 
 
 
 
 438 
 
 438 
 
 438 
 
 438 
 
 438 
 
 438 
 
 10 
 
 459 
 
 449 
 
 439 
 
 429 
 
 422 
 
 419 
 
 20 
 
 525 
 
 479 
 
 432 
 
 386 
 
 351 
 
 339 
 
 30 
 
 690 
 
 544 
 
 398 
 
 251 
 
 144 
 
 105 
 
 35 
 
 827 
 
 680 
 
 533 
 
 386 
 
 278 
 
 239 
 
 38 
 
 958 
 
 860 
 
 763 
 
 665 
 
 593 
 
 567 
 
 38 47' 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 Having considered the brightness of the various rings, we must 
 
104 TRANSPARENT ISOTROPIC PLATES. NEWTON S RINGS [CH. 
 now deal with their size. From the formulae above it appears 
 
 mr, where n is any integer, and ^jr is 
 
 
 that / is stationary when 
 given by the equation 
 
 If r be the radius of a ring, ^ the curvature of the refractive 
 surface, we have approximately c x = ^-/c 1 r 2 . Hence we get 
 
 p = r ^fafi'yXfjL = Vn sec 9', 
 
 the radius r being proportional to p. At the centre n = 0, and so 
 p = 0, and by putting n equal to 1, 2, 3, 4 etc. we get the radii of 
 the successive rings. It thus appears that the size of the rings is 
 the same in whatever azimuth the incident light is polarised, and 
 that it depends only on the angle of incidence within the film. 
 For a given incidence the radii of the different rings are propor- 
 tional to the square roots of the natural numbers, a law derived by 
 Newton from his experiments. Newton also made observations on 
 the dependence of the radii on the angle of incidence, the law 
 according to theory being that the radius is proportional to the 
 square root of sec <'. This law has been more exactly verified 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 105 
 
 since Newton's time, but we give his results in the following table 
 on account of their historical interest, and their accuracy con- 
 sidering the rough instruments of measurement at Newton's 
 disposal. 
 
 0' 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 p (theory) 
 p (Newton) 
 
 1 
 1 
 
 1-0077 
 1-0077 
 
 1-032 
 1-033 
 
 1-075 
 1-075 
 
 1-142 
 1-140 
 
 1-247 
 1-250 
 
 0' 
 
 60 
 
 65 
 
 70 
 
 75 
 
 80 
 
 85 
 
 p (theory) 
 p (Newton) 
 
 1-415 
 
 1-4 
 
 1-538 
 1-525 
 
 1-710 
 1-690 
 
 1-965 
 1-925 
 
 2-400 
 2-286 
 
 3-388 
 2-9 
 
 The discrepancy between theory and observation is appreciable 
 only for the larger angles of refraction, where a slight error in the 
 estimate of the incidence makes a considerable difference in the 
 radius. The values of p corresponding to the first bright ring for 
 different angles of incidence are given in the next table, calculated 
 for i = 1-596. 
 
 
 P 
 
 
 
 1 
 
 10 
 1-02 
 
 20 
 1-09 
 
 30 
 1-29 
 
 35 
 1-58 
 
 38 
 2-32 
 
 38 15' 
 2-56 
 
 38 30' 
 2-97 
 
 38 46' 
 5-70 
 
 38 47' 
 
 00 
 
 The radii of the other rings, which are alternately dark and 
 bright, are obtained by multiplying these values by the square 
 roots of the natural numbers 2, 3, ... n. It appears from the table 
 that the rings open out one after another on approaching the 
 critical angle (38 47' in this case), and that they expand slowly at 
 first, but afterwards very rapidly. The corresponding values of p 
 for the first ring in the case dealt with above, where the three 
 media are air, soap, and diamond, are as follows : 
 
 
 P 
 
 
 
 1 
 
 30 
 1-03 
 
 60 
 1-11 
 
 65 
 1-13 
 
 70 
 1-14 
 
 75 
 1-15 
 
 90 
 1-17 
 
106 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 Under such circumstances the opening out of the rings is a 
 slow process throughout. 
 
 The most striking feature of the rings is their colour. That 
 the rings should be coloured, when natural light is employed, is 
 evident from the formula for the radius 
 
 =\n sec 
 
 which shows that the size of the ring depends on the wave length, 
 and therefore differs for the different constituents of white light. 
 To investigate accurately the dependence of the size of a ring on 
 the colour of the light employed, we should need to know the 
 relation between X and //,, a relation that will be discussed in a 
 later chapter on dispersion. It will be found, however, that the 
 change in /JL/JJ,' in going from one colour to another is usually 
 small, so that we may, as a first approximation, regard /////, as 
 a constant, in which case the radius varies as the square root of 
 the wave length. Thus the red rings are bigger than the violet, 
 in accordance with the observation of Newton, who says, " I found 
 the circles which the red light made to be manifestly bigger than 
 those which were made by the blue and violet. And it was very 
 pleasant to see them gradually swell or contract according as the 
 colour of the light was changed." As a numerical example we 
 shall calculate the relative size of the first and second rings corre- 
 sponding to Fraunhofer's lines A, D, and H in the spectrum, A being 
 in the extreme red, and H in the violet. We have approximately 
 
 l'U and r h /r d 
 
 V \^/\ d = 0'81, 
 whence we get the following results for //, = 1*596. 
 
 
 = 
 
 = 20 
 
 = 38 
 
 ( Ph 
 
 081 
 
 0-886 
 
 1-89 
 
 First bright ring -\ p d 
 
 1-00 
 
 1-09 
 
 2-32 
 
 (Pa 
 
 1-14 
 
 1-24 
 
 2-63 
 
 ( Ph 
 
 1-15 
 
 1-26 
 
 2-68 
 
 First dark ring -j p d 
 
 1-41 
 
 1-55 
 
 3-29 
 
 (Pa 
 
 1-60 
 
 1-76 
 
 3-73 
 
 Having determined the radii of the rings, and the intensities 
 at these rings (p. 103), we can draw curves to represent the 
 
TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 107 
 
 intensities at different distances from the centre, for any given 
 incidence or azimuth. This is done in Figs. 40 and 41, which 
 show the intensities for the three colours red, orange, and violet, 
 corresponding to the lines A, D, H. Fig. 40 represents the case 
 of normal incidence, and Fig. 41 that when the angle of incidence 
 is 38, and the light is plane polarised in the plane of incidence. 
 
 r\ 
 
 
 2\ 
 
 1 
 
 
 \ 
 
 1-5 
 
 Fig. 40. 
 
 A 
 
 \ 
 
 \ 
 
 A 
 
 \ 
 
 \ 
 
 A 
 
 Y 
 
 An inspection of these figures shows the general effect of the 
 dependence of the size of the rings on the colour. Each bright 
 ring changes in colour from violet on the inside to red on the 
 outside, whereas in the dark rings the order of colours is reversed 
 
108 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 It also appears that, as the order of the rings increases, the red of 
 one ring tends to overlap the violet of the next. This diminishes 
 the sharpness of the rings and makes the first more conspicuous 
 than the others. 
 
 If the first medium be more highly refractive than the second 
 we may, by employing a prism, make the angle of incidence on the 
 film greater than the critical angle. Under these circumstances 
 it is still true that the transmitted system is complementary to the 
 reflected, and that the intensity of the latter is given in the most 
 general case by the formula of p. 95, viz. 
 
 I=siri 2 
 
 but we must, of course, employ expressions for r, r', and q ap- 
 propriate to the case of total internal reflection. When the first 
 and third media are the same, which is the case that has been 
 most carefully dealt with experimentally, we have 
 
 n + r 2 = = rV + r/, 
 
 and 
 
 COS 2 
 
 Icos 2 0. 
 
 The expressions for r, r', and q were obtained in Chap. ill. and 
 proved to be 
 
 
 
 r _ gt(2a-7r^ r _ g?2a ^ ^ = exp. 
 
 where X is the wave length in air, 
 
 tan a! 
 
 tana Vsnr< 
 
 /-r cos <p 
 
 Also we have 2c a /X = /c^/X = p~, where r is the distance from the 
 centre, and /^ the curvature of the refracting surface, so that p is 
 proportional to the distance from the centre. Thus 
 
 q = exp. [- /47T/0 2 Vsin 2 </> - I 
 
 and 
 
 /,= 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 109 
 
 where / x and I 2 represent the intensities of the reflected beam for 
 light polarised perpendicular and parallel respectively to the plane 
 of incidence. 
 
 A consideration of the formula for / shows that there are no 
 alternations of light and shade, as in the case of incidences below 
 the critical angle, so that there are no longer any rings. At the 
 centre q = 1, so that /= 0, and the central spot is perfectly black. 
 As we recede from the centre q diminishes slowly at first but 
 afterwards rapidly, so that / increases slowly, and then rapidly, 
 giving the impression of a dark spot of sensible magnitude with 
 its centre at the point of contact of the refracting surfaces. The 
 table below, giving the numerical values of B = 1 / for different 
 incidences when //, = T596, shows more precisely the manner in 
 which the blackness decreases from the centre. 
 
 If we wish to compare the intensity at a given distance from 
 the centre in the two cases, when the light is polarised perpendi- 
 cular and parallel to the plane of incidence, we note that I I >/ 2 , if 
 sin 2a < sin 2a', or, if B be the blackness, so that B = 1 I l} we have 
 B l >B 2 , if sin 2a > sin 2a'. We have seen that a and a' are zero 
 at the critical angle, whence they increase to 90, and that of the 
 two a is the greater. Hence near the critical angle 2a > 2a', and 
 both being less than 90 we have sin 2a > sin 2a', so that B l > B 2 , 
 i.e. the dark spot is more conspicuous in the case of light polarised 
 at right angles to the plane of incidence. This state of affairs 
 lasts until a and a' are complementary, where 
 
 sin 2 a = sin 2a', and B l B 2 . 
 
 At this point the difference of phase between the components 
 parallel and perpendicular to the plane of incidence is a minimum. 
 The corresponding angle of incidence when /z = T596 is = 48 40'. 
 Beyond this incidence the order of magnitude of B l and B 2 is 
 reversed, and the distinctness of the spots becomes more and more 
 unequal as the incidence increases, and the spots contract. The 
 following table gives the values of B l and B 2 for different angles of 
 incidence, and different distances from the centre, when //,= T596. 
 Stokes* obtained experimental results in complete harmony 
 with this theory, although he did not find it possible to make a 
 
 * Coll. Works, Vol. n. p. 56. 
 
110 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 sufficiently accurate measure of the intensity to render possible a 
 numerical comparison of the results of theory and observation. 
 The values of B^ and B 2 given in the table below are shown 
 
 
 
 ,=0 
 
 ,=0.5 
 
 ,=i 
 
 p = l-5 
 
 39 * 
 
 1 
 
 962 
 808 
 
 605 
 202 
 
 211 
 042 
 
 40 ^ 
 
 I 
 
 949 
 800 
 
 495 
 176 
 
 092 
 022 
 
 45 B! 
 
 I 
 
 846 
 766 
 
 135 
 086 
 
 002 
 001 
 
 50 j*i 
 
 \ 
 
 687 
 720 
 
 035 
 041 
 
 
 
 
 60 ^ 
 
 I 
 
 342 
 
 589 
 
 004 
 010 
 
 
 
 
 750 1: 
 
 I 
 
 057 
 245 
 
 
 001 
 
 
 
 
 85 *> 
 
 \ 
 
 006 
 037 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \\ 
 
 \ 
 
 1 1-5 
 
 Fig. 42. 
 
 5 1 1-5 
 
 Fig. 43. 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 111 
 
 graphically in Figs. 42 and 43. These indicate clearly how the 
 spot contracts in each case as the incidence rises, and how the 
 relative intensity varies in the two cases. 
 
 It will have been observed that a, a, and q, on which the 
 intensity depends, all involve JJL, and so vary with the colour of the 
 incident light. It would be easy to calculate the values of / for 
 different values of //., corresponding to the different constituent 
 colours in the light used ; but the factor that has a predominating 
 influence on the intensity is q, and the variation of q depends 
 chiefly on the presence of X in the formula for q. Thus we 
 may say, with sufficient accuracy for descriptive purposes, that 
 q = e~* lk , where K is a constant. Hence q increases with X, so 
 that / diminishes. Thus the intensity of the blue constituent is 
 
 Fig. 44. 
 
 rather greater than that of the red, so that a faint bluish tint 
 will be perceived round the dark spot a deduction from theory 
 that agrees with observation. 
 
 So far we have supposed that the ring system is new 
 directly after the light is reflected or transmitted, as the case may 
 be. Some interesting modifications of the results are obtained 
 experimentally when the rings are viewed through an analyser, 
 which destroys all vibrations except those confined to a particular 
 plane. It is found convenient in practice to work with the 
 transmitted system. We proceed to investigate the phenomena 
 in this case from the point of view of our theory. 
 
 Let 6 and & be the azimuths of the principal planes of the 
 polariser and analyser respectively, represented by OE and OF in 
 the figure. When the light emerges from the polariser the curl of 
 the displacement is along OE' perpendicular to OE, and on passing 
 
112 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 through the analyser it is along OF' perpendicular to OF. A wave 
 of unit amplitude polarised parallel to the plane of incidence is 
 transmitted through the plate as R'e^S, and if polarised at right 
 angles to the plane of incidence it is transmitted as Re** 1 , where 
 
 Fig. 45. 
 
 R, R', ^TU and ^r,' are given by the theory already expounded. 
 Hence a wave of unit amplitude polarised in an azimuth 6 will 
 emerge from the analyser as 
 
 R' cos cos 0V*-' +R$m6 sin 0V*', 
 and its intensity will be 
 
 / = R 2 sin 2 6 sin 2 & + R' 2 cos 2 6 cos 2 & 
 
 + 2RR' sin sin & cos cos 0' cos (^ - y/). 
 
 The consideration of this formula, with the values of R, R ', etc. 
 given by the theory, will enable us to account for all the main 
 features of the phenomena as revealed by experiment. 
 
 We have seen that the transmitted wave is represented by 
 
 ^ 1 2 , where q e~**/ 2 . It will make for brevity to write r for 
 1 + gifV 
 
 rjT s and s for Si8 z , and it will be noted that, on the theory of an abrupt 
 transition, r and s are both real, s being positive, and r being 
 positive, if the incidence is less than 1^ or greater than / 2 , but 
 otherwise negative. Here I I and / 2 denote, as on p. 97, the 
 polarising angles at the two interfaces. For light polarised 
 parallel to the plane of incidence, we replace r and s by r and s', 
 which are positive throughout. 
 
 For given values of 6 and & it is obvious that / is a maximum 
 or minimum along with R and R'. Thus the stationary points 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 113 
 
 occur, as before, when ty = n7r, and there is still a ring system. 
 In general we have 
 
 as se-W* 
 
 l+re-** \/l+r 2 + 2rcos^/r .._ 
 where <I> = ^ ^/2, and 
 
 tan % = (r sin ^)/(l + r cos i/r). 
 At the centre -\lr = 0, so that <E> = = <', and 
 
 
 r^n ^^ - ^-T> ~t / 
 
 1 + r ' 1 + r 
 
 At the next stationary point ^ = TT, so that <& = = ^^ and 
 
 ' 
 
 C1 " CY I " 
 
 Thus the difference of the intensities at these two stationary points, 
 which determines the brightness of the rings, is 
 D = [S sin 6 sin B f + ' cos 6 cos 0'] 2 
 
 - [S, sin (9 sin (9' + / cos cos dj 
 
 sn 
 
 ff 
 
 ss' (r + r') sin cos sin & cos 
 ' 
 
 0'1 
 
 We shall consider a few special cases of this formula. (1) When 
 & = 90 we have 
 
 D = - 4rV sin 2 <9/(l - r 2 ) 2 . 
 
 As this is negative, the rings are black centred. If we begin with 
 6 = 90 the rings are most conspicuous at the outset, any turning 
 of the polariser diminishing their brightness. The rings disappear 
 completely when 0=0 or 180, for then the formula makes D 
 vanish. (2) When & = 0, we have 
 
 D = _ 4rV 2 cos 2 <9/(l - r /2 ) 2 . 
 
 The rings are still black centred throughout, except when they 
 disappear, when the polariser is in the azimuth Q = 90 or 6 = 270. 
 (3) As an example of an intermediate azimuth of the analyser, 
 take ff = 45. We then have 
 
 D = -2[r 2 s 2 sin 2 0/(l-r 2 ) 2 
 
 + ss' (r + r') sin cos 0/(l - r 2 ) (1 - r" 2 ) + rV 2 cos 2 0/(l - r /2 )]. 
 M. L. 8 
 
114 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 This is negative when 6 = or 90. It vanishes, so that the rings 
 disappear when 
 
 tan = - s'(I - r 2 )/rs (1 - r' 2 ), 
 
 and when tan = - r's' (1 - r 2 )/r 2 s (1 - r' 2 ). 
 
 Between these two values of 6 the rings are bright centred, whilst 
 outside these limits they are dark centred. 
 
 At the polarising angles (/j or J 2 ) we have r = 0, and the 
 formula for D reduces to 
 
 4rycos0cos<9' ~ rV 
 
 C S COS * + S Sm S 
 
 This vanishes when or #' is 90, and also when 
 
 /y o 
 
 tan 0' = -- -- ;- cot 0. 
 5(1 r 2 ) 
 
 When 0' = 90, D and there are 110 rings. If 6' be slightly 
 less than 90 the rings appear with their centres bright or dark, 
 according as sin 20 is negative or positive. The rings disappear 
 again at the azimuth 0' determined by 
 
 tan 0'=- r's' cot 0/s(l- r' 2 ), 
 
 and after this there is a change from one system of rings to the 
 other. All these conclusions agree completely with observations 
 on the behaviour of the rings under different circumstances. 
 
 When the angle of incidence is beyond the critical angle the 
 intensity / is still given by the formula of p. 112, where R, R', ^ l} 
 A/T/ have the values appropriate to the case of total internal reflec- 
 tion. The phenomena were examined experimentally by Stokes* 
 with an angle of incidence slightly greater than the critical angle, 
 and with the incident light plane polarised at an angle of 45 to 
 the plane of incidence, and we shall consider this case more 
 particularly. Putting = 45 in the formula for /, we get 
 
 / = [R 2 sin 2 0' + R' 2 cos 2 & + 2RR' sin & cos 9' cos (^ - -f /)]. 
 At the centre we have R = R' = 1, and ^ i/r/ = 0, so that 
 
 / = sin 2 ((9'4-45 ). 
 
 Thus the intensity vanishes at the centre when & = 135. It also 
 
 vanishes for all values of 0' at considerable distances from the 
 
 centre where the thickness of the film is no longer very small, and 
 
 * Math, and Physical Papers, Vol. n. p. 56. 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 115 
 
 R and R are inappreciable. / cannot vanish in the interval, but it 
 may have certain maxima and minima as the tables and figures 
 that follow show. The calculations have been made for >Lt= 1*596, 
 and an angle of incidence of 39, which is 13' beyond the critical 
 angle. The numbers represent the intensities for different 
 
 \ 
 
 \ 
 
 45 
 
116 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 distances from the centre, and for different azimuths of the 
 analyser (0'). 
 
 e r - 90 
 
 P = 
 
 /3 = 0-5 
 
 P =l 
 
 p=l-5 
 
 
 
 -5 
 
 481 
 
 303 
 
 106 
 
 15 
 
 25 
 
 263 
 
 211 
 
 083 
 
 30 
 
 067 
 
 094 
 
 117 
 
 055 
 
 45 
 
 
 
 017 
 
 045 
 
 030 
 
 ff - 90 
 
 P = 
 
 />=0-5 
 
 P = l 
 
 /> = l-25 
 
 ,3=l-5 
 
 /> = l-75 
 
 45 9 
 
 
 
 017 
 
 045 
 
 042 
 
 030 
 
 009 
 
 50 
 
 008 
 
 022 
 
 030 
 
 021 
 
 023 
 
 006 
 
 55 
 
 030 
 
 030 
 
 020 
 
 013 
 
 017 
 
 004 
 
 60 
 
 067 
 
 055 
 
 016 
 
 008 
 
 013 
 
 002 
 
 08 
 
 06 
 
 04 
 
 02 
 
 60 
 
 ,45 
 
 551 
 
 \ 
 
 ^60" 
 
 1-5 
 
 Fig. 47. 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 117 
 
 An examination of the figures will indicate clearly the nature 
 of the phenomena. As ff increases the bright spot at the centre 
 becomes duller, and a bright ring is formed surrounding a central 
 dull patch. The darkness of the centre increases, and the bright 
 ring expands until at 6' = 135 the centre is absolutely black. 
 After this the brightness of the centre increases, and a dark ring is 
 formed which travels outwards until it is lost in the dark field 
 outside. All this is in complete agreement with Stokes' observa- 
 tions. 
 
 The results so far obtained have been reached by means of 
 Fresnel's formulae, which are strictly applicable only in the ideal 
 case of an abrupt transition from one medium to another. The 
 influence of the transition layer has been considered in the last 
 chapter, whence it appears that the effect of the layer is to make 
 T! and r 2 complex instead of real. We thus have r x = Rtf* 9 *, and 
 r 2 = R 2 e ie *, and as before <f = Qe~^, where R 1} R^ and Q are real. 
 Hence the intensity of the reflected beam, corresponding to 
 
 j- -^-^l ~1~ ^ J -*'2 ~T ^^ -*- t 'i JL * / 2 ^OS \ C/2 "^ "V^ ~~"" 
 
 IS 2 = =r 
 
 cos (ft ty + ft) ' 
 
 We have seen that the departure from Fresnel's laws is very 
 slight, except for light polarised at right angles to the plane of 
 incidence in the immediate vicinity of Brewster's angle. This 
 then will be the only region, where there can be any appreciable 
 departure from the laws already reached as to the character and 
 behaviour of the rings. As Brewster's angle is less than the critical 
 angle we have Q = 1, and putting R l 2 + R< 2 2 = a, 1 -f R?R 2 = a l , 
 and ( 2R 1 R 2 = b, for brevity, the formula for / becomes 
 
 ,. a + b cos (ft ^ ft) 
 ~ Oj + b cos (ft - ^+ ft) ' 
 
 For a given angle of incidence -^ is the only variable in this 
 expression, so that / is a periodic function whose period is 2?r. 
 There will thus be alternatives of bright and dark rings, the 
 stationary positions of / being given by the equation : 
 
 (a otj) cos ft sin (ft -\Jr) + (a + a x ) sin ft cos (ft -^r) + b sin 2ft = 0. 
 
118 TRANSPAKENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 Putting tan /3 = tan 6 l , 
 
 Otj ~~ Ctr 
 
 b sin 20i 
 and sin 7 = 
 
 V(a a - a) 2 + 4ad! sin 2 0j ' 
 we get sin (i|r + # - 2 ) = sin 7, so that ^ + /3 2 = WTT + (- l) w 7. 
 
 For light polarised in the plane of incidence 6 l and 2 are very 
 nearly zero or TT throughout, so that /3 and 7 are very small or 
 nearly TT, and ^ = imr very approximately, where m is any integer. 
 This is the formula obtained- on the hypothesis of an abrupt tran- 
 sition, so that the layer has a very slight influence on either the 
 size or the intensity of the rings. 
 
 For light polarised at right angles to the plane of incidence 
 b sin 20j is always small, for sin 20! is very small except near the 
 Principal Incidence, and in this neighbourhood b is very small. 
 Hence 7 is always very small, and we have very approximately 
 
 T|T + j3 2 = n-rr. 
 
 The intensity I is a maximum when n is even, in which case its 
 value is 
 
 a + b cos (0 T - /3) 
 
 while its minimum value occurs when n is odd, and is 
 
 a - b cos (0i - ff) 
 
 Owing to the smallness of b near the Principal Incidence, the 
 variations of the intensity are very slight, so that the rings in 
 this neighbourhood are faint, although not absolutely evanescent, 
 as would be the case with an abrupt transition. 
 
 From the foregoing discussion it will be apparent that the 
 layer influences the phenomena mainly in fivejways : 
 
 1. The centre is no longer one of the positions of maxima or 
 minima intensity. At the centre we have ^ = 0, and 
 
 T _ a + b cos (0 2 0i) 
 ~~ Oj + b cos (0 2 + 0!> ' 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 119 
 
 Near the first Principal Incidence (/j), 2 is nearly zero or TT, so 
 that the intensity at the centre is 
 
 a b cos 6 l 
 
 a a + 6 cos 0j 
 approximately. At the stationary points, on the other hand, 
 
 i 
 
 nately. At the 
 we have 
 
 g6cos((9 1 -/3) 
 
 and owing to the presence of ft this is not identical with the 
 intensity at the centre. There will be very little difference 
 between the two intensities, except in the neighbourhood of the 
 Principal Incidence, for outside that region ft is very nearly zero 
 or TT. 
 
 2. As b is not zero at the polarising angle the rings do not 
 disappear there, although they are very faint in this region, owing 
 to the smallness of 6. 
 
 3. The size of the rings is somewhat altered. We have seen 
 that the radii are given by the formula 
 
 p = r V/^/X = V(sec $') I/T/TT. 
 
 As i/r is no longer exactly equal to mr, there is a change in the 
 radii. ' This change, however, is appreciable only in the neighbour- 
 hood of the Principal Incidence, when we have ^ + ft = mr very 
 nearly and 
 
 p = \/(n ft ITT) sec </>'. 
 
 4. The rings do not all expand steadily when the angle of 
 incidence increases, as is the case with an abrupt transition ; but, 
 under certain conditions, they experience alternations of expansion 
 and contraction. The formula for p shows that 
 
 = sec f {(HTT - ft) tan f - ^,1 . 
 
 If ft diminishes with the incidence, ~, is negative and p increases 
 
 7/3 
 
 continuously with 6'. If, however, -~ be positive, the rings will 
 
 d<p 
 
 be stationary at the incidences determined by 
 
120 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 The equation for ft shows that the graph of ft will be very similar 
 to that of lt the gradient being smaller. Thus i , is very small 
 
 Ct(p 
 
 until the Principal Incidence is approached, when it rises rapidly 
 and falls again equally rapidly almost to zero. The graph of such 
 a function (see Fig. 48) will show that the equation 
 
 has one real root, two real, or none, the region of unreal roots 
 being approached as n increases, i.e. as the order of the rings 
 increases. In the case of real roots </ and < 2 ', the rings will 
 expand until </>' = </, then contract until </>' = </> 2 ', and beyond that 
 expand. If there are no real roots the rings continually expand 
 as the incidence rises. 
 
 We have seen that the contraction of the rings near the 
 Principal Incidence cannot take place for the larger rings unless 
 
 ^77 is very large, and that it cannot occur at all unless ^-r-. is 
 d<f) d(p 
 
 positive. To give more definiteness to the discussion, let us 
 suppose that the second medium is less refractive than the other 
 two as will be the case, for example, with air between two pieces 
 of glass, or air between glass and diamond. From the formula 
 for an abrupt transition 
 
 r, = - tan (< - </>')/tan (< + <'), 
 
 it appears that, since <j>' > <f>, i\ is positive until the Principal 
 Incidence I 1 is reached, after which it is negative. If then we 
 put T-L = R^ 1 we shall have 0, zero up to I ly and then a sudden 
 change to 1 = TT. The effect of the layer is to make this change 
 in O l gradual, so that l (and therefore also ft) increases continu- 
 
 ously from to TT. Thus -a is positive, and the contraction of 
 
 ct(b 
 
 the rings of lower order is to be expected in the neighbourhood of 
 the Principal Incidence. 
 
 We have dealt specially with the first Principal Incidence 
 (Jj), but very little modification is needed when dealing similarly 
 with the neighbourhood of 7 2 . In this region we have ty 2 = mr 
 very nearly, so that we have merely to replace ft by 2 . In the 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 121 
 
 special case mentioned above # 2 goes from TT to zero in passing 
 through J 2 , so that -TT> is positive, and the same phenomena are 
 
 to be expected near J 2 a s near I lt 
 
 By way of a numerical example we shall take the case where 
 the third medium is diamond and the second air, and consider the 
 nature of the phenomena in the neighbourhood of / 2 . The 
 equation determining the stationary points of the rings is 
 
 d' m 
 
 The angle 2 is the p of p. 74, and is given by the formula 
 
 sec <' 
 
 tan 9 = 
 
 Thus we get 
 
 - co 
 
 ,,dft 
 } d$ 
 
 sn2 
 
 cos 
 
 from which it appears that cot $' -~ is a maximum at the 
 Principal Incidence, where tan <' = //,, and its value is then 
 g (1 + I/A*- 2 ) 2 . Hence the greatest value of n for rings that 
 experience any contraction is given by 
 
 n + l/2 = (l + 1/p*)* /Sire!. 
 
 Taking //, = 2'434, and e t = 0'0241 for diamond we find that n 
 cannot be greater than eight, so that there are only nine 
 contracting rings, five of them being dark. The following table 
 
 gives the values of and ~ -j-f, , for various incidences. 
 
 7T 7T dd) 
 
 0' 
 
 60 
 
 65 
 
 66 
 
 67 
 
 I 2 =6740' 
 
 68 
 
 69 
 
 70 
 
 75 
 
 2 /T 
 
 935 
 
 846 
 
 780 
 
 647 
 
 5 
 
 420 
 
 243 
 
 156 
 
 045 
 
 cot 0' d8 2 
 
 23 
 
 1-24 
 
 2-30 
 
 4-55 
 
 9-01 
 
 5-33 
 
 2-72 
 
 1-26 
 
 12 
 
 ir d<f>' 
 
 A graph of these functions enables us to determine the 
 
122 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 stationary points of the rings, and the regions within which they 
 contract. 
 
 10 
 
 60 
 
 65 
 
 h 
 Fig. 48. 
 
 70 
 
 75 
 
 In Fig. 48 the curve BDB'D' represents the function 
 
 
 TT 
 
 ^, and the curves B n B n ', D n D n f represent n + OJir, for 
 
 odd and even values of n respectively, so that B n B n ' corresponds 
 to a bright ring, and D n D n ' to a dark one. The points of inter- 
 section of the curves indicate the incidences, where the rings are 
 stationary and the contracting process begins or ends. The first 
 bright ring corresponds to n = 0, and it appears from the figure 
 that once the contraction begins it continues until the incidence 
 is grazing. The region of contraction diminishes as the order of 
 the ring increases, and beyond the ninth ring there is no contraction 
 at all. 
 
 (5) The rings do not change suddenly from a dark to a light 
 centred system on passing through the Principal Incidence ; but 
 the transition from one system to the other is effected gradually. 
 Thus with the air-diamond combination discussed under (4) the 
 rings are dark centred to begin with at normal incidence, and expand 
 
V] TRANSPARENT JSOTROPIC PLATES. NEWTON'S RINGS 123 
 
 as </>' increases. As we approach Jj the bright ring surrounding 
 the centre contracts until it squeezes out the dark centre, so that 
 after passing I I the rings are bright centred. Then as 7 2 is 
 approached, the dark ring surrounding the centre contracts, and 
 continues this process until the incidence is grazing. Thus the 
 bright centre is soon squeezed out, and the rings appear to be dark 
 centred. 
 
 When considering the arrangement of colours in Newton's 
 rings, we observed that for the rings of higher order, where the 
 thickness of the films is a few wave lengths, the overlapping of the 
 different coloured rings is so great that the interference effects are 
 not observable. Thus for bright colours to appear, the thickness 
 of the film must be of the order of magnitude of a wave length. 
 However it should be noted that the fundamental formula from 
 which all the results of this chapter have been derived is true 
 whether the transparent plate or film be thin or thick. The 
 formula referred to is that obtained on p. 95, where it is shown 
 that the reflected beam is represented by 
 
 which reduces to 
 
 when the first and third media are the same. We shall have 
 occasion to apply this formula in cases where the plate is not thin. 
 We have seen that below the critical angle q is given by the 
 formula 
 
 ^2 ty p 2?# 
 
 U C/ c/ , 
 
 7T // 
 
 where x = ty/2 = . c a cos </> . 
 
 Hence we have 
 
 r 2 (1 - cos 'lx) 4r 2 srn 2 ^ 
 
 ~ 1 + r 4 - 2r 2 cos 2# ~ (I -r 2 ) 2 + 4r 2 sin 2*- ' 
 
 This formula refers to homogeneous light, of a definite wave 
 length, otherwise X, /i, $' and therefore r and # would not be 
 definite quantities. If natural light be employed, we have to deal 
 with different wave lengths, and r and x will be different for the 
 various constituents. The variation in r will usually, however, be 
 small ; but x will vary enormously for a plate of ordinary thick- 
 
124 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 ness, for which d/\ is very great. Hence the intensity of the 
 reflected light may be obtained in this case (when the incident 
 light is natural light of unit intensity) by regarding r as constant, 
 and giving as all possible values from to TT. Thus we find that 
 the intensity is 
 
 1 r ff 4r a sin a a?(fcc 2r 2 
 
 ~ TT J o (1 - r 2 ) 2 + 4?- 2 sin 2 ^ ~ 1+r 2 ' 
 
 The sum of the intensities of the different constituents of the 
 reflected beam (see Fig. 34) is easily seen to be 
 
 r 2 + r 2 (l -r 2 ) 2 [l +r* + r 6 +...]= 2r 2 /(l + r 2 ). 
 
 Hence the intensity of the reflected beam is the sum of the 
 intensities of its constituents. 
 
 The result just obtained enables us to estimate the intensity 
 of the reflected and transmitted beams when natural light is 
 incident on a pile of transparent plates, and to measure the degree 
 of polarisation of the transmitted light. We shall suppose the 
 plates to be all of the same material and thickness, and too thick 
 to give rise to the colours of thin films. Let R n and T n denote 
 the intensities of the reflected and transmitted light when there 
 are n plates. If we add one more plate behind the pile, the 
 quantity T n+l is made up of the light T n after this has been 
 transmitted through the single plate once, twice,... after successive 
 reflections from the pile, and in calculating T n+l we have merely 
 to find the sum of the intensities of the various constituents. 
 Thus we get 
 
 T n+l = T, T n (1 + R l R n + RfRj +...) = T,T n j(l - R,R n \ 
 
 Moreover since the plates are perfectly transparent there is no loss 
 of light, so that T n 4- R n = 1. Hence 
 
 or 
 
 Solving this difference equation, and determining the constant 
 by means of the relation jR, + 2\= 1, we get l/T n = I + nRJT^ 
 Hence 
 
 nR,), and R n = 1 - T n = ^ 1 /(2 7 1 + nR,). 
 
 It thus appears that when n is infinite T n = and R n = 1, so that 
 all the light is reflected. This helps to explain the whiteness of 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 125 
 
 snow, and other finely divided substances that are transparent in 
 mass. 
 
 The formulae for T n and R n just obtained give the intensity 
 in terms of the number of plates, and the intensities R^ and 2\ of 
 the reflected and transmitted light when there is only one plate. 
 For a single plate we have 
 
 J R 1 = E 2 = 2r 2 /(l + r 2 ), and T, = 1 - R, = (1 r 2 )/(l + r 2 ) 
 where r is the r or r' of Fresnel's formulae, according as the 
 incident light is polarised at right angles or parallel to the plane 
 of incidence. The following table, due to Stokes*, gives the 
 intensities at normal incidence, at the polarising angle /= tan" 1 /j,, 
 and at 0=7 + 2 when the refractive index is //, = T52. 
 
 n 
 
 0=0 
 
 <f> = I 
 
 = 1+2 
 
 R n =Rn 
 
 T T ' 
 L n~ ^n 
 
 Rn 
 
 T n ' 
 
 *j 
 
 *v 
 
 Rn 
 
 T n 
 
 1 
 
 082 
 
 918 
 
 271 
 
 729 
 
 300 
 
 700 
 
 001 
 
 999 
 
 2 
 
 151 
 
 849 
 
 426 
 
 574 
 
 459 
 
 541 
 
 002 
 
 998 
 
 4 
 
 262 
 
 738 
 
 598 
 
 402 
 
 628 
 
 372 
 
 004 
 
 996 
 
 8 
 
 416 
 
 584 
 
 749 
 
 251 
 
 771 
 
 229 
 
 008 
 
 992 
 
 16 
 
 587 
 
 413 
 
 856 
 
 144 
 
 870 
 
 130 
 
 016 
 
 984 
 
 32 
 
 740 
 
 260 
 
 922 
 
 078 
 
 931 
 
 069 
 
 032 
 
 968 
 
 00 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 We have seen in Chap. n. that natural light of unit intensity 
 
 can be represented by two streams, each of intensity 1/2, polarised 
 
 respectively at right angles and parallel to the plane of incidence. 
 
 The intensities of the transmitted streams are ^T n and \Tn, where 
 
 T, I-R, 1-r 2 
 
 l+(2n-l)r 2 ' 
 
 and 
 
 For brevity we shall put 
 
 (f) (f> / = p, <f> + <f>' = (r, cosec 2 p = C!, and cosec 2 cr 
 
 * Math, and Physical Papers, Vol. iv. p. 152. 
 
126 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 then from Fresnel's formulae we have 
 
 r 2 = tan 2 p/tan 2 <r = (c 2 l)/fa 1), 
 and r' 2 = sin 2 p/sin 2 cr = c 9 /Ci . 
 
 Hence denoting the degree of polarisation of the transmitted light 
 by P, we have 
 
 T n - T' 
 
 \ + Tn 1 + (n - 1) (/ 2 + r 2 ) - (2w - 1) rV 2 
 
 1 + (n - 1) [c 2 / Cl + (c 2 - l)/( Cl - 1)] - ('2n - 1) c 2 (c 2 - l)/ Cl fa - 1) 
 
 ~ d + (2n 1) c 2 n cosec 2 /> + (2n 1) cosec 2 a n' 
 
 The following table gives the values of P for different values of 
 n and 0, when //, = 1'52, the results being represented graphically 
 in Fig. 49. If we wish to determine the maximum value of P we 
 
 n 
 
 = 10 
 
 30 
 
 50 
 
 60 
 
 75 
 
 1 
 
 003 
 
 034 
 
 114 
 
 182 
 
 320 
 
 2 
 
 006 
 
 063 
 
 203 
 
 307 
 
 440 
 
 4 
 
 Oil 
 
 109 
 
 334 
 
 470 
 
 544 
 
 8 
 
 018 
 
 172 
 
 493 
 
 637 
 
 614 
 
 16 
 
 025 
 
 238 
 
 647 
 
 775 
 
 658 
 
 32 
 
 031 
 
 300 
 
 770 
 
 870 
 
 681 
 
 00 
 
 042 
 
 402 
 
 943 
 
 988 
 
 707 
 
 must make cosec 2 p + (2w 1) cosec 3 a a minimum. For this 
 purpose we note that, since sin <f> = //, sin <', we have 
 
 d(f) dcf) f _ dp _ da 
 
 tan <f> tan <' tan ^> tan <^> / tan < + tan cf>' 
 
 = cos </> cos (/>' . dO (say). 
 
 Hence dp=smpd0, and sincr.cft?. 
 
 Thus P is a maximum when 
 
 cosec 2 p cos p + (2n 1) cosec 2 a cos <r = 0, 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 127 
 
 and its maximum value is 
 
 Pl = (l+/n a )sin< ~ lm 
 
 The equation giving the position of the maximum may be 
 written in the form 
 
 cosec 2 p . cos p 1 
 
 2n-l = - 
 
 cosec 2 <r . cos <r 
 
 r 
 
 Since p is less than 90, cos p is positive, so that as Zn 1 is 
 positive we must have cos <r negative. Thus < + </>' is greater 
 
 Fig. 49. 
 
 than 90, and the angle of incidence is beyond Brewster's angle. 
 We have seen that the numerical values of r and r' increase with 
 the angle of incidence from Brewster's angle to grazing incidence. 
 Hence in this range 1/rr' diminishes, and consequently n 
 decreases from infinity to unity. At the same time the maximum 
 polarisation diminishes from 1 to (p? l)/(fj?+ 1). We see then 
 that as the number of plates increases the polarisation tends 
 to become complete, and the angle of incidence that gives the 
 greatest degree of polarisation approaches more and more nearly 
 to Brewster's angle. 
 
128 TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS [CH. 
 
 The following table gives corresponding values of n, P l} and <f> 
 calculated from the formulae 
 
 PI- 
 
 (1 + p?) sin 2 
 
 -1, 
 
 and 
 
 cosec 2 p cos p 
 cosec 2 a cos a 
 
 for the case already considered where //, = 1'52. 
 
 
 
 56 40' 
 
 60 
 
 65 
 
 70 
 
 75 
 
 80 
 
 85 
 
 90 
 
 n 
 
 00 
 
 30-372 
 
 9-775 
 
 4-921 
 
 2-913 
 
 1-944 
 
 1-330 
 
 1 
 
 PI 
 
 1 
 
 925 
 
 823 
 
 735 
 
 663 
 
 610 
 
 578 
 
 567 
 
 By representing these results graphically, as in Fig. 50, we can 
 calculate approximately the angles of incidence that give the 
 greatest degree of polarisation for any integral value of n, and 
 also obtain a numerical measure of the polarisation at that 
 incidence. 
 
 This figure gives the graphs both of P l and n, the former 
 corresponding to the dotted, and the latter to the continuous line. 
 The scale of n is indicated on the left, and that of P l on the right. 
 The values of P 1 and <f> estimated from these graphs for different 
 integral values of n are set out in the following table. 
 
 n 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 <t> 
 
 90 
 
 79 30' 
 
 74 45' 
 
 72 
 
 69 45' 
 
 68 15' 
 
 67 15' 
 
 PI 
 
 567 
 
 615 
 
 667 
 
 707 
 
 739 
 
 763 
 
 783 
 
 n 
 
 8 
 
 9 
 
 10 
 
 15 
 
 30 
 
 00 
 
 
 
 
 66 20' 
 
 65 30' 
 
 64 45' 
 
 63 
 
 60 5' 
 
 56 40' 
 
 
 Pl 
 
 798 
 
 814 
 
 827 
 
 862 
 
 922 
 
 1 
 
 
 The consideration of some other properties of thick plates 
 suggested by the subject matter of this chapter will be more 
 
V] TRANSPARENT ISOTROPIC PLATES. NEWTON'S RINGS 129 
 
 conveniently postponed until the theory of dispersion has been 
 discussed. 
 
 -* 
 
 60 
 
 65 
 
 70 
 
 75 
 
 Q<f 
 
 85 
 
 Fig. 50. 
 
 M. L. 
 
CHAPTER VI. 
 
 PROPAGATION OF LIGHT IN TRANSPARENT 
 CRYSTALLINE MEDIA. 
 
 WHEN dealing in Chapter in. with the propagation of light 
 in a material medium, we found it convenient mentally to replace 
 the matter by a continuum formed by smoothing out the atoms, 
 and spreading their influence uniformly throughout the body. It 
 was tacitly assumed that the material molecules were arranged 
 perfectly at random, so that the smoothed out representative 
 medium was isotropic. Experience, however, shows us in the 
 case of crystals that for matter of some kinds the molecules are 
 not arranged at random. Under such circumstances the repre- 
 sentative medium will not be the same in all directions. Hence it 
 will no longer be possible to express the potential energy per unit 
 volume in terms of a single constant C//A, and write W in the form 
 
 W-i will now be a general quadratic function of /, g, h. This 
 function will have certain planes of symmetry, and by taking the 
 axes of symmetry as coordinate axes we can express W l in the 
 form 
 
 i (a 2 / 2 + by + c 2 A 2 ). 
 
 The investigation of the propagation of light in such a medium 
 can be carried out on exactly the same lines as before with the aid 
 of the Principle of Action, and the formulae 
 
 and W = J j(a-f + 6y + <*) dr. 
 
OH. VI] TRANSPARENT CRYSTALLINE MEDIA 131 
 
 The dynamical equations are found as on p. 33, and prove to be 
 
 while the surface conditions require the continuity of 
 nb 2 g mc^h, lc*h na?f, and ma 2 / lb*g, 
 
 and of course that of f , 77, f. These boundary conditions may also 
 be expressed by saying that the displacement (f , 77, f) is continuous 
 at an interface, as are also the tangential components of the vector 
 (a 8 / b 2 g, c 2 A). 
 
 As we have to deal throughout with periodic functions of the 
 time we may put everything proportional to e ipt , and so replace | 
 by p 2 f, and so for rj and . The dynamical equations then 
 show that f , 77, f may be regarded as the curl of the vector 
 
 i(a'/ &y ctt). 
 
 Hence ^ + ^ + L? = 0> 
 
 c*^ 3y S^ 
 
 so that there is no compression involved in the motion. 
 
 If (?i> Vi> ?i) ^ e ^ ne vector of which (f, 77, f) is the curl we have 
 g l =fa?/p* ) and so for 77! and f,. The dynamical equations may be 
 written in terms of this new vector in the form 
 
 &=-*/, %=-% ?i = -^. 
 
 But we have 
 
 8f_a^ = _ + Af? + ?^ + 
 
 9y 9^ c)# \ 9 9y 
 
 Hence /= o'9/ - g , y = 6 2 ^ - 1 . & = c 2 V% - , 
 
 where = a , + b *%> + c ,f . 
 
 ox dy oz 
 
 We shall consider a progressive plane wave moving in the 
 direction (Z, w, w) with velocity v. Then we may take 
 
 ( 77, f) = (Z, Jf, JV). ^[ e i'(te+wy+-o = (^ M, N)Ae i<a say, 
 and (f,g,h) = (\,n,v)Be*<-*, 
 
 where L, M, N are the direction cosines of the displacement, and 
 
 92 
 
132 PROPAGATION OF LIGHT IN [CH. 
 
 X, fi, v those of the curl, while A and B are the amplitudes of 
 these two respectively. Since 
 
 we have IL + mM -f nN = ........................ (I), 
 
 so that the displacement is in the wave front. Also since 
 
 we have l\ + m/j, + nv = Q 
 
 so that the curl is also in the wave front. Substituting for/, g, h 
 
 in the equations 
 
 we get X (v 2 - a 2 ) + I (a?l\ + b 2 m/j, + c 2 rci>) = 0, 
 
 and two similar equations. Hence we have 
 
 * _X m _p n _v Tn 
 
 v 2 -a 2 tc' v 2 -b 2 K' v 2 -c 2 K" 
 where K = a 2 l\ + tymp + c 2 nv. 
 
 Multiplying the first of these by I, the second by m, and the third 
 by n, we get on addition by means of (II) 
 I 2 m 2 n 2 
 
 .(IV). 
 
 v 2 -a 2 v 2 -b 2 v 2 - c 2 
 Again multiplying 
 
 X (v 2 -a 2 ) + l (a 2 l\ 4 b 2 m/j, + c*nv) = 
 
 by X, and the other two similar equations by //, and v, and adding, 
 we get 
 
 v 2 = a 2 \ 2 + b 2 tJ 2 + c 2 v 2 (V). 
 
 Finally from (III) we have 
 
 I 1 , . m 1 , , . w 1 , . 
 
 _,__(,._.), _ = _ ic(t)2 _ n _ = _ K(J , 2 _ C2)) 
 
 hence ^-(6 2 -c 2 ) + - (c 2 - a 2 ) + (o 2 - fe 2 ) =0 (VI). 
 
 K fju v 
 
 These six equations (I VI) enable us to deal completely with 
 the problem of the propagation of light in transparent crystals. 
 It will be convenient at the outset to find the relation between 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 133 
 
 A and B the amplitudes of the displacement and its curl, and to 
 obtain a formula for the kinetic and potential energies in terms of 
 these amplitudes. Putting K'V = p, so that p/%7r is the frequency, 
 and substituting for (f , rj, f) and (f, g, h) in the dynamical equation 
 
 we get p*AL = BK (c*mv b*np). 
 
 But since the lines (I, m, n),(\, /*, z>),and (L,M, N) are rectangular, 
 we have I pN vM, m = vL \N> and n = \M pL. 
 Hence tfmv - frnp = &v (vL - \N) 6> (\M - 
 
 By (V) we have 
 
 also 
 
 a 2 X (mv rip) + &> (n\ - Iv) + c~v (lp raX) 
 
 l - (6 2 - c 2 ) + - (c 2 - a 2 ) 4- - (a 2 - 
 X p v 
 
 = by (VI). 
 
 Hence c 2 wz/ 6 2 rtyt = Lv 2 , 
 
 and j9 2 ^Z = 5/c x Zi; 2 =pvBL. 
 
 Thus J. = (v/p)B, which is the relation between amplitudes of 
 the displacement and its curl. 
 
 The kinetic energy per unit volume is 
 
 The potential energy is similarly 
 
 4 (a 2 / 2 + &y + c 2 h*) = J (a 2 X 2 
 
 Thus the kinetic energy is equal to the potential, and the intensity 
 is proportional to A z , whichever method of measuring the intensity 
 be adopted. 
 
 Equation (IV) above, viz. 
 
 tf _ J2) + W 8^ V 2 _ C 2) = 0, 
 
134 PROPAGATION OF LIGHT IN [CH. 
 
 gives the velocity of propagation of a wave in any given direction. 
 As it is a quadratic equation in v 2 there are in general two distinct 
 waves corresponding to any wave normal, the two waves travelling 
 at different speeds. Before dealing with the most general case 
 we shall consider the simpler one where b = c. We then have 
 
 l*/(v 2 - a 2 ) + (1 - / 2 )/<> 2 - c 2 ) = 0, since I 2 + m 2 + n 2 = 1. 
 This leads at once to 
 
 tf = c 2 , or v 2 = I 2 c 2 + a 2 (1 - 1 2 ). 
 
 The first root makes v a constant, so that for this case the 
 phenomena are the same as in an isotropic medium, and the 
 ordinary laws of refraction apply. The wave is therefore called 
 the ordinary wave. The alternative root is v 2 = I 2 c 2 + a 2 (1 I 2 ), so 
 that in this case v depends on the direction of propagation, and the 
 ordinary laws of refraction do not hold any longer. The corre- 
 sponding wave is consequently styled the extraordinary wave. 
 
 If, in the expressions just obtained for v 2 , we make the substi- 
 tutions v = 1/r, I = xjr, m yjr, and n = z/r, we get r = l/c for the 
 ordinary wave, and 1 = a 2 (y 2 + z 2 ) + c 2 x 2 for the extraordinary one. 
 These equations represent the two sheets of a surface called the 
 Index Surface from the fact that the radius vector in any direction 
 is proportional to the refractive index for a wave, whose normal is 
 in that direction. It is at once evident that the Index Surface 
 consists of a sphere and a concentric spheroid, with the axis of x 
 as a common axis. As the spheroid may be either oblate or 
 prolate, there should be two classes of crystals in which an ordinary 
 and an extraordinary wave may be propagated. This is found to 
 be the case, and the two types are usually distinguished by the 
 names positive and negative crystals. The common axis of sphere 
 and spheroid is called the optic axis of the crystal, and all crystals 
 of this kind are described as uniaxal, to distinguish them from 
 those of a kind dealt with later. It will be observed that in the 
 direction of the optic axis there is only one value of v, so that in 
 this direction there is only one wave propagated, and so only one 
 refracted wave corresponding to an incident one. For all other 
 directions, however, there are two waves propagated, and this 
 phenomenon is spoken of as double refraction. 
 
 We have seen that, for the ordinary wave, v and therefore also 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 135 
 
 fju (the refractive index) is a constant, independent of the direction 
 of propagation. This deduction from theory has been completely 
 verified by experiment, the differences in the estimates of the 
 refractive index from theory and experiment being, with careful 
 experiments, never greater than '00002, and well within the limits 
 of experimental errors. 
 
 For the extraordinary wave, if be the angle between the 
 wave normal and the optic axis, we have 
 
 cos = 1, so that v* = c 2 cos 2 6 + a 2 sin 2 0, 
 
 1 cos 2 sin 2 6 
 
 or = -- H . 
 
 P" Pi V* 
 
 Here p is the refractive index for a wave whose normal is in the 
 direction 0, and /^ and /u 2 are the principal refractive indices, i.e. the 
 indices for waves whose normals are parallel and perpendicular 
 respectively to the optic axis. This relation was put to the test 
 of experiment by Stokes and Glazebrook. Stokes found that the 
 difference between theory and experiment in the estimate of ft did 
 not exceed a unit in the fourth place of decimals, and this result 
 was confirmed by Glazebrook, who found that the mean dis- 
 crepancy in sixty experiments was '000055, irrespective of sign 
 Glazebrook experimented with different colours (the rays G, F&ndg) 
 and concluded that in all cases the differences between theory and 
 experiment were comparable with the probable errors of experi- 
 ment. The following are his results (p. 136) in the case of the ray F. 
 
 These results might be represented graphically, but unless a 
 very large scale were used the differences between theory and 
 experiment could not be made visible, since the radii vectores of 
 the two curves corresponding to theory and observation would not 
 differ by more than one ten-thousandth part of either. 
 
 Returning to the consideration of the general case where a, b, 
 and c are all different, we can find the equation of the Index 
 Surface by making the same substitutions as before. 
 
 We thus get 
 
 a?/(I - aV) + i/ 2 /(l - 6 2 r 2 ) + * 2 /(l - c 2 ?' 2 ) = 0. 
 
 The sections of this surface by the coordinate planes are easily 
 proved to consist of a circle and an ellipse in each case. Parts of 
 these sections are represented in Fig. 51, for the case in which 
 
136 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 a > b > c. We have seen that, in general, there are two different 
 velocities for each direction of the wave normal, or what is the 
 
 8 
 
 (theory) 
 
 /* 
 (exp.) 
 
 Differ- 
 ence 
 
 e 
 
 (theory) 
 
 i 
 (exp.) 
 
 Differ- 
 ence 
 
 2'40" 
 
 1-66779 
 
 1-66780 
 
 + 00001 
 
 4646' 2" 
 
 1-56645 
 
 1-56653 
 
 + -00008 
 
 4 19 58 
 
 1-66660 
 
 1-66663 
 
 -f -00003 
 
 49 23 10 
 
 1-55861 
 
 1-55876 
 
 + -00015 
 
 7 51 58 
 
 1-66387 
 
 1-66385 
 
 - -00002 
 
 5242 6 
 
 1-54902 
 
 1-54914 
 
 + -00012 
 
 11 23 12 
 
 1-65967 
 
 1-65978 
 
 + -00011 
 
 58 39 10 
 
 1-53303 
 
 1-53312 
 
 + -00009 
 
 17 826 
 
 1-64987 
 
 1-64996 
 
 + -00009 
 
 61 39 33 
 
 1-52570 
 
 1-52573 
 
 + -00003 
 
 2026 1 
 
 1-64279 
 
 1-64287 
 
 + -00008 
 
 63 9 6 
 
 1-52228 
 
 1-52241 
 
 + -00013 
 
 23 50 45 
 
 1-63451 
 
 1-63455 
 
 + -00004 
 
 66 14 27 
 
 1-51579 
 
 1-51571 
 
 - -00008 
 
 25 49 35 
 
 1-62934 
 
 1-62930 
 
 - -00004 
 
 72 18 55 
 
 1-50476 
 
 1-50475 
 
 - -00001 
 
 29 18 42 
 
 1-61965 
 
 1-61974 
 
 + -00009 
 
 75 36 18 
 
 1-50009 
 
 1-50005 
 
 - -00004 
 
 3448 
 
 1-60336 
 
 1-60336 
 
 00000 
 
 79 6 26 
 
 1-49612 
 
 1-49610 
 
 - -00002 
 
 35 58 47 
 
 1-59048 
 
 1-59058 
 
 + -00010 
 
 80 14 4 
 
 1-49507 
 
 1-49507 
 
 00000 
 
 40 49 21 
 
 1-58478 
 
 1-58487 
 
 + -00009 
 
 87 640 
 
 1-49112 
 
 1-49114 
 
 + -00002 
 
 45 45 57 
 
 1-57000 
 
 1-57014 
 
 + -00014 
 
 8949 6 
 
 I 
 
 1-49074 
 
 1-49074 
 
 00000 
 
 Fig. 51. 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 137 
 
 same thing, that any radius vector cuts the Index Surface in two 
 points. An inspection of the figure shows, however, that there are 
 exceptions to this general rule. In the plane y = the circle and 
 the ellipse intersect in the points F and F' , where 
 
 zjx= 
 
 and for each of the directions OF and OF' there is only one index 
 of refraction, and only one wave, and this wave moves with velocity 
 b. Moreover it is easy to prove that OF and OF' are the only 
 two directions in which the two velocities of wave propagation are 
 equal. For solving the equation 
 
 l 2 /(v 2 - a 2 ) + m 2 /(v 2 - b 2 ) + n 2 /(v 2 - c 2 ) = 0, 
 and putting 
 
 p (b 2 - c 2 ) = \ , m 2 (a 2 - c 2 ) = ^ , n 2 (a 2 - b 2 ) = ^ 
 for brevity, where X, //,, v are all positive, we get 
 
 2y2 /2 (2 + C 2) _|_ m 2 ( C 2 + ^2) + tf ^2 _ fr} + V(Xj - ft - I 
 
 The condition for equal velocities is 
 
 (\i-fjLi Vi) 2 + 4X 1 /x 1 = 0, 
 and since \ and ^ are positive this requires 
 
 \ lfjLl = 0, and Xj ^ v 1 = 0, 
 
 so that we must have ^ = 0, and \ l =v 1) i.e. m = 0, and 
 n/l = 
 
 The two lines OF and OF' are called the optic axes of the crystal, 
 or the axes of single wave velocity, and crystals with two such 
 axes are styled Iriaxal crystals. In Fig. 51 the axis of x is the 
 bisector of the acute angle between the optic axes, so that in this 
 case this bisector coincides with the least axis of the ellipsoid 
 
 In such circumstances the crystal is said to be negative, while if 
 the bisector coincides with the greatest axis the crystal is called 
 positive. If &) be the angle between the optic axes we have 
 
 tan to/2 = tan FOx = V(6 2 - c 2 )/(a 2 - &*), 
 so that the crystal is positive or negative as (b 2 c 2 )/(a 2 b 2 ) is 
 
138 
 
 PROPAGATION OF LIGHT IN 
 
 CH. 
 
 greater or less than unity, i.e. according as 26 2 is greater or less 
 than a 2 + c 2 . This formula for the angle between the optic axes 
 can be tested by experiment, for the optical constants a, 6, c as 
 well as the angle w can all be determined experimentally. The 
 constants a, b, c vary with the colour of the light employed, so 
 that the angle between the optic axes varies also with the colour, 
 a fact known as the dispersion of the optic axes. The following 
 table gives the value of co calculated from the above formula, and 
 compared with its measure obtained experimentally in the case of 
 aragonite for different rays of the spectrum. 
 
 Kay 
 
 w (observed) 
 
 w (theory) 
 
 Difference 
 
 Percentage 
 difference 
 
 B (red) 
 
 18 5' 
 
 17 58' 
 
 7' 
 
 0-6 
 
 E 
 
 18 17' 
 
 18 3' 
 
 14' 
 
 1-3 
 
 H (violet) 
 
 18 40' 
 
 18 27' 
 
 13' 
 
 1-2 
 
 A very slight error in the estimate of the optical constants 
 would account for the small differences between observation and 
 theory. The test is a delicate one, and the agreement satisfactory. 
 
 The experimental investigation of the form of the index 
 surface has been undertaken by Glazebrook, and he has compared 
 
 e 
 
 Mi 
 (theory) 
 
 Mi 
 (exp.) 
 
 Differ- 
 ence 
 
 e 
 
 M2 
 
 (theory) 
 
 M 2 
 (exp.) 
 
 Differ- 
 ence 
 
 
 
 1-68123 
 
 1-68119 
 
 - -00004 
 
 
 
 1-68506 
 
 1-68500 
 
 - -00006 
 
 2 9'14" 
 
 1-68124 
 
 1-68125 
 
 + -00001 
 
 311'24" 
 
 1-68578 
 
 1-68578 
 
 00000 
 
 4 14 15 
 
 1-68125 
 
 1-68130 
 
 + -00005 
 
 5 14 5 
 
 1-68567 
 
 1-68568 
 
 + -00001 
 
 7 1327 
 
 1-68123 
 
 1-68119 
 
 - -00004 
 
 7 12 12 
 
 1-68514 
 
 1-68517 
 
 + -00003 
 
 9 5943 
 
 1-68118 
 
 1-68123 
 
 + -00005 
 
 95926 
 
 1-68371 
 
 1-68376 
 
 + -00005 
 
 12 7 8 
 
 1-68096 
 
 1-68104 
 
 + -00008 
 
 11 43 33 
 
 1-68251 
 
 1-68226 
 
 - -00025 
 
 14 47 18 
 
 1-67893 
 
 1-67918 
 
 + -00025 1 
 
 14 4 34 
 
 1-68152 
 
 1-68122 
 
 - -00030 
 
 17 18 49 
 
 1-67574 
 
 1-67591 
 
 + -00017 
 
 17 44 11 
 
 1-68137 
 
 1-68134 
 
 - -00003 
 
 19 29 6 
 
 1-67247 
 
 1-67274 
 
 + -00027 
 
 18 59 25 
 
 1-68136 
 
 1-68127 
 
 - -00009 
 
VI] 
 
 TRANSPARENT CRYSTALLINE MEDIA 
 
 139 
 
 the results of theory and experiment for a number of arcs taken 
 from widely different portions of the surface. The table (p. 138) 
 gives some of his results, ^ and p^ being the refractive indices 
 of the inner and outer sheets of the index surface, and the angle 
 that the wave normal makes with a fixed line in the plane of the 
 section under discussion. 
 
 The greatest difference is less than 1 in 6000, so that the 
 discrepancy between theory and observation would not be made 
 manifest to the eye by drawing, unless a very large scale were 
 used. The average difference is very much less than this, and is 
 not far from the limit of experimental error, while most of the 
 differences are within that limit. 
 
 Another arc examined by Glazebrook gave the following 
 results : 
 
 I 
 
 Mi 
 (theory) 
 
 Mi 
 (exp.) 
 
 Differ- 
 ence 
 
 6 
 
 M2 
 
 (theory) 
 
 M-2 
 
 (exp.) 
 
 Differ- 
 ence 
 
 
 
 1-53014 
 
 1-53014 
 
 00000 
 
 
 
 1-68242 
 
 1-68218 
 
 - -00024 
 
 326'53" 
 
 1-53016 
 
 1-53010 
 
 - -00006 
 
 3 0'40" 
 
 1-68263 
 
 1-68254 
 
 - -00009 
 
 7 54 17 
 
 1-53019 
 
 1-53013 
 
 - -00006 
 
 656 11 
 
 1-68292 
 
 1-68289 
 
 - -00003 
 
 17 43 19 
 
 1-53028 
 
 1-53028 
 
 00000 
 
 10 18 37 
 
 1-68317 
 
 1-68336 
 
 + -00019 
 
 22 45 50 
 
 1-53035 
 
 1-53036 
 
 + -00001 
 
 13 842 
 
 1-68339 
 
 1-68363 
 
 + -00024 
 
 26 58 44 
 
 1-53043 
 
 1-53045 
 
 + -00002 
 
 18 45 47 
 
 1-68383 
 
 1-68421 
 
 + -00038 
 
 For the inner sheet the agreement is very close, the differences 
 being all within the limits of experimental error. The average 
 difference, irrespective of sign, is about one in 60,000, and the 
 greatest difference one in 25,000. For the outer sheet the 
 agreement is less satisfactory, the average difference irrespective 
 of sign being about one in 9000, and the greatest difference one 
 in 4000. 
 
 A third arc gave the following results : 
 
140 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 
 
 Mi 
 (theory) 
 
 (exp.) 
 
 Differ- 
 ence 
 
 
 
 /*2 
 
 (theory) 
 
 A) 
 
 Differ- 
 ence 
 
 
 
 1-68103 
 
 1-68099 
 
 - -00004 
 
 
 
 1-68533 
 
 1-68526 
 
 - -00007 
 
 312'50" 
 
 1-67714 
 
 1-67721 
 
 + -00007 
 
 7 9'10" 
 
 1 -68465 
 
 1-68454 
 
 - -00011 
 
 13 620 
 
 1-66298 
 
 1-66300 
 
 + -00002 
 
 17 2 40 
 
 1-68445 
 
 1-68448 
 
 + -00003 
 
 21 430 
 
 1-64607 
 
 1-64603 
 
 - -00004 
 
 25 050 
 
 1-68443 
 
 1-68452 
 
 + -00009 
 
 28 14 10 
 
 1-62824 
 
 1-62807 
 
 - -00017 
 
 32 10 30 
 
 1-68443 
 
 1 -68447 
 
 + -00004 
 
 35 29 20 
 
 1-60900 
 
 1-60897 
 
 - -00003 
 
 38 27 30 
 
 1-68444 
 
 1-68453 + -00009 
 
 45 14 50 
 
 1-58363 
 
 1-58365 
 
 + -00002 
 
 49 13 
 
 1-68445 
 
 1-68457 + -00012 
 
 60 1 30 
 69 37 40 
 
 1-55154 
 
 1-53784 
 
 1-58157 
 1-53774 
 
 + -00003 
 - -00010 
 
 63 59 30 
 
 73 35 50 
 
 i 
 
 1-68447 
 1-68448 
 
 1-68452 
 1-68444 
 
 + -00005 
 - -00004 
 
 In this case we have a large arc over which, for both the inner 
 and the outer sheets of the index surface, the agreement between 
 theory and experiment is remarkably close. In view of all the 
 results given, there can be no doubt that the theory represents 
 the facts, at any rate as an excellent first approximation, which is 
 all that need be claimed for it. 
 
 Feeling sure that we are on solid ground, in so far as this may 
 be tested by experiments such as those just referred to, we may 
 proceed to further developments of the theory. 
 
 (1) The equations (II) and (VI) above (p. 132) are those 
 obtained in Solid Geometry when investigating the direction 
 cosines (X, //,, v) of the principal axes of a central section of the 
 ellipsoid 
 
 made by the plane 
 
 Ix + my + nz = 0. 
 
 We see then that the curl of the displacement is in the direction 
 of the principal axes of this section. 
 
 (2) It follows from (1) that the two waves propagated in any 
 given direction are plane polarised, their planes of polarisation 
 being at right angles. The use of doubly refracting media is, in 
 fact, one of the most usual means of obtaining plane polarised 
 light. 
 
VI] 
 
 TRANSPARENT CRYSTALLINE MEDIA 
 
 141 
 
 (3) In the case of waves moving in the direction of an optic 
 axis the two velocities are the same, so that the principal axes of 
 the section of the ellipsoid aV 4- 6 2 y 2 + c*z 2 = 1 by the wave front 
 Ix + my -f nz = are equal. The section is therefore a circle, and 
 the optic axes are the normals to the circular sections of the 
 ellipsoid. 
 
 (4) Let R'PRQ be a section of the ellipsoid by the wave 
 front, ON the wave normal, OF and OF' the optic axes. If OR and 
 
 R' 
 
 Fig. 52. 
 
 OR be sections of the wave front by the circular sections normal 
 to OF and OF' respectively, then we have OR = l/b = OR. As 
 the principal axes of an ellipse bisect the angles between a pair 
 of equal radii vectores, we see from (1) that the directions of the 
 curls are the bisectors of the angles between OR and OR. 
 
 (5) Let >/r and i/r' be the angles between the optic axes and 
 the curl OP, 6 and 6' the angles between the wave normal ON 
 and the optic axes OF and OF', 2^ the angle between the planes 
 ONFandONF'. 
 
 N 
 
 Fig. 53. 
 
 It follows from (4) that PNF= PNF' = % . 
 
142 PROPAGATION OF LIGHT IN [CH. 
 
 The direction cosines of the optic axes are 
 
 (a 2 - 6 2 )/(a 2 - c 2 ), 0, V(6 2 - c 2 )/( a 2 - c 2 ). 
 Hence we have, from (V) p. 132, 
 = a 2 X 2 + &> 2 + cV = 6 2 + X 2 (a 2 - 6 2 ) - v 8 (6 2 - c 2 ) 
 
 = 6 2 + (a 2 - c 2 ) [X V(a 2 - & 2 )/(a 2 - c 2 ) -f i; V(6 2 - c 2 )/(a 2 - c 2 )] 
 
 [X \/(a 2 - 6 2 )/(a 2 - c 2 ) - v V(6 2 - c 2 )/(a 2 - c 2 )] 
 
 = 6 2 + (a 2 C 2 ) COS i/r COS fa. 
 
 Let Vj and v 2 be the two velocities of waves travelling in the 
 direction ON, and fa, fa, fa', fa' be the corresponding angles as 
 defined above. We then have 
 
 ^2 + V2 2 = 26 2 + (a 2 - c 2 ) (cos ^ cos i/r/ + cos fa cos -i/r/), 
 and VT* - v<? = (a 2 c 2 ) (cos fa cos fa cos i/r 2 cos fa'). 
 From Fig. 53 we see that 
 
 cos fa = sin 6 cos ^, cos fa' sin 0' cos %, 
 
 cos i|r 2 = sin 6 sin ^, and cos -\|r 2 ' = sin 6' sin ^. 
 Making use of these relations we get 
 
 ^2 _ v j = (a? - c 2 ) sin sin 0', 
 and v a 2 + v 2 2 = 26 2 + (a 2 c 2 ) sin sin 7 cos 2%. 
 
 The first of these relations represents a law first suggested by 
 Biot as a generalisation from his experimental results. 
 We also have 
 
 cos FF' = (a 2 - 6 2 )/(a 2 - c 2 ) - (6 2 - c 2 )/(a 2 - c 2 ) 
 
 = (a 2 - 26 2 4- c 2 )/(a 2 - c 2 ) = cos cos 6' + sin sin 0' cos 2%. 
 
 Thus finally we get 
 
 2^ 2 = a 2 + c 2 - (a 2 - c 2 ) cos (0 + 0'), 
 2v 2 2 = a 2 + c 2 - (a 2 - c 2 ) cos (0 - 0'), 
 or v 2 = c 2 + (a 2 - c 2 ) sin 2 (0 7 )/2, 
 
 which gives the velocities and refractive indices in a form con- 
 venient for comparison with experiment. 
 
VI] TKANSPARENT CRYSTALLINE MEDIA 143 
 
 (6) The results of the last section can also be very simply 
 obtained by the aid of the elementary geometry of sphero-conics. 
 If we regard v as a constant, and (I, ra, n) as the coordinates of 
 a point, then the equations 
 
 Z 2 + m 2 + n? = 1, 
 
 and Pl(v* - a 2 ) + m 2 /0 2 - 6 2 ) + n*/(v* - c 2 ) = 
 
 represent a sphere and a cone respectively, and their intersection 
 a spherical ellipse. Of this ellipse F and F' are the foci, C, the 
 middle point of FF', the centre, and 6 + 6' the axis major. As 
 the sum of the focal distances is constant for all points of the 
 ellipse, we have (6 + 0')/2 = CN, when N is in the plane containing 
 FF' t i.e. in the plane y = 0. Hence putting m = we get 
 
 / 2 /<> 2 - a 2 ) + n 2 /(v* - c 2 ) = 0, 
 
 or (v* - a 2 )/(c 2 - v*) = l 2 /n 2 = tan 2 (6 + ff )/2, 
 
 so that 2v 2 = a 2 + c 2 - (a 2 - c 2 ) cos (0 + 6'). 
 
 The other value of v 2 is obtained by interchanging a and c, and 
 replacing 6' by TT 6'. A third method of obtaining these results 
 will be employed when dealing with ray velocities (17 below). 
 
 (7) A geometrical construction for the direction of the curl 
 is given in (1) above. If we wish to obtain an analytical 
 expression for this when the velocity (v) of the wave, and the 
 direction (I, m, n) in which it is travelling are given, we have this 
 in the fundamental relation (III, p. 132), 
 
 X (a 2 - v z )/l = fj, (6 2 - v 2 )/m = v (c 2 - v*)/n, 
 
 which determines X, //,, v. 
 
 So far we have been occupied entirely with the consideration 
 of waves propagated in a given direction, and we have seen that 
 the index surface lends itself very readily to the discussion of 
 such problems. If, however, we wish to introduce the idea of 
 rays, it is more convenient to employ another surface intimately 
 related to the index surface. This is the Wave Surface described 
 in Chapter II. It is the envelope of all positions of the wave 
 front at any time. If Q be a point on this surface and ON the 
 perpendicular from the origin on the tangent plane at Q, then ON 
 is the wave normal for the wave corresponding to the point Q, and 
 
144 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 ON= v. A point P being taken on ON such that OP. ON = I 
 (taking the velocity in free ether as unity), we have OP l/v, so 
 that P is a point on the index surface as defined above. Thus 
 
 Fig. 54. 
 
 the wave surface is the polar reciprocal of the index surface with 
 respect to a unit sphere whose centre is 0. Its equation is easily 
 obtained by the methods usually employed for finding the polar 
 reciprocal of a given surface. We have to find the envelope of 
 
 las + my + nz v, 
 subject to the relations 
 
 I 2 + m 2 -f 7i 2 = 1, 
 
 and l*/(v 2 - a 2 ) + m*/(v 2 - 6 2 ) + w 2 /<> 2 - c 2 ) = 0. 
 
 Differentiating these equations we get 
 
 xdl + ydm 4- zdn dv = 0, Idl + mdm + ndn = Q, 
 ldl/(v z - a 2 ) + mdm/(v* - 6 2 ) + ndn/(v* - c 2 ) - vdv [7 2 /(t> 2 - a 2 ) 2 
 
 + ra 2 /(V - 6 2 ) 2 + ?i 2 /0 2 - c 2 ) 2 ] = 0. 
 From these, using undetermined multipliers, we obtain 
 
 x + Al + Bll(v* - a 2 ) = 0, y + Am + Bm/(v 2 - b 2 ) = 0, 
 
 and 1 + Bv [7 2 /0 2 - O 2 + m 2 /(u 2 - bj + rc 2 /0 2 - c 2 ) 2 ] = 0. 
 Multiplying the first of these equations by I, the second by m, 
 the third by n, and adding we get v + A = 0. Transposing the 
 third terms of these same equations, squaring and adding, and 
 putting x 2 + y* + z' 2 = r 2 , we get 
 
 r 2 + 2Av + A* = B 2 [ 2 /<y - a 2 ) 2 + m 2 /(fl 2 - 6 2 ) 2 + ^ 2 /(v 2 - c 2 ) 2 ], 
 so that r*-v*=-Bv. 
 
VI] 
 
 TRANSPARENT CRYSTALLINE MEDIA 
 
 145 
 
 Hence x = - A I - Bl/(v* - a 2 ) = lv + lv (r 2 - v 2 )/(v 2 - a 2 ) 
 
 = Iv . (r 2 - a 2 )/0 2 - a 2 ), 
 and similarly for y and z. 
 Thus we have 
 
 lv = x. O 2 - a 2 )/(r 2 - a 2 ) = x + a? O 2 - r 2 )/(r 2 - a 2 ), 
 and so for mv and nv. Hence 
 v z = v (Ix -f my + nz) 
 
 or 2 /(r 2 - a 2 ) + y 2 /(r 2 - 6 2 ) + * 2 /(r 2 - c 2 ) = 1, 
 
 which is the equation of the wave surface*. 
 
 The geometrical peculiarities of the wave surface have been 
 very thoroughly investigated, having attracted the attention of 
 a number of mathematicians of the first rank. It would, however, 
 be outside the purpose of this book to enter at any length into 
 this subject. We are concerned with the wave surface only so far 
 as its properties help us to understand the optical behaviour of 
 crystals, and for this end only the elements of the geometry of the 
 surface are required. 
 
 (8) We have seen that the wave surface is the polar reciprocal 
 of the index surface, and from this fact or from the equation just 
 
 Fig. 55. 
 
 obtained, the general form of the surface can be readily imagined. 
 It consists of two sheets symmetrical about the coordinate planes. 
 
 * This equation can also be readily obtained by means of a few elegant theorems 
 as to apsidal surfaces due to MacCullagh, and contained in the first paper of his 
 Collected Works, and also in Salmon's Geometry of Three Dimensions, Chap. xiv. 
 
 M. L. 10 
 
146 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 The sections of the surface by these planes are ellipses and circles, 
 which intersect in real points E and E' in the plane y = 0, when 
 a > b > c. The equations to the lines joining to these points 
 E and E' are 
 
 zjx = (a/c) . \/(6 2 - c 2 )/(a 2 - 6 2 ). 
 
 (9) The refracted rays corresponding to any incident ray are, 
 as explained in Chap, n., given by Huyghens' construction with 
 the aid of the wave surface. Suppose that a ray OA is incident 
 
 Fig. 56. 
 
 on a given plane which cuts the plane of the paper in OT. Let 
 this ray meet the sphere whose radius is V (the velocity of light 
 in the external medium) in A, then the tangent plane at A to 
 this sphere cuts the given plane in a line through T at right 
 angles to the plane of the paper. If through this line we draw 
 tangent planes to the wave surface whose centre is 0, these planes 
 will touch the surface in two points Q l and Q a , and Oft and OQ 2 are 
 the directions of the two refracted rays. 
 
 (10) From equation (III) p. 132, we have 
 
 l/\ = (a 2 - V*)/K, 
 
 where K. = a?l\ + b s mjj, + c*nv. 
 
 Also if L l ,M l , NI be the direction cosines of the ray corresponding 
 to the point (a, y, z) on the wave surface we have, by means of 
 the equations obtained above when finding the equation of the 
 wave surface, 
 
 Ljr = x = lv-lv(r*- v 2 )/(tt 2 -v*) = lv-v (r 2 - y 2 ) \/ K . 
 By elimination from this and the two similar equations, we get 
 
 L, I 
 
 m 
 n 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 147 
 
 which proves that the ray, the wave normal, and the curl are 
 coplanar. 
 
 (11) The direction of the curl in the wave corresponding to 
 the ray OQ is easily obtained. If N be the 
 foot of the perpendicular from on the wave Q\ iN 
 
 front, its coordinates are Iv, mv, and nv. Also 
 from the last section (10) we have 
 
 7 / o o\ -v / 
 
 Iv - x = v (r 2 - v 2 ) \IK, 
 
 so that the direction cosines of NQ, which 
 are proportional to Iv x, mv y, nv z, are 
 proportional to X, /z, v. Hence NQ is the 
 direction of the curl, and the curl is there- 
 fore directed along the projection of the ray on the wave front. 
 
 We see also from the above that 
 
 NQ = v (r 2 - v 2 )//c, 
 
 whence, since NQ = \/OQ 2 - ON' = vV 2 - v 2 , 
 
 we derive the relations 
 
 NQ = Vr 2 - v 2 = K/V. 
 
 (12) A useful formula for * 2 may be obtained as follows : 
 We have I 2 f(a 2 - v 2 ) + m 2 /(6 2 - v 2 ) + n?/(c 2 - v 2 ) = 0. 
 Putting a 2 v* = w we get a quadratic equation for w of the form 
 
 w 2 + Aw + l* (a 2 - b 2 ) (a 2 - c 2 ) = 0. 
 
 Hence if w-^ and w 2 be the roots of this equation, we have 
 I- = WjWzKa 9 - b 2 ) (a 2 - c 2 ) = (a 2 - v, 2 ) (a 2 - v*)/(a? - b 2 ) (a 2 - c 2 ). 
 
 Similar expressions for ra 2 and n 2 may be written down from 
 symmetry. 
 
 Also we have 
 
 (a 2 - v 2 ) \ = tcl, (6 2 -v~}fj, = /cm, (c 2 - v 2 ) v = /en. 
 Thus 
 
 [ll(a 2 - v 2 )] 2 + [m/(b 2 - v 2 )] 2 + [n/(c* - v 2 )] 2 , 
 
 102 
 
148 PROPAGATION OF LIGHT IN [CH. 
 
 so that we get 
 
 K * ( a _ 6 2 ) (a 2 - c 2 ) (ct 2 - *V 2 ) (6 2 - c 2 ) (6 2 - a 2 ) (6 2 - 
 
 c*-v 
 
 (c 2 -a 2 )(c 2 -6 2 )(c 2 -^ 2 ) 
 y* - vj 
 
 and 
 
 These formulae for K and K 2 can be put in another form by 
 means of the expressions for ^ and v 2 obtained on p. 142. From 
 these we have 
 
 ^2 _ ^2 = ( a 2 _ C 2) COS '2 (0 + 0^2, 
 
 and c 2 - v* = - (a 2 - c 2 ) sin 2 (0 + 6>')/2. 
 
 Whence we get 
 
 2/d = (a 2 - c 2 ) sin (0 + 0') cos ^, 
 and 2/c 2 = (a 2 - c 2 ) sin (0 - 0') sin % . 
 
 (13) The angle between the ray and the wave normal i 
 and we have 
 
 a 2 c 2 
 Hence tan NOQ l = *,/f^ = ~s i~ s ^ n (^ + ^') cos 
 
 and 
 
 (a 2 -c 2 ) sin (0 + 0') cosy 
 
 ~ a 2 4- c 2 - (a 2 - c 2 ) cos (0 + 0') 
 
 x . 
 a 2 + c 2 - (a 2 - c 2 ) cos (0 - ) 
 
 It should be noted that if N coincides with F or F' (Fig. 53), 
 the angle y is indeterminate, and these formulae become inde- 
 terminable. Also in the special case of a uniaxal crystal we have 
 = Q, and = 0', so that NOQ* = 0, and 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 149 
 
 (14) We have seen (p. 137) that, in general, any radius vector 
 to the index surface cuts that surface in two points. Recipro- 
 cating we see that in general there are two parallel tangent 
 planes to the wave surface for each direction, so that there are 
 two rays corresponding to a given wave normal, the directions 
 of these two rays being determined by the equations in the 
 last section (13). However, in the direction of an optic axis, 
 such as OF, there is only one radius vector to the index surface, 
 whence we see, by reciprocation, that for a wave normal in this 
 direction there is only one tangent plane to the wave surface. 
 On investigation it appears that this plane touches the surface 
 not in a single point, but in an infinity of points forming a circle. 
 Hence corresponding to a wave normal along the optic axis there 
 are not two rays, but an infinity of rays joining the origin to the 
 various points of this circle. This, which was first predicted from 
 theory*, has been verified by experiment, and is known as the 
 phenomenon of internal conical refraction. 
 
 From the investigation on p. 144 it appears that the co- 
 ordinates of a point on the wave surface may be written in the 
 form 
 
 x = lv (r- - tt 2 )/(y 2 - a 2 ), y = mv (r 2 - 6 2 )/(y 2 - 6 2 ), 
 
 z = nv (r 2 - c 2 )/(v 2 - c 2 ). 
 At the extremity of an optic axis we have y = and v = b, so that 
 
 #/(r 2 - a 2 ) = /6/(6 2 - a 2 ) = - 6/V(a 2 - 6 2 ) (a 2 - c 2 ), 
 and z/(r* - c 2 ) = nb/(b* - c 2 ) = b/\/(b 2 - c 2 ) (a 2 - c 2 ). 
 
 Hence the extremities of the rays, corresponding to a wave normal 
 along the optic axis, lie on the intersection of the spheres 
 
 r 2 - a 2 + x V(a 2 -6 2 )(a 2 -c 2 )/6 = 0, 
 and r 2 - c 2 - z V(6 2 - c 2 ) (a 2 - c 2 )/6 = 0. 
 
 The intersection of these spheres is a circle in the plane 
 
 = ! Thus the wave front touches the 
 
 , 
 
 c 2 b v a 2 c 2 
 
 surface at all the points of this circle. The cone of rays is formed 
 of lines from the origin to the points of intersection of the above 
 spheres and plane. The equation of this cone is obtained by 
 
 * By Sir William Hamilton. 
 
150 PROPAGATION OF LIGHT IN [CH. 
 
 forming a homogeneous equation of the second degree by the 
 aid of the equations of these surfaces. In this way we obtain 
 the equation 
 
 r 2 + - - (as Va 2 - 6 2 + z \/6 2 - c 2 ) 
 
 [x Va 2 - 6 2 + z V6 2 - c 2 ] 2 , 
 
 6 a (a 2 - 
 i.e. a 2 (fr 2 - c 2 ) x 2 + 6 8 (a 2 - c 2 ) ?/ 2 + c 2 (a 2 - &*) ^ 2 
 
 = xz (a 2 + c 2 ) V(a 2 -6 2 )(6 2 -c 2 ). 
 
 By considering the section -of this cone by the plane of sym- 
 metry y 0, we readily find that, if <I> be the vertical angle of the 
 cone, tan 3> = V(6 2 - c 2 ) (a 2 - 6 2 )/6 2 . 
 
 This result can also be obtained from an examination of a 
 section of the wave surface by the plane y = 0. This section 
 consists of the circle # 2 + 2 = 6 2 , and the ellipse a?/b* + z*/a?=l. 
 If x cos O l + z sin 1 = b be the common tangent to these two curves, 
 we have 6 2 = c 2 cos 2 6 l -f a 2 sin 2 0j, and therefore 
 
 tan^ = V(6 2 -c 2 )/(a 2 -6 2 ). 
 
 The coordinates of the point of contact of this tangent with the 
 ellipse are x = (c 2 /6) cos 6 l and z = (a-/b) sin l , so that if 2 be 
 the vectorial angle to the point of contact, we have 
 
 tan 2 = zjx = (a 2 /c) tan ^ = (a 2 /c 2 ) V(6^c 2 )/(a 2 - 6 2 ). 
 
 Hence tan <E> = tan (0 2 - ^) = V(6 2 "- c 2 ) (a 2 - 6 2 )/6 2 as before. The 
 angle ^E> can be measured experimentally, and the results obtained 
 are found to agree closely with those calculated from the formula 
 just obtained. In the original experiments made by Lloyd with 
 aragonite the angle of the cone was found ^^___^Q 
 
 to be 1 50'. Its magnitude as indicated f / >9 
 
 by the above theory was 1 55 X , the differ- 
 ence being within the limits of the experi- 
 mental errors. 
 
 The direction of the curl has been 
 proved in 1 1 above to be along the pro- 
 jection of the ray on the wave front. Fig. 58. 
 
 Hence the various rays forming the cone of internal conical 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 151 
 
 refraction are polarised in different azimuths NQ l} NQ ... (Fig. 
 58); so that if the incident light were plane polarised, there 
 would only be one refracted ray. The polarisation of the different 
 rays is found experimentally to be that predicted by the above 
 theory. 
 
 (15) The wave surface represents the confines of disturbances 
 that have travelled from the origin in a given time, so that its 
 radii vectores are proportional to the ray velocities. As the 
 surface consists of two sheets there are, in general, two ray 
 velocities for a given direction of the ray. An exception occurs, 
 however, for the rays OE and OE' (Fig. 55), the points E and E' 
 being singular points on the surface. Corresponding to each of 
 the rays OE and OE' there is only one ray velocity, and these 
 lines are consequently called the axes of single ray velocity, or 
 more briefly the ray axes. The angle EOF between a ray axis 
 and the corresponding optic axis is obtained from the formulae on 
 pp. 137 and 146. We have 
 
 a 2 - 6 2 ) (6 2 - c 2 ) 
 
 6* -foe 
 
 (16) Formulae for the ray velocities corresponding to those 
 for the wave velocities given on p. 142 can be found by methods 
 similar to those employed there. The equation of the wave 
 surface is a*x*/(r 2 - a 2 ) + b 2 f/(r* - 6 2 ) + c' 2 2/(r z - c 2 ) = 0, where 
 r 2 = x z + y 2 - -f z~. If we regard r as constant, these two equations 
 represent a cone and a sphere, and their intersection is a spherical 
 ellipse of which E and E' are the foci. If $ and $>' be the angles 
 that a ray makes with the axes of single ray velocity, then, by the 
 properties of a spherical ellipse, <p + <' is constant and equal to 
 the axis major. By taking the special case when y = 0, we get 
 
 a*a?l(i* - a 2 ) + c 2 ^ 2 /(r 2 - c 2 ) = 0, 
 
 r 2 - a 2 a 2 # 2 a 2 , , d> + <>' 
 whence - = - = tan 2 - y . 
 
 c 2 r 2 c 2 ^ 2 c 2 2 
 
 This gives 2/r 2 = I/a 2 + 1/c 2 + (I/a 2 - 1/c 2 ) cos (< + <f/), and the 
 other value of 2/r 2 is obtained by interchanging a and c, and 
 
152 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 replacing </>' by TT </>'. Also since the normal to a spherical 
 ellipse bisects the angle between the focal radii, we see that the 
 
 Fig. 59. 
 
 curls corresponding to the two rays in a given direction OQ lie in 
 planes that bisect the angles between the planes through OQ and 
 the ray axes OE and OE r (Fig. 59). 
 
 (17) The formula for r in the last paragraph can also be 
 obtained without the aid of the geometry of sphero-conics. If 
 L lt M lt N-L be the direction cosines of the ray we have, from the 
 equation of the wave surface, 
 
 a 2 i 1 2 /(r ! - a 2 ) + &M?l(i* - 6 2 ) + c 2 ^ 2 /(r 2 - c 2 ) = 0, 
 and if rf and r 2 2 are the roots of this equation 
 
 l/r 1 2 +l/r 2 2 = S(l/6 2 + l/c 2 )J r : i 2 , and I/rfrf = 2 (A 2 /& 2 c 2 ). 
 The direction cosines of the ray axes are 
 
 (c/6) V(a 2 - 6 2 )/(a 2 - c 2 ), 0, (a/6)\/(6 2 -c 2 )/(a 2 -c 2 ). 
 Hence we have 
 
 COS 
 
 and 
 
 (c/6) V(a^ - 6 2 )/(a 2 - c 2 ) + ^ (a/b) V(6 2 - c 2 )/(a 2 - c 2 ), 
 
 cos </>' = Z x (c/b) V(a 2 - 6 2 )/(a 2 - c 2 ) - 
 T _ cos 4- cos </>' 6 
 
 A ~" ^T~ "c 
 
 Thus 
 and 
 
 r x (a/6)\/(6 2 - 
 V - c 2 
 
 a 2 - c 2 ). 
 
 cos <ft cos eft 
 
 ' 6 /a 2 c 2 
 a\i W^' 
 
VI] TRANSPARENT CRYSTALLINE MEDIA 153 
 
 By the aid of these relations we get 
 
 1/n 2 + l/r 2 2 = (1/6 2 + 1/c 2 ) Lf + (1/c 2 + I/a 2 ) M? + (I/a 2 + 1/6 2 ) N^ 
 = (1/6 2 + 1/c 2 ) A 2 + (1/c 2 + I/a 2 ) (1 - A 2 - ^ 2 ) 
 
 = I/a 2 + 1/c 2 + (I/a 2 - 1/c 2 ) cos <f> cos <', 
 and 4/rM = 4/a 2 c 2 + (I/a 2 - 1/c 2 ) 2 (cos 2 <f> + cos 2 0') 
 
 + 2 (I/a 4 - 1/c 4 ) cos < cos 0', 
 
 so that 1/rj 2 l/r 2 2 = (I/a 2 1/c 2 ) sin < sin </>', which in conjunction 
 with the formula for 1/r^ 4- l/r 2 2 yields the same results as before. 
 
 (18) The direction of a ray on emerging from a crystal is 
 determined by the position of the tangent plane to the wave 
 surface at the point where the ray meets that surface. For an 
 ordinary point there is only one tangent plane, so that the 
 direction of the emerging ray is quite definite. The points E and 
 E r , however, are singular points on the wave surface. At such a 
 point there is not a single tangent plane and a single normal, but 
 an infinity of tangent planes enveloping a tangent cone and an 
 infinity of normals forming a normal cone. It follows, then, that 
 to a single ray within the crystal in the direction of a ray axis, 
 there corresponds a cone of emerging rays, each ray having a 
 different plane of polarisation. This prediction from theory has 
 been verified by experiment, and the phenomenon thus described 
 is known as external conical refraction. 
 
 To investigate the form of the normal cone we have to make 
 use of the formulae for the coordinates of any point of the wave 
 surface, 
 
 <c = lv (r 2 - a 2 )/(v 2 - a 2 ), y = mv(r>- b 2 )/(v* - 6 2 ), 
 and z = nv (r 2 - c 2 )/(v 2 - c 2 ). 
 
 At E a point on the ray axis we have y = 0, and 
 
 # = c \ / (a a -6 2 )/(a 2 -c 2 ), z = a V(6 2 - c 2 )/(a 2 - c 2 ). 
 Substituting these in the above equations for x and z, we get 
 v a - a 2 + (Iv/c) v ; (a 2 -6 2 )/(a 2 -c 2 ) = 0, 
 
 and v* - c 2 - (nv/a) V(6 2 - c 2 )/(a 2 - c 2 ) = 0. 
 
 On eliminating v from these two equations we obtain 
 
 Z 2 (a 2 - 6 2 ) + ?i 2 (6 2 - c 2 ) + In V(a 2 - 6 2 ) (6 2 - c 2 ) (a 2 + c 2 )/ac = a 2 - c 2 . 
 
154 
 
 PROPAGATION OF LIGHT IN 
 
 [CH. 
 
 In this equation I, m, n are the direction cosines of any normal 
 through E, so that the equation of the normal cone referred 
 to E as origin is 
 
 = xz V(a 2 - 6 2 ) (6 2 - c 2 ) (a 2 4- c 2 )/ac. 
 
 By considering the section of this cone by the plane of symmetry 
 (y = 0), we find that the angle "SP of the cone is given by the 
 
 This can also be obtained 
 
 formula tan ^ = \l(b^- c 2 ) (a 2 - 6 2 )/ac. 
 
 by an examination of the section of the wave surface by the plane 
 y = Q. The section consists of a circle and an ellipse, and the 
 angle of the cone is the angle between the normals to these two 
 curves at their point of intersection. 
 
 (19) The normal cone discussed in the last paragraph is not 
 the same as the cone of rays that emerge from the crystal. This 
 latter cone is obtained by means of Huyghens' construction and, 
 by way of illustration, we shall use this construction to find 
 the section of the cone of rays by the plane of symmetry. The 
 section of the wave surface in the crystal by this plane of symmetry 
 consists, as has been seen, of a circle and an ellipse intersecting at 
 E (Fig. 60). 
 
 Fig. 60. 
 
 Let OT^T<i be the section of the surface of the crystal by the 
 plane y = 0, and let the tangents at E to the circle and the ellipse 
 meet this line in Ji and T 2 . From these points tangents T^ and 
 T 2 Q 2 ar to be drawn to the circle of radius V, whose centre is at 
 0, this circle being the section of the spherical wave surface for the 
 waves in the external medium, where the velocity of propaga- 
 tion is V. Then, by the principles explained when describing 
 
Vl] TRANSPARENT CRYSTALLINE MEDIA 155 
 
 -Huyghens' construction, OQ l and OQ 2 are the emergent rays in 
 the plane under consideration, and as this plane is the plane 
 of symmetry the angle between these lines is the angle of the 
 cone of external conical refraction. It is thus a matter of ele- 
 mentary geometry to calculate the angle of this cone for any given 
 circumstances. Thus when the surface of the crystal is perpen- 
 dicular to the ray axis OE, we find the angle given by 
 
 sin x = FV(a 2 - 6 2 ) (6 2 - tf)/abc. 
 
 The polarisation of the different emergent rays is found experi- 
 mentally to agree with the theory, and there is also a good 
 agreement with reference to the angle of the cone of external 
 conical refraction. The mean of Lloyd's measurements on arago- 
 nite gave ^ = 2 59', and the calculated value was 3 I/, the 
 difference being within the limits of the errors of experiment. 
 
 (20) Before closing this chapter we shall investigate the 
 direction of the flow of energy at any point of the disturbance in a 
 crystalline medium. We have 
 
 ^=i/(f + i* + ?)dT, and W = $f(a*f* + &f + c ^ 2 ) dr - 
 As (f, g, h) is the curl of (f , 77, f) we may write W in the form 
 
 whence, integrating by parts, we get 
 
 n, (a?fy - %)] dS, 
 where (l l} m l , n^) are the direction cosines of the outward normal 
 to the element dS of the surface bounding the region under 
 consideration. 
 
 From the dynamical equations we have f = 6 2 TJ- c 2 z , so that 
 
 if P/%TT be the frequency, we have p 2 f = c 2 ^ -- ^ 2 p) ' 
 equations for 17 and f. Hence 
 
156 TRANSPARENT CRYSTALLINE MEDIA [CH. VI 
 
 If then E be the whole energy, so that E T+ W, we have 
 
 E = - i/R (%- ^-hrj) + Wl (c 2 A? - a 2 /?) + ^ (a 2 /?? - fyf )] rf& 
 From the formula on pp. 146 and 132 we have h^vr = v 2 / /eX, and 
 v 2 a 2 = ;//X, so that 
 
 <y 2 X = a 2 X tcl, v 2 fj, = 6 2 //, /cm, v^v = c 2 v ten. 
 Multiplying the second of these by N, the third by M, and 
 subtracting, we get 
 
 v 2 (fiN - vM) = 6>JV - c?vM + (^fn - #m). 
 
 But the lines (^, m, ri), (X, /it, i^), and (L, M, N) are mutually at right 
 angles, so that we have I =,^N vM, X = Mn Nm, and other 
 similar equations. Hence 
 
 tfl = b*pN-c*vM+K\, and h l w = lPpN-&vM. 
 If then, as before, A is the amplitude of (, 77, f), and 5 of the curl 
 (/, ^r, 7i), we have b z g - c^hrj = 45 (6>^ - c 2 ^if) = L.vrAB. We 
 thus get 
 
 JS=- ^jABvr (L& + Jlf^tj + NM) dS = - %/ABvr cos e. dS 
 where e is the angle between the ray (L lt M l} N^ and the normal 
 to the bounding surface (l lt m l , n^. The rate at which energy is 
 
 7 -ri 
 
 flowing out of this surface is -j- = ipE, so that this rate is 
 
 at 
 
 proportional to fABvr cose.dS. 
 
 Since the component of the flux along the normal to the 
 surface is proportional to the cosine of the angle that this 
 normal makes with the ray, we see that the energy may be 
 regarded as flowing along the ray. Thus, with crystalline as well 
 as with isotropic media, the two definitions of a ray suggested in 
 Chapter n. are really identical. 
 
CHAPTER VII. 
 
 REFLECTION AND REFRACTION WITH 
 TRANSPARENT CRYSTALS. 
 
 IN the last chapter we have dealt with the problem of 
 crystalline reflection and refraction in so far as we have obtained 
 geometrical relations between the wave normal, the refracted ray, 
 and the direction of displacement. The results thus obtained will 
 be required now, when we seek a fuller solution of the problem, and 
 attempt to find the intensity and azimuth of the displacement for 
 all the rays involved. This is solved in the same way as the 
 corresponding problem for isotropic media, by making use of the 
 boundary conditions suggested by the fundamental Principle of 
 Action. We have seen that these boundary conditions are satisfied 
 when (f, 77, f) and the tangential components of the vector 
 (a 2 /, b 2 g, crh) are continuous at an interface, and it will appear 
 on trial that these conditions lead to only four independent 
 equations. 
 
 If A be the amplitude of the displacement (f, rj, f), so that for 
 progressive waves A 2 is proportional to the intensity of the light, 
 we have, in the notation already adopted, 
 
 f = ALe iK ' (lx+m y +nz - vt) = ALe { , 77 = AMe iu >, and f = ANe*. 
 The vector (/, g, h) is, as we have seen, intimately associated with 
 the displacement, so that we have 
 
 (/ g, h) = (X, /,, ) BJ<"-*I*> = (X, f*, v ) (p/v) Ae*<"-*. 
 Let accented letters, such as A' and X', refer to the reflected wave, 
 and subscripts, such as A 1 and Xj or A a and X 2 , refer to the refracted 
 waves. The boundary conditions require f, 77, and f to be con- 
 tinuous at an interface for all values of t and at all points of the 
 interface. This necessitates the relations o> = &>' = a>j = w 2 , so that 
 
158 REFLECTION AND REFRACTION [CH. 
 
 there is no change of phase in the ideal case of an abrupt transi- 
 tion. These relations require 
 
 sin <f)/v = sin </>'/v = sin c^/^ = sin cf) 2 /v 2 = q say, 
 just as in the case of isotropic media, $ being the angle that the 
 wave normal makes with the normal to the interface. The equations 
 thus obtained give the laws of reflection and refraction as far as 
 the direction of the waves is concerned. The calculation of v 1 and 
 v 2 is carried out in the last chapter, as well as the discussion of the 
 relations between the wave normals and the corresponding rays. 
 The intensities in the different waves are obtained by means of the 
 boundary conditions. These boundary conditions involve the vector 
 (a 2 /, tfg, c 2 /i), and we begin by obtaining the components of this 
 vector in certain directions convenient for our purpose. 
 Resolving along the wave normal, we get 
 
 (p/v) Ae i(t -l' 2] (a?l\ + 6 2 m^ + tfnv) = (p/v) Ae* 1 "-". K 
 
 tan .^( w -W2) 
 
 (where % is the angle between the ray and the wave normal) 
 
 = (p/q) A sin <f> tan ^.e i(( *~ n & } . 
 Resolving along the direction of the curl, we get 
 
 (p/v) (a 2 X 2 + 6> 2 + cV) Ae i(ta - n !' 2) = pvAe i((a ~^ 
 
 = (p/q) A sin<f>e* (< -' r/2) , 
 
 and finally resolving along the direction of displacement, we get 
 
 (p/v) (a 2 \L + PpM + c 2 vN) Ae i( -^ = 0, 
 
 as was proved on p. 133. From these results it appears that the 
 vector (a 2 /, b*g, c z h) has a resultant (p/q) A sin </> sec ^e i(ta ~ nlz) at 
 right angles to the ray, in the plane of the ray, the curl, and the 
 wave normal. Hence if OB be any line in the interface, ON the 
 wave normal, and OP the direction of the curl, the boundary con- 
 dition requiring the continuity of the component of (a 2 /, b 2 g, c 2 h) 
 in the direction OB leads to the equation 
 A sin (f> cos BP + A' sin < cos BP' 
 
 = A sin </> : cos BP^ 4- A 2 sin < 2 cos BP 2 
 
 + A l sin 0j tan ^ cos B^ + A. 2 sin c 2 tan % 2 cos BN 2 *. 
 
 * x is h ere taken positively when the wave normal lies between the ray and the 
 positive direction of the curl. 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 159 
 
 In Fig. 61, so = represents the interface, z = the plane of 
 incidence, TOP the wave front, ON the wave normal, and OT the 
 
 direction of the displacement, OP that of the curl the arrow-heads 
 indicating the sense which is to be regarded as positive. Let 
 POz = 0, and xON = </> (the angle of incidence). The plane through 
 the wave normal and the direction of the displacement is the plane 
 of polarisation NOT, so that 6 is the angle between the planes of 
 polarisation and incidence. We have 
 
 L = cos 6 sin $, M = cos cos (f>, N = sin 0, 
 and if B be a point on the axis of y, cos BP = sin cos <, and 
 cos EN = sin 0. In the refracted waves we must replace </> by fa 
 (or fa), and by 6 l (or 2 )- In the reflected wave $ must be 
 replaced by TT <f> from the law of reflection, and 6 by 6'. That 
 6' is to take the place of 6 in the reflected wave is seen on 
 considering the case of direct incidence. In this case N and N' 
 are on opposite sides of the wave front, so that 6 and 6' are 
 measured in opposite senses, so that, with our convention as to 
 signs, if OT be above the plane z = 0, OT' is below it. 
 
 We are now in a position to set out the boundary conditions in 
 a convenient form. The continuity of f , 77, f gives us 
 
 and two similar equations, while a fourth independent condition 
 
160 REFLECTION AND REFRACTION [CH.. 
 
 is obtained by expressing the continuity of the component of 
 (a 2 /, b z g, c 2 h) along OB. We thus get the four equations 
 
 (A cos 6 + A' cos 0') sin < = A l cos X sin fa + J. 2 cos 2 sin fa, 
 (A cos A' cos 0') cos (/> = J.j cos 0j cos </>j + A 2 cos # 2 cos fa, 
 J. sin .A' sin 0' = A-^ sin # x + A 2 sin 2 > 
 
 (^. sin + A' sin 0') sin 2< = A-L (sin ^ sin 2^ 4- 2 tan ^ sin 2 ^) 
 
 + J. 2 (sin # 2 sin 2 fa + 2 tan ^ 2 sin 2 $ 2 ). 
 
 When the incident wave is completely specified A, 6, and < are 
 given, and thence lt # 2 , <i, </>2, %i, % 2 are known from the properties 
 of the wave surface dealt with in the last chapter. There are thus 
 just sufficient equations to determine 6' and the unknown ampli- 
 tudes A', A-i, and A 2 . 
 
 From these four equations we see that it will be possible by a 
 proper choice of the direction of displacement (0) in the incident 
 wave to make one or other of the refracted waves (A 1 or A z ) 
 disappear. Suppose, e.g., that we wish to make the wave A 2 
 disappear, then writing sin 2 ^ tan %i/sin 6 l =f L for brevity, we 
 have 
 
 A cos + A' cos 6' = A l cos 6^ sin fa/sin <, 
 
 A cos 6 A' cos 6' = A^ cos O l cos ^/cos </>, 
 A sin 6 A' sin 6' = A l sin lt 
 
 A sin 6 -f A' sin 6 = A 1 sin 6 (sin fa cos fa +yi)/sin < cos <. 
 These give 
 
 2J.cos0 = A l cos6 l sin(( + ^^/sin^) cos </>, 
 2^.' cos 6' = A l cos 6 l sin (<^> fa)/sm cos <^>, 
 2^L sin 6 A l sin 6 l [sin ($ + c^j) cos ($ ^) +/J/sin $ cos c/), 
 2J.' sin 0' = AI sin 1 [cos (<^> + c^j) sin (< fa) /J/sin cos (/>, 
 whence tan = tan 0! [cos (<f) fa) +/j/sin (< + fa)] 
 and tan 6' = tan ^ [cos (< + (^j) /I/sin (< fa)]. 
 
 These equations correspond to those obtained on pp. 44 and 46 for 
 isotropic' media, where ^ = 0, and therefore /j = 0. The equation 
 tan 6 == tan 1 [cos (</> fa) +/ 1 /sin(< + </>0] determines a definite 
 direction for the incident displacement in order that the second re- 
 fracted wave (A 2 ) may disappear, and by interchanging the subscripts 
 1 and 2 we should get a similar condition for the disappearance of 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 161 
 
 the other refracted wave. The two directions of the displacement in 
 the incident wave, corresponding to a single refracted wave, are 
 termed uniradial directions. The relative magnitudes of the 
 different displacements for a uniradial system are determined 
 by the equations obtained above. 
 
 We have seen that the displacement (f, 77, f) is a vector whose 
 square is proportional to the energy per unit volume in any 
 medium, and that in passing from one medium to another the 
 components of this vector are continuous. It follows from this 
 that the refracted displacements (, 77^ ^) and (f 2 , ??2, 2) are equi- 
 valent, in the mechanical sense, to the displacements in the first 
 medium, i.e. to the displacements in the incident and reflected 
 waves. As a special case it is to be noted that when the incident 
 displacement coincides with a uniradial direction then the refracted 
 displacement is the resultant, in the mechanical sense, of the other 
 two, so that the three displacements are in the same plane and are 
 connected by the parallelogram law. 
 
 We proceed to investigate the position of this plane that 
 contains the three displacements, and to prove that it coincides 
 
 Fig. 62. 
 
 with what is known as the polar plane of the refracted ray. 
 The polar plane of a ray is defined as follows. Let OQ (Fig. 62) 
 
 11 
 
 M. L. 
 
162 REFLECTION AND EEFRACTION [CH. 
 
 be the ray, ONP the corresponding wave normal, P a point on the 
 index surface, so that ON. OP = 1. Let OT be the direction of 
 the displacement and draw OR through parallel to PQ. 
 
 Then the plane HOT is called the polar plane of the ray. 
 In the special case in which the ray coincides with the wave 
 normal, as happens with an isotropic medium, or with the 
 ordinary ray in a uniaxal crystal, the polar plane becomes the 
 plane through the normal and the displacement, i.e. the plane 
 of polarisation of the ray. 
 
 The angle between the wave normal and the polar plane of 
 the refracted ray is OPQ, and tan OPQ = QN/NP. Also we have 
 ON = 0! = I/OP, and therefore ON/ OP = vf = sin 2 fa/sin 2 fa Hence 
 
 sin 2 ^ ON ON ON QN 
 
 where ^ is the angle between the ray and the wave normal. 
 Thus we get 
 
 tan OPQ = tan ^ . sin 2 fa/(sirf (f> - sin 2 fa) 
 
 =/i sin ^/sin (</> + fa) sin(cf) fa). 
 
 Now let z = poo + qy be the equation of the plane of the three 
 displacements when the incident displacement is in a uniradial 
 direction. This equation is satisfied by the coordinates of any 
 point on the vector representing the incident displacement, and 
 also any point on that representing the reflected displacement. 
 The coordinates of the former point are proportional to L, M, N, 
 i.e. to cos sin </>, - cos 6 cos $, and sin 6, while those of the 
 latter are proportional to L', M', N', i.e. to cos 9' sin fa cos 0' cos fa 
 and sin0'. Substituting in the equation of the plane, and 
 using the values of tan and tan & already obtained on p. 160 
 we get 
 
 p = tan l [sin fa + /, cos fa /sin (<j> + fa) sin (</> fa)] 
 
 - tan 0! sec fa sin (fa + fa), 
 and 
 
 q = - tan l [cos fa -/j sin fa /sin (</> + fa) sin ((/> - fa)] 
 
 = tan l sec fa cos (fa + fa), 
 where tan fa =/i/sin (0 + fa) sin ((/> - fa). 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 163 
 
 Hence the equation of the plane becomes 
 
 z = p.-K + qy = tan 1 sec ^ [x sin (fa + ^)-y cos (fa 
 
 so that the plane cuts the plane of incidence (z = 0) in a line 
 making an angle fa + ^ with the axis of x. 
 
 Fig. 63. 
 
 Next consider the polar plane R^OT^ (Fig. 63) of the refracted 
 ray, R l being the point on the unit sphere where the line OR of 
 Fig. 62 meets the sphere, and so for the other points. Let this 
 polar plane meet the plane z = in the line OK l . T l is the pole 
 of the plane N 1 Q 1 R 1 , and z is the pole of N l K l , hence Tr/2 6 l 
 = angle T l Oz = angle between the planes of which T l and z are 
 the poles = RJ$J^ Thus from the triangle R^N^ we get 
 sin 6 l = tan R l N l cot N l K l . 
 
 But tan R } N t = tan R.ON, = tan RON = tan OPQ 
 
 =f l sin 0J sin (0 4- <i) sin (< c^) = sin O l tan ^ 1} 
 
 so that we have sin X tan ^ = tan R^N^ sin t tan N l K l . Hence 
 JVj J^T! = ^j and 0^ makes an angle : + ^i with the axis of x. 
 
 We have thus proved that the polar plane of the refracted ray 
 and the plane of the three displacements have the line OK l in 
 common, and as they also have the line OT^ in common, the two 
 planes must coincide. As the displacement is in the wave front, 
 it follows from this theorem that the displacements in the incident 
 and reflected waves are along the lines of intersection of these 
 waves with the polar plane of the refracted ray. 
 
 112 
 
164 REFLECTION AND REFRACTION [CH. 
 
 These theorems with reference to a uniradial system enable us 
 to solve the most general problem even when the incident dis- 
 placement is not in a uniradial direction. For suppose we have 
 an incident displacement of amplitude A in any azimuth 6. We 
 
 .Fig. 64. 
 
 can resolve this displacement into two displacements whose 
 amplitudes are a a and 0% along the uniradial directions OE^ and 
 OE Z , the azimuths of these displacements being o^ and or 2 (say), 
 We may then apply the results already reached to obtain the 
 displacements in the reflected wave. Their directions are the inter- 
 sections of the reflected wave with the polar planes of the refracted 
 rays corresponding to each uniradial direction. Denoting their 
 lengths by a/ and a 2 ' and their azimuths by a/ and a 2 ', we can com- 
 pound them by the parallelogram law into a single displacement A' 
 in the azimuth 6'. Of this general construction there is one case 
 of special interest. This occurs when the reflected wave passes 
 through the line of intersection of the two polar planes. In these 
 circumstances there is only one possible direction of the reflected 
 displacement whatever be the azimuth of the incident displace- 
 ment, this direction being the line common to the two polar 
 planes. In this case, therefore, the reflected light is plane 
 polarised, and, as the reflected ray is perpendicular to its wave 
 front, the ray is at right angles to the intersection of the two 
 polar planes a relation corresponding to Brewster's law that the 
 incident and refracted rays are at right angles at the polarising 
 angle for isotropic media. When the two reflected displacements 
 coincide we have 6' /3', where 
 
 tan j3' = tan B 1 [cos (</> + <i) - /i cosec (0 - 00] 
 = tan 2 [cos (< + < 2 ) / a cosec (</> - </> 2 )]. 
 
 This is an equation to determine the polarising angle and also the 
 value of 6' at this angle. It thus appears that when a ray is 
 
VII] WITH TRANSPARENT CRYSTALS 165 
 
 polarised by reflection from a crystal the plane of polarisation 
 deviates by an angle ft' from the plane of incidence. The angle 
 ft' is therefore called the deviation. 
 
 From the equations on p. 160 we see that 
 
 a l __ cos a/ sin (0 + <bi) , a 2 _ _ cos a 2 ' sin (< 4- fa) 
 
 a/ ~ cos aj sin (</> ^) ' a 2 ' cos a 2 sin (</> fa) ' 
 
 Also, from the parallelogram law, we have 
 
 a 1 _sin(a 2 0) a/ _ sin (a/ 0') 
 a 2 ~~ sin (0 aj) ' a 2 ' ~~ sin (6' a/) * 
 From these equations we derive 
 
 sin ($ + fa) sin (c < 2 ) cos a/ cos a 2 
 
 a 2 a/ sn (<^> + </> 2 ) sin (^> ^ ) cos a a cos a 2 ' 
 _ sin (a. 2 6) sin (^' a/) 
 sin (a/ ^ / )sin(^ a^' 
 
 T> , cos a/ sin (^ a a ) _ cos ^ tan 6 tan atj 
 
 coso^ sin(^' a/) cos^' tan # x tana/ ' 
 , cos a/ sin (a 2 ^) _ cos 6 tan tan or 2 
 
 cos 2 sin (a a ' #') cos 7 tan X tan a/ ' 
 Whence 
 
 ^ _ sin (0 + fa) sin ((ft - <ft 2 ) _ tan 6 tan ot 2 tan 7 - tan q/ 
 . sin (< + <^ 2 ) sin (<f> - fa) ~ tan 6' tan a/ tan 6 - tan ^ 
 
 t'-t 2 ' t-t,' 
 
 where, for brevity, t is written for tan 0, and so for the others. 
 Thus we have 
 
 (1 _ (7) u' + (0/ _ ^') t + (Cfcj _ * 2 ) t' + t - Ct = 0, 
 which is of the form 
 
 pit' + qt + rt' 4- 5 = 0, 
 
 where p, 5, r, and 5 are constants for a given angle of incidence. 
 Thus there is a homographic relation between t and t', as might 
 have been anticipated from the fact that there is one value of 6' 
 for each value of 0, and vice versa. This homographic relation 
 shows that, for a given angle of incidence, the intersection of the 
 planes of polarisation of the incident and reflected rays describes 
 a cone of the second degree. 
 
166 REFLECTION AND REFRACTION [CH. 
 
 Just as with an ordinary homographic range the relation 
 between the conjugates can be simplified by a proper choice of 
 origins, so the homographic relation found above can be put in 
 a simpler form. It can in fact be replaced by 
 
 tan(0'-') = 7.tan(0-), 
 where 
 
 (7 tan 13' - tan /3)/p = (7 + tan fi tan j3')/q 
 
 = - (1 + 7 tan ff tan @')/r = (tan ' - 7 tan j3)/s. 
 From these we get 
 
 (^-s)/(r + 2) = tan(/3 + /3'), 
 and (p + s)l(r - q) = tan (/3 - '). 
 
 These give fi and j3' in terms of p, q, r, and s, and 7 is then 
 obtained from the equation 
 
 7 - (p tan /3' + s tan j3)/(p tan ft + s tan '). 
 It appears from the equation 
 
 tan(<9'-/3') = 7tan(0-/3) 
 
 just obtained, or from the geometrical discussion on p. 164, that 
 the azimuth (0') of the reflected displacement will, in general, 
 depend on the azimuth (0) of the incident displacement. If, 
 however, we have 7 = 0, then 6' = $' whatever be the incident 
 azimuth, and in this case the reflected ray will be plane polarised 
 in the azimuth ft'. On putting 7 = in the relations given above 
 connecting p, q, r, and s, and eliminating ft and fi 1 , we get ps = qr, 
 
 i.e. (1-0) (Mi' - CM) = (6V - O (Ct, - t,), 
 
 or C(t,-t,)(t l '-t 2 ')=0. 
 
 As C cannot vanish, and ^ is not equal to t 2 , we see that t^ t^ or 
 / = OL 2 f . Hence when y = we have 6' = $' = / = a/, which is 
 the relation obtained on p. 164, for determining the polarising 
 angle and the deviation. 
 
 The azimuths 9 and 6' are measured from the plane of 
 incidence ; but it appears from the equation 
 tan (d' - j3') = 7 tan (0 - /3) 
 
 that the formulae will be somewhat simplified by measuring the 
 incident azimuth from the line 6 = ft, and the reflected azimuth 
 from the line O' = p r . The relation between the azimuths then 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 167 
 
 takes the form tan 6' = 7 . tan 6, so that 6' = when = 0, and 
 0' = 90 when = 90. Everything then corresponds closely with 
 the case of an isotropic medium, the azimuths 9 = and 90 
 corresponding with directions parallel and perpendicular to the 
 plane of incidence in the simpler case. In particular the 
 investigation of p. 44 holds in the case of crystals, so that if 
 we vary the incident azimuth, but keep the amplitude constant, 
 the end of the line representing the reflected displacement traces 
 out an ellipse, and the area described by the reflected displacement 
 is proportional to that described by the incident displacement. 
 These relations between the azimuths of the displacement in the 
 incident and reflected light were compared with experimental 
 determinations of them by Neumann. His experimental methods 
 precluded very great accuracy, but the number of observations 
 at each incidence was very large. The differences between the 
 results of observation and the calculations from theory rarely 
 exceeded 20' and generally were less than 5', such differences being 
 well within the limits of errors in the experiments. 
 
 The formulae are, of course, somewhat simplified when the 
 crystal is uniaxal, and as this case is one of special interest to the 
 experimenter it will be worth considering in some detail. 
 
 Fig. 65. 
 
 In Fig. 65 let x = be the interface ; z = Q the plane of 
 incidence ; ON the normal to the incident wave, coinciding with 
 the incident ray; OQ l the extraordinary refracted ray; OA the 
 optic axis. The position of this axis is determined with reference 
 to the coordinate planes by the coordinates (I, X) or (u, v), where 
 
168 REFLECTION AND REFRACTION [CH. 
 
 AB being perpendicular to yz, yE l and AB = \, and AC being 
 perpendicular to xy t Cx u and AC v. 
 
 We shall use the suffix 1 when dealing with the extraordinary 
 refracted wave, and the suffix 2 when dealing with the ordinary 
 refracted wave. If &> be the angle that the wave normal makes 
 with the optic axis, we have from the formulae of last chapter, 
 
 vf = a 2 sin 2 CD, + c 2 cos 2 Wl = c 2 + (a 2 - c 2 ) sin 2 w^ 
 
 V 2 = C, 
 
 and I/sin </> = vjsm fa = vjsin fa = c/sin fa. 
 
 These equations determine fa and fa in terms of < and o^. For 
 the ordinary ray the wave normal and the ray coincide, so that 
 X 2 = 0, and therefore f z = 0. Hence the equations on p. 160, which 
 give the uniradial directions, reduce to 
 
 tan 6 = tan 2 cos (cf> fa), and tan Q' = tan 2 cos (< + fa). 
 
 For the extraordinary ray we have 
 
 tan ^ = (a 2 c 2 ) sin w^ cos cojvj 2 
 
 = (a 2 c 2 ) sin CO L cos a)j sin 2 </sin 2 </> a , 
 and 
 
 /! = sin 2 fa tan %i/sin 9^ = (a 2 c 2 ) sin a^ cos Wj sin 2 0/sin ^ . 
 Hence in this case 
 
 tan = tan X cos (<j> fa) 
 
 4- (a 2 c 2 ) sin ajj cos a)j sin 2 ^>/cos ^ sin (0 + ^j), 
 and 
 
 tan 0' = tan 0j cos (<f> + fa) 
 
 + (a 2 c 2 ) sin a^ cos w sin 2 c/>/cos 0! sin (</> fa). 
 
 To put these equations, giving the uniradial directions, in a form 
 suitable for comparison with experiment we must obtain the 
 relations between O lt 2 and fa, fa. The latter (fa and fa) can be 
 determined by observing the deviation produced by a prism. It 
 is obvious from symmetry that the two possible directions of the 
 displacement in the wave front are in the meridional plane of the 
 crystal and at right angles thereto. Hence if AM (Fig. 65) be 
 perpendicular to the wave front, the directions of displacement are 
 
VII] WITH TRANSPARENT CRYSTALS 169 
 
 along OM and at right angles to this. From the triangle AzM 
 
 we get 
 
 cos AzM = tan zM cot zA, 
 
 i.e. sin (fa ~ u) 4- cot 2 tan v, 
 
 and sin (fa ~u) = tan 0j tan v. 
 
 Also from the triangle ACN-^ we have 
 
 cos A N! = cos J. (7 cos CNj 
 
 i.e. cos &>! = cos v cos (^ ~ u), 
 
 whilst from AzM we have 
 
 cos AM cos ZJf = cos Az, 
 i.e. sin ^ cos 0j = sin v. 
 
 These equations give to lt &i and 6% in terms of fa and $ 2 when u 
 and v are known, i.e. when the position of the optic axis is known 
 with reference to the coordinate planes. 
 
 The azimuths of the displacements in a uniradial system were 
 observed by Neumann for a uniaxal crystal, Iceland spar. The 
 differences between theory and experiment varied from 8' to 4- 4', 
 and these differences were scarcely greater than the probable 
 errors of the experiments. The range of incidence, however, was 
 small, the values of (f> employed being 40, 45, 50, and 55. The 
 more modern and more accurate experiments of Glazebrook 
 confirm Neumann's results by showing a very good agreement 
 between theory and observation for certain incidences. For other 
 incidences, however, there is a small discrepancy, so that the 
 theory can be relied upon as colligating the experimental results 
 only as a close approximation. Instead of attempting a comparison 
 with theory by direct measurements of 6, Glazebrook proceeded as 
 follows. He allowed plane polarised light to fall on a prism of 
 spar, and varied the angle of incidence until only one ray emerged. 
 He then measured <f> and the deviation produced by the prism, and 
 from these observations fa or fa could easily be determined. He 
 thus had sufficient data to calculate 6 by means of the formulae 
 obtained above. The incident light was then made to pass 
 through a sugar cell before falling on the prism, and the incidence 
 was again varied until the same ray emerged as before, 6 being 
 once more calculated from theory, the change from the former 
 
170 
 
 REFLECTION AND REFRACTION 
 
 CH. 
 
 value gave the rotation produced by the cell. This rotation could 
 be determined accurately by experiment, and proved to be 
 4 6' 15' when only the ordinary ray was transmitted, and 
 4 4' 20" when only the extraordinary ray traversed the crystal. 
 The results obtained are given in the following tables, in which 
 the first line gives the incidence before and the second that after 
 the interposition of the sugar cell. The third line gives the 
 difference between the rotation as calculated from theory and as 
 found by experiment. Table I refers to the case where only the 
 ordinary ray traversed the crystal, and Table II deals similarly 
 with the extraordinary ray. 
 
 TABLE I. 
 
 
 
 24 42' 
 
 30 36' 
 
 36 5' 40" 
 
 41 41' 30" 
 
 46 50' 40" 
 
 52 16' 40" 
 
 57 54' 20" 
 
 (with cell) 
 
 17 45' 
 
 24 25' 
 
 30 18' 50" 
 
 36 4' 10" 
 
 41 30' 50" 
 
 47 1' 40" 
 
 52 24' 30" 
 
 Difference 
 between 
 theory & exp. 
 
 + 4' 55" 
 
 - 0' 5" 
 
 + 0'35" 
 
 + 6' 25" 
 
 - 0' 25" 
 
 -4' 35" 
 
 - 0' 55" 
 
 
 
 59 45' 
 
 64 12' 30" 
 
 66 6' 20" 
 
 70 57' 40" 
 
 73 43' 
 
 78 14' 
 
 80 47' 
 
 (with cell) 
 
 54 8' 30" 
 
 58 9' 30" 
 
 59 51' 50" 
 
 63 57' 50" 
 
 66 6' 20" 
 
 69 29' 
 
 70 59' 
 
 Difference 
 between 
 theory & exp. 
 
 - 0' 5" 
 
 + 3' 35" 
 
 + 4' 5" 
 
 + 9' 5" 
 
 + 11' 25" 
 
 + 18' 35" 
 
 + 15' 55" 
 
 TABLE II. 
 
 
 
 24 15' 
 
 29 40' 
 
 35 7' 30" 
 
 40 9' 20" 
 
 45 17' 
 
 50 18' 
 
 55 43' 
 
 (with cell) 
 
 17 45' 
 
 24 4' 
 
 29 50' 30" 
 
 35 4' 30" 
 
 40 17' 
 
 45 15' 
 
 50 27' 30" 
 
 Difference 
 between 
 theory & exp. 
 
 + 13' 50" 
 
 + 4' 50" 
 
 + 5' 20" 
 
 + 6' 20" 
 
 + 4' 20" 
 
 + 2' 0" 
 
 - 5' 10" 
 
 
 
 61 29' 40" 
 
 64 46' 50" 
 
 66 42' 
 
 69 3' 30" 
 
 73 4' 40" 
 
 77 24' 20" 
 
 82 0' 55" 
 
 <j> (with cell) 
 
 55 37' 
 
 58 17' 
 
 59 57' 
 
 61 37' 30" 
 
 64 57' 30" 
 
 67 
 
 69 21' 30" 
 
 Difference 
 between 
 theory & exp. 
 
 - 6' 20" 
 
 -6'0" 
 
 - 12' 0" 
 
 - 8' 10" 
 
 - 17' 50" 
 
 '-23' 2" 
 
 - 40' 10" 
 
VII] WITH TRANSPARENT CRYSTALS 171 
 
 Returning to the theory we see that we can determine the 
 polarising angle and the deviation (ft') by making 6' the same for 
 the ordinary and extraordinary rays. We thus get 
 
 tan /3' = tan # 2 cos (< + fa) = tan O l cos (<f> -f fa) 
 
 + (a 2 c 2 ) sin Wj cos a^ sin 2 </cos O l sin (c/> fa). 
 
 To solve this equation we must proceed by successive approxi- 
 mations, a process greatly simplified by the fact that in all actual 
 cases the difference between a and c is small. Thus fa and fa are 
 nearly equal. We have 
 
 sin fa = c sin fa and sin fa Vi sin </>, 
 and therefore 
 
 sin 2 fa sin 2 fa = (vf c 2 ) sin 2 $ = (a? c 2 ) sin 2 o^ sin- </>. 
 Hence fa fa = sin (fa fa) approximately 
 = (a 2 c 2 ) sin 2 a)! sin 2 ^>/sin (fa + fa) = (& 2 c 2 ) sin 2 ^ sin 2 ^>/sin 2 fa. 
 We have seen that in the case of isotropic media, for which 
 fa = fa } the polarising angle occurs when <f) + fa = 7r/2. Thus 
 when fa and fa are nearly equal we have $ + fa = 9 + fa = ?r/2 
 approximately. Hence to this order of approximation 
 
 ^ - fa = $ + fa - 2 fa = 7T/2 - 2</>! = 7T/2 - 2< 2 , 
 and 
 
 cos (< + 00 = cos (4> + fa+fa- fa) = cos (< + 2 ) - (fa - fa) 
 
 = cos (0 + < 2 ) ( 2 c 2 ) sin 2 o^ sin 2 </sin 2< 2 - 
 
 Moreover we have seen that the two waves corresponding to any 
 wave normal are polarised at right angles to one another, so that, 
 as the normals N! and N 2 (Fig. 65) are nearly coincident, we have 
 # 2 = 7T/2 4 O l nearly. Making these various substitutions in the 
 equation for the polarising angle, we get 
 
 cos (</> + fa) = (a 2 c 2 ) sin 2 </> sin a^ cos 2 [cos a^/cos 2 fa 
 
 + sin a*! cos # 2 /sin 2 fa] . 
 
 The ordinary ray ON Z is polarised in the principal section of the 
 crystal OAN 2 , so that 2 is the angle AN 2 C of Fig. 65. Hence 
 from the triangle AN 2 C of that figure we have 
 
 cos o> 2 = cos v cos (fa u), sin o> 2 cos # 2 = cos v sm ($2 ~~ M )- 
 Substituting in the above equation for cos (<f> + fa), and making 
 
172 
 
 REFLECTION AND REFRACTION 
 
 [CH. 
 
 use, in the terms containing the small factor (a 2 c 2 ), of the 
 approximations o> 2 = &>i and tan < 2 = cos $ = c, we obtain the 
 following equations to determine the polarising angle <f> = J: 
 
 sin < 2 = c sin J, 
 and 
 
 cos ( J + 2 ) = # cos 2 v (sin 2 ^ - sin 2 < 2 ) = ^T cos 2 v (sin 2 / - sin 2 u), 
 where K = (a 2 c 2 ) (1 + c 2 )/2c(l c 2 ). Solving these equations 
 by successive approximations, we get as the first approximation 
 J=I cot" 1 c, which is the polarising angle for an isotropic 
 medium, and as the second approximation 
 
 J = I K . cos 2 v (sin 2 1 cos 2 u), 
 
 where K = (a 2 c 2 )/2c(l c 2 ). As the value of J is not altered by 
 changing the signs of u and v, it follows that the polarising angle 
 remains the same when the reflecting face of the crystal is turned 
 through 1 80 in its own plane, and this in spite of the fact that 
 this process changes one of the angles of refraction and alters the 
 situation of the refracted rays with respect to the optic axes. This 
 is in accordance with Brewster's observations and the more accurate 
 measurements of Conroy. The following table gives the differences 
 observed by Conroy in the values of the polarising angle for 
 azimuths u and u + 180 in the case of reflection from Iceland 
 spar in air. 
 
 M 
 
 1 
 
 11 
 
 21 
 
 31 
 
 47 20' 
 
 51 
 
 61 
 
 71 
 
 91 
 
 Difference 
 in J 
 
 + 14' 
 
 + 5' 
 
 -1' 
 
 -9' 
 
 + 8' 
 
 
 
 -15' 
 
 -2' 
 
 + 11' 
 
 u 
 
 101 
 
 117 20' 
 
 131 
 
 141 
 
 151 
 
 161 
 
 167 20' 
 
 171 
 
 177 20' 
 
 Difference 
 in J 
 
 -7' 
 
 + 3' 
 
 
 
 -7' 
 
 + 6' 
 
 + 13' 
 
 
 
 + 5' 
 
 -13' 
 
 If we wish to use the coordinates I and \ instead of u and v, 
 we have merely to transform the equation for J by means of the 
 trigonometrical relations sin v = cos X sin I, and sin X = cos u cos v. 
 These give 
 
 cos 2 v (sin 2 / cos 2 u) (1 cos 2 X sin 2 1) sin 2 / sin 2 X 
 
 = sin 2 / sin 2 X cos 2 X sin 2 1 sin 2 / 
 
 cos 2 1 [sin 2 1 - sin 2 X] sin 2 1 cos 2 / sin 2 X. 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 173 
 
 Hence J = / - K cos 2 1 (sin 2 / - sin 2 X) + K sin 2 1 cos 2 / sin 2 X. If 
 then Jj and J 2 denote the values of J when the plane of 
 incidence is parallel and perpendicular respectively to the principal 
 section of the crystal, we get J = J l cos 2 1 + J" 2 sin 2 1, where 
 /! = / - K (sin 2 / - sin 2 X), and J 2 = / -f K cos 2 / sin 2 X. 
 The following table compares the deductions from these formulae 
 with the results of Seebeck's experiments on a crystal for which 
 I =58 55' and * = 01158 radians = 6 38'. 
 
 X 
 
 025' 
 
 27 2' 
 
 45 29' 
 
 64 1' 30" 
 
 89 47' 
 
 /! (theory) 
 
 54 3' 
 
 55 25' 
 
 57 25' 
 
 59 25' 
 
 60 41' 
 
 Ji (exp.) 
 
 54 12' 
 
 55 36' 
 
 57 22' 
 
 59 19' 
 
 60 33' 
 
 Difference 
 
 -9' 
 
 -11' 
 
 + 3' 
 
 + 6' 
 
 + 8' 
 
 J 2 (theory) 
 
 58 55' 
 
 59 17' 
 
 59 48' 
 
 60 23' 
 
 60 41' 
 
 <7 2 (exp.) 
 
 58 56' 
 
 59 4' 
 
 59 48' 
 
 60 15' 
 
 60 33' 
 
 Difference 
 
 1' 
 
 + 13' 
 
 
 
 + 8' 
 
 + 8' 
 
174 
 
 REFLECTION AND REFRACTION 
 
 [CH. 
 
 These results are shown graphically in Fig. 66, and from this 
 figure or from the table it appears that the approximate formulae 
 that have been employed in the calculations represented the facts 
 very closely. 
 
 The mean of Brewster's observations on a crystal with a natural 
 cleavage corresponding to X = 45 C 23' SO" gave J T = 57 25' and 
 J >2 = 59 38', while the values calculated from the above formulae 
 are J^ = 57 25' and J 2 = 59 48'. The following table gives the 
 values of J for different values of I, calculated from the formula 
 J J l cos 2 1 + Jg sin 2 1, and compared with Brewster's measurements. 
 
 I 
 
 
 
 22 30' 
 
 45 
 
 67 30' 
 
 90 
 
 J (theory) 
 
 57 25' 
 
 57 46' 
 
 58 36' 
 
 59 27' 
 
 59 48' 
 
 J(exp.) 
 
 57 20' 
 
 57 46' 
 
 58 34' 
 
 59 29' 
 
 59 51' 
 
 Difference 
 
 + 5' 
 
 
 
 + 2' 
 
 -2' 
 
 -3' 
 
 Such small differences between theory and observation are within 
 the limits of error of Brewster's experiments. 
 
 Although these approximate formulae have proved sufficiently 
 accurate for our purposes, it may be well to obtain a rigorous 
 solution of the problem under discussion in a case where such 
 a solution can be easily reached. This happens when the optic 
 axis lies in the plane of incidence. The crystal is then symmetrical 
 on opposite sides of this plane, and the uniradial directions are 
 obviously parallel and perpendicular thereto. We shall deal with 
 the latter direction as representing the extraordinary ray. 
 
 In Fig. 67 let the plane of the paper represent the plane of 
 incidence, OT the interface, and ON the wave normal. The in- 
 cident and refracted rays OQ and OQ X are obtained from Huyghens' 
 construction by drawing tangents to the wave surface, the section 
 of which, in this case, is a circle of unit radius OQ, and an ellipse 
 AQ 1 C of which the axis 00 makes an angle X with OT. The 
 argument employed on p. 50 for the case of an isotropic medium holds 
 equally well here, so that QQ l is parallel to OT. The expression 
 of this fact leads directly to a simple formula for the polarising 
 angle. 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 175 
 
 We have OQ = 1, OT=cosecfa and QM=cosfa where QM 
 is the perpendicular from Q on OT. Taking OA and OC as the 
 
 Fig. 67. 
 
 axes of x and y, and denoting the eccentric angle of the point Q l on 
 the ellipse by fa, the coordinates of Q l are acos^ and csin^, 
 and the length of the perpendicular from this point on the line 
 OT, whose equation is x cos X + y sin X = 0, is 
 
 a cos fa cos X 4- c sin fa sin X = QM = cos <. 
 The equation to Q 1 T, the tangent to the ellipse, is 
 ex cos fa -f ay sin fa = ac. 
 
 This line passes through T whose coordinates are sin X cosec <f> 
 and cosXsec</>, so that we have 
 
 a sin ^x cos X c cos fa sin X = ac sin $. 
 
 Squaring this equation, and also ex cos fa + ay sin fa = c cos $, and 
 adding, we get 
 
 a 2 cos 2 X + c 2 sin 2 X = cos 2 < + a 2 c 2 sin 2 </> = !- sin 2 < (1 - a 2 c 2 ), 
 
 whence sin 2 / = sin 2 $ = (1 - a 2 cos 2 X - c 2 sin 2 X)/(l - aV), which 
 determines the polarising angle. 
 
 Returning to the consideration of the approximate formulae, we 
 find the deviation ft' from the equation tan ft' = tan # 2 cos (<f> + fa), 
 where </> = /. But tan 2 = + tan v . sin (fa ~ u), and 
 
 cos (<f> + fa) = K cos 2 v sin (0! u) sin ((/> 2 + u). 
 
176 
 
 REFLECTION AND REFRACTION 
 
 [CH. 
 
 Hence tan /3' = K sin v cos v sin (^> 2 -f u), and since 
 u + $2 = u <f> + (j> 4- < 2 = 7T/2 (4>~ u) 
 
 nearly, we get tan ft' = K sin v cos v cos (</> u) = K sin v cos ^, 
 where ty = ^.JV (Fig. 65) is the angle between the incident ray and 
 the optic axis. Thus 
 
 tan ft' = K cos X sin I cos i/r 
 
 K cos X sin Z (sin X cos / + cos X sin I cos /). 
 
 The following table gives the results of Seebeck's observations of 
 the deviation for reflection from a face of natural cleavage, and also 
 for a face parallel to the optic axis. Fig. 68 represents the results 
 of theory and observation graphically, and shows that the argument 
 is, on the whole, very satisfactory. 
 
 
 1 
 
 
 
 22 30' 
 
 45 
 
 67 30' 
 
 90 
 
 
 ft' (theory) 
 
 
 
 2 11' 
 
 3 33' 
 
 3 31' 
 
 2 21' 
 
 Face of 
 
 natural cleavage 1 
 
 P' (exp.) 
 
 
 
 2 9' 
 
 3 38' 
 
 3 34' 
 
 2 30' 
 
 ( 
 
 Difference 
 
 
 
 + 2' 
 
 -5' 
 
 -3' 
 
 -9' 
 
 
 ft' (theory) 
 
 
 
 2 46' 
 
 3 54' 
 
 2 46' 
 
 
 
 Face parallel 
 to axis 
 
 ft' (exp.) 
 
 
 
 2 46' 3 57' 
 
 2 43' 
 
 
 
 
 Difference 
 
 
 
 -3' 
 
 1 
 
 + 3' 
 
 
 
 The polarising angle dealt with above is the angle of incidence 
 at which the reflected light is plane polarised, whatever be the 
 azimuth of the polarisation of the incident light. If, however, 
 this azimuth corresponds to a uniradial direction, there is only one 
 refracted ray, and we may inquire what the angle of incidence 
 must be in order that the reflected beam should be polarised in 
 the plane of incidence or at right angles thereto. This angle will 
 be called the uniradial polarising angle to distinguish it from the 
 complete polarising angle dealt with hitherto. Its magnitude is 
 determined by putting 6' = in the fundamental equation 
 
 tan 6' = tan 6 1 [cos (</> + ^) /I/sin (</> c^)] 
 
 of p. 160. We shall consider more particularly the case of a 
 uniaxal crystal. If the refracted ray be the ordinary ray we have 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 177 
 
 % 2 = 0, and therefore / 2 = 0, so that the above equation gives 
 cos (< + fa) = or <f)-\-fa = 7T/2, which is Brewster's law. This 
 result has been verified experimentally in the case of Iceland 
 
 \ 
 
 
 \ 
 
 10 20 30 4tf 50 6Cf 70 
 
 Fig. 68. 
 
 80 
 
 spar by Schmidt. He found the polarising angle for the ordinary 
 ray to be independent of the orientation of the plane of incidence, 
 and that for a series of observations in different azimuths the 
 greatest departure from Brewster's law was only 16'. If, on the 
 other hand, the refracted ray be the extraordinary one Brewster's 
 law no longer holds. We then have 
 
 cos (< + fa) sin (0 fa) =/! = sin 2 fa tan %i/sin lt 
 
 Also from Fig. 65 we have sinzM =tan AM .cot AzM, so that 
 sin 6 l tan w l = tan (u ** <pi). Moreover 
 
 tan ft = (a 2 c 2 ) sin o^ cos co^v^ = (a 2 c 2 ) sin 2 cojvf tan ^ 
 = (vi* c 2 )/V! 2 tan &>! = (sin 2 fa sin 2 < 2 )/sin 2 fa tan a) l 
 = sin (fa fa) sin (fa + </> 2 )/sin 2 fa tan w l . 
 
 M. L. 12 
 
178 REFLECTION AND REFRACTION [CH. 
 
 Hence 
 
 cos (<j> + fa) sin (< fa) = sin 2 fa tan %i/sin Z 
 
 = sin (fa fa) sin (fa + fa)/sm l tan o^ 
 
 = sin ((/>! fa) sin (</>i + $ 2 ) cot (u ~ fa). 
 
 This equation, combined with sin fa = c sin </>, enables us to find < 
 approximately by the method employed on p. 171. Since fa fa 
 is small, we see that cos (<j> 4- fa) is small, so that < + fa is nearly 
 7T/2, and the polarising angle differs by a small quantity from that- 
 given by Brewster's law. We have, approximately, 
 
 cos (</> + fa) = cos (</> + fa) - (fa - fa) sin (0 + fa) 
 
 = sin (fa fa) sin (fa + fa) cot (u ~ fa) /sin (< fa) 
 
 = (fa fa) sin 2< 2 cot (w ~ fa)/siu (< c^). 
 So that 
 
 cos (< + fa) = (fa - fa) [sin ($ + fa) -*- sin 2</> 2 cot (M~< 2 )/sin (<j>-fa)], 
 also fa fa = (a 2 c 2 ) sin 2 ojj sin 2 ^r/sin 2< 2 
 
 = (a 2 - c 2 ) sin 2 (/>[!- cos 2 v cos 2 (w - </> 2 )]/sin 
 
 Hence the polarising angle for the extraordinary ray is given by 
 the formula 
 
 < = /' = / - K [1 - cos 2 v sin 2 (u + /)] [1 + tan 21 tan O + /)], 
 
 where K' = (a 2 - c 2 )/2c (1 + c 2 ), and / = cot" 1 c is Brewster's angle. 
 This formula represents very accurately the results of Schmidt's 
 observations with the extraordinary ray, the difference between the 
 observed and the calculated values of /' being never greater than 
 18', and usually only a few minutes. 
 
 Influence of a Layer of Transition. 
 
 We must now consider to what extent these formulae are 
 changed when we regard the transition into the crystal to be 
 gradual, and not abrupt. The general character of the results 
 might be predicted from the analogy of the behaviour of isotropic 
 media, but without more minute examination it is not easy to 
 decide with certainty to what extent the crystalline structure 
 
VII] WITH TRANSPARENT CRYSTALS 179 
 
 modifies the results previously obtained. As nearly all the 
 experiments have been made with uniaxal crystals cut either 
 parallel or perpendicular to the optic axis, we shall consider 
 these two cases more particularly. Just as before it is convenient 
 to deal first with a uniradial system, and if the incident displace- 
 ment be not in one of the uniradial directions to resolve it into 
 components along these directions. 
 
 If A be the end of the optic axis, and AM the perpendicular 
 from A on the wave front (Fig. 65), then the uniradial directions 
 are along OM and OM ', the latter being at right angles to OM. 
 Of these OM corresponds to the ordinary and OM' to the extra- 
 ordinary ray. Whenever, then, the optic axis lies in the plane of 
 incidence, which happens among other cases when the crystal is 
 cut at right angles to the axis, the uniradial directions are in the 
 plane of incidence and at right angles thereto. Under such 
 circumstances the argument used in the case of isotropic media 
 may be applied to crystals with very slight modifications*. It 
 will be sufficient to indicate the procedure for a crystal cut at 
 right angles to its optic axis. As usual we derive everything from 
 the Principle of Action, taking W = ^f < 3>dr, where in the air 
 2O =/ 2 + g* + h 2 (the velocity in air being taken as unity), and 
 in the crystal 2O = a 2 / 2 + c 2 (g 2 + h 2 ), while in the layer of transi- 
 tion 2<l> = J. 2 / 2 4- C 2 (g z + A 2 ). Here A and C are functions of #, and 
 the continuity of the elastic properties of the medium requires that 
 A 1 and C 1 when x = 0, and A = a and C = c when x d, 
 where d is, as before, the thickness of the layer. 
 
 When the displacement is perpendicular to the plane of 
 incidence we have ' ( = rj = and %=ue ip(t ~ v y\ 
 
 r 
 
 while the boundary conditions are satisfied if f and ^ are con- 
 tinuous, i.e. if u and ^- are continuous. The dynamical equation 
 
 * Of course any incident displacement can be resolved in directions parallel and 
 perpendicular to the plane of incidence; but unless these are uniradial directions 
 each component will give rise to two waves in the crystal, and this will greatly 
 complicate the discussion. 
 
 122 
 
180 REFLECTION AND REFRACTION [CH. 
 
 is g = - (A*f) - - (B*g), i.e. - p*u = - A'tfifiu + - O 2 , 
 dy v y ; 9# 
 
 or - C* + df (1 - AW) u = 0. 
 
 do?, V ctei/ 
 
 Solving this by successive approximations, regarding &? as small, 
 we get as the first approximation 
 
 where JT- 
 
 and as the next approximation w = J. x + MBJC* + c?! 2 :v, where 
 
 On integrating this gives 
 
 dw/da?! = - ^.i^/C 8 - 
 and v = - ^! (a?Jlf - -fl/O/O 8 - B.M./C^ + ^L^ 2 ^ + B^My/C*, where 
 
 o o 
 
 When ^ = we have u = A l} dujdx^ = B l and when x 1 we have 
 u = A, + ^/c 2 + ^ 2 [A {(H - E)/c* + M] 4- (5,/c 2 ) {, 2 ^ - K/c*}], 
 and dw/da?! = ^/c 2 -4- ^ 2 [Jl {i^ 2 ^ 7 - 1/c 2 } + (A/c 2 ) {^ 2 /-^/c 2 }], exactly, 
 as before, p. 65, except that ^ is replaced by 1/c, and the constants 
 JE, F, H, etc. are altered. Hence all the results previously obtained 
 apply to this case, the refractive index that occurs in the formulae 
 being that for the ordinary ray. 
 
 Similarly, when the displacement is parallel to the plane of 
 incidence, we have f ' = ?; ' = 0, f ' = ue ip(t - vy) ; 
 
 e-w "=-!'' ?=0; /-M^M-^yr, 
 
 The boundary conditions are satisfied if w and du/dx are con- 
 tinuous, and the dynamical equations give ' C 2 V 2 f / = 0, i.e. 
 
 - z^ 2 ) ^ = 0. This equation is of the same form as that 
 
VII] WITH TRANSPARENT CRYSTALS 181 
 
 obtained when dealing with isotropic media (p. 69), and its dis- 
 cussion yields results identical in form with those obtained in that 
 case. 
 
 Turning now to the consideration of a crystal cut parallel to 
 its optic axis, we shall suppose that the axis of the crystal makes 
 an angle I with the plane of incidence. We then have 
 
 2$ = a 2 /' 2 + c 2 (g* + 7i /2 ) = (a 2 - c 2 )/' 2 + c 2 (/ /2 + g'* + &' 2 ). 
 
 Hence on referring <E> to the coordinate axes instead of the 
 principal axes of the crystal, we get 
 
 2< = (a 2 - c 2 ) (g cosl + h sin If + c 2 (/ 2 + g* + A 2 ) 
 
 = Af 2 + Bg* + Ch* + 2Fgh, 
 
 where A = c 2 ', B = c 2 + (a 2 - c 2 ) cos 2 1; C = c 2 + (a 2 - c 2 ) sin 2 1; 
 F=(a 2 -c 2 )smlcosl', BC - F* = a 2 c 2 . 
 
 The Principle of Action requires Ch + Fg and Eg + Fh to be 
 continuous at the interface and gives the dynamical equations 
 in the form : 
 
 In the air we have 
 
 ( 77, f ) = (L, M, N) Ae*P<t ~ * cos * - y sin *> (incident wave) 
 
 + (', Jlf', JV")^' e P(* + **- gin*) 
 
 (reflected wave). 
 In the crystal 
 
 (refracted wave), 
 
 there being only one refracted wave, as the incidence is uniradial 
 by hypothesis. In the layer we have (f, 77, f) = (u, v, w)e ip(t ~ vy) , 
 where u, v, w are functions of x only. Since 
 
 3f , drj d A du . 
 
 ^- + - + sr = 0, we have -= -- ipi^v = 0, 
 
 9# 9y 3^ rfa; r 
 
 and therefore v = --- =-. 
 
 vp dx 
 
182 REFLECTION AND REFRACTION [CH. 
 
 Also /= | = - ivpwe* (t ~ y y } \ g = - g = - e 
 
 A = ^_8| = _^r^_ 
 
 da? 8y i/p LM^ 
 
 Substituting in the dynamical equations we get 
 
 where w 2 = ipvw. 
 
 Putting x\d x^ d l =pd, as before, and w l d l w^p } these 
 equations take the forms 
 
 These equations are to be solved by successive approximations. 
 Neglecting df we get 
 
 anc -, i JL- i ., / i 7 \ j ~ f j i ' 
 
 ?! \ dx l / 
 
 The second of these gives 
 
 where 7 is a constant. 
 
 Substituting in the first we get 
 
 whence we derive 
 
 rx l 
 
 U OL + @Xi + yj, where J = I Idx^, 
 
 J o 
 
 and = 
 
-VII] WITH TRANSPARENT CRYSTALS 183 
 
 As the next approximation we put 
 
 u = a + (Sx-L 4- 7/+ dj 2 . u'; w^ = a x + 7 
 and the equations for u' and w' are 
 
 and 
 
 Before solving these equations we may use the boundary 
 conditions at x = to determine the relative magnitudes of the 
 constants a, /3, 7, and a l . Of the five boundary conditions only four 
 are independent, and these we may take to require the continuity 
 of f > *7> f an d -% + Fh. The continuity of f , 77, f involves that of 
 
 r/?y 
 
 w, v, w, and so of u, -y- , and w ; while that of Bg + Fh involves the 
 dx 
 
 continuity of 
 
 _ B <^_ F i^u_ \ 
 c?i dx \dsF I 
 
 When ^ = we have 
 
 u = a, w l = !, dufdx^ 0, dw^jdx-^ = 7, 5=1, F=0. 
 Hence the boundary conditions at x = give 
 
 a = AL + ^177 ; = ivd, (AM + A'M') ; 
 7 = i/^ 2 cos <t>(AN- AN') ; ttl = iVcfe (^+ A'N'). 
 
 As y8 and j are of the first order in d^ and 7 is of the second 
 order we may simplify the equations for u' and w' by omitting 
 terms containing these constants, since v! and w' are multiplied 
 by di* in the equations for u and w lt and we are neglecting powers 
 of di above the second. In this way we get 
 
 dx l |_ o?^ 2 ~ dx l \ dxj ' 
 
184 REFLECTION AND REFRACTION [CH, 
 
 Taking the simplest solutions possible we get 
 
 du r f xi ~] 
 
 = i/X - J Pt^ = a (wz?! - Q), 
 
 [,.2/v. 2 r^r, 
 
 ^-- Qt^ =a[*V/2- 
 ^ J o J 
 
 ' TT / r 
 
 = an. , w = a.1. 
 
 ; 
 da; l 
 
 Hence when x l = 1 we have 
 
 , 2 (i/'/2 - R) ; 
 
 in which all the integrals are made definite by putting ^ = 1 at 
 the upper limit, e.g., 
 
 Fdx, 
 - 
 
 If now we substitute these expressions in the boundary 
 conditions at a^ = l, and as the first approximation neglect terms 
 in d l of a higher order than the first, then, since BK FH= 1 and 
 BH FP = 0, we get the following equations : 
 
 A,L, = AL + A'L' + ivdi (AM+ AM'), 
 A.M, = A M + AM' - id, [I cos < (^J^ - AN") 
 
 A^Ni = AN+A'N' - ii [P cos (4 JV - ^L'^ 7 ) - (AL + A'L') I\v\ 
 sin $ [5^.^! cot <^! + FA^ cosec 2 ^J 
 
 = cos (/> (^IJV- ^l^ 7 ) + ^ sin 2 (.4 Jlf + 4'Jlf '). 
 Expressing Z, Jf, JV in terms of and </> as on p. 159, and 
 putting A cos 6 = x and A sin = y for brevity, we get 
 a?! sin 0J = (a? + a?') sin (/> ic^ sin </> cos </>( a?'), 
 #! cos </>! = (a? a?') cos (/> + idi [I cos ^ (y + 2/0 
 
 2/i = y - V - id i {.P cos (/> (y + y') - 1 (x + a?')] 
 cos fa+Fx^ sin 0/sin 0!= cos </> (y 4- y') ic^ sin 2 <^> cos <j>(x x'). 
 
VII] WITH TRANSPARENT CRYSTALS 185 
 
 Eliminating x l and y^ we get 
 
 x [sin (</> + fa) + idi {sin $ cos < cos ^ - sin fa (sin 2 </> - Q)}] 
 
 y id-L 1 cos sin fa 
 = - x [sin ($ fa) idi {sin < cos < cos (/>j + sin fa (sin 2 </> 
 
 + y idi / cos </> sin </>j , 
 and x [F sin 2 </> td^ {sin 2 < sin 2 fa cos < F sin 2 </> cos </> 
 
 BI sin </> sin fa cos c^}] 
 ?/ [sin fa (sin ^ cos <f> + B cos ^ sin <) 
 
 + idi BP sin < cos <f> sin ^>j cos fa] 
 = x[F sin 2 < 4- ic^ (sin 2 (/> sin 2 < 3 cos <f> F sin 2 < cos $> 
 
 + jB7 sin <^> sin fa cos </h}] 
 + y [sin </>! (sin c^j cos <j> - B cos fa sin </>) 
 
 + iWj BP sin (/> cos (/> sin fa cos 0J. 
 From these we get 
 
 y'\x =(p + idtf^Kq + id^) = ee^, 
 
 where e is the ratio of the amplitudes, and A the difference of 
 phase between the components parallel and perpendicular to the 
 plane of incidence. As d is very small we have y I =pfq very 
 nearly, unless p or q is very small. Hence as a rule e is very 
 nearly the same as if there were no layer and the difference of 
 phase A is very nearly zero or IT. This, however, will not be the 
 case in the neighbourhood of the uniradial " polarising angle," for 
 p is zero at this angle. 
 
 When p = we have ee 1 * = id^jq, so that the coefficient of 
 ellipticity is e^ = d^/q, and the difference of phase is Tr/2 or 
 7T/2 according to the sign of pjq. This, then, is the Principal 
 Incidence and in its neighbourhood we have, approximately, 
 tan A = d^pi/p = e^/p. With the aid of the above equations these 
 formulae give 
 
 , F sin 20 sin fa (sin 2 <f> + BI) + m tan 6 
 1 sin (</> fa) (sin fa cos $ + B cos fa sin <) ' 
 where 
 
 m = IF sin 2 </> cos </> sin </> cos </> cos fa {BP sin (< + fa) 
 
 + sin fa cos <j> B cos fa sin 0} 
 
 + sin fa (sin 2 < Q) (sin 0! cos </> - B cos ^ sin </>), 
 
186 REFLECTION AND REFRACTION [CH. 
 
 and 
 
 sin ($ fa) (sin fa cos <j> + B cos fa sin $) 
 1 
 
 At the Principal Incidence, tan A is infinite and 
 tan 6 = F sin 2$ sin </>/sin ((/> + fa) (sin ^ cos <j> B cos </>! sin </>). 
 
 The reflection is positive or negative according as A varies 
 from to TT, i.e. according as p^jq is negative or positive. 
 Hence the reflection is positive or negative according as 
 
 F sin 2< sin fa (sin 2 </> + BI) + m tan 
 is positive or negative, i.e. according as 
 JPsin 2< [sin fa (sin 2 < + 51) 
 
 4- m sin </sin ($ + fa) (sin (^ cos $ 5 cos (/>! sin 0)] 
 is positive or negative. Since F(a 2 c 2 ) sin I cos Z, the sign of F 
 will vary with the orientation of the crystal and the reflection will 
 change from positive to negative, as is found experimentally to be 
 the case. 
 
 For some purposes it is convenient to express x' and y in terms 
 of #! and y lt the angle #j being readily given in terms of fa and I by 
 the formulae on p. 168. Eliminating x and y from the equations 
 on p. 184, we get 
 
 x [2 cos $ + idi (2 cos 2 (f> - sin 2 < + Q)] = x l [- sin (< - <i)/sin < 
 + id-i }(1 - Q) sin </>!/sin < + /^sin c/>/sin fa}] 
 
 + idj i/j 75 sin $ cos ^>/sin X , 
 and 
 
 #/ [2^! sin 2 0] + y [1 + *<ii P cos 0] = ^ [F sin ^/sin 0! cos </> 
 
 - icZi {^P sin 0/sin 0! sin ^ sin fa - I sin fa/sin <}] 
 
 - 2/j [(sin fa cos ^ - 5 sin < cos $i)/sin fa cos ^ 
 + id l BP sin <^> cos fa/ 'sin ^J. 
 
 From these we derive 
 
 as before, where 
 
 6l = dtfjq = ra'c^/sin 0! cos <f> sin (</> </>a), 
 
 m' = 2 sin 3 </> cos sin 0! cos fa 2FP sin 2 (/> cos 2 </> + 2/sin 2 0! cos 2 </> 
 + -F sin 2 ^ (2 cos 2 < - sin 2 </> + Q) - sin </> tan ^ [25 sin </> cos 2 c/> cos fa 
 H- (2 cos 2 </> - sin 2 ^ + Q) (sin 0! cos ^ - 5 cos (^ sin </>)], 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 187 
 
 and tan A = 
 
 = 61 
 
 sn ! sn <> 
 
 ~ l 2 sin </> [(sin ^ cos <jf> B cos ^ sin </>) tan 6 l F sin <] ' 
 This formula for A is somewhat simpler than the equivalent 
 one found above in terms of tan 6 *. The following tables give the 
 values of A in the neighbourhood of the uniradial polarising angles 
 for Iceland spar at different orientations and compare the results 
 of theory with the experimental measurements made by Schmidt. 
 
 Ordinary ray (I = 78 52'). 
 
 
 
 57 23' 
 
 58 23' 
 
 59 23' 
 
 60 23' 
 
 62 23' 
 
 - A (theory) 
 
 0-06 
 
 0-16 
 
 0-73 
 
 0-90 
 
 0-94 
 
 - A (exp.) 
 
 0-06 
 
 0-16 
 
 0-72 
 
 0-90 
 
 0-94 
 
 Extraordinary ray (I = 44 39'). 
 
 </> . 
 
 56 32' 
 
 58 32' 
 
 59 32' 
 
 60 32' 
 
 62 32' 
 
 
 
 
 
 
 
 - A (theory) 
 
 0-04 
 
 0-11 
 
 0-63 
 
 0-90 
 
 0-96 
 
 -A (exp.) 
 
 0-04 
 
 0-12 
 
 0-62 
 
 0-90 
 
 0-96 
 
 These results are shown in Fig. 69. 
 
 If we wish to discuss the refracted ray, we eliminate x' and y 
 from the equations on p. 184 and so get 
 #! [sin (0 + </>i)/sin </> - id l (1 Q) sin c^/sin </>] + id^ I cos $y l 
 
 = 2x cos </>[!- id l cos $] + 2^ / cos </> . y, 
 and x l [F sin 0/sin </>j id l sin ^ sin <f> (sin < + / cosec </>)] 
 
 +2/ 1 [s 1 cosec< 1 id l P cos 2 </>] 
 = 2^j x sin 2 <f> cos < + 2y (1 ^ P cos<), 
 where ^ = cos </> + J5 sin (/> cot ^ . 
 
 * At the Principal Incidence we have 
 
 cot l = (sin 0j cos - B cos 0j sin 0)/F sin 
 
 which with cot d l = - cos 0j cot I for the ordinary ray, and cot 1 = sec X tan for 
 the extraordinary ray, enables us to calculate the uniradial polarising angles as 
 on p. 176 above. The velocities of the ordinary and extraordinary waves are 
 *JA and \/C + (B - A) cos a X respectively. 
 
188 
 
 REFLECTION AND REFRACTION 
 
 [CH. 
 
 From these we derive 
 
 , Sj sin (< 4- 00 tan 2.Psin $ cos c/> + id l m" sin </>j 
 ] 2$! cos (f) %idi cos (sj cos <^> /5 sin (^> cos <^! tan $) ' 
 
 _ g'g^A* gay^ 
 
 where 
 
 m" 2 cos 2 $[sin </>! sin <^> + 1 sin ^ cosec cf> + Fsm(f) cosec <i] 
 tan ^ [2 IF sin < cos ^> cosec fa + ^ (1 Q) cosec ^> 
 + P cos 2 <f> sin (^> -f </>j) cosec 6]. 
 
 The refracted wave is elliptically polarised, the major axis of 
 the elliptic orbit making an angle ^ with the plane of incidence, 
 where tan 2^ = tan 2a 1 cos A x and tan ^ = e' (cf. p. 84). As A' 
 is small, the ellipse is approximately a straight line, and the light 
 is very nearly plane polarised in the azimuth A/^. If the transition 
 were abrupt, the light would be plane polarised in an azimuth 
 #! where 
 
 sn 
 
 tan 2F sin < cos 
 
 tan #! = 
 
 so that the effect of the layer is to turn the plane of polarisation 
 through an angle ^i l . 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 189 
 
 Internal Reflection. 
 
 Returning to the ideal case of an abrupt transition, we shall 
 next consider the problem of internal reflection at the second 
 surface of a crystal. This problem may, as MacCullagh pointed 
 out, be reduced in certain cases to one that has already 
 been solved. Thus suppose that we have a crystalline plate with 
 parallel faces and that we wish to determine the amplitude and 
 azimuth of the reflected and refracted displacements corresponding 
 to an internal ray R : . This ray will, in general, give rise to two 
 
 Fig. 70. 
 
 reflected rays />/ and p 2 ' and a refracted ray R 3 . The ray J^ could 
 be produced by an incident ray R in a uniradial direction, and 
 corresponding to this we should also have a reflected ray R'. The 
 amplitude and direction of the displacement in R l being given, 
 we can determine those in R and R' very simply by MacCullagh's 
 theorem of the polar plane (p. 163). The directions of the dis- 
 placements are the intersections of the incident and reflected 
 waves with the polar plane of R l and their magnitudes are given 
 by the parallelogram law, since they are the components of a 
 vector of which the displacement in R is the resultant. If now 
 the direction of the displacement in R' were reversed and the ray 
 turned back so as to become JR/, it would give rise to a reflected 
 ray R and two refracted rays R and RJ, and it follows from 
 the dynamical principle of reversion that p-f arid p 2 ' are the same 
 as R[ and R% except for the change of sense. The displace- 
 ments in RI and R 2 ' can be determined by methods already 
 discussed, so that those in /o/ and p 2 ' can be obtained. The dis- 
 
190 REFLECTION AND REFRACTION [CH. 
 
 placement in R 3 will be equal and parallel to the resultant of 
 those in R and R. 
 
 The method indicated above gives the complete solution of the 
 problem of partial reflection ; but if we wish to make a numerical 
 measure of the amplitudes, and in particular if we wish to deal 
 with the case of total reflection, it is better to abandon the 
 geometrical method of MacCullagh and to proceed straight- 
 forwardly by writing down the boundary conditions exactly as on 
 p. 160 for the special problem in hand. Thus, let the incident 
 ray 1 give rise to two reflected rays 2, 3 and a refracted ray 4 
 (Fig. 71). As before, denote the amplitude of a displacement 
 
 CRYSTAL 
 
 Fig. 71. 
 
 by A and its direction cosines by L, M, N. For brevity we shall 
 use the symbol S to denote the quantity 
 
 sin 6 sin </> cos <f> 4- tan % sin' 2 (/> 
 (see p. 160). The boundary conditions then give 
 
 A^L^ + AJj^ + A 3 L 3 = A 4 L 4 = A 4 cos 4 sin fa ; 
 
 A 1 M 1 + A 2 lf 2 4- A 3 M 3 A 4 M 4 = - A 4 cos 4 cos fa ; 
 A^ + AJf* + A 3 N 3 = A 4 N 4 = A 4 sin 4 ; 
 A-iSi H- -4.2^2 + A 3 S 3 = A 4 8 4 = A 4 sin 4 sin < 4 cos fa. 
 These are to be coupled with 
 
 sin <f> l /v 1 = sin (j) 2 /v 2 sin <j> 3 /v B = sin < 4 /v 4 = sin fa ; taking v 4 = 1. 
 
 These last equations, together with the properties of the Index 
 Surface, determine fa, fa and fa in terms of fa and then the equa- 
 tions representing the boundary conditions are just sufficient to 
 determine A 2 ,A 3 ,A 4 and 4 in terms of A and 0j. If we wish to 
 consider particularly the reflected rays we eliminate A 4 cos 6 4 from 
 
VII] WITH TRANSPARENT CRYSTALS 191 
 
 the first two equations, and ^ 4 sin0 4 from the last two, and so 
 obtain 
 
 J. x (L! cot fa + M,) + A a (L 2 cot fa + MI) + A 3 (Z, cot fa 
 and 
 A l (Ni sin fa cos fa $) + -4 a (^2 sin $ 4 cos </> 4 $ 2 ) 
 
 + A s (N 3 sin < 4 cos fa $ 3 ) = 0, 
 whence J. 1 /A 23 = ^ 2 /A 31 = J. 3 /A ]2 , ' 
 
 where & pq = L p cot < 4 + M p L q cot < 4 4- M q 
 
 N p sin fa cos < 4 $p N q sin < 4 cos $ 4 S q 
 
 In case of partial reflection fa is real, and in this case the A's are all 
 real. Thus A 2 and A 3 are real, and the change of phase produced 
 by reflection is zero or TT, as is found experimentally to be the 
 case when the layer of transition at the surface of the crystal is 
 negligible. If, however, fa be not real, then A z and A 3 are in 
 general complex, so that there will be a change of phase on 
 reflection. If the outer medium were absorptive then, as we shall 
 see later, v 4 would be complex, so that sin fa would be complex, 
 and a change of phase would always be expected. If the outer 
 medium be transparent, sin fa must be real, but it may be greater 
 than unity (as will happen when sin fa > y : ), so that cos fa and 
 cot fa will be purely imaginary. Under such circumstances fa is 
 imaginary and the reflection is total. There is no difficulty in 
 obtaining formulae for the amplitude and phase of each of the 
 reflected rays. 
 
 Putting sin fa=q= cosh x, we have 
 
 & pq = (iL p tanh x -I- M p ) (iN q sinh x cosh x S q ) 
 
 (iL q tanh x + M q ) (iN p sinh x cosh x S p ) 
 
 = (N p L q - L p N q ) sinh 2 x + (8 P M q - M p S q ) 
 
 + i tanh x [(S p L q - L p S q ) + (M p N q - N p M q ) cosh 2 x} 
 
 = D pq e^. 
 Thus 
 
 A,/ A, = A^ = (Ai/As) eW-*-\ and AJA, = (D/D m ) <(->. 
 
 The change of phase in the two cases is 6 sl 6& and 12 # 23 , and 
 the difference of phase between the two reflected rays is # 12 n . 
 
192 REFLECTION AND REFRACTION [CH, 
 
 It thus appears that all that is unknown may be expressed in 
 terms of L, M, N, and S. These quantities, in turn, depend on 
 0, 0, and %, which are given by the theorems of the last chapter. 
 The resulting formulae for the amplitude and phase in the reflected 
 waves are very complicated in the general case. There is, however, 
 a considerable simplification when the crystal is uniaxal, and a 
 still further lightening of the labours of calculation from the fact 
 that in practice the difference between a and c is always small. 
 
 We shall take the same axes and adopt the same notation as 
 on p. 167, and shall suppose the incident ray 1 to be ordinary, 
 and the reflected rays 2 and 3 to be ordinary and extraordinary 
 respectively. We then have 
 
 sin fa/c = sin 2 /c = sin (f> 3 /v 3 = sin fa = q = cosh x ; 
 v 3 2 = a? (a 2 c 2 ) cos' 2 &> 3 ; tan ^ 3 = (a 2 c 2 ) sin a> 3 cos w 3 /v 3 2 ; 
 
 cos o> 3 = cos v cos (03 u)\ ^ = ^2 = 0; 
 sin (fa t+*u) = cot l tan v ; sin (0 2 ~ u) = cot # 2 tan v ; 
 
 sin (03 %/) = tan # 3 tan v. 
 From these we get 
 
 2 = TT - 0! ; sin 2 03 - sin 2 2 = <f (v./ - a 2 ) = - q 2 (a* - c 2 ) cos 2 o> 8 ; 
 so that 3 2 = sin (0 3 2 ) approximately 
 
 = (f (a 2 - c 2 ) cos 2 a> 3 /sin 20! = y, 
 
 where y is small. There are two special cases worth dealing with : 
 (1) when the face of the crystal is perpendicular to the optic axis, 
 and (2) when it is parallel to that axis. 
 
 In the first case we have u = v = Q, & 3 = Tr/2, 
 
 o> 3 = 03 = TT - 0! + y ; y = (q z /2) (a 2 - c 2 ) cot fa ; 
 
 tan x.3 = - q 2 (a 2 - c 2 ) cot fa=-2y. 
 
 The sign to be attributed to tan % 3 in the formula for S 3 has not yet 
 been determined. It is fixed by the convention of signs indicated 
 in Fig. 61 and the note on p. 158 that tan % is taken positively 
 when the wave normal lies between the ray and the positive 
 direction of the curl. In the present case, if tan^ 3 be taken 
 positively, the extraordinary ray is nearer the optic axis than the 
 ordinary ray, which Huyghens' construction shows to be the case 
 with a positive crystal. Hence tan ^ 3 must be taken positively or 
 
VII] WITH TRANSPARENT CRYSTALS 193 
 
 negatively according as the crystal is positive or negative. The 
 equations on which the solution of the problem depends may now 
 be written down. We have Z^ = sin fa = L z ; M l = cos fa = M% ; 
 N, = N. 2 = S 1 = S 2 = L 3 = M 3 = Q; N 3 = l; S 3 = - sin fa cos fa 4- 2-, 
 where 2 = y if the crystal be positive, and z = y (1 4 sin 2 fa) if 
 the crystal be negative. These make A 12 = 0, so that A 2 =Q, and 
 there is no extraordinary ray. If the incident ray were extra- 
 ordinary we should take 1 = 3 = 0, # 2 = T/2, and this would make 
 A 31 = 0, and so A 2 = 0. Hence with a crystal cut at right angles to 
 the optic axis an ordinary ray gives rise to an ordinary ray and an 
 extraordinary ray to an extraordinary one, as might have been 
 anticipated from the perfect symmetry with respect to the axis. 
 Returning to the case when the incident ray is ordinary we get 
 
 tan 31 = tanh x cot fa + z tanh #/sin 2 fa (q 2 sin 2 fa) 
 
 = tanh x [cot fa + zl(jj? 1) sin 4 fa], where p = I/a ; 
 and 
 
 tan #23 = tanh x [cot fa O 2 + 1) sin 2 fa/ {2 - (^ + 1) sin 2 fa] 
 
 + (^-I)z/{2-( fjL * + l)sin*fa}^ 
 
 which determines the change of phase. It is easy to verify that 
 Ai = As, so that A 2 = A l} as should be the case with total re- 
 flection, when there is no loss of light. 
 
 We shall next consider the case when the crystal is cut parallel 
 to the optic axis. To deal with this we must put u = 7r/2 in the 
 general formulae of p. 192. This gives 
 
 cos o> 3 = cos v sin fa ; y = (q 2 /2) (a 2 c 2 ) cos 2 v tan fa ; 2 = 6 l , 
 as is obvious geometrically ; 
 
 tan 3 = cot 6. 2 cos fa sec fa = cot # 2 (1 + y tan fa), 
 so that 3 = 7T/2 -\-6z-y sin # 2 cos 2 tan fa 
 = 7T/2 #! + y sin X cos 6 tan (/>! . 
 Also in this case we have 
 
 AzM = TT- fa = fa; 
 7r/2 &> 2 = 7T/2 o> 3 (approximately) ; 
 7r/2 + 0. 2 = 7T/2 - 0! ; A* = Tr/2 - w; 
 and hence cos X = cot </>j cot o> 3 ; 
 
 cos 6 tan ^ 3 /i/ = 2 cos l tan &> 3 cot fa 2 cot 2 </>j. 
 M. L. 13 
 
194 REFLECTION AND REFRACTION [CH. 
 
 In this case, as before, tan % 3 must be taken positively if the 
 crystal be positive. We thus get 
 
 Zj = cos O l sin fa = L 2 ; M! = cos 1 cos fa = - M 2 ; N t = sin l = N 2 ; 
 
 $, = sin #! sin fa cos fa S. 2 ; 
 L 3 =s sin 0j sin fa y sin ^ sec ^ (1 sin 2 0j sin 2 </>!) ; 
 M 3 = sin ^ cos fa + y sin 3 6 l sin fa ; 
 -A r 3 = cos l y siri 2 1 cos ^ tan 0! ; 
 
 $ 3 = cos ^ sin fa cos fa y cos ^ (cos 2^ sin 2 1 sin 2 c^) 
 
 + 2y cos 2 fa sec 0j , 
 
 the upper or lower sign being taken according as the crystal 
 is positive or negative. 
 These equations give 
 
 A JV 2 L 2 Ni = M 1 8. 2 M Z S 1 = 0, and hence 12 = ?r/2 ; 
 tan 3l 
 
 also 
 
 tanha? 
 
 cos %6 l cos : (g 2 - sin 2 a ) + ?/ sin ^ (of sin 2 ^ cos 2^ + cos 2 6 l cos 20j 
 
 - sin 2 6) sin 2 0, cos 2^ - sin 2 0, T 2 cos 2 0j) 
 
 - sin 2 X ) + T/ cos X {sin 2 ^ x (1 + sin 2 X - g 2 ) - cos 2 6 1 cos 20 a 2 cos 2 X }- 
 
 = cot <i cot 2^ + (y/yL6 2 - 1) cosec 4 fa [(q* cos 2 fa) sin 2 ^ cos 20j 
 -f (sin 2 </>i 2 cos 2 fa) (cos 2 ^ cos 20! sin 2 ^) 
 -I- cos 2 0! cos 2 </>! cos 20 X cos 20J . 
 
 If & be the difference of phase between the two reflected rays, 
 we have 8 = 12 3 i = Tr/2 3l , so that 8 is determined by the 
 formula first obtained. With quartz the birefringence is so small 
 that y is practically negligible, and the difference of phase is given 
 very approximately by cot 8 = tanh x cot fa cot 2^. The following 
 table compares the results of theory and experiment for the ray F 
 incident at an angle of 45 for various values of v. 
 
 The agreement is as close as could be expected, the differences 
 being within the limits of experimental errors. With a crystal of 
 spar, where the birefringence is not negligible, it has been found 
 that for <k = 45 and v = 22 48' the value of S/TT is 0'184 for 
 the ray D and 200 for the ray 6 X . The values given by the 
 above theory in these two cases are 0192 and 0'200, so that the 
 agreement is excellent. 
 
VII] 
 
 WITH TRANSPARENT CRYSTALS 
 
 195 
 
 V 
 
 5/7T (theory) 
 
 5/7T (exp.) 
 
 Difference 
 
 15 47' 
 
 0-852 
 
 0-840 
 
 + 0-012 
 
 28 46' 
 
 0-674 
 
 0-668 
 
 + 0-006 
 
 31 15' 
 
 0-614 
 
 0-612 
 
 + 0-002 
 
 44 50' 
 
 0-286 
 
 0-284 
 
 + 0-002 
 
 53 18' 
 
 0-186 
 
 0-194 
 
 - 0-008 
 
 68 50' 
 
 0-126 
 
 0-142 
 
 -0-016 
 
 Reflection at a twin plane of a crystal. 
 
 If the media on each side of an interface are each portions of 
 the same uniaxal crystal, and the optic axes of the two portions 
 are in a plane at right angles to the 
 interface and make angles a and a 
 with the interface, then the interface 
 is called a twin plane of the crystal. 
 The features of the reflection at such 
 a twin plane can be discussed by 
 
 Fig. 72. 
 
 means of the principles used in this chapter. In general the 
 potential energy is given by an expression of the form 
 
 W = J |(4/ 2 + Bf + Ch* + 2Fgh + 2Ghf+ 2Hfg) dr. 
 
 In the present case we shall take x = as the interface, and z 
 as the plane containing the optic axes of the two portions of the 
 crystal. There is then perfect symmetry with respect to the 
 plane z 0, and hence we must have F=G = 0. In the upper 
 medium 
 
 Ch*) dr y 
 
 W = i (Af* 
 and in the lower medium 
 
 132 
 
196 REFLECTION AND REFRACTION [CH. 
 
 Applying the Principle of Action to the first medium we get the 
 dynamical equations 
 
 while the boundary conditions require the continuity of 
 
 n ( Hf+ Bg) - mCh, ICh-n (Af + Eg), 
 and m (Af+ Eg) - 1 (Ef+ Bg). 
 
 Since, in the present case, x = is the interface, the boundary 
 conditions are satisfied by the continuity of Ch and Ef+Bg. 
 In the lower medium H must be replaced by H, hence if 
 f\ , <7i> ^i refer to the first medium and / 2 , g z , h 2 to the second, the 
 boundary conditions require f, 77, f to be continuous, h to be con- 
 tinuous, and Hfi 4- Bg l = Hf 2 4- Bg 2 . 
 
 We shall consider two special cases of the general problem of 
 reflection at the twin plane, (I) when the plane of incidence 
 coincides with 2 = 0, the plane of the optic axes, and (II) when it 
 is at right angles to that plane. 
 
 I (a). First let us suppose that the curl (/, g, h) is perpen- 
 dicular to the plane of incidence, so that f=g = } and h is 
 independent of z. The dynamical equations are 
 
 $ = -cf, v = cf, -0, 
 
 dy dx 
 
 so that h (7V 2 h. This equation is the same in each medium, 
 so that if we take h to be the same in each medium the dynamical 
 equations will be satisfied, f, TJ and f will be continuous at the 
 interface, and, since f = g = Q, the other boundary conditions will 
 be satisfied. There is thus no discontinuity at the interface and 
 the wave passes on without any reflection or other change. 
 
 I (6). Next suppose that the curl is parallel to the plane of 
 incidence, so that h = 0, and, as before, everything is independent 
 of z. We then have 
 
VII] WITH TRANSPARENT CRYSTALS 197 
 
 The dynamical equations give 
 
 in the first medium, and the same equation with the sign of H 
 changed in the second. 
 
 For plane waves we may take 
 
 g l = e npt+ix+my) (incident wave) + re i(pt + l ' x+m y ) (reflected wave), 
 and f a = 5e *(p*+*i*+y) (refracted wave). 
 
 The boundary conditions require the exponential factors to be 
 the same for all values of t and y when x = 0, so that p and ra are 
 the same for all the waves. The dynamical equations give 
 
 and p* = A m 2 - 2Hl'm + Bl /z . 
 
 Equating the two values of p 2 we get 
 
 so that either I' I or B (l f + l) = 2#ra. The relation I' = I must 
 be rejected, as this would make the reflected ray coincide with the 
 incident one. Hence we have I' I + 2mH/B. 
 In the second medium we have similarly 
 
 and hence B (I* - If) = 2Hm (I + IJ, 
 
 so that either 1 1 = 1 J or B (I /j = 2Hm. 
 
 The relation l^ l is inadmissible, as this would represent a 
 wave travelling towards the interface in the same direction as the 
 incident wave. The alternative is ^ = I 2mH/B. 
 At the interface X = 0, we have 
 
 where 
 
 co=pt + my ; / 2 = imse iu> ; g l = ^ = i (I + I'r) e iu , g^~ ilise i<a , 
 
 so that 1 : r : s are the ratios of the amplitudes of the curls in the 
 three waves. Also the boundary conditions give 1 + r = s, and 
 
 Hm (1 + r) - B (I + I'r) = -Hms - Bl,s. 
 
198 REFLECTION AND REFRACTION [CH. 
 
 Substituting the values of V and h found above, the second of these 
 equations reduces to 
 
 (Hm - Bl) - (Hm -Bl)r = (Hm - Bl) s, 
 
 i.e. to 1 r = s. Since then 1 + r = s and 1 r = s, we must have 
 r = 0, so that there is no reflection. 
 
 II. We shall next consider the case when the plane of inci- 
 dence is y 0, perpendicular to that of symmetry. 
 * For the incident wave we have 
 
 ( v , f) = (L, M } N) leW+n+M = (L t M, N) le*", 
 
 and similarly for the other, waves, where I 2 + n 2 = p z /v 2 , v being the 
 velocity of the wave, and so a function of the direction cosines of 
 the wave normal. Since 
 
 % + * + %+* 
 
 90 8y dz 
 
 we have LI + Nn = 0, so that L/n = N/( I), and we may repre- 
 sent the incident wave by 
 
 This gives /= - -*? = - inMIe iti> ; g = i (p 2 /v 2 ) Ie ita ; h = UMIe ita ; 
 
 oz 
 
 Substituting for /, g, h in the dynamical equations we get 
 p* = (tfltf) B-nH=l*C+ n*A - n (p*/v*) H/M. 
 
 As the reflected waves are receding from the surface we 
 represent them by 
 
 (f, 7?, f) = (Z', if', ^) I' e *(-r*+*+pv = (w, Tltf 7 , Z') /V', 
 where Z x is positive. The dynamical equations then give 
 p* = (pa/t,'*) B - nM'H = l'*C +n*A-n (p^v^ H/M'. 
 
 Putting p^/v' 2 i'* + n 2 and eliminating M' from the equations just 
 obtained, we get 
 
 <> 2 - l'*B - n 2 B) (p 2 - l' 2 C -n*A)=- n*H 2 (I' 2 + n 2 ), 
 
 i.e. l'*BC - I' 2 {(B +C)p 2 -(BC + AB + H 2 ) ri*} 
 
 + p* -p 2 n 2 (A+B)+n*(AB + H 2 ) = 0. 
 
VII] WITH TRANSPARENT CRYSTALS 199 
 
 We should get exactly the same equation by eliminating M 
 from the equation 
 
 (p*/v 2 ) B-nH=PC + n*A - n (p z /v 2 ) H/M 
 
 obtained above when considering the incident wave. Hence one 
 of the roots of the quadratic in I' 2 is I' 2 = I' 2 , and as I' is positive, 
 this gives I' I and so v' = v. As, however, the equation in I' 2 is a 
 quadratic, there will be another possible value of I', and conse- 
 quently another reflected wave. From the theorem as to the 
 products of the roots of a quadratic we see that the other value of 
 I' is 
 
 Thus the two reflected waves are represented respectively by 
 
 and (f , 77, f ) = (n, M f , I') IV 
 
 where V has the value just obtained. 
 
 If we deal in exactly the same way with the transmitted waves 
 we find them represented by 
 
 and (f , 77, f) = 0, - 3//, - T) / 2 
 
 The boundary conditions, of which only four are independent, lead 
 to these equations : 
 
 / _ /' _ f'l'/i = /!+/, I'll ;/-/'- Fl'M'IlM = -/,-/, I'M'jlM ; 
 from which we obtain 
 
 7 = ~ (Of - IM') (I'M' - IM) ' and T = (Of - IM')(1'M' - IM) ' 
 
 These equations are quite general ; but it will be convenient 
 to deal with them in their simpler form when the doubly refracting 
 power of the crystal is small. In such circumstances we have 
 C = B = A nearly, H is small, and ^ and v 2 are nearly equal. We 
 have seen in the last chapter that for a given wave velocity in 
 any crystal there are only two possible directions of the displace- 
 ment (or of its curl), and that these two are at right angles. 
 
200 REFLECTION AND REFRACTION [CH. 
 
 Hence as v l = v 2 nearly, the lines (I, M, n) and (If, M', ri) are nearly 
 at right angles, and as the first approximation we may put 
 
 and therefore - MM' = IV + n~ = I 2 + n 2 (nearly) = p^/v 2 . 
 Also from the equations already obtained we have 
 
 -pa = (I* + n 2 ) A - nHM = (l r * + n) A - nHM' ; 
 and thus (Z /2 - 1 2 )/(M' - M) = nH/A, 
 
 and (I' - l)l(M' -M) = nH/2Al 
 
 approximately. Hence, to this order of approximation, we have 
 
 r MM' nH T MM + M' nH 
 
 I~ M-M'Al I M-M' 
 
 Let us suppose now that we have a crystal, such as has been 
 under discussion, imbedded in an isotropic medium the surface of 
 which is parallel to the twin plane of the crystal. We shall then 
 have two waves in the isotropic medium in directions (I, 0, n) and 
 (l' t 0, n) and these two will be incident on the twin plane. We 
 shall consider the components of the curl perpendicular and 
 parallel to the plane of incidence, denoting these components 
 respectively by P and Q. 
 
 We then have 
 
 Q = / 2 + A 2 = i *p + n*. MIe*" = i (p/v 
 so that, omitting factors that are common throughout, we may 
 put P = (p/v) I and Q MI. Thus for the two incident waves 
 we have P = (p/v) (I + J) ; Q = MI + M'J. In dealing with the 
 reflected waves corresponding to J, it must be noted that J' and 
 J" are to be obtained from the formulae for I' and I" by replacing 
 / by J, and interchanging I and l' t M and M '. Thus for the 
 reflected wave we get 
 
 P' = (p/v) [/' + /" + /' + /"] 
 
 = (p/v) (MI + M'J) nH/2Al* = (p/v) Q . nH/2AP, 
 and Q' = M [I' + J'] + M' [I" + /"] 
 
 = _ [/ + J] MM'nHjZAV = (p/v) P . nH/2A l\ 
 
 It thus appears that whatever be the polarisation of the incident 
 light, the intensity of the reflected light is proportional to that 
 of the incident ; that if P = Q and there is no permanent phase 
 
VII] WITH TRANSPARENT CRYSTALS 201 
 
 relation between the components, then the same holds for P' and Q', 
 in other words that if the incident light is unpolarised so also is 
 the reflected. Further we see that if Q = 0, then P' = 0, while if 
 P = 0> Q = 0, so that if the incident light be polarised parallel or 
 at right angles to the plane of incidence, the reflected light is 
 polarised in the opposite manner. 
 
 In the latter part of this investigation we have tacitly assumed 
 that the crystalline stratum imbedded in the isotropic medium 
 is thin, for we have disregarded any retardation of phase in 
 crossing the stratum to and from the twin plane. This retarda- 
 tion will not polarise any light originally unpolarised, so that 
 if the stratum is no longer thin, it will still remain true that 
 if the incident light is unpolarised, so also is the reflected. If, 
 however, the incident light be polarised the passage through the 
 crystalline stratum will tend to depolarise it, and the character of 
 the emergent beam will depend on the thickness of this stratum. 
 Let us suppose that the incident light is polarised so that we have 
 only one wave (/) incident on the twin plane. Then the ratio of 
 the intensities of the two reflected waves (/' and I") is 
 (F + M ' 2 + n 2 ) I' 2 : (I'* + M /2 + n 2 ) I" 2 
 
 = (Jf 2 - MM') / /2 : (M' 2 - MM') /" 2 = - MM f : (M + MJ. 
 Now by eliminating p 2 from the equations 
 
 pa = (pi/v 2 ) B-nH = PC + n 2 A - n (p 2 /v 2 ) H\M 
 obtained on p. 198, we get 
 
 nHM 2 + M[(A-B)n 2 + (C-B) I 2 ] - Hn (I 2 + n 2 ) = 0. 
 
 If we disregard the difference between I and I' we may look upon 
 this as a quadratic equation from which the two values of M are 
 to be found. We thus get 
 
 MM' = - (I 2 + n 2 ), and M + M' = {(A - B) n 2 + (C - B) l 2 }/Hn. 
 
 If then (f> be the angle of incidence so that cos <f> = l/*l* + n 2 , the 
 ratio of the square roots of the intensities of the two reflected 
 waves is 2H sin < : [(A - B) sin 2 <f> + (C-B) cos 2 </>]. This is not 
 negligible unless </> be very small, so that it is only when the 
 angle of incidence is small, and when the incident light is polarised 
 parallel or at right angles to the plane of incidence, that the 
 reflected beam is fully polarised. 
 
202 REFLECTION AND REFRACTION [CH. VII 
 
 The above theory has been applied by Lord Rayleigh to 
 explain the peculiar internal coloured reflection exhibited by 
 certain crystals. In some cases the reflection appears to be 
 almost total. The coloured light is not polarised, and if one 
 of the crystalline plates be turned round in its own plane, with- 
 out altering the angle of incidence, the reflected light vanishes 
 twice in a revolution, viz. when the plane of incidence coincides 
 with the plane of symmetry of the crystal. " It is certainly very 
 extraordinary and paradoxical," says Stokes, in describing the 
 phenomenon, " that light should suffer total or all but total 
 reflection at a transparent stratum of the very same substance, 
 merely differing in orientation, in which the light had been 
 travelling, and that independently of its polarisation." It appears, 
 however, from the above investigation that the theory accounts 
 for the facts observed by Stokes. It also predicts certain laws 
 not previously suspected, but afterwards confirmed experimentally 
 by Lord Rayleigh. " Theory shows," he says, " that in the act of 
 reflection at a twin plane, the polarisation is reversed. If the 
 incident light be polarised in the plane of incidence, the reflected 
 light is polarised in the perpendicular plane, and vice versa. 
 When I first obtained this result, I thought it applicable without 
 reservation in the actual experiment, and on trial was disap- 
 pointed to find that the reflected light was nearly unpolarised, 
 even when the incident light was fully polarised, whether in the 
 plane of incidence or in the perpendicular plane. When, however, 
 the angle of incidence was diminished, the expected phenomenon 
 was observed, provided that the original polarisation were in, or 
 perpendicular to, the plane of incidence. If the original polarisa- 
 tion were oblique, the reflected light was not fully polarised, even 
 though the angle of incidence were small." 
 
CHAPTER VIII. 
 
 TRANSPARENT CRYSTALLINE PLATES. 
 
 THIS chapter will be devoted to a discussion of the chief 
 features of the phenomena observed when polarised light is 
 transmitted through a crystalline plate and subsequently passed 
 through an ' analyser/ which damps all displacements except those 
 confined to a particular orbit. 
 
 Let OA (Fig. 73) be a wave front incident on the crystal. 
 The incident wave will give rise to two waves within the plate, 
 
 the fronts of these waves being OB and 00. These, on emer- 
 gence, will produce two waves BE and OF parallel to OA. The 
 emergent waves will thus differ in phase by an amount 
 
 A = (2-7T/X) . BN = (27T/X) . BO sin < = (2ir/\) . sin </> . (BM - CM) 
 
 = (2?re/X) sin <f> (cot ^ cot < 2 ), 
 
 where e is the thickness of the plate, and cf> represents as usual the 
 angle that the wave normal makes with the normal to the interface. 
 
204 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 If v l and v 2 be the velocities of the two waves in the crystal, that 
 in air being unity, we have sin </> = sin (f> 1 /v 1 = sin </> 2 /^ 2 , so that we 
 may also express A in the form A = (27re/\) [cos fafa cos </> 2 /u 2 ], 
 which is convenient for some purposes. 
 
 When light passes through such a crystalline medium it 
 follows from the principles explained in Chapter vi. that the 
 
 Fig. 74. 
 
 emergent waves are polarised in two directions OX and OF at 
 right angles to one another. If the plane polariser be so placed 
 that its principal plane makes an angle a with OX, and if A be 
 the amplitude of the incident displacement, we may conveniently 
 resolve this displacement into two components whose amplitudes 
 are A cos a along OX and A sin a along Y. Each of these 
 components will give rise to a separate wave within the crystal, 
 the waves travelling with different speeds and emerging from the 
 plate with a difference of phase A. There will also be a certain 
 diminution of amplitude in the transmitted beams due to the loss 
 of light by reflection ; but in all cases to which the theory will be 
 applied this diminution is small and is almost exactly the same 
 for the two waves which are always very close together. Thus, as 
 it is the relative amplitude of the two waves which mainly 
 concerns us at present, we may with sufficient accuracy repre- 
 sent the emergent waves by the components A cos a e ipt and 
 A sin ae i(pt ~* } , or, confining ourselves to the real parts of these 
 expressions, by A cos a cospt and A sin a cos (pt A). On resolv- 
 ing the displacements along OA, the direction of the principal 
 plane of the plane analyser, and putting AOX = j3 we find that 
 the displacement in the beam that emerges from the analyser is 
 A [cos a cos & cos pt 4- sin a sin /3 cos (pt A)]. Thus the intensity 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 205 
 
 of the emergent beam is / = A 2 [cos 2 (/3 a) sin 2a sin 2/3 sin 2 A/2]. 
 In a plate of uniform thickness all the quantities in this expression 
 are constant, so that the plate appears uniformly illuminated. 
 
 The above investigation applies primarily to the case of a 
 single incident wave ; but if white light be employed we have to 
 deal not with a single wave of definite amplitude A and wave 
 length X, but with an infinite number of such waves differing in 
 amplitude and wave length. The total illumination will, in this 
 case, be obtained from the above formula by summing the 
 intensities due to the different waves that make up the white 
 light. We thus find 
 
 / = cos 2 (ff - a) 2A 2 - sin 2a . sin 2 . 2A 2 sin 2 A/2 
 = 7 - sin 2a sin 2 2A 2 sin 2 A/2, 
 
 where / is the illumination when A = 0, i.e. when the crystalline 
 plate is removed and the light merely passes through the analyser. 
 The colours produced by transmitting light through crystalline 
 plates were compared by Biot with those seen in Newton's rings 
 formed by reflection from an isotropic medium. The theory of 
 these rings has been discussed in Chapter v., and from this it 
 appears that the intensity at normal incidence is given by the 
 formula 
 
 4R 2 sin 2 2W/X 
 
 ' 1 + E 4 - 2JR* cos 4W/X ' (1 - R 2 ) 2 + 4# 2 sin 2 2W/X 
 (^ 2 -l) 2 sin 2 27r67X 
 
 '4 A < 2 + O 2 -l) :i sin 2 
 
 where A is the amplitude of the incident wave, X the wave length, 
 I* the corresponding coefficient of refraction, and e' the thickness 
 of the film of air. In the case of glass for which p = 1/5, the term 
 (/-I 2 l) 2 sin 2 27re7X is small compared with 4/A 2 , so that we have 
 approximately / = A 2 sin 2 2ire / /\ . (//, 2 - 1) 2 /V 2 . This represents 
 the intensity due to a single wave ; but if white light were 
 employed we should have I = %A 2 sin 2 Zire j\ . (/x 2 l) 2 /4//, 2 . The 
 refractive index /A depends on the wave length, but the variation 
 within the visible spectra is not great. If, as an approximation, 
 we neglect the variation of //,, we have / proportional to 
 "A 2 sin 2 27re/X. Biot's experiments were made at normal incidence 
 with a = 45 and /3 a = 90. In this case the formula found 
 
206 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 above for the intensity after passing through a crystalline plate 
 gives / = 2A a sin a A/2. Hence the colours should be approximately 
 the same as those seen in Newton's rings, provided 
 
 2W/X = A/2 = (7re/X) (l/v, - l/v t ) ; 
 
 where v l and v 2 as before are the velocities of the two waves in the 
 crystal, the velocity in air being unity. This makes 
 
 which in many crystals is independent of the wave length to the 
 same order of approximation as before. If then we deal with 
 crystals that do not cause a considerable dispersion of the waves, 
 we should expect the ratio of the thicknesses required to produce 
 a particular colour to be nearly the same for the various colours. 
 The following table gives the results of Biot's measurements for 
 ten different colours. 
 
 e 
 
 38-5 
 
 55-5 
 
 60-2 
 
 60-5 
 
 62-0 
 
 65-5 
 
 73-6 
 
 77-15 
 
 93-8 
 
 153-1 
 
 e x 9/36-5 
 
 9-50 
 
 13-68 
 
 14-83 
 
 14-92 
 
 15-29 
 
 16-15 
 
 18-15 
 
 19-00 
 
 23-13 
 
 37-70 
 
 e' 
 
 9-70 
 
 13-55 
 
 14-67 
 
 15-10 
 
 15-10 
 
 16-25 
 
 18-17 
 
 18-70 
 
 23-20 
 
 37-50 
 
 If the above formula were exact and not merely approximate, 
 then 9 : 36'5 would represent the constant ratio of e' to e and the 
 last two rows of the table would be identical. There is thus 
 a good agreement between theory and observation as far as the 
 influence of the thickness of the plate on the colour is concerned. 
 
 Returning to the general formula 
 
 / = cos 2 (/3 - a) 2.A 2 - sin 2a . sin 2/3 2 ^ sin 2 A/2 
 
 = / - sin 2a sin 2/3 2 A' 2 sin 2 A/2, 
 
 we see that of the two terms in this expression for / the second 
 alone depends on the lengths of the waves forming the incident 
 beam. It is the presence of this term in the formula for / that 
 accounts for any colour that may be observed, and so it is 
 sometimes called the colour term. The colour term vanishes 
 when sin 2 . sin 2/3 = 0, which is the condition for achromatism. 
 The lines joining points in the field of view at which this condition 
 is satisfied are called achromatic or neutral lines. In crossing one 
 of these lines the colour term changes sign, so that the quantity 
 
VII I] TRANSPARENT CRYSTALLINE PLATES 207 
 
 in / that was formerly added to 7 is now subtracted and vice 
 versa. Hence the crossing of such a line involves a change from 
 one colour to the complementary one. If, however, the two factors 
 of sin 2a . sin 2/3 vanish simultaneously the neutral lines corre- 
 sponding to the two factors coincide, and there is no change of sign 
 or colour in crossing them. Such lines are called double neutral 
 lines. 
 
 If we are dealing with homogeneous light there will be a family 
 of lines in the field of view, the retardation A being constant 
 along any member of the family. With most crystals the form of 
 these lines is approximately independent of the wave length of the 
 light. Hence if we use white light and keep to one of these lines 
 the colour will be the same along any section of the line not 
 crossed by a " single " neutral line. These lines are consequently 
 spoken of as isochromatic lines. 
 
 A third family of lines is sometimes employed to describe the 
 phenomena. The angle POX in Fig. 74 represents the angle 
 (a.) between the principal plane of the polariser and the direction 
 of displacement in one of the waves (say the quicker wave) within 
 the crystal. Points in the field of view for which this angle is the 
 same will therefore correspond to points for which the plane of 
 polarisation has been rotated through the same angle by the 
 crystal. Hence the lines joining these points are called isogyric 
 lines. 
 
 The last set of lines that may usefully be considered in forming 
 a picture of the field of view are the lines of equal intensity, 
 corresponding to / = constant. 
 
 In any given position the intensity is a maximum when 
 A = 2tt7r and a minimum when A =(2n 4 I)TT, provided 
 sin 2a . sin 2/3 be positive ; but if sin 2a . sin 2/3 be negative the 
 conditions for maxima and minima are interchanged. When 
 ft a =90, / vanishes and the colouring is most marked. We 
 then have / = sin 2 2a . 2 A 2 sin 2 A/2 and there are consequently four 
 positions of maximum intensity given by sin 2a = + 1, and four 
 positions of zero intensity given by sin 2a = 0. As / vanishes with 
 sin 2a we see that, in this case, the neutral lines coincide with 
 the lines of zero intensity. Also for any given wave length / 
 vanishes when A = 2mr (n being any integer), so that the lines of 
 
208 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 zero intensity are special cases of the isochromatic lines. When 
 ft = a the analyser and polariser are parallel, and 
 
 / = 2A 2 - sin 2 2cA* sin 2 A/2, 
 
 so that when sin 2a = we have / = 2 A 2 , representing white light. 
 On replacing J3 by 90 + @ in the formula for / we see that, in 
 general, the intensity in any position of the analyser is comple- 
 mentary to that in the perpendicular position. 
 
 We proceed to investigate the form of some of the families 
 of lines spoken of above in certain cases that have been examined 
 experimentally, contenting ourselves with approximations suffi- 
 ciently accurate to colligate the facts of experience. 
 
 We begin with the case of a uniaxal crystal for which, in the 
 notation hitherto employed, we have 
 
 sin (f> = sin <t>i/Vj = sin <f> z /v z ; v l = c ; V 2 9> = a 2 sin 2 a> 2 + c' 2 cos 2 co. 2 . 
 If the crystal be cut at right angles to the optic axis 
 
 a>i=</>i and &> 2 ==( />2> 
 
 so that sin 2 < 2 cosec 2 (f> = v 2 2 = a 2 sin 2 < 2 + c 2 cos 2 <p 2 . 
 Hence we have cosec 2 <f> = a 2 + c 2 cot 2 <f> 2 , 
 
 and cot </> 2 sin < A/ 1 a 2 sin 2 <f>/c. 
 
 Similarly we should find 
 
 cot $! sin (j> = v 1 c 2 sin 2 <j>/c ; 
 and so obtain A in the form 
 
 A = sin $ (cot </>! cot ^ 2 ) 2?re/X 
 
 = (Vl - c 2 sin 2 - Vl - a 2 sin 2 </>) 27re/Xc. 
 
 The isochromatic curves correspond to A = constant, and 
 therefore to </> = constant. If the incident light consist of a small 
 pencil of rays proceeding from a point G (Fig. 75) at 
 a considerable distance d from the plate, the waves 
 near will be approximately plane and the above d \ 
 
 investigation will be applicable. If CQ be any in- OQ 
 
 cident ray, the foot of the normal from C on the 
 plate, and OQ = r, then cf> = constant corresponds to 
 r constant, and the isochromatic curves are a series of concentric 
 
VIII] 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 209 
 
 circles of which is the centre. If the pencil is so small that 
 sin 4 <f> may be neglected, we have 
 
 A = (a 2 - c 2 ) sin' 2 c/> . -rre/Xc = -rre (a 2 - c 2 ) r*/\cd 2 , 
 
 which expresses the relation between the radii of the rings, the 
 thickness of the plate and the wave length of the light. The 
 quantities a and c vary with the wave length ; but this variation 
 is slight in most cases, so that for a given retardation the radii of 
 the rings are approximately proportional to the square root of the 
 wave length and inversely to the square root of the thickness of 
 the plate. When A = 2mr, where n is an integer, we have r 2 /X 
 proportional to the natural numbers, which is the same law as 
 with Newton's rings. This special case is important because, as 
 we have seen above, these isochromatic lines coincide with the 
 lines of zero intensity when the polariser and analyser are at right 
 angles. In this case, too, the lines of zero intensity also include 
 the neutral lines given by sin 2a = 0, which represents two straight 
 lines at right angles to one another, parallel and perpendicular 
 respectively to the plane of the polariser. Fig. 76 represents the 
 
 Fig. 76. 
 
 central lines of the curves of zero intensity and is drawn from a 
 photograph of what is actually seen. It will be observed from this 
 that there is a good agreement between theory and observation in 
 this matter. 
 
 When the polariser and analyser are parallel it is found that 
 the effect is complementary to that seen in the last case. This ia 
 M. L. 14 
 
210 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 in accordance with the theory as stated above, as is also the fact 
 that when /3 a is neither zero nor 90, the colour along an 
 isochromatic circle changes into its complementary on crossing 
 one of the neutral lines corresponding to a = 0, a = 90, = 0, 
 fi = 90. Also when ft - a is zero or 90 the neutral lines are 
 double and there is no change of colour in crossing the lines. 
 
 Turning to the case where the crystal is cut parallel to the 
 optic axis, we see that there are now no neutral lines. For in this 
 case the displacements in the two waves are parallel or perpen- 
 dicular to the optic axis. Thus the displacements are parallel for 
 all points of the field, so that there could be no achromatic lines 
 unless the polariser or analyser had their principal planes parallel 
 or perpendicular to the optic axis. If, however, this were the case 
 the waves transmitted through the instrument would be all of one 
 species, i.e. either all ordinary or all extraordinary, so that there 
 could be no interference. 
 
 To obtain the isochromatic lines we have cos &) 2 = cos v sin <f> 2 ; 
 
 sin 2 < 2 cosec 2 </> = v 2 2 = a 2 - (a 2 - c 2 ) cos 2 a> 2 = a 2 - (a 2 - c 2 ) cos 2 v sin 2 </> 2 , 
 and therefore 
 
 sin 2 2 = a 2 sin 2 </{! + (a 2 - c 2 ) sin 2 (f> cos 2 v}. 
 Thus we get 
 A = sin (cot </>! cot </> 2 ) 27re/X 
 
 = [Vl c 2 sin 2 </>/c - Vl - sin 2 < (a 2 sin 2 v + c 2 cos 2 v)/a] 27re/A, 
 = [2 (1/c - I/a) + {(a 2 sin 2 v + c 2 cos 2 v)/a - c} sin 2 </>] 7re/X 
 
 to the same order of approximation as before. 
 
 In Fig. 77 the surface of the crystal is represented by the plane 
 
 Fig. 77. 
 
Vlll] TRANSPARENT CRYSTALLINE PLATES 211 
 
 z = 0, the axis of the crystal is along the axis of #, and CQ is any 
 incident raj. Then COQ is the plane of incidence and xOQ the 
 angle that the optic axis makes with this plane, an angle hitherto 
 denoted by v. Hence if OC= d, as before, the coordinates of P are 
 given by the equations x = d tan </> cos v = d sin < cos v, and 
 y = d tan $ sin v = d sin < sin v, to our order of approximation. 
 The isochromatic curves correspond to A = constant, i.e. to 
 (ac a 2 sin 2 v c 2 cos 2 v) sin 2 $ constant, i.e. to ex 2 ay' 2 = constant. 
 This represents a series of hyperbolas, which are very nearly 
 rectangular, since the difference between a and c is small. 
 The formula for A may be expressed thus : 
 
 A = A + K (a - c) (ay 2 - ex 2 ), 
 where A = (1/c I/a) 27re/X, 
 
 and K = 7re/\ad 2 . 
 
 In the case of normal incidence x and y are zero and A = A , so 
 that A is the retardation of phase when the incidence is normal. 
 By superposing one plate on another it is possible to make the 
 retardation of phase vanish for normal incidence, and when this 
 happens the plates are said to be compensated. When the axes of 
 the two plates are parallel, an ordinary wave in the first remains 
 ordinary in the second. The total difference of phase produced 
 by the plates is 
 
 A + A' = [e (1/c - I/a) + e' (1/c' - I/a')] 2?r/X 
 
 + [e (af - ex 12 ) (a - c)/a + e (ay 12 - c'a?) (a - c)/a'] 7r/\d 2 , 
 
 for, the plates being thin, the difference between d and d' may be 
 neglected in the terms of the second order of small quantities. The 
 plates will be compensated if 
 
 e (1/c -I/a) =e'(l/a'-l/c'), 
 
 and the isochromatic curves will be ellipses or hyperbolas 
 according as 
 
 e (a c) e' (c' - a'), 
 
 and e'c (c a)/a tc (a c)/a 
 
 have the same or opposite signs. 
 
 When the plates are crossed so that their axes are at right 
 angles, the ordinary wave in the first plate becomes the extra- 
 
 142 
 
212 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH, 
 
 ordinary wave in the second and vice versa. In this case v in the 
 formulae above must be replaced by 90 + v, so that x is replaced 
 by y and y by x. Thus the retardation produced by the plates is 
 
 A + A' = [e (1/c - I/a) - e' (1/c - I/a')] 27T/A, 
 
 + [e O# 2 - c# 2 ) (a - c)/a - e' (a V - c'?/ 2 ) (a - c)/a] vr/Xd 2 . 
 
 If the plates be of the same material and thickness they will be 
 compensated for normal incidence and the retardation will be 
 
 (a 2 - c 2 ) (f - # 2 ) Tre/Xad 2 . 
 
 Hence the isochromatic curves consist of a series of rectangular 
 hyperbolas whose equations are # 2 ?/ 2 = constant. When the 
 analyser and polariser are at right angles we have seen that the 
 lines of zero intensity are the neutral lines given by sin 2 2 = 0, 
 i.e. the two lines a = and a = 90, and the isochromatic lines 
 A = 2mr which, as we have just seen, are the rectangular 
 
 hyperbolas 
 
 x* y 2 _ constant. 
 
 Fig. 78 represents these dark lines drawn from a photograph and 
 
 Fig. 78. 
 
 shows that there is a close agreement between theory and experi- 
 ment. 
 
 So far we have been concerned with uniaxal crystals cut either 
 at right angles or parallel to their optic axes ; but the methods 
 employed in these special cases will enable us to solve the more 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 213 
 
 general problem when the crystal is cut in any manner whatever. 
 The position of the optic axis with reference to the interface and 
 the plane of incidence may be determined as in the last chapter 
 by means of the coordinates u, v or I, \ (see p. 167). We can 
 transform from one set of coordinates to the other by means of the 
 relations 
 
 sin v = cos X sin I. 
 
 and sin X = cos u cos ?;, 
 
 while a). 2 may be expressed in terms of u and v by the formula 
 
 cos <w 2 = cos v cos (</> 2 - u), 
 and in terms of / and X by the formula 
 
 cos ft> 2 = cos 2 sin X -f sin </> 2 cos X cos /. 
 Taking the latter formula we get 
 
 sin 2 02 cosec 2 < = v 2 2 = a 2 (a 2 c 2 ) cos 2 2 
 
 = a 2 (a 2 c 2 ) (cos c/> 2 sin X -h sin </> 2 cos X cos Z) 2 , 
 so that 
 
 cosec 2 (f> = a 2 (1 + cot 2 </> 2 ) - (a 2 c 2 ) (cot </> 2 sin X -f cos X cos If, 
 
 and 
 
 (sin <f) cot 2 ) 2 [a 2 cos 2 X + c 2 sin 2 X] 
 
 2 (sin (f) cot < 2 ) [( 2 c 2 ) sin X cos X cos sin 0] 
 - [1 - a 2 sin 2 < + (a 2 - c 2 ) cos 2 X cos 2 Z sin 2 <j>] = 0. 
 This is a quadratic equation in sin < cot t/> 2 , and the product of 
 the roots is negative. Hence the equation has a positive and 
 a negative root, of which we must take the positive one. Thus 
 sin <f) cot 2 is given definitely in terms of known quantities, and 
 sin (/> cot (/>j may be found similarly, or deduced by making a = c. 
 The retardation A is then obtained by means of the fundamental 
 formula 
 
 A = (sin cot 0j sin </> cot 2 ) 2 < 7re/X / , 
 
 where X 7 is the wave length, the accent being used temporarily 
 to distinguish the wave length from the angle X. If we wish 
 to obtain the equation of the isochromatic curves we have to 
 remember that I is the angle between the plane of incidence and 
 that through the normal to the plate and the optic axis. Hence 
 we have, to the same order of approximation as before, 
 
 x = d . sin <f> cos I, 
 and y = d . sin sin /, 
 
214 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 so that the equation for sin (f> cot fa becomes 
 (sin < cot fa) 2 [a 2 cos 2 X + c 2 sin 2 X] 
 
 2 sin </> cot ( 2 [sin X cos X . (a 2 c 2 ) ac/d] 
 
 - [1 - a 2 O 2 + ?/ 2 )/d 2 + cos 2 X . (a 2 - c 2 ) # 2 /d 2 ] = 0. 
 
 The discussion of this equation is somewhat simplified by in- 
 troducing the quantity v 0) the velocity of an extraordinary wave 
 moving parallel to the face of the crystal. We have 
 
 so that the equation for sin </> cot fa gives 
 v 2 sin cot fa 
 
 = [(a 2 - c 2 ) sin X cos X . x/d + \V - a 
 Hence . A = (sin (/> cot fa sin </> cot fa) 2-Tre/X' 
 
 = (4 + Bx + <7# 2 + Df) 27re/X', 
 where 
 
 A = I/c- l/v ; 5 = sin 2X . (c 2 - a 
 
 (7 = (a 2 c 2 /v 3 - c)/2c^ 2 ; D = (a 2 /?; - c)/2^ 2 . 
 
 The special cases already considered are obtained from these 
 formulae by putting X = 90 or 0, i.e. v = c or a respectively. In 
 general the isochromatic curves are conies whose nature depends on 
 the coefficients B, C, and D. As the optic axis moves from the 
 normal the angle X decreases from 90 to zero, and the curves 
 change from ellipses to hyperbolas. The transition takes place 
 when (7 = 0, i.e. when 
 
 sin 2 X = (1 - c 2 / 3 /a 2/3 )/(l - c 2 /a 2 ), 
 
 in which case the curves are parabolas. 
 
 Just as in the special cases considered above we can obtain 
 some interesting modifications of the forms of the isochromatic 
 curves by superposing one plate upon another. If the two plates 
 be of the same material and thickness, similarly cut, and held 
 with their axes and principal sections parallel, the retardation is 
 double that for a single plate. If now one plate be turned through 
 two right angles in its own plane we must replace I by 180 + 1 
 when dealing with the second plate, and so replace x by x. The 
 retardation is thus 
 
 e/X / , 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 215 
 
 and the isochromatic curves are ellipses or hyperbolas with their 
 centres at the foot of the normal from the radiant point to the 
 plate. When the analyser and polariser are crossed a series of 
 black spots should appear at the points of intersection of the two 
 systems of isochromatic lines relative to each of the plates. Thus 
 the lines will seem to be discontinuous. Fig. 79, drawn from 
 
 a photograph, represents the actual appearance in one case, and 
 indicates a good agreement with the predictions of the theory. 
 Again, if the two plates be crossed we must replace I by 90 4- 1 
 when considering the second plate, and this will change x into y 
 and y into x. Moreover as the waves change their species in 
 passing from one plate to the other, the complete retardation is 
 the difference of the retardations produced by each plate. Hence 
 in this case we have 
 
 A = [(A + Bx + Ctf + Dz/ 2 ) - (A - By + Cy* + Dx*)] 2ir/V 
 = [B(x + y) + (C- D) (x* - f)] 2-Tre/X'. 
 
 The isochromatic curves are therefore hyperbolas, approximating 
 closely to straight lines, represented by x + y constant, in the 
 neighbourhood of the origin. These lines are a well known 
 feature of Savart's polariscope, which consists essentially of two 
 plates crossed in the fashion here described. 
 
 The corresponding phenomena in the case of biaxal crystals 
 
216 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 may be dealt with in exactly the same way by aid of the formulae 
 in Chapter vi. Thus suppose that we have a crystalline plate 
 cut at right angles to the bisector of the angle between the optic 
 axes, when this angle is nearly 180. The phenomena, as might 
 be expected, are very like those produced by a uniaxal plate cut 
 parallel to the optic axis, and discussed on p. 210. We have 
 seen that if v 1 and # 2 be the two wave velocities corresponding to 
 a given direction of the wave normal, then 
 
 vf = 6 2 + (a 2 c 2 ) cos I/TJ cos ^r/ = 6 2 + (a 2 c 2 ) sin 6 sin 6' cos 2 ^, 
 and v 2 2 = 6 2 - (a 2 - c 2 ) sin sin 6' sin 2 ^. 
 
 It should be observed that the two waves in the crystal have not 
 the same normal ; but in the present case the difference of 
 directions is so small that we may neglect it to the order of 
 approximation with which we are being satisfied. We may then 
 take Vi and v 2 given by the above formulae as representing the 
 velocities of the two waves in the crystal. Let OA l and OA 2 be 
 the optic axes, ON a wave normal in the crystal, OC the normal to 
 
 Fig. 80. 
 
 the plate. NA^ and NA 2 are what have previously been denoted 
 by 6 and 6' respectively. Let the angle A-fJN-^ (Fig. 80) be 7, 
 
 A,C = A 2 C = a)/'2 = 7T/2 - f , 
 
 where f is a small quantity. Neglecting powers of f and < above 
 the second we get 
 
 cos 6 = cos (j> cos to/2 + sin </> cos 7 sin o>/2 = f + c cos 7 ; 
 
 cos & = f < cos 7 ; sin 6 sin 6' = 1 f 2 c/> 2 cos 2 7 ; 
 
 cosx=0; sin% = l; b' 2 -c 2 = (a 2 - b 2 ) f 2 ; 
 
Vlll] 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 217 
 
 Hence, to this order of approximation, 
 
 v l = 6, and sin < cot fa = (1 - 6 2 </> 2 /2)/6, 
 
 V 2 2 = &2 + ( S _ J2) (J + 2) (1 - f - ^2 CQS 2 7 ) 
 
 = 6 2 + (a 2 - 6 2 ) (1 - c/> 2 2 cos 2 7) 
 = a 2 - (a 2 - 6 2 ) </> 2 2 cos 2 7 ; 
 sin $ cot 2 = I/a - < 2 (a 2 sin 2 7 + 6 2 cos 2 7)/2a. 
 Making the substitutions 
 
 x = d . tan (j) cos 7 = a 7 . < cos 7 and y = d . (f> sin 7 
 as before we get 
 
 A = (sin </> cot </>! sin < cot < 2 ) 27re/X 
 = (a - 6) [l/a& + (ai/ 2 - 6^ 2 )/2a^ 2 ] 27re/X. 
 
 Thus the isochromatic curves are hyperbolas, as in the case of 
 a uniaxal plate cut parallel to the optic axis. In this case, too, 
 there are not, as a rule, any achromatic lines. For if we take O^V, 
 the normal to the plate, as axis of z t the optic axes are nearly 
 coincident with Ox and Ox (Fig. 81). The directions of dis- 
 placement at any point are approximately along Ox and Oy, i.e. 
 
 Fig. 81. 
 
 are nearly parallel for all points in the field, and these directions 
 will not, in general, be parallel or perpendicular to the principal 
 planes of either polariser or analyser, OP or OA. 
 The formulae 
 
 Vi = 6 2 + (a 2 c 2 ) cos ^ cos A/T/ = b 2 + (a 2 c 2 ) sin 6 sin 6' cos 2 %, 
 and v 2 2 = 6 2 (a 2 c 2 ) sin 6 sin 6' sin 2 ^, 
 
 lead to some other results of importance. From them it appears 
 
218 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 that whenever 0, or 6', or a 2 c 2 is small, we have v l and v nearly 
 equal to b, and as the first approximation we may put 
 
 (a 2 - c 2 ) sin 6 sin 
 
 When a wave is incident nearly normally on a crystal, it gives rise 
 to two waves with different velocities v and v', and the difference 
 of phase on emergence is 
 
 It must be observed, as before, that these two waves have not the 
 same normal ; but in all cases to which we apply the formulae 
 the error in neglecting the difference of directions of the two 
 waves within the crystal is very small. Hence, as an approxima- 
 tion sufficiently close for most purposes, we may put 
 
 A - (a 2 - c 2 ) sin sin 6' (7re/\6 3 ) 
 
 whenever either 0, or 6', or a 2 c 2 is small. In such cases the 
 isochromatic curves are given by sin 6 sin 6' = constant. 
 
 As an example of this we may take the case when the optic 
 axes are inclined at a small angle and the crystal is cut at right 
 angles to the bisector of this angle. With the same notation as 
 before we have (Fig. 82) NA l = 0, NA 2 = 6' t and both these 
 
 quantities are small. Thus the equation sin 6 sin 6' = constant, 
 corresponds approximately to NA-^.NA^ constant, so that the 
 locus of N is a lemniscate of which A^ and A 2 are the foci. 
 
VIII] 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 219 
 
 Owing, however, to refraction the apparent positions of A l} A 2 and 
 N as seen from outside will be different from the real positions. 
 These points will be shifted to A , A and N', and as the refrac- 
 tive index will be very nearly constant for the different 
 directions represented by CA lf CA Z , and CN in the figure, the 
 triangle A^'A^N' will be very nearly similar to A l A Z N. Hence 
 NA-[ . NAz will be very nearly constant, and the isochromatic 
 curves will appear as lemniscates whose foci are the apparent 
 positions of the ends of the optic axes. 
 
 The neutral or achromatic lines are also readily obtained in 
 this case. We have seen that in general they are determined by 
 the condition sin 2a . sin 2/3 = 0, so that an achromatic line is the 
 locus of a point in the face of the crystal such that the displace- 
 ments in the two waves in the crystal are at that point parallel or 
 perpendicular to the principal plane of the polariser or analyser. 
 In Chapter vi. it appears that the displacement in a wave whose 
 normal is ON is at right angles to ON, and in a plane bisecting 
 the angle between the planes AfiN and A^ON. In the present 
 case ON is nearly at right angles to the interface, so that the dis- 
 placement will be very nearly in the face of the crystal along one 
 or other of the bisectors of the angle A^NA Z . Let us take rect- 
 angular axes Ox and Oy (Fig. 83) parallel and perpendicular to the 
 
 O i 
 
 Fig. 83. 
 
 principal plane of either polariser or analyser, and the origin at 
 the middle point of A 1 A 2> and let x lt y be the coordinates of A l 
 and #j, yi those of A z . Expressing the fact that if NN l be 
 
220 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 parallel to Oy the angle A^N^ is equal to the angle AJtfN lt we 
 get 
 
 <X - x)j(y - y,) = (as l + x)j(y + y,), 
 
 so that xy = x^. Hence the achromatic lines are rectangular 
 hyperbolas passing through the ends of the optic axes, and having 
 their asymptotes parallel and perpendicular to the principal plane 
 of either analyser or polariser. If x l = y l , so that the plane of the 
 optic axes is inclined at an angle of 45 to the plane of the 
 polariser or analyser, the foci of the isochromatic lines (A l and A z ) 
 are the vertices of the achromatic lines. This case is represented 
 in the left side of Fig. 84, which is drawn from a photograph and 
 shows a close agreement between theory and observation. If 
 either ^ or y x be zero, the plane of the optic axis is parallel or 
 perpendicular to the plane of the polariser or analyser and the 
 achromatic lines reduce to the straight lines xy = 0. The right 
 
 Fig. 84. 
 
 side of Fig. 84 is drawn from a photograph, showing the actual 
 appearance of the achromatic lines in these circumstances. 
 
 It is interesting to note that if we vary the position of the 
 analyser or polariser the achromatic and the isochromatic lines 
 constitute ' conjugate ' systems of curves*. Taking as the 
 origin, and the axis of x along OA we have x c + iy = i\e iB \ and 
 
 * If u + iv =f (x + iy) , the curves u = constant and v = constant represent families 
 of curves having many important relations, and styled ' conjugate.' 
 
VIIl] TRANSPARENT CRYSTALLINE PLATES 221 
 
 x + c -t- iy = r%e ie *, where OA l = c and x and y are the Cartesian 
 coordinates of any point P, (r l} 0^ and (r 2 , 2 ) its polar coordinates 
 referred to A l and ^4 2 as pole and the axis of x as initial line. 
 Hence if 
 
 u = log (x + iy c)(x + iy + c) = log ?' 1 r 2 + i (^ + 2 ), 
 we see that the curves r^\ constant, and 6 1 4- 6. 2 = constant are 
 conjugate. The first represents the isochromatic and the second 
 the achromatic lines, for when 6 l 4- 2 is constant, the bisector of 
 the angle A 1 PA 2 is fixed in direction. 
 
 Another application of the formulae of p. 217 is met with 
 when considering a plate cut at right angles to an optic axis OA a . 
 Let ON be any wave normal in the crystal near the axis OA lt and 
 put = A l N=fa = r,0' = A t N, NA,A 2 = ^ (Fig. 85). We have 
 
 Fig. 85. 
 
 also A 1 A Z = a), the angle between the optic axes, an angle given in 
 terms of the optical constants of the crystal by the formula of 
 p. 137. From the figure we see that 
 
 cos 0' = cos r cos CD + sin r sin co cos ^r = cos co + r sin co cos i|r, 
 approximately, and hence 
 
 sin 0' = sin co r cos &> cos ^. 
 
 The phase difference A is then given by the formula 
 A = (a 2 - c 2 ) sin . sin 6' . 7re/X6 3 
 
 = (a 2 c 2 ) [r sin co r 2 cos a> cos -\/r] 7re/X& 3 , 
 and the isochromatic curves are represented by 
 
 r sin w ? v2 cos co cos -fy = constant. 
 
 These curves can be readily drawn from this equation, from which 
 it is evident that the isochromatic lines are symmetrical with 
 respect to the plane of the optic axes, and that for equal in- 
 crements of A they are nearly equidistant circles in the immediate 
 vicinity of A lt but become oval shaped at greater distances from 
 this point. The investigation on p. 219 as to the form of the 
 
222 
 
 TRANSPARENT CRYSTALLINE PLxlTES 
 
 [CH. 
 
 neutral lines applies also to the case now under discussion, the 
 only difference being that the points of incidence in this case are 
 
 Fig. 86. 
 
 all near the end of one of the optic axes. A neutral line will thus 
 consist of a portion of that branch of a rectangular hyperbola that 
 passes through the end of the optic axis dealt with, the asymptotes 
 of the curve being, as before, parallel and perpendicular to the 
 principal plane of either polariser or analyser. The neutral line 
 and the isochromatic curves for one position of the polariser and 
 analyser are shown in Fig. 86 drawn from a photograph. 
 
 The possibility of most of the approximations that we have 
 used in this chapter arises from the fact that the two waves, and 
 also the two rays, within the crystal are so close to one another 
 that they may be regarded as coincident. The results obtained 
 with respect to the isochromatic lines may be presented in a more 
 geometrical and somewhat more vivid manner with the aid of 
 what is called an Isochromatic Surface. The exact form of 
 presentation depends on whether we prefer to deal with waves or 
 with rays, and the method in each case being only approximate we 
 obtain slightly different results in the two cases. 
 
 If we are dealing with waves and take a series of wave normals 
 OP, OP' etc. through a given point in the crystal, then the 
 difference of phase between the two waves that have travelled 
 a distance p in the direction OP is A = p (l/v 2 1/Vj), the differ- 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 223 
 
 ence being expressed in time. The surface for which A is constant 
 is an isochromatic surface. Its equation in Cartesian coordinates 
 may be obtained by eliminating I, m, n, v lt v 2 , and p from the 
 above equation for A, and the equations 
 
 combined with the fundamental equation 
 
 l*/(v* - a 2 ) + ra 2 /0 2 - 6 2 ) + n 2 /(V - c 2 ) = 0, 
 
 of which v l and v 2 are the roots. It will be convenient, however, 
 to obtain an approximate equation from which the general form 
 of an isochromatic surface will be much more readily perceived 
 than from the exact equation. In practically all crystals whose 
 optical constants have been measured the quantity a 2 c 2 is small, 
 so that, unless great accuracy be required, we may use the 
 approximations of p. 218, and so put 
 
 A = p (a 2 - c 2 ) sin sin 0'/26 3 . 
 
 To this order of approximation the equation to an isochromatic 
 surface takes the very simple form p sin 6 sin 0' = /c (a constant). 
 This makes p infinite when or 6' is zero or TT, so that the surface 
 is asymptotic to the optic axes. In the neighbourhood of such 
 an axis we may put 6' = w (the angle between the optic axes), and 
 so get p sin 6 = K cosec co, which represents a circular cylinder 
 whose radius is K cosec &>, and whose axis is along the optic axis. 
 The distances p l and p 2 of the vertices of the surface from the 
 origin are obtained by putting 9 = 6' = ay/2, and ^ = (TT o>)/2, 
 8' = (ir + o>)/2 respectively. These give 
 
 p l = K cosec 2 w/2, and p 2 = K sec 2 w/2. 
 The general form of the surface is shown in Fig. 87. 
 
 Fig. 87. Fig. 88. 
 
224 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 In the case of a uniaxal crystal 6 and 6' coalesce, and the 
 isochromatic surface becomes a surface of revolution, formed by 
 turning the curve p sin 2 = K round the optic axis. The vertex is 
 at a distance K from the origin, and the curve approximates to a 
 parabola for large values of p. The shape of the surface is 
 indicated in Fig. 88. 
 
 We have now to see in what way the isochromatic surface 
 enables us to find the form of the isochromatic lines in any given 
 
 Fig. 89. 
 
 circumstances. For this purpose let a wave normal RQ (Fig. 89) 
 in the plate be refracted so as to be received by an eye at E. 
 Draw EOC at right angles to the plate, and OP parallel to RQ, 
 and let <f> and </>' be the angles that EQ and QR make with EOG. 
 The difference of phase between two waves after traversing the 
 crystal along RQ as wave normal is 
 
 RQ (l/v. 2 - l/ Wl ) _ OP (l/v a - 1M) = p (l/v 2 - l/ Vl ), 
 and this, therefore, represents the difference of phase between the 
 two waves on reaching E. If Q be any point on an isochromatic 
 curve as seen from E, this phase difference must be constant for 
 all points on the curve. From the figure we have 
 OQ/CP = (OE/OC) (tan </>/tan <') 
 
 The factor OE/OC is a constant, and the ratio tan $/tan </>' is very 
 nearly constant for small values of <, even although in crystals 
 sin$/sin<' is not strictly constant. Hence the ratio OQ/CP is 
 nearly constant, so that the locus of P is similar to that of Q. 
 But P is a point of intersection of the isochromatic surface with 
 the second surface of the crystalline plate. Hence the isochromatic 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 225 
 
 curves are similar to the sections of the isochromatic surface by 
 the face of the crystal, except for a slight distortion when 
 tan(/tan</>' is not constant. By taking sections of the isochro- 
 matic surface as depicted in Figs. 87 and 88 we obtain the circles, 
 hyperbolas, and lemniscates that were obtained above in the 
 analytical investigation of the form of the isochromatic curves in 
 different cases. 
 
 If we prefer to deal with rays rather than with waves, we make 
 an approximation similar to the above by supposing the two rays 
 in the crystal to coalesce, and the light to travel along them with 
 different ray velocities r x and r 2 . The difference of phase for a ray 
 of length p is then A = p(l/r 2 I/PI), and the Cartesian equation 
 of an isochromatic surface is obtained by eliminating L 1} M l} N l} 
 r lt r. 2 , and p from the above equation for A, combined with 
 x = ^p, y = J/i/o, z = N^p, x* + y 2 + z 2 = p 2 , and the fact that 
 T! and r 2 are the roots of the equation 
 
 a 2 i 1 2 /(r 2 - a 2 ) + &M?l(i* - 6 2 ) + clR^r 8 - c 2 ) = 0. 
 This last equation gives 
 
 l/'n 2 + i/r 2 2 = 2 (1/6 2 + l/c 2 ) L* = 2 O 2 2 + ^) A 2 , 
 
 and 1/rM = 2 L^/frc* = 2 ^> 3 2 A 2 , 
 
 where /A! I/a, yu 2 = l/&, /4 3 = l/c, so that /i l5 ^ 2 , and /4 3 are the 
 
 principal refractive indices. Substituting in the equation 
 
 we get 
 
 [O 2 2 + /"3 2 ) x* + (^ + tf) 2/ 2 + (tf + /W 2 ) 2* - A 2 ] 2 
 
 = 4 (a? + 2/ 2 + z 2 ) (/u 2 >3 2 ^ 2 + /* 3 >i 2 ;?/ 2 + Ati> 2 2 ^ 2 ). 
 
 The isochromatic surface is therefore a surface of the fourth degree, 
 whose form and properties can be discussed from this equation. 
 Fig. 89 and the accompanying argument on p. 224 may be 
 employed equally well in this case, on replacing wave normals by 
 rays, and from this it appears that an isochromatic curve is 
 obtained by taking a section of the isochromatic surface by the 
 second face of the crystalline plate. The Cartesian equation just 
 obtained lends itself very readily to this discussion; but if we 
 wish to get an idea of the form of the surface it is convenient to 
 use approximations similar to those on p. 223. From the formulae 
 
 M. L. 15 
 
226 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 of pp. 151 and 153, it appears that when a 2 c 2 is small we have 
 r*! 2 and n> 2 each approximately equal to l//x 2 2 , and 
 
 IM 2 - 1/7Y 2 = (tf - ^) sin $ sin <', 
 
 where (/> and <' are the angles that the ray makes with the ray 
 axes. Thus we have 
 
 A = p (1/r, - 1/n) = p (l/rf - l/ 
 
 = P Os 2 - Mi 2 ) sin </> sin < 72^2 . 
 
 To this order of approximation the equation to an isochromatic 
 surface takes the form p sin <f> sin $>' = K. This is the same as the 
 equation on p. 223, except that the angles involved are measured 
 from the ray axes and not from the optic axes. In the case of 
 uniaxal crystals the ray axis and the optic axis coincide, so that 
 the two approximations to the form of the isochromatic surface are 
 identical. With biaxal crystals, however, the two approximations 
 do not lead to exactly the same results, although the difference is 
 small. The angle between the ray and the optic axes is given by 
 the formula of p. 151. Using the values of the constants found 
 experimentally with sodium light, this angle proves to be 29' for 
 mica and 53' for aragonite, and these numbers will give some idea 
 of the errors that may be introduced into the calculations by the 
 use of the approximations here employed. 
 
 As another example of the errors made by disregarding the 
 difference of directions of the two rays in the crystal, let us 
 consider the case of nearly normal incidence on a uniaxal plate cut 
 parallel to the axis. If </> be the angle of incidence we have 
 A = [(cos r 2 )/v 2 (cos r^jv{\ 27re/\ = [/Lt. 2 cos r 2 - ^ cos rj 2-Tre/X 
 
 = [(^ - fr) + (l/^ - 1K> ^/2] 27T6/X 
 
 = A + </> 2 . (fr - /*!> Tre/X^/^. 
 On regarding the two rays as coincident the difference of phase is 
 
 A' = (/*,, - ^) 27T6/X COS fa = (fr - fr) (I + C^/2/^) 27T6/X 
 
 = A + < 2 (/4 2 - ^i) 7T6/X/V. 
 
 Hence (A - A')/(A - A ) = 1 - /^/Mi = 0'04 for quartz, and - 012 
 for Iceland spar. This indicates the error that may be made in the 
 Order of the isochromatic fringe, and shows that; such approxima- 
 tions cannot be safely employed when great accuracy is required. 
 At the same time it does not alter the fact that the isochromatic 
 
VIIl] TRANSPARENT CRYSTALLINE PLATES 227 
 
 surface is most useful in enabling us to form rapidly and easily an 
 idea of the shape of the isochromatic curves for a crystalline plate 
 cut in any manner. 
 
 Hitherto we have supposed that the incident light is plane 
 polarised. The phenomena are, of course, somewhat modified if 
 circularly or elliptically polarised light be employed, but as the 
 general principles of solution are the same in all cases it will not 
 be necessary to do more than indicate the chief modifications to be 
 looked for. 
 
 Taking axes OX and OF, as on p. 204, the displacements in 
 the incident light after it has been passed through a circular 
 polariser may be represented by x A cos pt, and y = A sin pt (as 
 was shown in Chapter II.), where 2J. 2 is the intensity of the 
 incident light*. The passage through the crystalline plate will 
 produce a relative retardation A as before, so that on emergence 
 we shall have displacements represented by x = A cos pt, and 
 y = A sin (pt A). Resolving along OA, the principal plane of 
 the plane analyser, we get a displacement 
 
 A [cos /3 cos pt -f- sin /3 sin (pt A)], 
 so that the intensity of the emergent beam is 
 I = A*(I - sin 2j3 . sin A). 
 If the light used be not homogeneous we have, by summation, 
 
 / = S^ 2 - sin 2/3 . 2A 2 sin A = / - sin 2/3 . 2A* sin A, 
 where 2/ is the intensity of the incident light. A comparison of 
 this formula with that on p. 205 will show the changes introduced 
 by using circularly instead of plane polarised light. The main 
 modifications are these : 
 
 (1) The neutral or achromatic lines are given by sin 2/3 = 0, 
 instead of sin 2a . sin 2/3 = 0. The absence of the lines correspond- 
 ing to sin 2a = 0, reduces their number by one-half in the general 
 case. 
 
 (2) In the colour term sin 2 A/2 is replaced by sin A. Thus 
 the dependence of the colour on the thickness of the plate follows 
 a different law and the arrangement of colours is no longer like 
 that in Newton's rings. 
 
 * This represents right-handed circular polarisation. To deal with left-handed 
 polarisation we have merely to change the sign of p and therefore of y. 
 
 152 
 
228 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 (3) As a further consequence of this change of sin 2 A/2 into 
 sin A, there is a change in the position of the lines of maximum 
 and minimum intensity. If sin 2/3 be positive the maximum 
 intensity occurs when A = 2?i7r Tr/2, and the minimum when 
 A = 2n7r -I- 7T/2, and if sin 2/3 be negative the positions of the 
 maxima and minima are interchanged. Comparing this with the 
 previous result we see that there is a difference of ?r/2 in the 
 expressions for A in the two cases. For circularly polarised light 
 the isochromatic lines will be intermediate in position between 
 the corresponding lines for plane polarised light. As an illustra- 
 tion of this, let us take the case of a uniaxal plate cut at right 
 angles to the optic axis. The neutral lines form a rectangular 
 grey cross whose intensity is one-half that of the incident light, 
 and this cross divides the plane into four quadrants as in Fig. 90. 
 The isochromatic lines are circular, as 
 
 with plane polarised light; but they 
 are not continuous in crossing the 
 neutral lines, being pulled in towards 
 the centre in one pair of quadrants, 
 and pushed out by the same amount 
 in the other. The different appear- 
 ance in the various quadrants affords, 
 as a matter of fact, a convenient 
 method of testing whether a crystal 
 is positive or negative. Sin A is posi- 
 tive for crystals of one kind and 
 negative for those of the other kind, so that the bright rings 
 produced by one are replaced by dark rings with the other. The 
 two views presented will be that of Fig. 90, and the same figure 
 turned through a right angle. 
 
 (4) As I is half the intensity of the incident light, the 
 colour term never stands alone, and there is always a wash of 
 white present to the view. 
 
 We shall obtain still further modifications if we replace the 
 plane analyser by a circular one, i.e. by an instrument that damps 
 all displacements except those whose orbits are circular, right- 
 handed, or left-handed as the case may be. After the light has 
 passed through the circular polariser the displacement may be 
 
 Fig. 90. 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 229 
 
 represented by x = A cospt, and y = A sinpt ; but it is convenient 
 to replace each of the rectilinear oscillations by a pair of circular 
 ones thus : 
 
 x = A cospt is equivalent to x l A/2 . cospt, y l A/2 . smpt, 
 and # 2 = A/2 . cos pt, y 2 = A/2. smpt. 
 
 In the same way the displacement along OY on emergence from 
 the plate, viz., y = A sin (pt A) is equivalent to 
 
 #/ = A/2. cos (pt - A), y/ = A/2 . sin (pt - A), 
 and a? 2 ' = - A/2 . cos (pt - A), y a ' =A/2. sin Qrt - A). 
 It will be observed that (a^, yj and (a?/, y/) represent right-handed, 
 while (# 2 , y 2 ) and (a? 2 ', yi) represent left-handed circular orbits. The 
 displacement in one or other of these orbits will be damped by 
 the analyser. Let us suppose that the second is destroyed, then 
 the final displacement is represented by 
 
 x = # x -1- #/ = A cos A/2 . cos (pt A/2), 
 and y y l + y/ = A cos A/ 2 . sin (pt A/2). 
 
 The amplitude is therefore A cos A/2, and the intensity ul 2 cos 2 A/2. 
 Thus for light that is not homogeneous we have / = X^L 2 cos 2 A/2, 
 so that there are no neutral lines, and the isochromatic lines are 
 of the same form as when a plane polariser and analyser are 
 employed. Thus with a uniaxal crystal cut at right angles to 
 the axis, the isochromatic lines are circles whose radii are given 
 by the same law as on p. 209. The main differences in the 
 appearance presented in the two cases is that with circularly 
 analysed light there is no cross corresponding to the neutral 
 lines, and the brightness of each ring is uniform along its circum- 
 ference. 
 
 If elliptically polarised light be used the formulae are, of 
 course, more complicated, as we then have to deal with the general 
 case of which those previously discussed are but special examples. 
 The usual method of producing elliptical polarisation is to pass 
 plane polarised light through a crystalline plate, the thickness of 
 which is chosen so that the relative retardation of the two waves 
 that pass through the crystal is tr/2, or a quarter of a wave length. 
 Such a plate is called a quarter-wave plate. It is obvious that 
 this arrangement gives rise to elliptical polarisation. For if the 
 
230 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. 
 
 plane of polarisation of the incident light make an angle a with 
 the plane of least retardation of the plate, we may conveniently 
 resolve an incident displacement whose amplitude is a into two 
 components whose amplitudes are a cos a and a sin a. After 
 passing through the plate the second component has its plane 
 changed by Tr/2, so that the components on emergence are 
 represented by f = a cos a cos pt, and 77 = a sin a sin pt. On elimi- 
 nating t, we get 
 
 f 2 /a 2 cos 2 a + 77 2 /a 2 sin 2 a = 1, 
 
 so that the orbit is elliptical. 
 
 We proceed to investigate the intensity of a polarised wave, 
 which is analysed after passing through a crystalline plate, and in 
 order to make the resulting formula as general as possible we 
 shall suppose that both polariser and analyser are elliptical. Let 
 OA (Fig. 91) represent the plane of polarisation of the quicker 
 
 Fig. 91. 
 
 wave in the crystal, 00 the plane of the plane polariser, OB that 
 of least retardation in the quarter- wave plate that forms part of 
 the polarising apparatus, OB' and 00' similar planes in the 
 analyser. Let the angles between these lines be represented by 
 ij a 2, s, and a 4 as in the figure. For brevity it will be convenient 
 to denote cos a by c, sin a by s, cos 2a by C, and sin 2a by S. 
 If we represent the incident wave of unit amplitude in the usual 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 231 
 
 way by e ipt , then a retardation of phase A is represented by the 
 introduction of the factor e~* A , and, as a special case of this, the 
 factor i represents a retardation Tr/2. 
 
 Omitting the factor e ipt throughout for the present, the wave 
 that has emerged from the plane polariser may be represented by 
 the components d and s l along and perpendicular to OB. After 
 passing through the quarter- wave plate, these components become 
 d and is 1 . Resolving these along OA and at right angles thereto, 
 we get CiC-z + iSiSz and CiS^ is^. After traversing the crystalline 
 plate these become dc 2 + is^ and (ds 2 isiC. 2 ) e~^. Then resolving 
 along OB' and at right angles thereto, we get 
 
 (dd + is^z) c 3 (d&s - **iC 2 ) s 3 e~^ } 
 and (d c 2 + is 1 s 2 ) s 3 + (d s 2 is l c 2 ) c 3 e~ iA 
 
 respectively. After passing through the quarter- wave plate of the 
 analyser these become 
 
 (dd + is l s 2 ) c 3 (ds 2 - IS&) s 3 e~ * A , 
 and - i (dc 2 + is^) c 3 (c^ is 
 
 Finally, resolving these along OC", we get 
 c 3 - d$ 2 - ^iC 2 se~^ 
 
 Hence the emergent beam is represented by 
 
 where A = c^c^ s^s^ ; B = dc 3 s z s 4 + c a c 4 5 1 s 3 ; 
 
 O ^ ^03^1 4 ~~ d ^4^ 2 ^3 } -*-' ^ CiCg5 2 54 "p 5^53C 2 C4 
 
 Thus the intensity / is given by 
 
 / = P + Q sin A + R cos A = J + Q sin A + N sin 2 A/2, 
 
 where 
 
 2P = A* + B* + (7 2 + ^ 2 = 1 + C\C,C 3 C 4 ; 
 
 2Q = 2 (^D - BG) = SAC, + S^Ci ; 
 
 2.R = 2 (AC + ^D) = ftft - QC&S, ; N=-2R; 
 
 = cos 2 (! or 4 ) cos 2 (or 2 + s) + sin 2 (ttj + 4 ) sin 2 ( 2 + 3 ). 
 It is to be observed that the expression for I contains a term 
 
232 TRANSPARENT CRYSTALLINE PLATES [CH. 
 
 sin 2 A/2 as well as sin A, whereas in the special cases previously 
 considered one or other of these terms was absent. 
 
 The maximum and minimum values of the intensity are found 
 by differentiating I with respect to A. This leads to 
 
 tan A = Q/R, and 27 = 1 + C&C& (l - CfCf) (1 - <7 3 2 C 4 2 ). 
 
 All possible arrangements could be discussed by the aid of 
 these formulae ; but it will be sufficient to deal with a few simple 
 cases that have not yet been considered. In the first place we 
 shall suppose that the quarter-wave plates are parallel so that 
 a 2 + a 3 = 0. We then have 2Q = $ 2 sin 2 ( 4 o^), so that Q vanishes 
 when <! = ! or when or 4 = j + 90. When a 4 = i we get / = 1, 
 N=-(l-Ci*C 9 *), and 1= 1 - (1 - C*C?) sin 2 A/2, while when 
 4 = otj + 90, we get 
 
 / = 0, N=l-C 1 *Cf, and / = (l-a i 2 C 8 ) sin 8 A/2, 
 
 so that the intensity in this case is complementary to that in 
 the other. Next let us suppose that the quarter-wave plates are 
 crossed so that a z + 3 = 90. We then have 
 
 2Q = S 2 sin 2 ( 1 + a 4 ), 
 
 and Q vanishes when 4 = c^ or when a 4 = ^ + 90. In the 
 first case we have / = (! C^C 2 2 ) sin 2 A/2, and in the second the 
 complementary expression /= 1 (1 C-?C<?) sin 2 A/2. In each of 
 these four cases, then, the intensity is either (1 Of Of) sin 2 A/2 
 or its complement, so that a discussion of one case practically 
 serves for all. If the incident light be not homogeneous, but be 
 made up of different waves of amplitude a, we get for the intensity 
 
 I = 2 a 2 [l - (1 - C^CJ) sin 2 A/2] = 2a 2 - (1 - ft 2 ^ 2 ) Sa 2 sin 2 A/2. 
 
 Since 1 C-fCf does not vanish, the colour term is never absent, 
 and there are no achromatic lines. The isochromatic curves 
 correspond to A = constant, and are bright if A be an even multiple 
 of TT and black if it be an odd multiple. The colour along one of 
 the bright lines is most marked when 2 is 45 or 135, and least 
 marked when 2 is zero or 90. The appearance presented with a 
 uniaxal crystal cut at right angles to its axis is very similar to 
 that described on p. 209, except that there is no cross. 
 
VIII] TRANSPARENT CRYSTALLINE PLATES 233 
 
 We have seen that, in the general case, the minimum intensity 
 is given by the formula 
 
 21 = 1 + C&C& - J(l-CfCf) (\-CfCf). 
 
 In order that this intensity may be zero, we must have 
 
 (?!#,+ 0,04-0. 
 
 If 6 be the angle between the quarter-wave plates, we have 
 2 4. a 3 = 0, and the condition for zero intensity takes the form 
 
 sin 2 2 (sin 20 cos 2 4 ) + cos 2a 2 (cos 2a x + cos 20 cos 2a 4 ) = 0. 
 
 To satisfy this for all values of a 2 , we must have sin 20 = and 
 cos 2! + cos 20 cos 2a 4 = 0. The first condition requires to be 
 zero or 90, so that the quarter-wave plates must be parallel 
 or crossed if black lines are to be seen. If = 0, we have 
 cos 2j + cos 2a 4 = 0, so that cc 4 = 90 a, or 4 = 90 + a lt while if 
 8 = 90, we have cos 2^ cos 2a 4 = 0, so that 4 = a, or a 4 = j . 
 There are thus four cases, and only four, in which black lines are 
 possible, and it will be found that these four group themselves 
 naturally in pairs. If = and 4 = 90 + a 1 , or if = 90 and 
 4 = ! , we find Q = 0, and the equation tan A = Q/R shows that 
 in this case A = mr. The formula for A in the case of a uniaxal 
 plate cut at right angles to the axis is given on p. 208, from which 
 it appears that A = *r 2 . 2?r/X, where K is a quantity depending on 
 the thickness and optical constants of the crystalline plate. Thus 
 we have /cr*/\ = 2n, so that the black lines are circles, following 
 the same law as the curves described on p. 209, the centre (corre- 
 sponding to n = 0) being black. 
 
 In the two other cases corresponding to = 0, 4 = 90 a lt 
 and = 90, 04 = 011, we get 2Q = 28,8,0, and 2R = S,*-C,*Sj > 
 
 2 tan A/2 
 
 tanA = 
 
 Solving this equation in tan A/2, we get 
 
 tan A/2 = sin 2 2 cot 2a lt or tan A/2 = - tan 2j cosec 2a 2 . 
 
 The first corresponds to the maxima values of /, and the second 
 to the minima with which we are concerned at present. Thus 
 the form of the black lines is determined from the equation 
 tan A/2 = - tan 2^ cosec 2a 2 . Putting A = tcr 2 . 2ir/\ as before, we 
 
234 
 
 TRANSPARENT CRYSTALLINE PLATES 
 
 [CH. VIII 
 
 get tan 7r/cr 2 /\ = tan 2 a 1 cosec2o 2 , which is the polar equation of 
 the lines. In this equation ! is constant while oc 2 varies with the 
 plane of incidence of the light. Its general solution is of the form 
 #r 2 /X =y (o,j) + n, where n is an integer, and from this we see that 
 the curves approximate more and more closely to circles as their 
 order (n) increases. The maxima and minima values of r occur 
 when sin2 2 = 1, i.e. when a 2 = 45, and the curves are sym- 
 metrical with reference to these two lines. Their form depends on 
 the value of the constant a lf the accompanying figure being drawn 
 to scale for the case a x = 90. 
 
 Fig. 92. 
 
 All these results as to circular and elliptical polarisation and 
 analysation are in thorough accord with experiment, just as in the 
 case of plane polarisation. 
 
CHAPTER IX. 
 
 PROPAGATION OF LIGHT IN ABSORBING ISOTROPIC MEDIA. 
 METALLIC REFLECTION. 
 
 A GENERAL method of dealing with the propagation of waves in 
 an absorbing medium has been introduced into dynamics by Lord 
 Rayleigh. The method consists in modifying the variation equation 
 &f(T W)dt = Q, which expresses the Principle of Least Action, 
 by introducing a Dissipation Function to take account of the loss 
 of energy due to absorption. If the frictional stress between two 
 particles of the medium be proportional to their relative velocity, 
 and the coordinates of the particles considered be x^ ft, z l} and 
 X 2, 2/2 > ^a> then the components of the stress between them are 
 
 0*i - ff a )/i, (ft - 2 )/ 2 , (*i - z 2 )/ S) 
 
 where f lt f 2 , and / 3 are functions of the coordinates. Hence the 
 virtual work in any small displacement of these particles is 
 
 (a?! - # 2 ) 8 (a?! - # 2 ) ./ + (ft - ft) 5 (ft - ft) ./ 2 + ( ^ - z 2 ) S (^ - * 8 ) ./ 
 and the virtual work for the whole system is obtained from this 
 by summation for all the particles. Now if we vary the expression 
 
 F = i SK^ - ^ 2 ) 2 / 1 + (fa - y 2 ) 2 / 2 + fr - z^f s \ 
 
 with respect to the velocities alone and replace the variations of 
 the velocities by those of the corresponding coordinates, we obtain 
 the expression just indicated for the virtual work of the frictional 
 stresses. Hence the Principle of Least Action leads to 
 
 where & denotes that the variation is with respect to velocities 
 only, and that, after variation, the differentials of velocities are to 
 be replaced by those of the corresponding coordinates. Of course 
 the Dissipation Function F may be expressed in terms of any 
 
236 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 coordinates that may be convenient for the problem in hand, and 
 its variation &' found by the rule here indicated. 
 
 The only condition that limits the generality of this procedure 
 is that the method is explicitly confined to cases where the 
 dissipative forces are linear functions of the relative velocity of 
 neighbouring particles ; but experiment shows that this condition 
 must be satisfied in all cases to which the theorem will be applied. 
 If the viscous forces were not proportional to the relative velocity, 
 the period of vibration would be a function of the amplitude, and 
 a beam of homogeneous light passing through the medium would 
 emerge as a mixture of different colours. This is not the case 
 with light that passes through absorbing crystals or thin metallic 
 plates or prisms, and as these are the only cases to which we shall 
 have occasion to apply the method, we may do so with confidence 
 in the basis on which it rests. 
 
 However, before any further progress can be made towards the 
 solution of the optical problem of absorbing media, it will be 
 necessary to specify the Dissipation Function F. And here, just 
 as with the potential energy function W, various forms may suggest 
 themselves, and we have to look to the results to guide us in the 
 selection. In the present scheme the medium is of such a nature 
 as to resist rotation only, and not distortion or compression, so that 
 it seems natural to assume that F as well as W depends on the 
 curl (/, g, h). In this case, in an isotropic medium, we have F of 
 the form 
 
 so that VF = /a' 2 (/Bf+gBg + h Sh) dr. 
 
 Hence putting everything proportional to e ipt , we get 
 
 dr 
 
 , 1 1 . a' 2 
 
 where - = + ip . 
 
IX] ISOTROPIC MEDIA. METALLIC REFLECTION 237 
 
 Thus the dynamical equations and boundary conditions are the 
 same as for a transparent medium, the only difference being that 
 fj, is replaced by //. It will be noted that /*' is a complex of the 
 form yu, ia. 
 
 This analysis is sufficient to prove that when dealing with the 
 propagation of light in an absorbing medium we have merely to 
 replace //, by a complex, and proceed exactly as in the case of 
 a transparent medium; but there is one important feature not 
 brought out by the discussion. When the theory is applied to the 
 problem of metallic reflection, and the optical constants /x and a 
 are determined by a comparison between theory and observation, 
 it turns out that fi is such that its real part is invariably negative. 
 This fact was a fundamental difficulty with the elastic solid theory 
 of the ether as formerly developed, and although we have nothing 
 to do with that theory it is none the less incumbent on us to show 
 how the real part of yit 2 may be negative in a metal. This will 
 necessitate the introduction of some of the terminology of electrical 
 theory ; but for present purposes little more is required than the 
 fact that what we have called (/, g, h) is the ' displacement ' of 
 electrical theory. With this before us the relation between light 
 and electricity appears to be of the most intimate character ; the 
 same medium and the same dynamical principles and notation are 
 required for the elucidation of both classes of phenomena. The 
 peculiarity of a metal, or any electrical conductor, is that it is 
 a medium in which there are a number of free electrons whose 
 motion constitutes the electric current. These electrons, as they 
 move about, carry their atmosphere of strain with them, and they 
 may be described in mathematical language as point singularities, 
 in the neighbourhood of which the strain (f, g, h) is proportional 
 to (d/dx, d/dy, d/dz) (1/r), where r is the distance from the point 
 x, y, z. When such point-singularities are present, the integral 
 
 taken over any closed surface containing them will no longer be 
 zero. If then, as on p. 37, we carry out the ideal process of 
 smoothing out the electrons and spreading their influence uni- 
 formly throughout the medium, the vector (/, g, h) will no longer 
 be circuital, and so cannot be regarded as the curl of the displace- 
 
238 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 ment. However, when there are no electrons present, the rate of 
 change of the electric 'displacement' is represented by (/, g, h\ 
 and if we suppose that the drift of electrons added on to this 
 ' displacement ' is just sufficient to make the circuit complete (an 
 hypothesis forced upon us by electrical considerations), we get a 
 new vector (f+x l ,g-\-y l ,Ji + z^) which is circuital, x lt y lt z^ being 
 coordinates determining the position of an electron. In free ether 
 we have seen that W =c 2 /(/ 2 + # 2 + A 2 ) dr, so that if we write $W 
 in the form f(FSf+ GSg + HSh) dr, we shall have 
 
 (F,G,H)=c*(f,g,h). 
 
 In this notation F, G, IT" are generalised components of force, to 
 be identified with the electric force of electrical theory. This 
 force in the metallic conductor will be used up partly in acce- 
 lerating the motion of the electrons, and partly in maintaining 
 their steady drift against the resistance to migration through the 
 medium. Thus we have /= mx + ax, where m and a are constants. 
 Putting all the variables proportional to e ipt , we get 
 
 so that 
 
 x =f/(mip + a) f\ip (mip + a) 
 
 = /[- m/(<7 2 + m*p 2 ) - ia-jp (a 2 + m 2 /) 2 )]. 
 Hence f+x = yu, 2 /, 
 
 where yu 2 = 1 - m/(cr 2 + m 2 p 2 ) - ia/p (a 2 + m 2 /) 2 ) *, 
 
 and similarly for the other components. In this way we obtain 
 a vector ^ 2 (/, g, h) which is circuital, and which replaces the 
 (/> 9> ^) f f ree e t ner - Thus the ideal process of smoothing out 
 the electrons leads us, in this case, to a continuum in which the 
 displacement being (f, 77, ) as before, we have 
 
 and 
 
 BW = J(FBf+ GSg + H&h) dr = c 2 J(/x 2 fSf+ n*g*g + 
 
 Since yit 2 (/, g, h) is the circuital vector that replaces the ( f, g, h) 
 of free ether we have yu, 2 (/, g, /t) = curl(f, 77, f), and therefore 
 
 * When dispersion is taken into account it may be necessary to add some other 
 terms to /A 2 , as will appear later. 
 
IX] 
 
 ISOTROPIC MEDIA. METALLIC REFLECTION 239 
 
 and so for the other 
 
 (fi> ffi> AI) = curl (f, 77, f), where yi 
 components. Hence we have 
 
 where (/ 15 #, ^) = curl ( 77, ), 
 
 so that the dynamical equations and boundary conditions obtained 
 from the Principle of Least Action are of exactly the same form as 
 with a transparent medium. The only difference is that with the 
 absorbing medium //, is complex, being given by the equation 
 
 If the second term be greater than unity, the real part of /j? will 
 be negative. 
 
 Our first application of these principles will be to the 
 interesting and important case of refraction through a metallic 
 prism. Owing to the absorption of light by the metal, the angle 
 of the prism must be extremely small in order that the intensity 
 of the light transmitted may be appreciable, so that the special 
 case of a prism of small angle (6) is the only one that need be 
 dealt with. 
 
 We shall suppose that the incident ray is in the plane z = 0, 
 the principal plane of the prism, and when considering the first 
 
 Fig. 93. 
 
240 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 refraction we shall take the axis of x normal to the surface, and 
 that of y along the surface towards the edge of the prism. We 
 shall take <f> to be the angle of incidence, and adopt the same 
 notation as before when dealing with problems of reflection and 
 refraction. Then, whatever be the nature of the polarisation of 
 the incident wave, the components of the displacement in it are 
 proportional to 
 
 e ip [t - (x cos <f> + y sin *)/e] 
 
 and those in the refracted wave to 
 
 The dynamical equations for motion in the metal require 
 
 n 2 + sin 2 c/> = /z 2 . 
 
 Before dealing with the second refraction it will be convenient to 
 turn the axes through an angle 6, so as to measure x and y' 
 perpendicular and parallel respectively to the second surface. 
 We thus have 
 
 x = x' cos 6 y' sin = x y' .6, 
 
 and y = x sin + y' cos 6 = x . 6 + y', 
 
 since 6 is small. Hence the components of the displacement 
 within the prism are proportional to 
 
 e ip\_t-{n(x'-y'.e)+sm<t>(x'e+y')}/c]^ } e Q e ip[t-{x'(n+esm<f>)+y'(sin<}>-n6)}lc\ 
 
 On refraction at the second surface the components of the 
 displacement are proportional to 
 
 gip [t - (x' cos $+y' sin >^)/c] 
 
 and the condition of continuity of the displacement at all points 
 of the surface x requires that sin ^ should be equal to the 
 real part of sin c/> nd. Hence if n be the real part of n, we have 
 
 sin i/r = sin < 0n Q . 
 Solving this by approximations we get 
 
 ^lr = <l) n 6 sec </>, 
 so that the deviation D is 
 
 D = ( f ) -^-0 = (n sec 0-1). 
 
 If, then, accurate measurements of D could be made for two 
 different angles of incidence we should have sufficient equations 
 
IX] 
 
 1SOTROPIC MEDIA. METALLIC REFLECTION 
 
 241 
 
 to determine both the real and the imaginary part of /z. Most of 
 the experiments have been conducted at nearly normal incidence, 
 for which we have n = //., so that n = yu. the real part of //., and 
 D = Q (^ 1). The following table gives some of the values of /* 
 obtained by this method for different metals in sodium light. 
 
 Silver 
 0-27 
 
 Gold 
 0-58 
 
 Copper 
 0-65 
 
 Platinum 
 1-70 
 
 Iron 
 1-73 
 
 Nickel 
 2-01 
 
 Bismuth 
 2-26 
 
 Cobalt 
 2-76 
 
 The most striking feature of this table is the small value of the 
 ' refractive index ' obtained for silver, gold, and copper. In each 
 case this is less than unity, indicating that in these metals light 
 travels faster than in air. 
 
 In order to deal with the problem of metallic reflection we 
 have merely to modify Fresnel's formulae given on pp. 39 and 40 
 by replacing //, by a complex quantity. For light polarised at. 
 right angles to the plane of incidence we have 
 
 r = - tan (0 - 0')/tan (<f> + <')> 
 
 and for light polarised parallel to that plane we have 
 r' = - sin (0 - <')/sin (< + </>'). 
 
 Putting IJL in the forms fjL ia = Me~ ia ) then /x is the 'refractive 
 index ' of the medium, i.e. the ratio of the velocity of light in air 
 to that in the medium, and a\^ is the coefficient of absorption. 
 Since 
 
 sin </> = p sin <f>' = (^ ia) sin <', 
 
 the quantity <$>' is complex, and we may put 
 cos 0' = ce~ iu = (1 - M~ z sin 
 With this notation we get 
 
 _ 7? io __ 
 
 ~ 
 
 M. L. 
 
 ~~ P cos <l> __c 
 
 cos 
 
 COS 
 
 cos </>' -f fju cos (f) c-\- M cos </>.e~ i(a ~ u) 
 
 c M cos </> cos (a u) + i Jf cos < sin (a u) 
 c + M cos < cos (a u) iM cos <f> sin (a u) ' 
 
 16 
 
242 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 \x . 2Mc cos ( cos (a u) 
 
 Whence * = , where a- 
 
 2Mc cos cf> sin (a 
 
 and tan = --- ^ , , , 
 
 M 2 cos 2 < c 2 
 
 Similarly 
 
 / 7?' i0' _ cos ^ "" A 6 cos </>' _ cos $ ~" 
 = 
 
 cos 9 4- /-t cos <p cos 9 + 
 cos < Me cos (a 4- u} -*- { Me sin (a 4- u) 
 ~ cos <f> + Me cos (a 4- u) iMc sin (a 4- u) ' 
 
 1 - of , , 2 Jf c cos <6 cos (a 4- u) 
 
 whence R ' 2 = ^ , , waere # = ,. r-r , 
 
 1 4- as M*c* 4- cos 2 $ 
 
 nl 2Mc cos d> sin (a 4- u) 
 
 and tan = - - / ^r- 
 
 .M 2 c 2 cos- 9 
 
 If Jlf and a be given, these equations suffice to determine 
 R, R, 6, and 0' completely. 
 
 For some purposes we are mainly interested in the ratio R : R', 
 and in the difference of phase 6 0' between the light polarised 
 at right angles and parallel to the plane of incidence. These are 
 readily obtained from the above formulae, for we have 
 
 R' 6 ~ r'~~ tan (< 4- <') ' sin (</> </>') cos (< </>') 
 
 cos <f) cos (>' sin <k> sin <// Me cos </> . g-*( a+ ) sin 2 
 
 Whence 
 
 
 
 COS <j 
 
 /jzy 
 Uv 
 
 !> cos <' 4- sin <j) sin <// Jfc cos </> . e l ^ +u ^ + sin 2 ^> 
 1 y ^ 2Mc cos <b sin 2 cos (a 4- w) 
 
 1 4- y M-tf cos 2 ^> 4- sin 4 <f> 
 
 The last equation shows that as < increases from to 90, 
 0' 6 increases from to TT ; and that we have 6' 6 = TT/% when 
 M 2 c* cos 2 (j) = sin 4 $ ; and this equation accordingly determines the 
 Principal Incidence. For this angle we have 
 
 R \ 2 1 cos (a 4- u) 
 
 (-Y-- 
 Uv i 
 
 4- cos (a 4- w) 
 where /3 = (a 4- 1/)/2, and is the Principal Azimuth. 
 
IX] ISOTROPIC MEDIA. METALLIC REFLECTION 243 
 
 Before comparing these formulae with the results of experi- 
 ments on metallic reflection, it will be convenient to make some 
 transformations that will be useful for other purposes. When the 
 optical constants M and a are determined from experiment it 
 appears that for the metals M' 2 is always large. This enables 
 us to expand some of the above functions in ascending powers 
 of l/M' 2 and so to obtain approximate formulae that are sufficiently 
 accurate for many purposes. 
 
 We have 
 
 cos =(l M *sm^ 
 
 sin 2 (j> . e 21 " sin 4 <b . e 4ia sin 6 . e 6ia 
 
 If we multiply each side of this identity by e ia and by e~* a , and 
 equate real parts, we get 
 
 sin* <f> sin 4 <f> 
 
 c cos (a u) = cos a Wifi* cos ^ a l/r cos oa ~" * 
 
 sin 2 <f> sin 4 <f> 
 
 c cos (a + u) = cos a ^-~ cos a [ cos 3a . . . 
 
 Also we have 
 
 2 sin 2 (j> cos 2a 
 
 [ 2 si 
 
 _ sin 2 (/> cos 2a sin 4 c^> sin 2 2a sin 6 </> sin 2a sin 4a 
 J/ 2 217 4 ~ 4>M* 
 
 Let us now consider how R varies as < increases from to 90. 
 For brevity we shall put M cos $=pi, and we then have 
 
 cos (a u) 
 
 _ 
 
 As If 2 is large we have, as the first approximation, c = 1 and 
 u 0, so that 
 
 x = 2p : cos a./ (pi 2 + 1) = 2 cos a/(_pi 
 
 This is a maximum when ^ = 1, so that R is then a minimum. 
 Hence for light polarised at right angles to the plane of incidence 
 the intensity of the reflected light diminishes as < increases, until 
 it reaches a minimum in the neighbourhood of p l 1, which thus 
 
 162 
 
244 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 determines what is known as the quasi-polarising angle. When 
 p 1 = 1 we have x cos a (approximately) and 
 
 R = V(l - ?)/(! + x) = tan a/2. 
 
 For a certain class of steel we have M 2 = 13 and a = 53 42'. In 
 this case jpi = l, or cos(f> = M~ l } gives < = 73 54' as the quasi- 
 polarising angle, and J? = 0*5062 as the minimum value of the 
 amplitude of the reflected wave. This approximation is somewhat 
 rough as we have neglected squares and higher powers of 1/Jf 2 . 
 Proceeding to a higher order we get 
 
 _ 2p l (cos a - %M~ Z sin 2 cos 3a - ^M~* sin 4 < cos 5a) 
 
 " 
 
 _ 
 
 In the small terms we may put the results of the first approxima- 
 tion p l = 1 and sin 2 < = 1 - M~ 2 . This gives 
 
 _ 2pj (cos a ^if~ 2 cos3a + \M~* cos3a - ^M~ 4 cos5a) 
 p* + 1 - M-> cos 2a + M~* cos 2a + J M~* sin 2 2a ' 
 
 From this we see that R is a minimum when 
 
 pf = 1 - i~ 2 cos 2a + 3/~ 4 cos 2a + J ^~ 4 sin 2 2a. 
 
 With the values of M and a given above for steel, this makes the 
 quasi-polarising angle </> = 73 43'. 
 
 For light polarised parallel to the plane of incidence we have 
 
 2Mc cos (f> cos (a -I- u) 
 
 _ 
 
 As </> increases the denominator alters little, as cos 2 c is always 
 small compared with ^f 2 c 2 , while the numerator steadily decreases. 
 Thus R' increases steadily and has no maximum or minimum. 
 
 We have seen that the Principal Incidence is determined by 
 the equation sin 4 (/> = if 2 c 2 cos 2 $. Putting 
 
 c = 1 - \ M~ 2 sin 2 $ cos 2a + J Jf~ 4 sin 4 </> (2 sin 2 2a - cos 2 2a) 
 
 we get as an approximate equation to determine the Principal 
 
 Incidence, 
 
 sec < = M - \ M' 1 sin 2 < cos 2a + i/~ 3 sin 4 < (2 sin 2 2a - cos 2 2a) 
 
 + Jf- 1 (1 + J if" 2 sin 2 c cos 2a). 
 
 This is most simply solved by successive approximations. The 
 first approximation gives sec cf> M, so that p l M cos = 1. To 
 
IX] ISOTROPIC MEDIA. METALLIC REFLECTION 245 
 
 this order of approximation the Principal Incidence and the quasi- 
 polarising angle are the same, so that as a rule the Principal 
 Incidence will be very near the quasi-polarising angle. The 
 second approximation gives sec <f> = M + M~ l (1 \ cos 2a), and the 
 third 
 
 sec </> = M + M~ l (1-4 cos 2a) + M~ 3 {cos 2a + T V (1-3 cos 4a)}. 
 
 For most of the metals the second approximation is sufficiently 
 accurate for comparison with the results of experiment. 
 
 From the formulae hitherto derived it will appear that the 
 two optical constants of a metal, M and a, can be calculated if the 
 Principal Azimuth and Principal Incidence are known from 
 experiment. At the Principal Incidence we have 7lfc = sm 2 </cos< 
 and a + u = 2/5. Also c 2 sin 2u = M~ 2 sin 2 < sin 2a. Hence 
 
 sin 2u tan 2 < = sin 2a = sin (4/3 2w), 
 so that tan 2u = sin 4/3/(tan 2 < + cos 4/9). 
 
 If the Principal Incidence ((/>) and the Principal Azimuth (/3) be 
 known, the last equation determines the value of u at the Principal 
 Incidence, and the equation a + u = 20 then enables us to calculate 
 the constant a. Moreover we have 
 
 cot 2u = (M 2 sin 2 < cos 2a)/sin 2 < sin 2a, 
 
 so that J/ 2 = sin 2 sin 4/8 cosec 2&, which determines M. Thus 
 Conroy found for steel that the Principal Incidence was 76 20' 
 and the Principal Azimuth was 28 29'. This would make 
 
 a = 55 23', M 2 = 15'67 ; 
 fjLo = M cos a = 2'249, and a = J/sina = 3'257. 
 
 Having obtained a and M, we can calculate c and u for any value 
 of <f> from the formulae 
 
 cot 2u = M z cosec 2 cosec 2a cot 2a, 
 and c 2 = J/~ 2 sin 2 $ sin 2a cosec 2 w. 
 
 Owing, however, to the smallness of u the latter formula is not a 
 very good one from which to determine c 2 , since the variations of 
 cosec 2u are very rapid, so that a small error in u will affect c 2 
 considerably. This difficulty may be avoided by using the formula 
 
 c 4 = 1 - 2M~* sin 2 < cos 2a + M~* sin 4 <. 
 
246 
 
 PROPAGATION OF LIGHT IN ABSORBING 
 
 [CH. 
 
 As a numerical example of the application of these formulae 
 we shall calculate R and R for various values of $ in the case of 
 steel, and compare the deductions from theory with Conroy's 
 measurements. The results are set out in the following table 
 and illustrated graphically in Figs. 94 and 95. 
 
 <t> 
 
 u 
 
 c 
 
 R (theory) 
 
 Difference 
 from exp. 
 
 R' (theory) 
 
 Difference 
 from exp. 
 
 30 
 
 026' 
 
 1-003 
 
 0-726 
 
 + 0-018 
 
 0-789 
 
 + 0-009 
 
 40 
 
 042' 
 
 1-005 
 
 698 
 
 018 
 
 811 
 
 009 
 
 50 
 
 059' 
 
 1-006 
 
 656 
 
 016 
 
 839 
 
 006 
 
 60 
 
 116' 
 
 1-009 
 
 597 
 
 013 
 
 873 
 
 010 
 
 70 
 
 129' 
 
 1-012 
 
 527 
 
 012 
 
 911 
 
 004 
 
 75 
 
 134' 
 
 1-012 
 
 508 
 
 003 
 
 932 
 
 005 
 
 80 
 
 137' 
 
 1-014 
 
 540 
 
 014 
 
 954 
 
 004 
 
 \ 
 
 <g IOP 20P 30 40 50 60 70 80 
 
 Fig. 94. 
 
IX] 
 
 ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 247 
 
 An inspection of the table or the figures will show that 
 although the theory represents very well the general trend of the 
 
 R' 
 
 yf 
 
 70* 
 
 Fig. 95. 
 
 quantities R and R', the numerical agreement between theory and 
 observation is not always very close. The values of R given by 
 theory are almost uniformly between 1 and 2 per cent, larger than 
 those actually found by experiment, while the values of R' are 
 larger by quantities varying between \ and 1 per cent. The 
 experimental difficulties of measuring R and R with great 
 accuracy are very considerable, and the differences just referred 
 to might be ascribed to experimental errors were it not that the 
 fact of them being all of the same sign renders this improbable, 
 and that there are other indications that the theory of an abrupt 
 transition when applied to the actual conditions is only an 
 approximation to the truth. 
 
 The difference of phase between the components of the dis- 
 
248 
 
 PROPAGATION OF LIGHT IN ABSORBING 
 
 [em- 
 
 parallel and perpendicular to the plane of incidence 
 is A = #' 6, and this is given by the formula of p. 242, viz., 
 _ 2Mc cos < sin' 2 < sin (a + u) 
 
 .M 2 c 2 cos 2 </>-sm 4 </> 
 
 The following table gives the values of A calculated from this 
 formula and compares the results with those obtained experi- 
 mentally by M. de Senarmont. 
 
 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 75 
 
 80 
 
 A (theory) 
 
 038 
 
 069- 
 
 121 
 
 199 
 
 342 
 
 462 
 
 661 
 
 A (experiment) 
 
 037 
 
 067 
 
 115 
 
 178 
 
 317 
 
 449 
 
 649 
 
 Difference 
 
 + 001 
 
 + 002 
 
 + 006 
 
 + 021 
 
 + 025 
 
 + 013 
 
 + 012 
 
 These results are exhibited graphically in Fig. 96. 
 
 Here, again, it will be observed that the theory of an abrupt 
 transition indicates correctly the march of A with the incidence, 
 but that there is a difference between observation and theory 
 
IX] 
 
 ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 249 
 
 which is always in the same direction. Experiments with 
 reflection from other metals, such as those of M. Meslin on 
 gold, show still more clearly that the discrepancies in the 
 estimates of A cannot be ascribed to experimental errors. In 
 fact all careful experiments show that the character of the reflected 
 light, as regards both amplitude and phase, depends not only on 
 the metal employed as a reflector, but also on the condition of its 
 surface. Hence we cannot expect to obtain a very close agreement 
 between theory and observation unless we take into account the 
 nature of the transition in passing into the metal just as was 
 found to be necessary with reflection from transparent media. 
 Before, however, turning to the consideration of the layer of 
 transition, we may note that the theory as developed hitherto 
 represents the facts sufficiently closely to enable us to obtain an 
 approximate measure of the reflecting power of a metal at normal 
 incidence, and the change of phase produced by reflection in that 
 case. At this incidence we have 
 
 where x = 2Mcosa/(M 2 +l)=2/j, Q /(M*+I), 
 
 and tan 6 = tan 0' = - 2Msm a/(M* - 1) = - 2a/( M 2 - 1). 
 
 The following table gives the values of R, jR 2 , and calculated 
 from these formulae, the constants a and //- being those found by 
 Drude* for the reflection of sodium light from metallic surfaces 
 that were as clean as possible. The quantity 6 is expressed as a 
 fraction of the half wave length. 
 
 
 Silver 
 
 Gold 
 
 Copper 
 
 Platinum 
 
 Steel 
 
 Nickel 
 
 Bismuth 
 
 Cobalt 
 
 a 
 
 3-67 
 
 2-82 
 
 2-62 
 
 4-26 
 
 3-40 
 
 3-32 
 
 3-66 
 
 4-03 
 
 Mo 
 
 0-18 
 
 0-37 
 
 0-64 
 
 2-06 
 
 2-41 
 
 1-79 
 
 1-90 
 
 2-12 
 
 E 
 
 977 
 
 923 
 
 856 
 
 837 
 
 765 
 
 787 
 
 808 
 
 822 
 
 R 2 
 
 953 
 
 851 
 
 732 
 
 701 
 
 585 
 
 620 
 
 652 
 
 675 
 
 -e 
 
 169 
 
 214 
 
 222 
 
 121 
 
 125 
 
 148 
 
 137 
 
 123 
 
 * Drude, Wied. Ann. xxxvi. p. 885, 1889; xxxix. p. 481, 1890. 
 
250 PROPAGATION OF LIGHT IN ABSORBING [CH, 
 
 It thus appears that the reflecting power of these metals is 
 high, which agrees with observation, and that in the case of silver 
 the amplitude of the reflected wave is very nearly equal to that of 
 the incident. If then we employ a silver reflector we shall produce, 
 approximately, the stationary waves described on p. 36. The 
 first loop will occur at a distance L from the reflector, where 
 A = -X0/47r = 0-04;v, the first node at a distance Z 2 = 0'29X, and 
 the distance between successive loops or successive nodes will be 
 A./2. The difficulties of demonstrating the existence of these loops 
 and nodes were overcome by Wiener*. He allowed a beam of light 
 from an electric arc to fall normally on the silver coating of a plate 
 of glass on which was placed an extremely thin film of sensitized 
 collodion, the thickness of the film being only about one-thirtieth 
 of a wave length. The film was very slightly inclined to the 
 silver surface and the photograph when developed was crossed by 
 alternately bright and dark bands, the dark bands being due to 
 the deposit of the silver at the places where the intensity of the 
 light was greatest. The intensity vanished at the loops of the 
 stationary waves, indicating that the intensity as measured by 
 the photographic effect is proportional to the mean potential 
 energy. (See pp. 18 and 37.) 
 
 The modifications introduced by a layer of transition may be 
 derived from the formulae of Chapter IV. by making fju a complex 
 quantity. If we neglect squares and higher powers of the thickness 
 of the layer, the formulae of pp. 66 and 70 give 
 
 //, cos <f> cos </>' + id l (jjL~ l F sirf (f> //, + E/n 2 cos $ cos <') 
 fju cos <f> + cos (/>' ididjr 1 F sin 2 < //- Ey? cos <f> cos <') ' 
 and 
 
 , _ cos (ft //, cos <jj 4- id l (//. cos <ft cos <// E/J? + sin 2 j>) 
 cos (f> + /j, cos $' + id l (fjb cos < cos <' + E^ sin 2 (/>) ' 
 
 Here E and F are complex constants defined by the equations 
 
 f 1 f 1 
 
 E = ^\ i#dxi and F = ^\ ^~~dx^ The equations for r and r' 
 
 Jo .' o 
 
 i . A i /. A l -\-a l , , AI+OI' i 
 
 may be written in the forms r = A - and r = -r-, -- ,, where 
 
 * 
 
 0. Wiener, Wied. Ann. XL. p. 203, 1890. 
 
IX] ISOTROPIC MEDIA. METALLIC REFLECTION 251 
 
 quantities such as a denote the small terms containing the factor 
 d lt and representing the corrections to the earlier formulae due to 
 the presence of the layer of transition. We thus have 
 
 and 
 
 . 
 
 //, 2 COS 2 (/> -COS 2 < J 
 
 To this order of approximation the corrections involve two complex 
 constants, viz., d l (El)fjL 2 and d l (FE). As each complex 
 requires two quantities to define it we see that the complete 
 specification of the layer will require four constants, depending 
 on the thickness and the law of distribution of /a 2 within the layer. 
 However the term (FE) is associated with the factor sin 2 <, so 
 that the product will be negligible for small angles of incidence ; 
 and even for larger values of <j> the term (FE)sm*<l> may be 
 expected to be small compared with (El)/j?, owing to the large 
 value of the modulus of p? for most of the metals. Hence as the 
 first approximation we may take 
 
 A l f acos6.e iw ~] 
 f = j^ -j- 
 
 A 2 \_ /jJ 2 cos 2 (f) cos 2 (p' J 
 and T' = 
 
 where ae iw = < 2id 1 (E - 1) ^ 2 . 
 
 In the small terms we may employ the approximations c = 1 
 and u = 0, so that cose/) 7 1, and we then have 
 
 {0 
 r = Re" 
 
 and 
 
 where R, R f , 6 and 0' have the same values as on p. 242. If the 
 modulus of r be R -f p and that of r' be R'+ p', then p and p are 
 the corrections to the amplitudes due to the layer of transition. 
 They are given by the formulae p = Rp cosX and p = R'p' cosX'. 
 Putting fji cos </> != B^e***, and /* cos </> + != B z e iv * t we have 
 
252 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 p = a cos ^/#i J9 2 and X = w (^ + v 2 ). The quantities 5 and w are 
 determined by the formulae 
 
 B* = M* cos 2 < + 1 - 2M cos </> cos a; 
 5 2 2 = M' 2 cos 2 04-1+ 2.cos cos a; 
 
 tan ^ = - M cos < sin a/(M cos a cos < 1); 
 
 tan v 2 = M cos </> sin aj(M cos a cos <j> 4- 1) ; 
 whence cos ^ = (1 - Jf cos a cos ^)/5 x ; 
 
 cos # 2 = (1 + -M" cos a cos <f>)/B 2 ; 
 
 sin Vj = M cos </> sin a/^ ; 
 
 sin v^ M cos (/> sin a/B 2 . 
 Making these substitutions in the formulae for p we get 
 
 p -- P2 p 2 [cos w M 2 cos 2 (f> . cos (2a + w)] . 
 
 &\ &2 
 
 This makes p vanish when <f> 90 and also when 
 
 cos (f> = M~^ Vcos w sec (2a + w). 
 
 In considering the law of variation of p, it will be convenient 
 to ascertain the position of its maxima and minima. Putting 
 Mcoscf)=p l , we have p proportional to pJH (p* b)jB^B^, where 
 b cos w sec (2a -f w). Also 
 
 B*B<? = (1 +^ 2 ) 2 - bp* cos 2 a = 1 - 2^ cos 2a + pf, 
 
 .and E = V(l a?)/(l + x), where ^ = 2p l cos a/(l -f ^j 2 ). Hence we 
 
 have 
 
 I dR _ 1 dx_ 2 cos a (1 - p*) 
 5 ^ ~ 1 -x 2 dp l ~ 1 -2p 1 2 cos2a + p 1 4 ' 
 
 and the equation to determine the position of the maxima and 
 minima of p is 
 
 3pf b _2cosa(l p^ 2 ) + 4pj (pf - cos 2a) 
 
 i.e. ^ 6 2 cos a . pf + (2 cos 2a - 36) p/ + 2 (1 + 6) cos a . pf 
 
 + (26 cos 2a - 3) p? - 26 cos a . p x + 6 = 0. 
 
 With the values of a and w adopted lower down for reflection from 
 steel this equation becomes 
 
 - 0-0783^ 
 
 -f- 0-0666 = 0. 
 
IX] 
 
 ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 253 
 
 Solving this, by Horner's process, we obtain two real roots, 
 p 1 = 0'37 and p t = 1*8. The former corresponds to < = 84 6', and 
 determines the position of the minimum, and the latter corresponds 
 to < = 60, and determines the position of the maximum. The 
 march of p is indicated for steel in Fig. 97. 
 
 o* 
 
 2ff 
 
 30 
 
 40 50 
 
 Fig. 97. 
 
 70 
 
 In dealing similarly with p we put /z cos </> = B^e* and 
 //, + cos </> = Bie iv *, and we then have p = a cos ^JB-^B^ and 
 X' = w - + v z '). The quantities B' and v' are determined by 
 
 the formulae 
 
 jB/ 2 = M* + cos 2 </> 2M cos </> cos a; 
 
 .# 2 /2 = Jf 2 + cos 2 < + 2if cos </> cos a; 
 tan Vi = M sin a/(M cos a cos (/>); 
 tan Vz M sin a/( Jf cos a cos <) ; 
 
 R'a cos 
 
 where 
 
 p = 
 
 r ... /6> 
 
 [cos 2 </> cos w - Jw 2 cos (2a + 
 
 This makes p' vanish when <f> = 90 and also when 
 
 cos (j> = M Vcos (2a + w;) sec w, 
 but the latter equation has not, as a rule, any real root. 
 
254 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 In the formula for p the term cos 2 <j> cos w will usually be 
 negligible in comparison with M 2 cos (2a + w), so that we shall 
 have approximately p R'a cos (2a + w) cos $jBt*Bp. Now 
 
 B?Bp = cos 4 + If 4 + 271/ 2 cos 2 0(1-2 cos 2 a) ; 
 
 and for almost all the metals a is greater than 45, so that 
 1 2 cos 2 a is positive. Thus B^B^ is greater than M 4 and p 
 is less than 2.M~ 2 .R'acos cos(2a + w). Thus p' diminishes with 
 </> and is always small owing to the presence of the factor M~ 2 . If 
 we investigate the position of its maxima and minima in the same 
 manner as with p, we are led to the following equation : 
 (4/|f-e_ 3^-8)^6 _ 2 jif-6 cosa . p* + (Gif-^ cos 2a - 4>M-*b + bM~*)pf 
 + (2 M-* cos a - 47lf- 4 cos 2a + 2bM~ 6 cos a)^ 8 
 - (3 - 26 M~* cos 2a) p? - 2bM~* cosa.p l -6 = 0. 
 
 Since p l = M cos < we see that p cannot be greater than M, and 
 the above equation in p 1 has no real roots less than M, so that 
 there are no maxima and minima. The march of the function p for 
 steel is shown in Fig. 97. It will be seen from this figure that the 
 range for which p and p' are appreciable is much larger than in 
 the case of a transparent medium, where the influence of the 
 transition layer is practically confined within a few degrees of the 
 polarising angle. 
 
 For some purposes it is convenient to have formulae for the 
 ratio of the amplitudes and the difference of phase between the 
 components of the displacement perpendicular and parallel to 
 the plane of incidence. These quantities e and A are given by 
 the formula 
 
 r' 
 Hence if rM* = l + ^ = s&ix > 
 
 where s = Vl + 2q cos 7 + <? 2 , and tan % = q sin y/(l + q cos 7), we 
 have e = sRjR' and A = 6 6' + ^. To our order of approximation 
 we have 
 
 q e iy = pe& -p' e ^' = a cos < . e iw [l/(yu 2 cos 2 <-!)- I/(p? - cos 2 <)] 
 = a cos . e iw (1 + tf) sin 2 </(yit 2 cos 2 0-1) (^ - cos 2 </>), 
 
IX] ISOTROPIC MEDIA. METALLIC EEFLECTION 255 
 
 from which q and 7 may be found in terms of the optical constants 
 M, a, a and w. 
 
 Since the modulus of p* is large the term (/u, 2 4-1 )/(//,* cos 2 <) 
 will differ very little from unity, so that as an approximation we 
 may simplify the calculations by putting this term equal to unity 
 in the formulae for the small corrections. In this way we get 
 
 q&y = a cos $.e iw sin 2 (j>/(fj? cos 2 </> - 1) = a sin 2 <j> cos <f>.e iw /B l B 2 e i(v * +v * ) . 
 
 Hence q = a sin 2 < cos (j)/B 1 B, 2 , 
 
 and 7 = w (^ 4- v 2 ) = \ = 2a 4 w 4- 7', 
 
 where tan 7' = sin 2ot/(M' 2 cos 2 </> cos 2a). 
 
 To determine the maximum value of q we put Mcos<f> = p 1 as 
 before, and we then have to make 
 
 P! (1 -pf/M*)/*/! - Zpf cos 2a + pf 
 a maximum. This requires 
 
 (Pi - ^Pi cos 2a + 1) (1 - 3j9 1 W- 2 ) 
 
 - 2^ 2 (1 - p*M-*) (p* - cos 2a) = 0. 
 
 On solving this by approximations, we get p = 1 as the first 
 approximation. This, as we have seen, is the first approximation 
 to the quasi-polarising angle. The second approximation gives 
 pf = 1 - SM~ Z sin 2 a. Thus if M 2 = 13 and a = 53 42', which are 
 the values of the constants derived from a consideration of 
 Coriroy's experiments on reflection from steel, the first approxi- 
 mation gives <f> = 73 54' and the second (/> = 75 52', which is very 
 near the Principal Incidence. 
 
 The correction to the difference of phase A is %, where 
 
 tan ^ = q sin 7/(l + q cos 7). 
 
 As q is small the correction is small, being greatest when q is 
 greatest, i.e. in the neighbourhood of the Principal Incidence. It 
 must be observed, however, that even although the correction to the 
 change of phase be small, it may make an appreciable difference to 
 the position of the Principal Incidence. At the Principal Incidence 
 we have 6 - 0' + % = Tr/2, so that cot (0 - 6') - tan ^, and hence 
 
 sin 4 < - l/ 2 c 2 cos 2 (f> = 23/c cos < sin 2 $ sin (a + u) tan ^. 
 
256 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 This is the equation to determine the Principal Incidence, and 
 owing to the presence of Mm the term on the right-hand side, that 
 term may be appreciable even although ^ be small. This being 
 the case, we cannot derive the optical constants M and a from 
 observations of the Principal Azimuth and Principal Incidence alone 
 by the formulae of p. 242. We must now proceed by successive 
 approximations. The true values of M and a will be smaller than 
 those obtained by neglecting the layer of transition, but the 
 method of p. 242 will yield results that will serve as a good 
 first approximation. We can then calculate the other constants a 
 and w from observation of the Principal Azimuth (/3) and the 
 Principal Incidence 
 We have 
 
 , _ 
 
 1 + q cos 7 * 2Mc cos $ sin' 2 </> sin (a + u) ' 
 
 The second of these equations determines %, and we then have 
 
 and q = R'/R . 
 
 These equations give us 7 and q, and then the constants a and 
 w are obtained from the formulae q = a sin 2 ^ cos<j)/B l B, i , and 
 y = 2a + w + 7', where tan 7' = sin 2a/(M 2 cos 2 </> cos 2a). Having 
 obtained approximate values for a and w, we may use them as a 
 basis for finding a closer approximation to the values of M and or, 
 and continue this process as far as may be thought desirable. On 
 applying this method to the consideration of Conroy's experiments 
 on reflection from steel we find that the quantities /4 and a (where 
 p ia fju = Me~ ia ) are 2*134 and 2*906 respectively, instead of 
 2*249 and 3'257 as given by the theory of an abrupt transition; 
 while from Meslin's experiments on gold we get similarly 0*9 and 
 2*47 instead of 0*135 and 3*31. Thus the neglect of the layer of 
 transition may introduce very considerable errors in the estimate 
 of the optical constants of a metal. 
 
 The values of p and p given by the method just indicated are 
 
IX] 
 
 ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 257 
 
 set out in the following table, and exhibited graphically in 
 Fig. 97 for the case of reflection from steel dealt with experi- 
 mentally by Conroy. 
 
 <p 
 
 P 
 
 P' 
 
 
 
 P 
 
 P' 
 
 
 
 0-016 
 
 0-016 
 
 60 
 
 0-021 
 
 0-009 
 
 30 
 
 017 
 
 014 
 
 70 
 
 019 
 
 007 
 
 40 
 
 018 
 
 013 
 
 75 
 
 015 
 
 005 
 
 50 
 
 019 
 
 Oil 
 
 80 
 
 006 
 
 004 
 
 With the aid of these numbers we get the following table, 
 giving the amplitudes R + p and R' 4- p and comparing them with 
 Conroy's measurements. 
 
 4> 
 
 R + P 
 (theory) 
 
 R + P 
 (exp.) 
 
 Difference 
 
 R' + p' 
 (theory) 
 
 R' + p' 
 (exp.) 
 
 Difference 
 
 30 
 
 0-711 
 
 0-708 
 
 + 0-003 
 
 0-777 
 
 0-779 
 
 - 0-002 
 
 40 
 
 682 
 
 680 
 
 + 0-002 
 
 800 
 
 801 
 
 -0-001 
 
 50 
 
 640 
 
 640 
 
 
 
 830 
 
 833 
 
 - 0-003 
 
 60 
 
 582 
 
 584 
 
 - 0-002 
 
 865 
 
 863 
 
 + 0-002 
 
 70 
 
 517 
 
 515 
 
 + 0-002 
 
 906 
 
 907 
 
 -0-001 
 
 75 
 
 504 
 
 505 
 
 - 0-001 
 
 928 
 
 928 
 
 
 
 80 
 
 543 
 
 525 
 
 + 0-018 
 
 951 
 
 950 
 
 + 0-001 
 
 These results are represented in Figs. 98 and 99. 
 
 It will be seen that there is a very close agreement between 
 theory and experiment, the only appreciable difference being for 
 R + p at an incidence of 80, and there is good reason to doubt the 
 accuracy of the experiments in this case. 
 
 The difference of phase between the components of the dis- 
 placement that are parallel and perpendicular to the plane of 
 incidence is A = 0' %, where % is determined by the formulae 
 above. Applying these formulae to the experiments of M. de 
 M. L. 17 
 
258 PROPAGATION OF LIGHT IN ABSORBING [CH. 
 
 8 
 
 9 
 8 
 
 7 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ,x 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^x 
 
 x 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^\ 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 M^v 
 
 
 
 
 
 ' 
 
 ^^" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 99. 
 
ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 259 
 
 Senarmont, referred to on p. 248, we obtain the following results, 
 which are set out in Fig. 100. 
 
 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 75 
 
 80 
 
 A (theory) 
 
 037 
 
 068 
 
 117 
 
 193 
 
 331 
 
 447 
 
 646 
 
 A (exp.) 
 
 037 
 
 067 
 
 115 
 
 178 
 
 317 
 
 449 
 
 649 
 
 Difference 
 
 
 
 + 001 
 
 + 002 
 
 + 015 
 
 + 014 
 
 -002 
 
 -003 
 
 1-0 
 8 
 6 
 
 4 
 z 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^Jt 
 
 , ' 
 
 ^ 
 
 
 
 
 
 
 
 
 
 -MM-? 
 
 
 
 
 
 . H 
 
 ^ -- 
 
 - 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 STfrT? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 K 
 
 
 
 Fig. 100. 
 
 The method of determining the constants M, a, a, and w, 
 explained on p. 255, supposes that we have before us the results 
 of a series of experiments giving the Principal Incidence and 
 Azimuth and the values of the amplitudes of the displacements in 
 the reflected waves for various incidences. We may also obtain 
 these constants from observations of the difference of phase A, at 
 various angles of incidence. Theoretically we could obtain the 
 four constants from any four observations of A ; but the equations 
 are much too complicated to be solved as a set of four simultaneous 
 equations. Practically we must proceed by a series of approxima- 
 
 172 
 
260 
 
 PROPAGATION OF LIGHT IN ABSORBING 
 
 [CH. 
 
 tions and in this way consistent results are soon obtained, unless 
 very great accuracy be aimed at. From the formula it appears 
 that % (the correction to the difference of phase due to the layer) 
 vanishes at normal incidence and is small when the angle of 
 incidence is small. We can, therefore, obtain the first approxima- 
 tion to M and a from two observations of the difference of phase 
 when the angle of incidence is not large. As the first approxima- 
 tion in such circumstances, we may neglect % altogether, or give 
 to it any small value that seems reasonable. Knowing M and a 
 (approximately) we can then calculate 0' 9, q/a, and 7' from the 
 formulae above. Having obtained 0' 9 by calculation and 
 9' 9 x from observation for any angle of incidence, we thus 
 obtain for that incidence. Also we have 
 
 so that 
 
 tan % = <? sin 7/(l + q cos 7), 
 
 I/a = (q/a) sin (7 %) cosec % = (qja) sin (w + 2a 4- 7' - %) cosec %. 
 
 In this equation everything is known (approximately) except a and 
 w, so that by using our knowledge of A at two angles of incidence, 
 other than those employed for obtaining the approximate values of 
 M and a, we obtain the unknown quantities a and w. Having 
 thus obtained approximate values of the four optical constants, we 
 can calculate % in terms of them, and proceed if necessary to 
 higher orders of approximation. If the experimental results give 
 us the difference of phase for more than four angles of incidence, 
 we may use the ordinary rules for finding the most probable values 
 of the optical constants, and estimate the probable errors. The 
 values of ^, for different angles of incidence, obtained in this way 
 from Meslin's experiments on gold are given in the following table. 
 
 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 74 
 
 80 
 
 85 
 
 X 
 
 0-009 
 
 0-017 
 
 0-027 
 
 0-041 
 
 0-055 
 
 0-061 
 
 0-051 
 
 0-027 
 
 A (theory) 
 
 0-056 
 
 0-103 
 
 0-176 
 
 0-270 
 
 0-423 
 
 0-5 
 
 0-663 
 
 0-815 
 
 A (exp.) 
 
 0-064 
 
 0-112 
 
 0-176 
 
 0-260 
 
 0-420 
 
 0-5 
 
 0-648 
 
 0-828 
 
 Difference 
 
 -0-008 
 
 - 0-009 
 
 
 
 + 0-010 
 
 + 0-003 
 
 
 
 + 0-015 
 
 -0-013 
 
ISOTROPIC MEDIA. METALLIC REFLECTION 
 
 261 
 
 The table also compares the observed and calculated values for A, 
 a comparison which is also brought out in Fig. 101, which repre- 
 sents the same results graphically. 
 
 0-9 
 08 
 
 07 
 06 
 05 
 
 0-4 
 03 
 
 0-2 
 
 01 
 
 f: 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 
 y 
 
 / 
 
 
 
 
 
 
 
 
 
 
 s 
 
 7 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 
 ^s 
 
 X* 
 
 "7 
 
 
 
 
 
 
 
 i 
 
 ^^ 
 
 ^ 
 
 ~ 
 
 
 
 
 
 
 
 
 [ 
 
 ^- 
 
 
 
 
 
 
 
 
 
 
 
 >0 o + Q 400 50 60 70 80 9 
 Fig. 101. 
 
 It has been supposed in all the formulae of this chapter that 
 the medium into which reflection takes place is air. If this be not 
 the case, the formulae must be somewhat modified. Thus, if the 
 medium in contact with the metal be of refractive index fjL 1} we 
 have to replace M in the formulae by M/fa, keeping a as before. 
 We have seen that for an abrupt transition the Principal Incidence 
 is given by the formula sin 4 < = M*c? cos 2 </>, which is approximately 
 equivalent to sec <f> = M 4- M~ l (1 -J cos 2a). If, however, there 
 be a surface layer of transition we have 
 
 sin 4 <f> = M 2 c* cos 2 < 4- 2Mc cos (f> sin 2 <f> sin (a + u) tan %. 
 Since tan % is always small we may in the last term put 
 sin 2 <f> = Me cos $>, 
 
262 ABSORBING ISOTKOPIC MEDIA [CH. IX 
 
 and we then get sin 4 <f> = M 2 c 2 cos 2 </>(! + 2/e), where K is a small 
 quantity given by the formula K sin (a + u) tan %. Thus the 
 effect of the layer is to replace M z by Jf 2 (l + 2/c) or M by 
 Jf(l + /e), while the effect of replacing air by a medium of 
 refractive index ^ is to change M into M/ft. Hence, if there 
 be a layer of transition between a medium /^ and a metal, the 
 Principal Incidence will be determined by the equation 
 
 secc/) = if / + lf- 1 (l - Jcos2a), where M' = M (I + K)/^. 
 
 An increase in /^ will diminish M ' and so diminish </>, the Principal 
 Incidence. This agrees with Conroy's measurements of the 
 Principal Incidences with gold and silver in contact with different 
 media. Thus he found the Principal Incidences to be 71 43', 
 67 39', and 66 36' respectively for the reflection of yellow light 
 from gold in air, in water, and in carbon bisulphide, and to be 
 74 37', 72 15', and 71 39' respectively for reflection from silver 
 in air, in water, and in carbon tetrachloride. A change in /^ 
 will also affect the Principal Azimuth (/3). We have 
 
 tan ft = (1 + q cos 7) tan (a + u)/2, 
 the angle u being given by the equation 
 
 cot 2u = (If//^) 2 . cosec 2 </> cosec 2a cot 2a. 
 
 An increase in ^ will diminish cot 2u and therefore increase u, so 
 that as a rule fi will be slightly increased, although in some cases 
 the increase of tan(a + w)/2 may be counterbalanced by the 
 diminution of the factor 1 + qcosy. This is also in accord with 
 Conroy's measurements. He found the Principal Azimuths 
 corresponding to the Principal Incidences given above to be 
 41 4', 41 15', and 41 41' for gold, and 43 22', 44 9 7 , and 
 43 40' for silver. 
 
CHAPTER X. 
 
 ON NEWTON'S RINGS FORMED BY METALLIC REFLECTION 
 AND ON SOME OPTICAL PROPERTIES OF THIN METALLIC 
 PLATES. 
 
 AFTER what has been done in the last chapter we are in a 
 position to discuss some special problems presented by the propa- 
 gation of light in metals. Of these one of the most interesting 
 is that of the peculiarities of Newton's rings when formed by 
 reflection from a metal. The main features of this phenomenon 
 are thus described by Sir G. Stokes. " When Newton's rings are 
 formed between a lens and a plate of metal, as the angle of 
 incidence is increased the rings, which are at first dark-centred, 
 disappear on passing the polarising angle of the glass, and then 
 reappear white-centred, in which state they remain up to a 
 grazing incidence, when they can no longer be followed. At a 
 high incidence the first dark ring is much the most conspicuous 
 of the series. 
 
 "To follow the rings beyond the limit of total internal reflection, 
 we must employ a prism. When the rings formed between glass 
 and glass are viewed in this way, we know that as the angle of 
 incidence is increased the rings one by one open out, uniting with 
 bands of the same respective orders which are seen beneath the limit 
 of total internal reflection ; the limit or boundary between total and 
 partial reflection passes down beneath the point of contact, and the 
 central spot is left isolated in a bright field. 
 
 " Now when the rings are formed between a prism with a 
 slightly convex base and a plate of silver, and the angle of 
 incidence is increased so as to pass the critical angle, if 
 common light be used, in lieu of a simple spot we have a ring, 
 which becomes more conspicuous at a certain angle of incidence 
 
264 NEWTON'S RINGS FORMED BY [CH. 
 
 well beyond the critical angle, after which it rapidly contracts and 
 passes into a spot. 
 
 "As thus viewed the ring is, however, somewhat confused. To 
 study the phenomenon in its purity we must employ polarised 
 light, or, what is more convenient, analyse the reflected light by 
 means of a Nicol's prism. When viewed by light polarised in the 
 plane of incidence, the rings show nothing remarkable. They are 
 naturally weaker than with glass, as the interfering streams are so 
 unequal in intensity. They are black-centred throughout, and, as 
 with glass, they open out one after another on approaching the 
 limit of total reflection and disappear, leaving the central spot 
 isolated in the bright field beyond the limit. The spot appears to 
 be notably smaller than with glass under like conditions. 
 
 " With light polarised perpendicularly to the plane of incidence, 
 the rings pass from dark-centred to bright-centred on passing the 
 polarising angle of the glass, and open out as they approach the 
 limit of total reflection. The last dark ring to disappear is not, 
 however, the first, but the second. The first, corresponding in 
 order to the first bright ring within the polarising angle of the 
 glass, remains isolated in the bright field, enclosing a relatively, 
 though not absolutely bright spot. At the centre of the spot the 
 glass and metal are in optical contact, and the reflection takes 
 place accordingly, and is not total. The dark ring, too, is not 
 absolutely black. As the angle of internal incidence increases by 
 a few degrees, the dark ring undergoes a rapid and remarkable 
 change. Its intensity increases till (in the case of silver) the ring 
 becomes sensibly black, then it rapidly contracts, squeezing out, as 
 it were, the bright central spot and forming itself a dark spot, 
 larger than with glass, isolated in the bright field. When at its 
 best it is distinctly seen to be fringed with colour, blue outside, red 
 inside (especially the former). This rapid alteration taking place 
 well beyond the critical angle is very remarkable. Clearly there 
 is a rapid change in the reflective properties of the metal, which 
 takes place, so to speak, in passing through a certain angle 
 determined by a sine greater than unity. 
 
 " I have described the phenomenon with silver, which shows it 
 best ; but speculum-metal, gold, and copper show it very well, 
 while with steel it is far less conspicuous." 
 
X] METALLIC REFLECTION. THIN METALLIC PLATES 265 
 
 In discussing the problems suggested by this description we 
 shall employ the same method and notation as in Chapter v. It 
 was there proved (p. 95) that the reflected beam is represented by 
 (r*! + <f r. 2 )/(l +q 2 r l r 2 ), where in general r l} r 2 , and <f are complex. 
 Putting r l = R l e i0 ', r 2 = R 2 e^ 2 , and q* = Qe~ i *, the intensity of 
 the reflected light is 
 
 .fr 2 + Q 2 R 2 * + 2QRA cos (ft - ^ - ft) 
 ~ 1 + (pltfMf + 2QR.R, cos (ft - ^ + ft) ' 
 
 It will be necessary to distinguish two cases first, when the angle 
 of incidence is less than the critical angle of the glass, and second 
 when it is greater than that angle. In each case vibrations 
 parallel and perpendicular to the plane of incidence must be dealt 
 with separately. 
 
 If </> be the angle of incidence, </>' that of refraction, and /z x the 
 refractive index of the glass, we have sin <' = /i^ sin <. The surface 
 of separation being the plane x = 0, and the plane of xy being that of 
 incidence, the components of the displacement are proportional to 
 
 (>ip [t - (x cos <f>' + y sin </') /c]_ 
 
 Putting x d, the thickness of the film of air, we see that 
 
 q _ gipCiC,os$'lc __ Q i2ircj\ . cos 4>' 
 
 where A, is the wave-length in air. Below the critical angle <f>' is 
 real, so that we have Q = 1, and ty = 27rc 1 /X . cos <$>'. Hence 
 
 ,. _ a + b cos (ft T/T ft) 
 
 dj + 6 cos (ft ^ + ft) ' 
 
 where a = R^ + jR 2 2 , b = 2^^, and c^ = 1 + R?R 2 \ For a given 
 angle of incidence R lt R 2) 6 l and ft are constants, and the only 
 variable in the expression for I is o/r, which depends on the thickness 
 of the film of air between the glass and the metal. Differentiating 
 / with respect to i/r, we find that it is stationary when 
 
 (a otj) cos ft sin (ft ^) + (a 4- a^ sin ft cos (ft ^) 
 
 + 6 sin 2ft = 0. 
 
 Putting - 1 tan ft = tan 0, 
 
 ftj ~~ G& 
 
 6 sin 2ft 
 
 and - = sin 7, 
 
 v(a! - or) 2 
 
266 NEWTON'S RINGS FORMED BY [CH. 
 
 we get sin (^ + /3 - 2 ) = sin 7, so that i|r + fi 6 2 = UTT + (- l) n . 7, 
 and 27TCJ/A, = sec $' [WTT + (- I) 71 7 -f 2 - ft]. By giving different 
 integral values to n we see that there will be rings of maximum 
 and minimum intensity. 
 
 Formulae suitable for the calculation of all the quantities 
 R 1} R 2 , #1, and # 2 have already been obtained. It has been seen 
 that they all depend to some extent on the character of the layer 
 of transition from glass to air and from air to metal. The influence 
 of the layer on the phenomena now under discussion is, however, 
 very slight, and we shall obtain an approximation sufficiently 
 accurate for present purposes by regarding the transition as 
 abrupt. In that case i\ and r 2 are given by Fresnel's formulae, 
 and the modifications thereof obtained by making the refractive 
 index complex for reflection from metal. 
 
 For light polarised at right angles to the plane of incidence we 
 have 
 
 r 1 = - tan (</> - 0')/tan (< + <') = tan (</>' - </>)/tan (</>' + c/>). 
 
 When (f> lies between zero and the polarising angle tan" 1 !///,!, this 
 gives R l = tan ((/>' <)/ tan (</>' + </>) and 0j = 0, while when c/> is 
 greater than the polarising angle it gives 
 
 R, = - tan (</>' - <)/tan ($' + <), and 0, = TT. 
 In both cases = 7 = 0, and / is stationary when ^r 2 = mr. 
 The maximum value of / is (a + b)/(a, + 6) = (R, + J? 2 ) 2 /(l + RA}\ 
 and the minimum is (a - 6)/(aj - 6) = (R^ - J2 2 ) 2 /(l RAJ- When 
 (/> is less than the polarising angle / is a maximum when n is even 
 and a minimum when n is odd ; but this state of affairs is reversed 
 when (f> is greater than the polarising angle. At the polarising 
 angle 6 = 0, so that / is constant and there are no rings. 
 
 For the determination of the radii of the rings and their 
 intensity nothing remains but the calculation of the constants 
 ./?!, R 2 , and 0. 2 . For this purpose we shall take ^ = 1*596, which 
 corresponds to a polarising angle of 32 V, and a critical angle of 
 38 47'. For the metals we shall take Drude's estimates of /u and 
 a for sodium light reflected from silver and steel respectively, viz. : 
 // = 0*18 and a = 3'67 in the case of silver, and /i = 2'41 and 
 a = 3*40 in the case of steel. The values of R 2 and # 2 are obtained 
 from the formulae of p. 242. 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 267 
 
 With the aid of these numbers there is no difficulty in calcu- 
 lating the intensity / at any point. For our purposes, however, 
 
 0... 
 
 
 
 10 
 
 20 
 
 30 
 
 35 
 
 38 
 
 38 15' 
 
 38 30' 
 
 38 46' 
 
 ft, 
 
 230 
 
 220 
 
 175 
 
 052 
 
 121 
 
 454 
 
 526 
 
 625 
 
 881 
 
 R.} silver 
 
 976 
 
 975 
 
 971 
 
 962 
 
 952 
 
 953 
 
 957 
 
 965 
 
 989 
 
 -R 2 steel 
 
 764 
 
 759 
 
 709 
 
 655 
 
 566 
 
 517 
 
 545 
 
 608 
 
 859 
 
 2 silver 
 
 329 35' 
 
 328 18' 
 
 323 39' 
 
 310 24' 
 
 290 10' 
 
 249 0' 
 
 236 52' 
 
 223 38' 
 
 19226' 
 
 2 steel 
 
 337 26' 
 
 336 41' 
 
 333 5' 
 
 32226' 
 
 303 18' 
 
 251 20' 
 
 239 17' 
 
 223 46' 
 
 19142' 
 
 it will be sufficient to obtain the maximum and minimum values, 
 and the intensity at the centre. The maxima and minima for 
 silver and steel are given in the following table. 
 
 
 
 
 
 10 
 
 20 
 
 30 
 
 35 
 
 38 
 
 38 15' 
 
 38 30' 
 
 38 46' 
 
 I (max.) 
 silver 
 
 969 
 
 967 
 
 958 
 
 932 
 
 926 
 
 962 
 
 974 
 
 982 
 
 998 
 
 I (min.) 
 silver 
 
 924 
 
 923 
 
 918 
 
 917 
 
 882 
 
 773 
 
 753 
 
 732 
 
 704 
 
 I (max.) 
 steel 
 
 715 
 
 703 
 
 618 
 
 468 
 
 413 
 
 619 
 
 692 
 
 798 
 
 981 
 
 I (min.) 
 steel 
 
 420 
 
 419 
 
 372 
 
 390 
 
 228 
 
 007 
 
 001 
 
 001 
 
 008 
 
 The value of the intensity at the centre cannot be derived 
 from these results, for at that point there is no air space between 
 the glass and metal, so that we have direct reflection from the 
 metal. The intensity in this case is most simply obtained from 
 the formulae of p. 242, by replacing yu- and a by yuo/^i an d a//*i 
 respectively. In this way we get the following values of the 
 intensity for different values of <f> ranging from zero to 90. 
 
 0... 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 I (silver) 
 
 931 
 
 930 
 
 929 
 
 924 
 
 918 
 
 910 
 
 905 
 
 896 
 
 925 
 
 1 
 
 I (steel) 
 
 442 
 
 438 
 
 425 
 
 406 
 
 362 
 
 316 
 
 266 
 
 215 
 
 346 
 
 1 
 
268 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 The size of the rings is easily obtained as on p. 104. We have 
 seen that at any point on a ring of maximum or minimum intensity 
 ^ # 2 = mr. Also we have -v/r = 2-TTCi/X cos </>' = TTK^/X . cos $', 
 where X is the wave length in the glass, not in the air. Hence 
 p = r ^j\ = \/(rc + 2 /7r)sec</>'. By putting n = - 1, 0, 1, 2, 3, 
 etc., we get the radii of the successive rings. For the rings of high 
 order the radii are very nearly proportional to the square roots of 
 the natural numbers. The following tables give the values of 
 pi, p2, PS and /o 4 corresponding to the first four rings, for different 
 values of ranging from zero to the critical angle. 
 
 
 
 SILVER 
 
 STEEL 
 
 Pi 
 
 P-2 
 
 Ps 
 
 At 
 
 Pi 
 
 P-2 
 
 Ps 
 
 P4 
 
 
 
 0-912 
 
 1-35 
 
 1-68 
 
 1-96 
 
 0-935 
 
 1-37 
 
 1-70 
 
 1-97 
 
 10 
 
 0-925 
 
 1-38 
 
 1-71 
 
 2-00 
 
 0-952 
 
 1-40 
 
 1-73 
 
 2-01 
 
 20 
 
 0-976 
 
 1-47 
 
 1-83 
 
 2-13 
 
 1-01 
 
 1-53 
 
 1-88 
 
 2-17 
 
 30 
 
 1-10 
 
 1-69 
 
 2-13 
 
 2-48 
 
 1-15 
 
 1-72 
 
 2-15 
 
 2-51 
 
 35 
 
 1-23 
 
 2-00 
 
 2-55 
 
 3-00 
 
 1-31 
 
 2-05 
 
 . 2-58 
 
 3-03 
 
 38 
 
 1-41 
 
 2-72 
 
 3-58 
 
 4-26 
 
 1-43 
 
 2-73 
 
 3-59 
 
 4-27 
 
 38 15' 
 
 1-44 
 
 2-93 
 
 3-89 
 
 4-66 
 
 1-44 
 
 2-95 
 
 3-90 
 
 4-67 
 
 38 30' 
 
 1-463 
 
 3-31 
 
 4-45 
 
 5-35 
 
 1-45 
 
 3-31 
 
 4-45 
 
 5-35 
 
 38 46' 
 
 1-496 
 
 5-89 
 
 8-19 
 
 9-98 
 
 1-45 
 
 5-88 
 
 8-19 
 
 9-97 
 
 38 47' 
 
 1-5 
 
 00 
 
 00 
 
 00 
 
 1-45 
 
 00 
 
 00 
 
 00 
 
 From these results it appears that the rings open out as the 
 angle of incidence increases. The expansion of all the rings except 
 the first is very rapid when nearing the critical angle, and the 
 radius of each increases to infinity when the critical angle is 
 reached. With the first ring, however, there is very little change 
 as the critical angle is approached, and the limiting value of 
 r V/Cj/X is 1*5 for silver and 1*45 for steel. Our formulae cannot, in 
 strictness, be applied when $ is exactly equal to the critical angle, 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 269 
 
 for then | q*r 2 r 3 = 1, and the fundamental series on p. 95 loses 
 its convergence. We may, . however, approach the critical angle 
 as near as we wish. The angle 2 is obtained from the formula 
 of p. 242, viz.: tan 2 = - 2Mc cos</>' sin (a - u)/(M 2 cos 2 <' - c 2 ). 
 When </>' is nearly 90, tan 2 is very small, and we have 
 
 # 2 TT = tan (0 2 TT), approximately, 
 
 = tan 2 = - 2Jfc cos </>' sin (a - u)/'(M* cos 2 </>' - c 2 ). 
 Hence (0 2 TT) sec<' = 27lfcsin(a w)/c 2 in the limit. For the 
 two metals, silver and steel, M* is 13'5 and 17*37 respectively, so 
 that the error in neglecting M~* and higher powers of M~ z is very 
 small. To this order of approximation we find from the formulae 
 of p. 242 that, when <f>' = 90, we have 
 
 2Mc sin (a - u)/c* = 2a (1 - 1/2 Jlf 2 ). 
 
 Hence p l = Vsec <j>' (0 2 /7r - 1) = \/(l - l/2M 2 )2a/7r which gives the 
 limiting values recorded above for silver and steel. 
 
 From the formula r V/^/A, = *J(n + ^/TT) sec </>', we see that r is 
 a function of X so that, unless homogeneous light be employed, the 
 rings will be coloured. <' and consequently 2 are functions of X, 
 but the change in r due to this fact is usually small, so that 
 approximately r varies as the square root of the wave length. 
 The following table gives the values of p corresponding to the 
 
 
 = 
 
 :0 
 
 = 
 
 20 
 
 = 
 
 38 
 
 
 Silver 
 
 Steel 
 
 Silver 
 
 Steel 
 
 Silver 
 
 Steel 
 
 ( Ph 
 
 0-741 
 
 0-760 
 
 0-793 
 
 0-821 
 
 1-15 
 
 1-16 
 
 First bright ring -\ p d 
 
 0-912 
 
 0-935 
 
 0-976 
 
 1-01 
 
 1-41 
 
 1-43 
 
 \Pa 
 
 1-04 
 
 1-06 
 
 Ml 
 
 1-15 
 
 1-60 
 
 1-62 
 
 f Ph 
 
 1-10 
 
 I'll 
 
 1-19 
 
 24 
 
 2-21 
 
 2-22 
 
 First dark ring J p d 
 
 35 
 
 1-37 
 
 1-47 
 
 53 
 
 2-72 
 
 2-73 
 
 (Pa 
 
 53 
 
 1-56 
 
 1-67 
 
 74 
 
 3-08 
 
 3-10 
 
 f Ph 
 
 37 
 
 1-38 
 
 1-49 
 
 53 
 
 2-91 
 
 2-92 
 
 Second bright ring 4 p d 
 
 68 
 
 1-70 
 
 1-83 
 
 88 
 
 3-58 
 
 3-59 
 
 \Pa 
 
 91 
 
 1-93 
 
 2-06 
 
 2-13 
 
 4-06 
 
 4-08 
 
 f Ph 
 
 1-59 
 
 1-60 
 
 1-73 
 
 1-76 
 
 3-46 
 
 3-47 
 
 Second dark ring -| p d 
 
 1-96 
 
 1-97 
 
 2-13 
 
 2-17 
 
 4-26 
 
 4-27 
 
 \ 
 
 ( Pa 
 
 2-22 
 
 2-24 
 
 2-42 
 
 2-46 
 
 4-84 
 
 4-85 
 
270 
 
 NEWTON S RINGS FORMED BY 
 
 [CH. 
 
 lines A, D, and H in the spectrum, calculated on the hypothesis 
 that p is proportional to the square root of \, an hypothesis that 
 will serve for illustrative purposes. 
 
 The general effect of this dependence of the radius on the wave 
 length is that each bright ring changes in colour from violet on the 
 inside to red on the outside, whereas in the dark rings the order of 
 colours is reversed. In the case of silver, however, the difference 
 between the maximum and the minimum intensities is nowhere 
 great, so that the rings are not sufficiently conspicuous to show the 
 colours well. Another effect, well illustrated by the tables above, is 
 that, for all the rings except the first, the red of one ring overlaps 
 the violet of the next. This diminishes the sharpness of the rings, 
 and makes the first appear more conspicuous than the others. 
 
 Most of these matters are illustrated graphically in the accom- 
 panying figures, which embody the results obtained in Chapter v,, 
 
 Fig. 102. 
 
 Fig. 103. 
 
 \ 
 
 /> 
 
 =3 
 
 Fig. 104. 
 
 Fig. 105. 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 271 
 
 when the metal is replaced by glass of the same refractive index 
 as the first piece. 
 
 Figs. 102, 103, 104 and 105 represent the intensities as far as 
 the second dark ring for various angles of incidence below the 
 polarising angle, viz. < = 0, 10, 20, and 30 respectively. It is 
 evident from the figures that the rings are all dark-centred, that 
 they gradually expand as cf> increases, and that they are much less 
 distinct with silver than with steel or glass. The rings are most 
 distinct with glass, but the difference between that and steel is 
 not very marked. Fig. 105 shows that as the polarising angle is 
 approached the rings rapidly disappear. After passing that angle 
 the rings pass from dark-centred to light-centred in the case of 
 the metals ; but with glass they remain dark-centred throughout. 
 This is clearly brought out in Figs. 106, 107, 108, and 109 which 
 follow. 
 
 X 
 
 Fig. 106. 
 
 Fig. 107. 
 
 
 -V 
 
 \ 
 
 Fig. 108. 
 
272 
 
 NEWTON S RINGS FORMED BY 
 
 [CH. 
 
 The rings are still very much more distinct with glass and 
 steel than with silver. There is a close resemblance between the 
 
 \/ 
 
 * 3 4 
 
 Fig. 109. 
 
 rings with glass and with steel, the main difference being that the 
 rings with steel are bright-centred, but the brightness at the 
 centre is not very appreciable compared with that at the rings. 
 The rings with steel are slightly more distinct than those with 
 glass and slightly larger. After passing the polarising angle 
 the rings have all expanded so much that it is convenient to use 
 a smaller scale to represent the radius, and in the figures 106 109, 
 the scale of p is half that in the earlier figures 102 105. 
 
 The colour effects are indicated in Figs. 110 and 111, the first 
 of which deals with the case where <f> = 0, and the second with 
 that where </> = 38, i.e. beyond the polarising angle. The dotted 
 curve represents the intensity for red light, and the continuous 
 curve that for violet. 
 
 . ~ 
 
 
 
 ~^ 
 
 o> 
 
 ^ 
 
 *S 
 
 o.'.. 
 
 ^--''\ 
 
 H 
 
 
 
 
 
 01 
 
 wei 
 
 
 
 $= 
 
 
 
 
 
 
 St 
 
 eel 
 
 
 
 
 
 
 /, 
 
 5 
 
 \ 
 
 A 
 
 
 / 
 
 
 
 
 /" 
 
 
 \ 
 
 
 V 
 
 
 
 
 \ 
 
 ^ 
 
 s 
 
 
 
 
 J 
 
 \ 
 
 d 
 
 rtolefc 
 
 \ 
 
 57 
 
 Fig. 110. 
 
 Fig. Ill, 
 
XI 
 
 METALLIC REFLECTION. THIN METALLIC PLATES 
 
 273 
 
 The discussion of the phenomena when the incident light is 
 polarised parallel to the plane of incidence may be conducted on 
 exactly similar lines. In this case we have 
 
 =^ 1 and l = 0. 
 
 The maxima and minima are given by the same formulae as before 
 (p. 266), and occur as before when ty 2 = nir. The rings are 
 dark-centred throughout instead of passing from dark-centred to 
 bright-centred as with light polarised at right angles to the plane 
 of incidence. The values of R ly R 2 , and 2 are as follows : 
 
 
 
 
 
 10 
 
 20 
 
 30 
 
 35 
 
 38 
 
 38 46' 
 
 *| 
 
 230 
 
 243 
 
 283 
 
 393 
 
 530 
 
 743 
 
 952 
 
 R 2 (silver) 
 
 976 
 
 976 
 
 980 
 
 985 
 
 9 ( .)2 
 
 996 
 
 999 
 
 R. 2 (steel) 
 
 764 
 
 773 
 
 799 
 
 852 
 
 911 
 
 952 
 
 999 
 
 2 (silver) 329 35' 
 
 330 48' 
 
 334 36' 
 
 341 49' 349 52' 
 
 354 25' 
 
 360 
 
 0o (steel) 
 
 337 26' 
 
 338 21' 
 
 341 12' 
 
 346 23' 
 
 353 0' 
 
 355 48' 
 
 360 
 
 
 | 
 
 
 
 
 
 
 From these results we derive the following values for the 
 intensity at the bright and dark rings. 
 
 < 
 
 
 
 10 
 
 20 
 
 30 
 
 35 
 
 38 
 
 38 46' 
 
 I (max.) silver 
 
 969 
 
 971 
 
 976 
 
 987 
 
 996 
 
 999 
 
 1 
 
 I (min.) silver 
 
 924 
 
 924 
 
 929 
 
 935 
 
 949 
 
 959 
 
 1 
 
 / (max.) steel 
 
 715 
 
 731 
 
 779 
 
 869 
 
 945 
 
 987 
 
 1 
 
 / (min.) steel 
 
 420 
 
 426 
 
 445 
 
 476 
 
 542 
 
 551 
 
 1 
 
 The value of the intensity at the centre, obtained as on p. 267, 
 is as follows : 
 
 0... 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 9.0 
 
 I (silver) 
 
 931 
 
 932 
 
 936 
 
 941 
 
 949 
 
 957 
 
 967 
 
 978 
 
 989 
 
 1 
 
 I (steel) 
 
 443 
 
 449 
 
 467 
 
 484 
 
 541 
 
 600 
 
 674 
 
 765 
 
 873 
 
 1 
 
 M. L. 
 
 18 
 
274 NEWTON'S RINGS FORMED BY [GEL 
 
 For the radii of the rings we have the following values : 
 
 
 
 SILVER, 
 
 STEEL 
 
 Pi 
 
 Pa 
 
 P.3 
 
 P4 
 
 Pi 
 
 Ps 
 
 Pa 
 
 P4 
 
 
 
 0-912 
 
 1-35 
 
 1-68 
 
 1-96 
 
 0-935 
 
 1-37 
 
 1-70 
 
 1-97 
 
 10 
 
 0-934 
 
 1-39 
 
 1-72 
 
 2-00 
 
 0-957 
 
 1-40 
 
 1-73 
 
 2-01 
 
 20 
 
 1-01 
 
 1-49 
 
 1-85 
 
 2-15 
 
 1-03 
 
 1-50 
 
 1-86 
 
 2-16 
 
 30 
 
 1-22 
 
 1-99 
 
 2-19 
 
 2-54 
 
 1-24 
 
 1-79 
 
 2-20 
 
 2-55 
 
 35 
 
 1-53 
 
 2-20 
 
 2-71 
 
 3-13 
 
 1-55 
 
 2-21 
 
 2-71 
 
 3-14 
 
 38 
 
 2-29 
 
 3-2(5 
 
 3-91 
 
 4-63 
 
 2-30 
 
 3-27 
 
 4-01 
 
 4-63 
 
 38 41' 
 
 00 
 
 00 
 
 00 
 
 00 
 
 oo 
 
 00 
 
 oo 
 
 00 
 
 The graphical representation of these results is undertaken in 
 the next three figures. It appears from these that the rings with 
 silver are very indistinct. With steel, on the other hand, the 
 rings begin, when c/> is small, by being almost as marked as with 
 glass. They remain fairly distinct as $ increases, but the rings 
 with glass gain in intensity much more conspicuously than those 
 with steel. In all cases the rings are dark-centred ; but, as will 
 be seen from Fig. 114, as the critical angle is approached the 
 difference between the intensity at the centre and at the first 
 maximum is very slight in the case of the metals, so that there 
 appears to be a bright spot at the centre. 
 
 \J 
 
 Fig. 112. 
 
 Fig. 113. 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 275 
 
 When a glass prisin is employed and the incidence is beyond 
 the critical angle, we must make use of the formulae obtained in 
 
 V 
 
 Fig. 114. 
 
 Chapter in. for the change of amplitude and phase in the case of 
 total reflection. When the incident light is polarised at right 
 angles to the plane of incidence we have seen (p. 56) that r = e i ^-^ > 
 where 
 
 tan ! = fjij 2 sec < Vsin 2 <f> l//-^ 2 . 
 
 Hence R = 1, and 6 l = 2^ TT. To find the values of R 2 and 2 we 
 must consider the reflection at the metal of the " surface waves " 
 due to total reflection. In the air we have 
 
 where n = -i Vsin 2 </> - 1//V, 
 
 the first term corresponding to the incident and the second to the 
 reflected wave. In the metal we have similarly 
 
 _ S gip [t- (mx + y sin 0) MI/C], 
 
 In order to satisfy the dynamical equations in the metal we must 
 have 
 
 (m 2 + sin 2 <#>) IJL* = it = (/Lto - iaf, 
 
 so that m is always complex, of the form m^ in^ = Ne~ i t. The 
 
 boundary conditions require the continuity of f and of g/jj?, i.e. of 
 
 i p\ OP 
 f and of ^ . Hence we must have 1 - r 2 = 5, and 
 
 182 
 
276 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 and these give 
 m 
 
 ' 2 sn a - 
 
 ' 2 cos 2a) 
 
 ~~ m 4- n/j? ~ m l p^M* sin 2a i (TCJ j^M 2 cos 2a) 
 where /*<> * = Me~ ia , 
 
 and Pi = Vsin 2 </> - l//^ 2 . 
 
 Thus ^ 2 and 2 are determined from the equations 
 
 where 
 
 x 
 
 a sin 2a - r^ cos 2a) _ Zp 
 
 sin (2 - 7) 
 
 and 
 
 tan ft, = 
 
 (m^ cos 2a + ^ sin 2a) _ 
 
 (2a - 7) 
 
 We thus derive the following table : 
 
 
 
 38 47' 40 
 
 45 
 
 50 
 
 60 
 
 70 
 
 80 
 
 ^+180 
 
 
 
 51 16' 
 
 99 32' 
 
 120 24' 
 
 143 40' 
 
 158 8' 
 
 169 46' 
 
 # 2 (silver) 
 
 1 
 
 2-04 
 
 7-37 
 
 35-4 
 
 6-78 
 
 4-48 
 
 4-09 
 
 2 (steel) 
 
 1 
 
 1-40 
 
 1-93 
 
 2-11 
 
 2-14 
 
 2-08 
 
 2-04 
 
 2 (silver) 
 
 180 1 177 41' 168 56' 
 
 75 11' 10 29' 
 
 6 50' 
 
 6 14' 
 
 2 (steel) 
 
 
 
 180 
 
 156 50' 
 
 124 54' 
 
 106 7' 
 
 84 31' 
 
 73 46' 
 
 68 38' 
 
 We have seen (p. 265) that the intensity is given by the 
 
 formula 
 
 J R 1 2 
 
 " 
 
 1 + Q'RfRf + 2QRA cos (6> 2 - f + 00 ' 
 In the present case we have ^ = 1,^ = and 
 
 Q _ 
 
 where X is the wave length in the glass, and r is the distance 
 from the centre. W T e thus have 
 
 1 = 
 
 cos 
 
 The numerator is the square of the resultant of two vectors 
 
METALLIC KEFLECTION. THIN METALLIC PLATES 
 
 277 
 
 1 and QR 2 inclined at an angle ft ft. This resultant will be least 
 when ft ft = 180, and its least vahie will be zero if 1 = QR Z . 
 Hence, if we are seeking for places of absolute blackness we must 
 take ft ft = 180 and 1 = QR 2 . The former equation, if it can 
 be satisfied at all, will confine us to a particular angle of incidence 
 (f> ; the latter will tie us to a ring of definite radius. It may be, 
 however, that these conditions cannot both be satisfied, but there 
 may still be, at different angles of incidence, a dark ring where 
 the intensity is a minimum. For a given angle of incidence 
 R, fa and ft are fixed, so that to find the position of the dark ring 
 we have to differentiate / with respect to Q. In this way we learn 
 that / is stationary when Q = l/R 2 . The corresponding value of 
 / is a minimum if cos (ft - ft) cos (ft + ft) be negative, which 
 is always the case since / cannot be greater than unity. The 
 
 minimum 
 
 f T . 1 + COS (ft 0,) mi i i i 
 
 value of I is - ~. The black ring occurs 
 
 1 + cos (ft -4- ft) 
 
 when ft = ft + 180, and its position is determined by drawing 
 graphs of the functions ft and ft 4- 180, and finding where they 
 intersect. This is done in Fig. 115, from which it appears that 
 
 iscr 
 
 150P 
 
 J20 
 
 90^ 
 
 60 
 
 30' 
 
 40 
 
 Sff 
 
 Bff 70 
 
 Fig. 115. 
 
 8CT 
 
 9rf 
 
278 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 with silver the black ring corresponds to an incidence of <> = 48 40', 
 and with steel to < = 47 55'. 
 
 The figure also shows that when the incidence is near that 
 giving an absolutely black ring, the angle O l 2 varies very much 
 more rapidly with silver than with steel, so that the change of 
 intensity at the dark ring will be much more marked with the 
 first metal than with the second. The distinctness of a ring 
 depends not only on the intensity of the light there, but also on 
 the difference of intensities at the ring and near its centre. Very 
 near the centre we have Q = 1, so that the intensity is 
 
 ~ 
 
 cos ! - 
 
 cos 
 
 The following table gives the minimum value of I, i.e. its value 
 at the dark ring, as well as that near the centre, for different 
 incidences. 
 
 
 
 38 47' 
 
 40 
 
 45 
 
 50 
 
 60 
 
 70 
 
 80 
 
 I (min.) silver 
 
 1 
 
 962 
 
 631 
 
 138 
 
 880 
 
 978 
 
 980 
 
 I (min.) steel 
 
 1 
 
 674 
 
 056 
 
 018 
 
 292 
 
 558 
 
 783 
 
 I (near centre) silver 
 
 1 
 
 966 
 
 900 
 
 908 
 
 946 
 
 987 
 
 988 
 
 I (near centre) steel 
 
 1 
 
 689 
 
 164 
 
 161 
 
 402 
 
 623 
 
 813 
 
 The radius of the dark ring is obtained from the formula 
 
 R 2 = l/Q 
 which gives 
 
 p = r >*, = 
 
 This gives the following values of p for different incidences with 
 silver and steel. 
 
 
 
 40 
 
 45 
 
 50 
 
 60 
 
 70 
 
 80 
 
 p (silver) 
 
 546 
 
 745 
 
 925 
 
 628 
 
 534 
 
 506 
 
 p (steel) 
 
 380 
 
 427 
 
 423 
 
 396 
 
 374 
 
 360 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 279 
 
 Fig. 116 embodies these results, and shows in a graphic way 
 how the ring expands and contracts as the angle of incidence 
 
 increases. 
 
 \ 
 
 2 
 
 40 
 
 50 
 
 60 70 
 
 Fig. 116. 
 
 80 
 
 It will be observed that with silver the ring contracts very rapidly, 
 shortly after the incidence corresponding to the black ring has 
 been reached, and that the contraction is not nearly so marked in 
 the case of steel. Another difference between the behaviour of 
 the two metals is that the difference between the intensity at the 
 ring and that near the centre is much greater for silver than for 
 steel. This adds greatly to the distinctness of the ring, which 
 with steel is, as a rule, scarcely distinguishable from the dark 
 patch that it encloses. Some of the results in the above tables 
 are illustrated in Figs. 117, 118, and 119, in which the ordinates 
 represent the blackness B = 1 / for different angles of incidence. 
 When the metal is replaced by glass the values of B are given on 
 p. 110, and these are used in the figures in order to represent to 
 the eye the differences in the phenomena presented by glass, steel 
 and silver. 
 
280 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 Fig. 117 represents the state of affairs when < = 45. The 
 ring with silver is quite conspicuous, but the blackness is much 
 
 ft 
 
 ^^ 
 
 S 
 
 vXR 
 
 Fig. 117. 
 
 \ 
 
 s 
 
 \ 
 
 \ 
 
 \ 
 
 \ 
 
 76 
 
 Fig. 118. 
 
 46 
 
 Fig. 119. 
 
 less than with steel. The centre of the ring with steel is, 
 however, almost as dark as the ring, so that the general effect is 
 rather that of a dark central patch very like that with glass. In 
 the next figure the angle of incidence has increased by five 
 degrees. The ring with silver is now much more pronounced, 
 
X] METALLIC REFLECTION. THIN METALLIC PLATES 281 
 
 being larger and more intense ; but there is no very marked 
 change with the ring in the case of steel. Fig. 119 represents, 
 the change that has taken place when <j> = 60. The ring with 
 silver has contracted so much as to squeeze out its bright centre 
 almost completely, and both rings now present the appearance 
 of a dark spot at the centre, larger than with glass, but less 
 intense. 
 
 As the radii of the rings depend on X and on the optical 
 constants, they will be different for different colours. The 
 general character of the chromatic effects is indicated in Fig. 120 > 
 where the intensities are represented for different colours. 
 
 Fig. 120. 
 
 It appears from the figure that the black ring is coloured red 
 on the inside and violet on the outside. The intensity of the 
 violet on the outside is greater than that of the red on the 
 inside, so that the outside colour is more marked, as was pointed 
 out by Stokes in the description with which this chapter opened. 
 
 Turning to the case where the incident light is polarised 
 parallel to the plane of incidence, we make use of the results- 
 obtained on p. 57. We have i\ = e' 2la/ , where 
 tan of = p l sec <f> = tan OL^/JL^, 
 
 so that ^ = 1 and l = 2a'. To find the values of R 2 and # 2 we 
 have 
 
 '_ gip [t-(nx+y sin 0) /n^c] _j_ r ^ip [t + (nxy sin <J>) /*,/c] 
 
 in the air, and 
 
 ' _ se ip [t-(mx + ysin<t>) MI/C] 
 
 in the metal, where n and m have the same values as before. The 
 boundary conditions give l + r 2 = s, and n(l r 2 ) = ms, so that 
 r 2 = (n-m)/(n + m). Thus E. 2 and 2 are determined from the 
 equations R 2 * = (1 #)/(! + #), where 
 
282 NEWTON'S RINGS FORMED BY [CH. 
 
 The values derived from these formulae are as follows : 
 
 
 
 38 47' 
 
 40 
 
 45 
 
 50 
 
 60 
 
 70 
 
 80 
 
 0, 
 
 
 
 21 20' 
 
 49 46' 
 
 68 54' 
 
 100 12' 
 
 127 36' 
 
 154 18' 
 
 2 (silver) 
 
 1 
 
 951 
 
 895 
 
 860 
 
 820 
 
 796 
 
 786 
 
 R. 2 (steel) 
 
 1 
 
 972 
 
 938 
 
 917 
 
 891 
 
 874 
 
 865 
 
 6% (silver) 
 
 180 
 
 180 8' 
 
 182 55' 
 
 183 49' 
 
 184 
 
 185 31' 
 
 185 49' 
 
 2 (steel) 
 
 180 
 
 180 39' 
 
 181 27' 
 
 181 56' 
 
 182 33' 
 
 182 55' 
 
 183 8' 
 
 We have seen that the intensity / is stationary when Q=I/R 2 . 
 In the present case, since sin 7 is positive, x cannot be negative, 
 so that R 2 cannot be greater than unity. Hence as cannot be 
 greater than unity, / is nowhere stationary. So that there can be 
 no rings. The intensity is least near the centre, where its' value is 
 given by putting Q 1, so that 
 
 The intensity increases rapidly with the distance from the centre, 
 and the appearance presented is very similar to that when the 
 metal is replaced by glass, except that there is not absolute black- 
 ness at the centre. The following table compares the intensities 
 at different distances from the centre in the case of glass, silver 
 and steel respectively. 
 
 p 
 
 
 
 0-5 
 
 1 
 
 2 
 
 ( Q 
 
 1-000 
 
 0-597 
 
 0-127 
 
 0-0003 
 
 
 o-ooo 
 
 0-339 
 
 0-919 
 
 0-9997 
 
 ^ = 40 j I (silver) 
 
 0-978 
 
 0-991 
 
 0-998 
 
 0-9997 
 
 ( I (steel) 
 
 0-886 
 
 0-953 
 
 0-997 
 
 0-9997 
 
 ( Q 
 
 1-000 
 
 0-391 
 
 0-023 
 
 3 x 10-7 
 
 fino J I (glass) 
 
 0-000 
 
 0-197 
 
 0-914 
 
 1-000 
 
 *=' 1 I (silver) 
 
 0-891 
 
 0-930 
 
 0-995 
 
 1-000 
 
 I I (steel) 
 
 0-929 
 
 0-951 
 
 0-998 
 
 1-000 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 283 
 
 These results are represented in Fig. 121, where the ordinates 
 correspond to the blackness 1 /. It will be seen that there is 
 little difference between the phenomena as presented by the two 
 
 \ 
 
 40 
 
 M 
 
 \ 
 
 \ 
 
 60 
 
 Fig. 121. 
 
 metals silver and steel, and that with each of these the intensity 
 of the black spot is much less marked than in the case of glass. 
 
 It thus appears that all the peculiarities of Newton's rings 
 formed by metallic reflection may be explained by means of the 
 principles laid down in the last chapter. As another example of 
 the use of these principles we shall discuss some of the optical 
 properties of thin metallic plates. Adopting the same notation 
 as before, the reflected and transmitted beams are represented by 
 
284 NEWTON'S RINGS FORMED BY [CH. 
 
 r (1 - 2 2 )/(l - 2 r 2 ) and q(I- r 2 )/(l - ^V 2 ) respectively. The wave 
 first refracted into the metal is 5e W -(&*+# sin 40/c^ wnere 
 
 6 2 -f sin 2 (f> = /A 2 . 
 Hence, if Cj be the thickness of the plate, we have 
 
 q = gipbcjc _ g-rZrri^/A^ 
 
 where X is the wave length of the incident light. Putting 
 
 /ji = p ia = Me~ ia , 
 we have b =me~ ix where 
 
 m* M 4 + sin 4 (f> 2M~ cos 2a sin 2 $, 
 and tan 2%= M* sin 2a/(M' 2 cos 2ot sin 2 <^>). 
 
 We thus get q 2 = Qe~^ ) 
 
 where Q = e-^/x.wsin^ 
 
 and -v/r = 47TCJ/X . m cos ^. 
 
 These relations suffice to determine Q in terms of the optical con- 
 stants of the metal and the thickness of the plate. The quantity 
 r is also a complex, given by the formulae on p. 241. 
 
 If the incident wave be of unit amplitude, then the reflected 
 beam is represented by R^ 6 ^ where R^ is the amplitude and X the 
 change of phase produced by reflection. We have 
 
 We thus get 
 
 and 01=0 + '-, where = TT - y/2, 
 
 and O' is obtained from O by putting jR = 1 and = 0. As the 
 thickness of the plate increases, Q approaches the limit zero, so 
 that R l and 0j approach asymptotically to the values R and 0, as 
 is to be expected. The values of R^ and l are given below for 
 silver in the case of direct incidence, the optical constants 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 285 
 
 employed being a = 3'67 and yLt = 0'18, and 1 being expressed as a 
 fraction of the half wave length. 
 
 , l\ Indef - 
 
 c i/ X small 
 
 0-001 
 
 0-005 
 
 0-01 
 
 0-02 
 
 0-05 
 
 0-1 
 
 0-2 
 
 00 
 
 Rf . 
 
 002 
 
 061 
 
 163 
 
 430 
 
 826 
 
 938 
 
 952 
 
 952 
 
 6 1 i -529 
 
 542 
 
 591 
 
 641 
 
 717 
 
 806 
 
 829 
 
 831 
 
 831 
 
 These results are represented graphically in the curves marked 
 1 in Figs. 122 and 123. 
 
 05 
 
 15 
 
 Fig. 122. 
 
 From the table or the figures it is evident that there is very 
 little change either in the intensity or the phase of the reflected 
 light after a thickness of one-tenth of the wave length has been 
 reached. It appears too that over sixty per cent, of the change of 
 phase due to a thick plate is produced by an indefinitely thin 
 plate. It must be borne in mind, however, that the intensity of 
 the reflected light for such thicknesses is vanishingly small, so that 
 it would not be practicable to measure the change of phase even 
 if such thin plates were available. And, in any case, we should 
 
286 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 not be justified in pushing our analysis to the extreme of vanish- 
 ing thinness, for when we approach molecular dimensions the 
 
 2 
 
 -01 
 
 05 
 
 15 
 
 Fig. 123. 
 
 fundamental hypothesis that the granular structure of reality can 
 be replaced for optical purposes by a continuum ceases to be 
 tenable. 
 
 Turning to the transmitted beam we have 
 
 so that 
 
 1 + Q 2 ^ 4 - 2QR* cos (i/r + 7 ) ' 
 
 and # 2 = <3>" <l> ^, where <> has the same value as before, and 
 <E>'' is obtained from O by putting Q=l and ty = 0. As the 
 thickness of the plate increases Q diminishes rapidly, so that the 
 intensity of the transmitted light diminishes rapidly to zero. 
 Moreover as the thickness increases <l> rapidly diminishes, and ^v/r 
 increases, while <1>" is a constant. As <I> diminishes at first more 
 rapidly than ^ increases, 2 begins by increasing, but it after- 
 wards decreases and ultimately becomes negative. (It must be 
 remembered that meanwhile Q diminishes rapidly, and the 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 287 
 
 intensity of the transmitted light becomes vanishingly small.) 
 The values of R<? and 2 for silver are as follows : 
 
 Cj/X 
 
 Indef. 
 small 
 
 001 
 
 005 
 
 01 
 
 02 
 
 05 
 
 1 
 
 2 
 
 V 
 
 1 
 
 961 
 
 871 
 
 742 
 
 464 
 
 1 
 
 009 
 
 
 
 *i 
 
 
 
 012 
 
 064 
 
 111 
 
 187 
 
 271 
 
 279 
 
 246 
 
 These results are represented in Figs. 124 and 125. From the 
 first of these it appears that a silver plate whose thickness is 
 one-tenth of the wave length is practically opaque. 
 
 \ 
 
 \ 
 
 \ 
 
 15 
 
 Fig. 124. 
 
 Few accurate measurements have been made on the intensity 
 and change of phase in the case of reflection from a thin metallic 
 plate with air surrounding it. In fact extreme tenuity can be 
 obtained only by depositing the metal chemically or electrically on 
 some other material such as glass. We shall therefore consider to 
 what extent the presence of the glass modifies the character of 
 the reflected beam. The result could be obtained by using the 
 general formulae of p. 95, but as we have already calculated the 
 
288 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 amplitude and phase of the reflected and transmitted beams for 
 a metal plate surrounded by air, it is convenient to utilise these 
 
 3 
 
 -2 
 
 -01 
 
 05 
 
 15 
 
 Fig. 125. 
 
 results. To enable us to do this we may suppose the metal 
 and glass to -be separated by an indefinitely thin layer of air. 
 
 Air 
 
 
 
 \ 
 
 Metal 
 
 Air 
 
 Glass 
 
 
 
 4 
 
 
 ^> 
 
 
 
 ^ 
 
 j 
 
 Fig. 126. 
 
 An incident wave of unit amplitude gives rise to a reflected 
 wave Rtf*^ and a transmitted wave R 2 e ie *. The latter wave is 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 289 
 
 immediately reflected by the glass, giving a wave <rR 2 e iez . This 
 wave <rR 2 e ie * is in turn transmitted through the metal plate and 
 emerges into the air as a wave aR^.R^ = Ae** say. The 
 complete reflected wave is thus the resultant of the two com- 
 ponents R^ and Ae**. If we denote this reflected wave by 
 R^e ie i we get 
 
 R^ = R* + A 2 + 2R l A cos (0j - ), 
 
 and 
 
 Q , R! sin l + A sin v 
 tan 0j = fj- * . 
 
 R l cos 0j + A cos % 
 
 If the incidence be direct, then <r is given by Fresnel's formula 
 O- = (/JL l)/(/x. + 1), where ft is the refractive index of the glass. 
 Thus 
 
 and = 7T + 20. 
 
 For small thicknesses R l is very small, while R 2 is considerable. 
 In these circumstances jR/ is very nearly equal to A and #/ to ^. 
 As the thickness increases, R 2 rapidly diminishes, so that R^ 
 and #/ approximate more and more closely to R l and lt Thus, 
 except for very thin plates, the results will be the same as if the 
 metal were surrounded by air instead of being deposited on glass. 
 With very thin plates, however, the two cases will be quite 
 different. Taking /<i=r54, we obtain the following values for 
 ^ 2 and 0'. 
 
 , j Indef. 
 CI/A small 
 
 001 
 
 005 
 
 01 
 
 02 
 
 05 
 
 1 
 
 2 
 
 00 
 
 RS* -045 -042 
 
 107 
 
 241 
 
 501 
 
 853 
 
 941 
 
 952 
 
 952 
 
 0/ -095 
 
 1 
 
 390 
 
 536 
 
 680 
 
 801 
 
 828 
 
 831 
 
 831 
 
 The curves marked 2 in Figs. 122 and 123 above give a 
 graphical representation of these results. A comparison of curves 
 1 and 2 shows the influence of the glass. The results thus 
 obtained are in close agreement with those found experimentally 
 for the change of phase produced by reflection from thin films of 
 silver deposited chemically on glass. 
 
 The same principles will enable us to calculate the ratio of the 
 amplitudes and the difference of phase between the components of 
 
 19 
 
 M. L. 
 
290 NEWTON'S RINGS FORMED BY [CH. 
 
 the displacement perpendicular and parallel to the plane of 
 incidence for reflection at any angle of incidence. We have 
 
 R.e^ =r(l -q~)/(l-fr z ) 
 for light polarised at right angles to the plane of incidence, and 
 
 for light polarised parallel to that plane. Hence 
 
 ) = R l 
 
 ~R I 
 
 Thus if e be the ratio of the amplitudes, a) the azimuth, and A the 
 difference of phase, we have 
 
 1 + Q*R'* - 2QR' 2 cos (^ + >/) 
 
 tan 2 w = e 2 = 
 
 cos 
 
 and A = 6' - + 3> - *', where tan * = 7 
 
 1 Q/2 2 COS (A/T + 7) 
 
 <!>' is given by a similar formula. 
 
 If the metal is deposited on glass, we must replace 
 
 r(l-2 2 )/(l-2 2 r>) by ( r -q*p)/(l-q*rp), 
 
 where r is the same as before, and p is the corresponding quantity 
 when a ray of light goes from glass to metal instead of from air to 
 metal. Thus in calculating p from the formula for r we must replace 
 M by M/fju and <f> by < , where sin <f> //, sin (/> and p is the coefficient 
 of refraction of the glass. Putting p = Pe i(T we have 
 
 p - qrp r - 
 
 Hence 
 
 2 
 
 ' 2 + Q 2 P' 2 - ZQR'P' cos (</ - B 1 - W 
 
 1 + Q*R' 2 P' 2 - ZQR'P' cos (6 f + o- 7 - i/r) 
 
 
 cos ((9 + o- - 
 and A = % / % + O ^> 7 , where 
 
 sin - QP sin (a- - ^) 
 
 n 
 
 and tan <E> = 
 
 . 
 
 COS (6 1 + O- - >|r) 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 291 
 
 The following table gives the values of A for different incidences 
 derived from these formulae for the case of silver deposited on 
 glass of refractive index 1'54. 
 
 c,/X 
 
 = 20 
 
 40 
 
 60 
 
 65 
 
 70 
 
 75 
 
 001 
 
 044 
 
 183 
 
 466 
 
 573 
 
 699 
 
 857 
 
 
 
 
 
 
 
 
 005 
 
 030 
 
 119 
 
 313 
 
 397 
 
 496 
 
 629 
 
 01 
 
 025 
 
 101 
 
 273 
 
 348 
 
 445 
 
 574 
 
 02 
 
 021 
 
 090 
 
 249 
 
 320 
 
 416 
 
 542 
 
 05 
 
 020 
 
 088 
 
 240 
 
 306 
 
 392 
 
 509 
 
 1 
 
 021 
 
 091 
 
 240 
 
 302 
 
 382 
 
 486 
 
 2 
 
 021 
 
 091 
 
 241 
 
 302 
 
 380 
 
 483 
 
 The corresponding values of the azimuth (&>) are as follows : 
 
 C] /X 
 
 0-20 
 
 40 
 
 GO 
 
 05 
 
 70 
 
 75 
 
 001 
 
 44 47' 
 
 44 25' 
 
 43 20' 
 
 42 37' 
 
 41 29' 
 
 40 13' 
 
 005 
 
 43 42' 
 
 39 51' 
 
 32 6' 
 
 29 5' 
 
 26 3' 
 
 22 46' 
 
 01 
 
 43 28' 
 
 37 37' 
 
 28 22' 
 
 25 50' 
 
 22 40' 
 
 19 54' 
 
 02 
 
 43 21' 
 
 38 38' 
 
 31 19' 
 
 29 34' 
 
 28 15' 
 
 26 41' 
 
 
 
 
 
 
 
 05 
 
 44 29' 
 
 42 59' 
 
 40 28' 
 
 40 1' 
 
 40 9' 
 
 41 22' 
 
 1 
 
 44 51' 44 27' 
 
 43 44' 
 
 43 22' 
 
 43 23' 
 
 43 24' 
 
 2 
 
 44 54' 
 
 44 36' 
 
 44 5' 
 
 43 23' 
 
 43 46' 
 
 43 41' 
 
 Fig. 127 represents A as a function of <f> for different thicknesses. 
 The curve marked corresponds to an indefinitely thin film of 
 silver on the glass, the other curves to the different thicknesses 
 mentioned in the table above. The values of the Principal 
 Incidence for each thickness may be obtained from the figure, 
 and the corresponding azimuth, which is the Principal Azimuth, 
 may then be found from a similar graphical representation of the 
 
 192 
 
292 
 
 NEWTON S RINGS FORMED BY 
 
 [CH. 
 
 azimuth w. In this way we find the following values of the 
 Principal Incidence and the Principal Azimuth. 
 
 Cl /\ 
 
 001 
 
 005 
 
 01 
 
 02 
 
 05 
 
 1 -2 
 
 P.I 
 
 61 35' 
 
 70 12' 
 
 72 
 
 73 25' 
 
 74 45' 
 
 75 40' ; 75 48' 
 
 P. A. 
 
 43 15' 
 
 25 40' 
 
 21 20' 
 
 27 10' 
 
 41 15' 
 
 43 30' ; 43 50' 
 
 70 75 
 
 Fig. 127. 
 
 The polarising angle of the glass is 57, and we thus see that 
 we may pass from vitreous reflection to metallic reflection through 
 all the intermediate states by the aid of layers of silver gradually 
 increasing in thickness. One of the most striking features of these 
 results is the very rapid change in the Principal Incidence produced 
 by a slight thickening of the silver film when the film is very thin. 
 A film of only one-thousandth of the wave length increases the 
 Principal Incidence from 57 to 61 35'. This indicates how con- 
 
X] METALLIC REFLECTION. THIN METALLIC PLATES 293 
 
 siderable would be the influence of a surface layer of any kind on 
 the position of the Principal Incidence, and serves to explain the 
 observation of all careful experimenters that the Principal Incidence 
 alters with different conditions of the reflecting surface. It is also 
 in good agreement with an experiment by Mascart on the subject. 
 A silver film was deposited on glass by electrolysis. The film was 
 so thin that its existence could not have been suspected without 
 previous knowledge. It was impossible to estimate the thickness 
 exactly, but it was certainly not greater than five-thousandths of a 
 wave length. Even with this thin film the properties of the light 
 reflected from the glass were considerably modified, and the 
 Principal Incidence had increased from 57 to 59 45'. Conroy 
 made various experiments with silver films on glass. The films 
 were of such a thickness that d/A, varied from '069 to '153, a range 
 that lies between curves 5 and 7 of Fig. 127. One graph makes 
 the Principal Incidence increase in this range from 74 45' to 
 75 48', while Conroy found an increase of 2 4'. Although the 
 Principal Incidence changes rapidly at first, it soon tends to a 
 constant value, and there is little further alteration when the 
 thickness exceeds '00003 mm. Quincke made a large number of 
 observations on the reflection of red light from silver films. He 
 found that the value of the Principal Incidence increased with the 
 thickness, tended to a constant value and changed very little when 
 the thickness of the silver film exceeded '00002 mm. 
 
 As the last example of these principles we shall consider the 
 character of the beam reflected from a metallic film deposited on 
 glass, when the condition of the surface is such that the influence 
 of the layer of transition into the metal is appreciable. We have 
 seen in the last chapter (p. 251) that the effect of the layer is to 
 replace r by r (1 +_pe zA ), where p and A, are quantities depending on 
 the thickness and nature of the layer of transition; and the method 
 of determining the various constants has been indicated on p. 260. 
 We can thus calculate, by the aid of the formulae already obtained, 
 the amplitude and phase of the light reflected from and transmitted 
 through a metallic plate surrounded by air. The method described 
 on p. 288 enables us to derive from these results the amplitude and 
 phase of the reflected light when the metal is deposited on glass. 
 The cases of light polarised parallel and perpendicular to the plane 
 
294 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 of incidence may be dealt with separately, and the ratio of the 
 amplitudes and the difference of phase thus obtained. On using 
 the optical constants derived from Meslin's experiments on reflection 
 from gold, referred to on p. 260, and taking T54 as the refractive 
 index of the glass, we find the following values of the amplitude of 
 the reflected wave, when the light is polarised at right angles to 
 the plane of incidence. 
 
 10 
 
 
 
 c 1 /\='01 
 
 02 
 
 05 
 
 1 
 
 2 
 
 30 
 
 221 
 
 385 
 
 687 
 
 854 
 
 899 
 
 40 
 
 201 
 
 362 
 
 667 
 
 843 
 
 887 
 
 50 
 
 179 
 
 338 
 
 638 
 
 812 
 
 857 
 
 60 
 
 180 
 
 316 
 
 573 
 
 723 
 
 749 
 
 70 
 
 278 
 
 379 
 
 618 
 
 743 
 
 750 
 
 80 
 
 529 
 
 599 
 
 756 
 
 842 
 
 850 
 
X] 
 
 METALLIC REFLECTION. THIN METALLIC PLATES 
 
 295 
 
 These results are represented in Fig. 128. The amplitude 
 diminishes rapidly with the thickness. It diminishes as the 
 incidence increases, reaches a minimum, and then increases to 
 unity. The "quasi-polarising" angle, i.e. the angle of incidence 
 for which the amplitude of the reflected light is least, increases 
 slowly with the thickness of the film. For the thicknesses repre- 
 sented in the above table and figure this quasi-polarising angle has 
 the values 58, 60, 62, 63 and 64. 
 
 The azimuth (o>) is given in the following table, and represented 
 in Fig. 129. 
 
 
 
 Cj/X = -01 
 
 02 
 
 05 
 
 1 
 
 2 
 
 30 
 
 38 31' 39 7' 
 
 42 14' 
 
 43 37' 
 
 44 3' 
 
 40 
 
 32 59' 
 
 34 53' 
 
 40 22' 
 
 42 52' 
 
 43 22' 
 
 50 
 
 25 42' 
 
 30 8' 
 
 37 43' 
 
 41 0' 
 
 41 50' 
 
 60 
 
 21 2' 
 
 25 40' 
 
 33 52' 
 
 37 50' 
 
 38 16' 
 
 70 
 80 
 
 24 36' 
 33 49' 
 
 26 17' 
 34 15' 
 
 34 11' 
 37 58' 
 
 37 48' 
 38 1' 
 
 37 56' 
 
 38 58' 
 
 50 
 
 40 
 
 50? 60 
 
 Fig. 129. 
 
 70 
 
 80 
 
 The crosses indicate the position of the Principal Azimuth, 
 which increases through the following values : 21, 26 45', 33 30', 
 37, 37 30'. Finally, the phase difference (A) has the following 
 values, represented in Fig. 130. 
 
296 
 
 NEWTON'S RINGS FORMED BY 
 
 [CH. 
 
 
 
 Cj/\=-01 
 
 02 
 
 05 
 
 1 
 
 2 
 
 30 
 
 008 
 
 033 
 
 046 
 
 055 
 
 058 
 
 40 
 
 028 
 
 086 
 
 105 
 
 104 
 
 105 
 
 50 
 
 079 
 
 190 
 
 214 
 
 190 
 
 178 
 
 60 
 
 483 -317 
 
 335 
 
 280 
 
 273 
 
 70 
 
 887 
 
 493 
 
 525 
 
 440 
 
 428 
 
 80 
 
 990 
 
 755 
 
 805 
 
 695 
 
 665 
 
 I O 
 
 0-9 
 
 0-6 
 o-7 
 06 
 
 04 
 0-5 
 
 0-2 
 
 \ 
 
 
 7 
 
 50 
 
 50 00 
 
 Fig. 130. 
 
 70 
 
 80 
 
 A comparison with Fig. 127 will show some marked differences. 
 In the present case the difference of phase increases with the 
 thickness for angles of incidence less than 40. For larger angles, 
 however, the law is completely different. When < is greater than 
 60, the difference of phase at first diminishes rapidly with the 
 
METALLIC REFLECTION. THIN METALLIC PLATES 
 
 297 
 
 thickness, then increases for a while, and afterwards diminishes 
 again till it reaches the value appropriate to a thick plate. The 
 curve (3) corresponding to Ci/X = "01 has a point of inflexion, and 
 cuts all the other curves. As might be expected, its form is 
 similar to that of the curve representing the difference of phase in 
 the case of reflection from glass or any other transparent medium 
 when account is taken of the layer of transition. The Principal 
 Incidence, instead of increasing steadily with the thickness, as is 
 the case with gold alone, increases rapidly at first, diminishes for a 
 time and then increases again. Its values for the thicknesses here 
 discussed are 61 30', 70 30', 67 45', 73 and 74. 
 
 The graphical representation of Meslin's results in Fig. 131 will 
 
 10 
 
 09 
 
 facilitate a comparison between theory and experiment, and shows 
 that the agreement is a close one. 
 
CHAPTER XI. 
 
 PROPAGATION OF LIGHT IN ABSORBING CRYSTALLINE MEDIA. 
 
 WE have seen in Chapter IX. that optical problems connected 
 with absorbing media are, from the mathematical point of view, 
 formally identical with the corresponding problems for transparent 
 bodies, the only difference being that constants that are real in 
 the one case are complex in the other. If then we pass from 
 isotropic to crystalline media we should expect to derive everything 
 as before from the fundamental Principle of Least Action, on 
 taking the potential energy function of the form W = f^dr, where 
 
 2^ = 4 n / 2 + A^ + A^h? + 2A as gh + 2A 3l hf+ 2A w fg, 
 
 the coefficients being complex quantities. If we separate the real 
 from the imaginary terms by putting A n = a n + ia n ', and similarly 
 for the other coefficients, we get ^ = ty + ity', where ty and ty' are 
 both expressions of the second degree in /, g, h, each involving six 
 constants. This will be the case if the coordinate axes be taken 
 at random ; but by a proper choice of axes it will always be possible 
 to reduce either 2^ or 2i/r' to the form a 2 / 2 + b*g 2 + c 2 /* 2 or 
 a' 2 / 2 + b' 2 g z 4- c' 2 A 2 respectively. As a rule, however, the principal 
 axes of the functions -fy and >// will be in different directions, and 
 where it is necessary to distinguish them the axes of ty may be 
 called the displacement axes and those of ^r f the absorption axes. 
 In general, then, there will be nine constants required to specify 
 the function Mf, the number being reduced to six when the dis- 
 placement and the absorption axes coincide. 
 
 When the potential energy function is known the Principle of 
 Action leads immediately to the boundary conditions and the 
 
CH. Xl] ABSORBING CRYSTALLINE MEDIA 299 
 
 dynamical equations. For the former it requires the continuity 
 
 of 
 
 m (A y J + A 32 g + A^h) - n (A 2l f+ A^g + A 23 h), 
 
 i.e. of m ^ n -= , and of two similar quantities, where Z, m, n 
 ofi eg 
 
 are the direction cosines of the normal to the interface. This 
 indicates that the tangential components of 
 
 ' dg ' Jh 
 
 must be continuous at an interface. The dynamical equations. 
 prove to be 
 
 and two similar equations. If, as before, we introduce the vector 
 (fi> Vi} ?i) f which (f, T;, f) is the curl, we obtain 
 
 dy dz dz\dg dy \dh ' 
 and two similar equations. These are all satisfied by 
 
 Thus the boundary conditions require the tangential compo- 
 nents of (, ??!, fO to be continuous at the interface. Moreover we 
 have (/ g, h) = (curl) 2 (, Vl , f^, so that 
 
 and the dynamical equations may be expressed in the form 
 
 f-V*W- 
 
 J ~ df dx' 
 
 and two similar equations, where 
 
 8 2 ^ 8 2 ^ 8 2 ^ 
 
 Q __ I __ _i ___ 
 
 ' 
 
 In obtaining these equations we have imposed no restrictions 
 as to the directions of the coordinate axes, which may form any 
 orthogonal system whatever. Before attempting to make any 
 deductions from the dynamical equations, it is convenient to- 
 
300 
 
 PROPAGATION OF LTGHT 
 
 [CH 
 
 choose the axes so as to simplify the expression for ^ as much as 
 possible. We have seen that the displacement and absorption 
 
 axes do not in general coincide, so that 
 we cannot put 2^ in the canonical 
 form ^ 2 ^ 2 + B 2 G 2 4- C*H 2 by means of a 
 real transformation. This, however, may 
 always be done by employing coordinates 
 (X, Y, Z) connected with the real coor- 
 dinates (as, y, z) by means of the accom- 
 panying orthogonal scheme of transfor- 
 mation. In this scheme (I, -m, n) denote complex direction cosines 
 connected by *the relations common to all orthogonal systems, viz. 
 
 and so forth. 
 
 The problem of obtaining the constants A, B, C, I, m, n, is the 
 familiar one of finding the magnitudes and directions of the 
 principal axes of a conicoid whose general equation is 
 
 
 X 
 
 Y 
 
 Z 
 
 X 
 
 *i 
 
 Ilij 
 
 i 
 
 y 
 
 t 
 
 m, 
 
 a 
 
 z 
 
 * 
 
 m s 
 
 "3 
 
 We have 
 
 = A u a? + 
 
 4- 
 
 and since the transformation is orthogonal we also have 
 23> = a,- 2 4- 7/ 2 + ^ 2 --= Z 2 4- F 2 4- Z 2 . 
 
 Hence on equating the discriminants of the two expressions 
 for ^ X^>, we get 
 
 A = j A n -\ 
 
 -Ll 12 
 -"13 
 
 A. 2l 
 
 -it 22 
 
 so that A\ B\ <7 2 are the roots of A = 0. 
 
 To find the direction cosines of the principal axes we make use 
 of the fact that the normal at the end of such an axis is in the 
 direction of the axis, so that 
 
 d /l = // - // 
 
Xl] IN ABSORBING CRYSTALLINE MEDIA 301 
 
 Hence on putting x, y, z proportional to l l} L 2 , l s we get 
 
 A ls l, + ^23/2 + (^33 ~ X) / 3 = 0. 
 
 On eliminating ^ : l z : 1 3 from these equations we get A = 0, so 
 that X is equal to A 2 , B 2 or (7 2 , and these being known the above 
 equations combined with 1 2 + / 2 2 + l/ = 1, determine l lt 1 2 , and / 3> 
 and the other direction cosines may be found similarly. 
 
 Returning to the optical problem and putting 
 
 (F, G,H) = (A,M, 
 (A,M, 
 
 where L^ = l^L 4- l*M + 1 3 N, and similarly for M l and NI, while 
 A, M, N, L, M, N, and F may all be complex, the problem becomes 
 formally identical with that discussed at the beginning of Chap. vi. 
 Thus, as on p. 132, we obtain the following relations : 
 
 A : M : N = ,/( F 2 - A 2 ) : MJ( F 2 - 5 2 ) : N,/( F 2 - C 2 ). . .(II), 
 
 L?l( F 2 - A*) 4- M?l( F 2 - B*) + N?l( F 2 - <7 2 ) = 0. . .(IV). 
 
 Before entering upon any discussion of these equations it will 
 be well to consider the meaning of the formula 
 
 where A, M, N and are complex. The result has been indicated 
 on p. 25. On putting A = X 4- i\f, similarly for the other complex 
 quantities, and proceeding as on p. 25, we get 
 
 F= Ae~ 6 ' (\ cos 6 X' sin 6), 
 
 and so for G and H. Solving any two of these equations for sin 
 and cos and eliminating 0, we get an equation representing an 
 elliptic cylinder if F t G, H be regarded as current coordinates. 
 Further, if we have 
 
 l\ -I- iitfji + nv = and X' 4- m/jf 4- nv = 0, 
 then IF + mG 4- nH = 0, so that (F, G, H) lies on a plane. Thus 
 
302 PROPAGATION OF LIGHT [CH. 
 
 the point (F, G, H) moving on a plane section of an elliptic 
 cylinder describes an ellipse, and formulae such as 
 
 (F, G, H) = (A, M, N)Ae i9 
 
 indicate that the polarisation is in general elliptical. Moreover 
 we have 
 
 r* = F 2 +G*+H* = A 2 e-* 6 ' [(X 2 4- ^ 4- i> 2 ) cos 2 6 
 
 4- (X/ 2 4- p'* + i/ 2 ) sin 2 6 - (XX' + fjLf/ 4- OT/) sin 20]. 
 
 Thus the maximum and minimum values of ?^ 2 are proportional 
 to 
 
 X 2 + yu- 2 + v 1 and to X /2 + // 2 + z/ 2 , 
 
 in other words (X, //,, v) and (X', /a', i/) are proportional to the 
 direction cosines of the major and minor axes of the ellipse. 
 
 Let us consider next the expression Lx + My + Nz that occurs 
 in the formula for (F, G, H). On putting L 1 + il', where I and 
 I' are real, and making a similar substitution for M and JV, we get 
 
 (Ix 4- my + nz) 4- 1 ('# + m'y 4- w's). 
 From this it appears that unless 
 
 l'jl = m'jm = w'/w = - ic (say), 
 
 the displacement at any moment will not be the same at all 
 points of a wave front las + my + nz = constant. We shall limit the 
 discussion to cases in which all points of the wave front are in a 
 similar condition at any time. We may then put 
 
 L = l(I- itc), M=m(l- ifc), N = n(l- IK), 
 and so have (F, G, H) in the form 
 
 (F, G, H) = (A, M, N)Ae i *>v- x + m y +n *^ l - iK W, 
 where l t m, n, K and v are real. The light will thus be elliptically 
 polarised, and the waves will be plane. A wave will move in the 
 direction (I, m, n) with velocity v, and with an amplitude diminish- 
 ing in the ratio of e~v*l v to unity, as the wave moves through unit 
 distance. The quantity /c/v, on which the rate of absorption 
 depends, may be called the index of absorption*. 
 
 Since L, M, N are proportional to I, m, n we see from the 
 equation ZA + MM + NN = of p. 301 that A 4- raM 4- riN = 0, 
 and, on separating the real and imaginary parts, that 
 X + m/jb + nv=0 and ZX 7 + ra// 4- nv = 0. 
 * This quantity is sometimes called the coefficient of absorption. 
 
Xl] IN ABSORBING CRYSTALLINE MEDIA 303 
 
 Thus both axes of the elliptic vibration are at right angles to the 
 wave normal, and the displacement lies entirely in the wave front. 
 These results have been obtained by means of a complex 
 scheme of transformation, but some important features of the 
 propagation may be brought into view without any such trans- 
 formation. Thus let us consider a plane wave whose normal is 
 along the axis of z, which may be in any direction desired. With 
 the limitations that we have already imposed, we may put 
 
 /= \A&P<t-*W ; g = nA&<t-*m ; h = 0. 
 The dynamical equations yield 
 
 i.e. (An-V'^X + A^^O, and A ia \ + (A K - F 2 )/* = 0. 
 On eliminating the ratio \ : //,, we get 
 
 From this we see that there are two values of V and therefore two 
 of v and K, so that in any direction two waves may be propagated, 
 the index of absorption being different for the two. Moreover, on 
 eliminating V 2 - from the above equations, we get 
 
 Au (A a + (A* - A u ) (\/fi) -A u = 0. 
 
 Hence, unless A 12 = 0, we have two values of A/yu,, both of which 
 are complex owing to the complex character of A u , A 12 , and A^. 
 The product of the two values of A,///, is 1, so that if the roots 
 be Xi///,i and \z/fi a we have \ l /p 1 = /* 2 /X 2 - From what has been 
 said above, it appears that the fact that A and /JL are complex 
 indicates that in each of the two waves the light is elliptically 
 polarised ; and, as we shall see, the relation X,/^ = fJ, 2 /\2 indicates 
 that the elliptic orbits in the two waves are similar and have their 
 major axes at right angles. To prove this, we may take the axis 
 of a along the major axis of one ellipse, and put 
 
 f l = A l cos ! , and #1 = iA 1 sin a 1} 
 
 omitting the factor g^(*-*/ F ), where tan j is the ratio of the axes 
 of the ellipse. If the major axis of the second ellipse make an 
 angle 6 with the axis of x, we shall have similarly 
 
 f 2 cos + g 2 sin 6 = A* cos a 2 , 
 and f. 2 sin 6 + g. 2 cos = iA s sin <z,. 
 
304 PROPAGATION OF LIGHT [CH. 
 
 From these we obtain 
 
 X 2 _ / 2 _ cos 2 cos 6 + i sin a 2 sin 6 _ P\ _ _9\ _ * sm i 
 u-2 2 cos 2 si n i sin a 2 cos X x /i cos j 
 
 whence 
 
 cos 6 cos ( 2 4- i) + * sin # sin (a j) = 0, 
 
 so that we must have cos 6 cos ( 2 + c^) = as well as 
 sin sin (a 2 a i) = 0. 
 
 These conditions require either = and 2 + a x = Tr/2, or 6 = ?r/2 
 and 2 = a i- Both conditions express the same fact, that the two 
 ellipses are similar and have their major axes at right angles. 
 
 We thus have an interesting generalisation of the result 
 obtained when dealing with the propagation of light in transparent 
 crystals. We found in that case that, in general, two waves could 
 be propagated in any direction, and that these two waves were 
 polarised rectilinearly, the planes of polarisation being at right 
 angles. With absorbing crystals we also have two waves in any 
 direction, but these are elliptically polarised and the major axes of 
 the elliptic orbits are at right angles. It should be observed that 
 in the special case when J. 12 = 0, the dynamical equations yield 
 (^ n -F 2 )/=0 and (A^ - V*)g = 0, so that either /= and 
 F 2 = A^_ or g = and F 2 = A u . In this case the waves are plane 
 polarised, the planes of polarisation being at right angles. 
 
 The equations obtained on p. 301 give the complete solution 
 of the problem of the propagation of light in an absorbing 
 crystalline medium. The quantities involved are all complex and 
 the separation of the real and the imaginary terms will make each 
 of these equations give rise to two others. When this process is 
 carried out the formulae lose their resemblance to the simple ones 
 of Fresnel, and become much more complicated than these. In 
 their general form they are almost intractable, but they simplify 
 greatly when the absorption is small. Unless this is the case the 
 light will be so quickly absorbed that there will be no optical 
 phenomena to observe, so that the case of weak absorption is one 
 of special importance. 
 
 Taking the coordinate axes along the displacement axes we have 
 
 2^ = Auf* + A,, g* + A ss h* + 2A m gh + 24 81 A/+ 2A l9 fg, 
 
XI] IN ABSORBING CRYSTALLINE MEDIA 305 
 
 where A u = a 2 + m' 2 , A 23 = ia w ', and similarly for the other 
 coefficients. We shall proceed to obtain an approximate solution 
 of the general problem by supposing the absorption so weak that we 
 may neglect squares and higher powers of the coefficients due to 
 the absorption, viz. a' 2 , a 23 etc. In carrying out this process we 
 must, as indicated above, first find the lengths and directions of the 
 principal axes of an ellipsoid whose equation is of the form 
 (abcfgh^xyzf = \, where /, g, h are small quantities of the first order. 
 If /, g, h were zero the equation of the ellipsoid would be referred 
 to the principal axes, so that these axes (X, Y, Z) must lie very 
 near the coordinate axes (x, y, z). It is thus obvious geometrically 
 that we make an error of the second order of small quantities 
 if, in calculating the lengths of the principal axes, we take 
 aX 2 + bY 2 + cZ* = 1 as the equation of the ellipsoid when referred 
 to those axes. Moreover the direction cosines of the new axes will 
 be very nearly (1, 0, 0), (0, 1, 0), (0, 0, 1), so that if we denote them 
 by (l l} m 3 , 7^), (1 2 , 77i 2 , w a ), (1 3 , m 3) n s ) as before, we shall have l l} m 2 , 
 and ?? 3 equal to unity (to our order of approximation), while the other 
 direction cosines will be small quantities of the first order. To 
 find these quantities we have 
 
 + hi, + gljfa = (hi, + bl, +fk)jk = (gk + A + d 3 )/l 3 = a, 
 
 with similar equations for m and n. Solving these and remembering 
 that / 2 and 1 3 are of the first order of small quantities, we get 
 
 I, = 1 , / 2 = hi (a -b), 1 3 = gj(a - c). 
 Similarly 
 
 m 1= = &/(&-), m 2 =l, m 3 =//(6-c); n 1 =g/(c-a), n z =fj(c-b), w 8 =l. 
 
 Hence in the optical problem we have 
 
 A = /] L + l*M + 1 3 N = (1 - IK) (Hi + mk + nl 3 ) 
 
 with similar expressions for M, and N,. 
 Also we have 
 
 to our order of approximation. 
 
 M. L. 20 
 
306 PROPAGATION OF LIGHT [CH. 
 
 Substituting in (IV) p. 301 and separating the real from the 
 imaginary terms, we get 
 
 l*/(v* - a 2 ) + m 2 /(v~ - 6 2 ) + n 2 /(v* - c 2 ) = 0, 
 and 
 
 - a 2 ) 2 + w a /(v 8 - & 2 ) 2 + n 2 /(v 2 - c 2 ) 2 ] 
 a' 2 Z 2 /(?; 2 - a 2 ) 2 + b' 2 m 2 /(v 2 - 6 2 ) 2 + c V/(t? - c 2 ) 2 
 + 2a, 3 'mn/(v 2 - b 2 ) (w - c 2 ) + 2a sl 'nl/(v 2 - c 2 ) (w a - a 2 ) 
 8 - a 2 v 2 - 6 2 . 
 
 The first of these is Fresnel's equation, and shows that the 
 velocity of propagation in. any direction is not altered by weak 
 absorption. The second determines the index of absorption for 
 each of the waves travelling in a given direction. 
 
 Again from (II) p. 301 we have 
 
 X + i\' = A = 
 
 F 2 - 
 
 where Z 7 and l{ are small. 
 Hence 
 
 X : p, : v = l/(v 2 - a 2 ) : m/(v 2 - b z ) : n/(v* - c 2 ). 
 
 This determines the direction of the principal axis of the elliptic 
 orbit in a given wave, and shows that 
 
 (6 2 - c 2 )l/\ + (c 2 - a a )m//i + (a 2 - b*)n/v = 0. 
 Further from (III) p. 301 we deduce 
 v 2 (1 + 2i) = (a 2 + ia*) (X 2 + 2i\V) 
 
 + (6 2 + ib'*) (fji* + 2ifAfA) + (c 2 + ic' 2 ) (v 2 + 2iW), 
 
 whence v 2 = a 2 \ 2 + 6> 2 -I- cV 3 
 
 and 2fcv z = a /2 X 2 + 6 V + c/v + 2a ' 2XX/ + 2 &>/*' + 2cVz// - 
 
 The first of these equations gives the velocity of the wave in 
 terms of the direction of the principal axis of the elliptic orbit, and 
 the second gives similarly the index of absorption. We may obtain 
 a different form for 2/cv 2 by eliminating I, m, n from the equations 
 
 - a 2 ) : ml(v 2 - b 2 ) : n/(v* - c 2 ), 
 
XI] IN ABSORBING CRYSTALLINE MEDIA 307 
 
 and 
 
 2fcv 2 [l 2 /(v 2 - a 2 ) 2 + m 2 /(v 2 - b 2 ) 2 + n 2 /(v 2 - c 2 ) 2 ] 
 
 = a' 2 l 2 /(v 2 - a 2 ) 2 + b' 2 m 2 /(v 2 - 6 2 ) 2 + c' 2 n 2 /0 2 - c 2 ) 2 
 
 4- 2a*'mn/(tf- b 2 ) (v 2 - c 2 ) + 2a sl 'nZ/(^- c 2 ) (v-a) 
 - a 2 ) (v 2 - 6 2 ). 
 
 This gives 
 
 2x;*; 2 = a /2 X 2 + 6 V + c'V + 2a SB >i> + 2a 31 'z/X + 2a M 'X/*. 
 
 Adding to these equations the equations l\ + TW/A + nv and 
 l\' + m/i/ + wz/ = 0, which were obtained on p. 301, we have a 
 series which give us the complete solution of the problem of 
 propagation in weakly absorbing crystals. A comparison with the 
 equations of Chapter vi. on propagation in transparent crystals 
 will show that the equations not involving K are identical in the 
 two cases, the only difference being that (X, //,, v) for absorbing 
 media no longer represents the direction of the curl of the dis- 
 placement, but that of the major axis of the elliptic orbit. It 
 follows, of course, that all the theorems previously derived from 
 these equations apply equally to the problem at present in hand. 
 
 There is one case in which the equation 
 
 - a 2 ) 2 + ?/i 2 /0 2 - 6 2 ) 2 + w 2 /0 2 - c 2 ) 2 ] 
 = a' 2 Z 2 /(v 2 - a 2 ) 2 4- 6 /2 m 2 /(v 2 - 6 2 ) 2 + c /2 n 2 /(i; 2 - c 2 ) 2 
 
 - 6 2 ) (v 2 - c 2 )+ 2a 3l 'nl/(v*- c 2 ) (v 2 - a 2 ) 
 
 cannot be employed to calculate the index of absorption. This 
 occurs when the wave normal is along an optic axis, when v 2 = 6 2 , 
 and V z B 2 becomes a small quantity of the first order, viz. 
 
 Equation (IV) p. 301, written in the form 
 
 L* ( F 2 - &)( F 2 - C 2 ) + M, z ( F 2 - C 2 ) ( F 2 - A 2 ) 
 
 then yields i (2*&* - 6 /2 ) [I 2 (6 2 - c 2 ) - n 2 (a 2 - 6 2 )] = ; but as the 
 factor I 2 (b 2 c 2 ) - n 2 (a? b 2 ) is zero, this equation cannot be 
 employed to determine K. It thus appears that the absorption 
 in the direction of an optic axis is indeterminate, unless the 
 
 202 
 
308 PROPAGATION OF LIGHT [CH. 
 
 direction of the polarisation be known. When this is given, K 
 may be calculated from the formula 
 
 ' 2 
 
 + c 4- a^ JJLV a 3l v + a 12 yu. 
 
 If natural light be incident normally on a crystal cut at right 
 angles to an optic axis, we may replace it by two beams polarised 
 at right angles. As the absorption for these two beams will be 
 different, one wave may be damped while the other passes through 
 the crystal, so that the emergent light will be polarised. The 
 formula just quoted indicates in what way the absorption depends 
 on the direction of the elliptic orbit (X, //,, v). By varying the 
 state of the polarisation we vary the absorption, so that if natural 
 light be transmitted through a crystal and subsequently analysed, 
 the emergent beam will show a colour depending on the plane of 
 analysation. This is a well known fact of experience. 
 
 For purposes of comparison with experiment it is convenient 
 to express K and v in terms of the angles that the wave normal 
 makes with the optic axes. The formula for v has already been 
 obtained (p: 142), viz. v* = c 2 + (a 2 - c 2 ) sin 2 (0 + 0')/2. To find 
 the corresponding formula for K we have to express (X, JJL, v) in 
 terms of 6 and 6'. The direction cosines of the optic axes are 
 Z , ra , n , where 
 
 Z = \/(a 2 - 6 2 )/(a 2 - c 2 ) = cos to/2 ; m =0; 
 n,= V(6 2 - c 2 )/(a 2 - c 2 ) = sin <o/2. 
 
 Using the notation and figure of p. 141, OP now representing the 
 direction of the major axis of the elliptic orbit, we have 
 
 1 \ + n v = cos ^ = sin 6 cos ^ ; 
 X n v = cos i/r/ = sin 6' cos ^ ; 
 X' + n i/ = cos i/r 2 = sin 6 sin ^ ; 
 
 1 Q \' n v' = cos -v/r./ = sin & sin ^. 
 Hence 
 
 X = cos x (sin 6 + sin 6')/2l ; v = cos % (sin 6 - sin <9')/2rc ; 
 
 ft = - l- X 2 -; 
 and X' = sin ^ (sin - sin tJ')/2J ; 
 
 v' = sin x (sin + sin / )/2/? ; /*' = Vl - X /2 
 
Xl] IN ABSORBING CRYSTALLINE MEDIA 309 
 
 while ^ is given by cos co = cos cos 0' 4- sin sin 6' cos 2%, or 
 
 _ sm 2 (04-0')/2-sin 2 ft)/2 _ sin 2 (0 + 0')/2 - (ft 2 - c 2 )/(q 2 - c 2 ) 
 S " % ~ sin sin 0' sin sin 0' 
 
 Thus all the quantities entering into the formula for K are expressed 
 in terms of and '. 
 
 There is a considerable simplification when the displacement 
 and absorption axes coincide, in which case a^ = a 31 ' = a 12 ' = 0. We 
 then have 
 
 2/; 2 = a' 2 A 2 + &V 2 + c V = 6' 2 + X 2 (a' 2 - 6' 2 ) - v* (b' 2 - c /2 ). 
 Hence 
 
 = 6 , 2 (a'-^Bin^+jn/g-^-c-) 
 sin sin 
 
 ""^'2 7y*> /J i ZJ' Z) Zl ' 7 /*> ^9 Z) i /I ' ^J ^' 
 
 B " t/"T"(/ f/ u ~ C C7"t~i/ . (/ C/ 
 
 a^Tp Sm2 ~~2~~ C s2 ~2 b*-c* C S2 ~T~ Sm * ^~ 
 
 and 
 
 2/tot; 9 2 = 6 /2 
 
 sm sin 
 
 . 0-6> / 9 + (9 x 6' 2 -c /2 
 
 sm - cos ' cos sm 
 
 -2- 
 
 As a special example of the use of the formula for /c, we shall 
 find the index of absorption when light is incident nearly at right 
 angles to the face of a biaxal crystal cut perpendicularly to one of 
 the optic axes. Let ON be any wave normal in the crystal near 
 the axis OA lf A l N=0 = r (say) where r is small, A Z N=0', 
 
 Fig. 132. 
 
 1 2 = ^. If we neglect powers of r above the first we have 
 2^ + ^ = 180; so that cos% = sm^lr/2 and 8^1^ = 008^/2. On 
 putting 1 cos ft) cos \fr = 2p and 1 + cos ft) cos ^r = 2^, we have 
 these relations : 
 
 sin 6' = sin o> r cos ft) cos -v/r ; 
 
 sin + sin 0' = 2 (sin &)/2 . cos <w/2 + pr) ; 
 
 sin sin 0' = 2 ( sin to/2 . cos to/2 + qr). 
 
310 PROPAGATION OF LIGHT [CH. 
 
 Hence 
 
 X = (sin a)/2 4- pr sec to/ 2) sin 
 
 X 7 = ( sin co / 2 4 qr sec to/ 2) cos yfr/2 ; 
 
 yn = cos ^/2 4 r cos -\Jr/2 . tan 2 i/r/2 . (p tan co/2 q cot to/ 2) ; 
 
 /A' = - sin ^r/2 -f ?" sin i|r/2 . cot 2 ^/2 . (p cot o>/2 q tan o>/2) ; 
 
 v = ( cos o>/2 4- qr cosec eo/2) sin ^r/2 ; 
 
 i/ x = (cos co/2 4- pr cosec co/2) cos -\Jr/2. 
 
 Hence the formula for 2/cv 2 gives 
 
 2* 1 1 2 = sin 2 T/r/2 . (a' 2 sin 2 w/2 + c' 2 cos 2 o>/2) 4- &' 2 cos 2 i|r/2 
 4- 2023' cos to/2 . sin ^r/2 . cos i/r/2 
 
 - 2a 31 ' sin to/2 . cos co/2 . sin 2 i/r/2 - 2a 12 ' sin to/2 . sin ^r/2 . cos ^r/ 
 + 2r [p (a' 2 +6 /2 ) tanw/2 . sin 2 ^/2 
 
 a^ {g cosec w/2 . sin -v/r/2 . cos 
 
 + cos to/2 . cos i/r/2 . tan 2 ^/2 . (p tan w/2 - g cot to/ 2)} 
 
 + a 3 / cos &) cos >/r sin 2 ^r/2 
 
 4- a 15{ ' {sin w/2 . sin ^r/2 . cos \|r/2 . tan 2 ^r/2 . ( ^ tan o>/2 - g cot to/2 ) 
 
 p sec to/2 . sin i/r/2 . cos ^/2}] 
 = A + -Bif (say), 
 
 and 
 
 2A: 2 v 2 2 = cos 2 \/r/2 . (a' 2 sin 2 to/2 4- c /2 cos 2 o>/2) 
 
 + 6 /2 sin 2 yjr/2 - 2a 23 / cos to/2 . sin i/r/2 . cos ^/2 
 
 2a 31 ' sin to/2 . cos <o/2 . cos 2 i/r/2 4- 2a 12 ' sin co/2 . sin ^r/2 . cos >/r/2 
 
 - 2r [g (a /2 - 6 /2 )tan to/2 . cos 2 ^/2 + j9(6 /2 - c /2 )cot co/2 . cos 2 i/r/2 
 + ^ {p cosec co/2 . sin ^r/2 . cos -^r/2 
 
 4- cos co/2 . sin ^/2 . cot 2 i/r/2 . (g tan o>/2 -_p cot w/2)} 
 
 a 31 ' cos co cos ^r cos 2 ^/2 
 
 + a 12 ' {sin to/2 . sin -\/r/2 . cos-v/r/2 . cot 2< ^/2 . (p cot a>/2 ^ tan to/ 2) 
 + ^ sec to/2 . sin -^/2 . cos 
 
 These determine K and /c 2 , since, as was seen in Chapter vin., 
 we may put v l =v 2 = b ) to our order of approximation. 
 
 The formulae just obtained cannot be employed when the wave 
 normal coincides with an optic axis, for then ty becomes indetermi- 
 
XI] IN ABSORBING CRYSTALLINE MEDIA 311 
 
 nate. In this case, if the major axis of the elliptic orbit be parallel 
 to the plane of the optic axes we have 
 
 X = sin o>/2, fi ' = 0, v cos o>/2, 
 so that 
 
 2/c/6 2 = 2/c^ 2 = a' 2 sin 2 to/2 + c' 2 cos 2 to/2 - 2a 31 ' sin to/2 cos to/2. 
 
 If, however, the major axis be perpendicular to the plane of the 
 optic axes, we have \ = v 0, and 
 
 Since B and r are small quantities we may, as an approximation, 
 neglect their product, and we then obtain ^ and K 2 in terms of KI 
 and KZ thus : 
 
 KI = KI ' sin 2 ijr/2 + K 2 ' cos 2 i/r/2 + >' sin -*|r, 
 and /e 2 = KI cos 2 ^/2 + * 2 ' sin 2 ^/2 // sin -v/r, 
 
 where 26 2 p' = a^' cos o>/2 a 12 ' sin a>/2. 
 
 We must next inquire into the modifications of the formulae 
 when the crystal is uniaxal. The results cannot be derived by 
 simply putting b = c in the formulae already obtained, for when 
 this condition is satisfied some of the steps in the process adopted 
 require alteration. The conicoid (A n A i2 A^A^A^ l A l ^xyz}' i = 1 
 is now very nearly an ellipsoid of revolution about the axis of x, 
 so that although one principal axis must lie near the axis of x, 
 the others need not be near the axes of y and z. 
 
 As, however, the first axis lies near the axis of #, and the other 
 two are nearly at right angles thereto, we see that 1 2 , 1 3 , m 1 and ^ 
 are small quantities of the first order. Putting 
 
 we find the directions of the axes given by the following 
 equations : 
 
 I, = 1 ; 1 2 = ia l2 '/(a* - c 2 ) ; Z, = ta n 7(a - c 2 ) ; 
 m l = i (a 12 ' cos 7 + a sl ' sin y)/(a? c 2 ) ; ra 2 = cos 7 ; m 3 = sin 7 ; 
 
 i = i (a 12 ' si n 7 3/ cos 7)/(a 2 c 2 ) ; n 2 = sin 7 ; n 3 = cos 7. 
 Whence 
 
 /(a 2 - c 2 )] 
 
 : [m cos 7 + n sin 7 il (a 12 ' cos 7 4- a 3 / sin 7)/(a 2 c 2 )] 
 : [ m sin 7 + n cos 7 4- *7 (a 12 ' sin 7 a 31 ' cos 7)/(a 2 c 2 )]. 
 
312 PROPAGATION OF LIGHT [CH. 
 
 Substituting in (IV), p. 301, the real part of the equation 
 
 gives us 
 
 p/(& - a 2 ) + O 2 4- n 2 )/(v 2 - c 2 ) = 0, 
 
 which is Fresnel's law as before. From this we get 
 
 v* = Pc 2 + a 2 (1 - I 2 ), and v 2 2 = c 2 , 
 
 the first referring to the extraordinary and the second to the 
 ordinary wave. 
 
 Taking the ordinary wave first, we see that V 2 B 2 and V 2 C 2 
 are small quantities of the first order, so that equation (IV), which 
 is equivalent to 
 
 L*(V* - B 2 )(V 2 - C 2 ) + M? (F 2 - O 2 ) (F 2 - A 2 ) 
 
 reduces, to our order of approximation, to 
 
 M, 2 (V* - C 2 ) + N 2 (V 2 - B 2 ) = 0. 
 This gives 
 
 (ra cos 7 -f- n sin y) 2 (2/cc 2 -c' 2 ) + ( msmy + n cos y) 2 (Ztcc 2 - b' 2 ) = 0, 
 whence 
 
 2fc 2 c 2 (m 2 + n 2 ) = b' 2 ( m sin 7 + n cos y) 2 + c' 2 (m cos 7 + 71 sin 7) 2 , 
 so that on putting 
 
 I = cos 0, m = sin cos <, and n = sin sin <, 
 we get 2/e 2 c 2 = b /2 sin 2 (^ - 7) + c' 2 cos 2 (</> - 7). 
 
 For the extraordinary wave we get, similarly, 
 
 9 tf f ^ 2 , m2 + ^ 2 1 = 2l(m>a>v + was/) 
 L(tf - a 2 ) 2 (v 2 - c 2 ) 2 J (v 2 - a 2 ) (v 2 - c 2 ) 
 
 2 a /2 6 /2 (m cos 7 -f ?i sin 7) 2 -f- c /2 ( m sin 7 + r& cos 7) 2 
 f (t^^y + (v 2 -c 2 ) 2 
 
 whence, putting in the value of v 2 , we find 
 
 2* x [7 2 c 2 + a 2 (1 - Z 2 )] = - 21 (ma 12 ' + na 31 ') + a /2 (1 - I 2 ) 
 + \b' 2 (m cosy + n sin 7)* + c 2 (-msmy + n cos 7) 2 } Z 2 /(l - I 2 ), 
 
 i.e. 2! [c 2 cos 2 + a 2 sin 2 0] = - sin 20 (a 12 ' cos $ + a 3 / sin 0) 
 
 + a /2 sin 2 + cos 2 [b' 2 cos 2 (< - 7 ) + c /2 sin 2 (</> - 7)]. 
 
 In the special case where there is complete symmetry about 
 the optic axis we have b' = c', a 12 ' = a 3 / = 0, and thus get 
 
 2* 2 c 2 = c /2 , and 2/d [c 2 cos 2 <9 + a 2 sin 2 (9] = c' 2 cos 2 + a /2 sin 2 0, 
 
XI] 
 
 IN ABSORBING CRYSTALLINE MEDIA 
 
 313 
 
 where 6 is the angle that the wave normal makes with the optic 
 axis. If K O be the value of * T when 6 = 0, we have 
 
 * t _ 1 + (a'/cj tan 2 d 
 
 K O ~ 1 + (a/c) 2 tan 2 6 ' 
 
 On putting in the values of a/c and a'/c' found experimentally for 
 tourmaline, we are enabled to compare the results of theory and 
 experiment. This is set out in the following table, and shown 
 graphically in Fig. 133. 
 
 e... 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 *!/* 
 
 (theory) 
 
 1 
 
 990 
 
 960 
 
 914 
 
 857 
 
 798 
 
 742 
 
 697 
 
 667 
 
 657 
 
 KI/KO (exp.) 
 
 998 
 
 987 
 
 943 
 
 898 
 
 834 
 
 814 
 
 734 
 
 719 
 
 661 
 
 657 
 
 Difference 
 
 + 002 
 
 + 003 
 
 + 017 
 
 + 016 
 
 + 023 
 
 -016 
 
 + 008 
 
 -012 
 
 + 006 
 
 
 
 10 
 
 20? 
 
 3CT 
 
 40 5CT 
 
 Fig. 133. 
 
 60 
 
 7<f 
 
 We are now in a position to see in what way the index of 
 absorption VKJV varies with the direction of propagation of the 
 wave (I, m, n). From Fresnel's equation it appears that there are, 
 
314 
 
 PROPAGATION OF LIGHT 
 
 [CH. 
 
 in general, two velocities for a given direction, and from the 
 equations for K we see that K has a different value for the two 
 velocities. Hence there are two values of v for each direction. If 
 then through any point we draw radii vectores whose directions 
 represent the directions of propagation of the wave, and whose 
 lengths are equal to v, the surface so formed will consist of two 
 sheets. It may be called the Absorptive Index Surface, on the 
 analogy of the Index Surface of Chapter v. Its equation in polar 
 coordinates (r, 0, <) is obtained by putting I = cos 6, m = sin 6 cos $, 
 n = sin sin 0, and eliminating K and v from the equations K/V = r, 
 and those already obtained involving v z and 2/cv 2 . In the case of 
 uniaxal crystals we thus get 
 
 2rc s = b' 2 sin 2 (< - 7) + c' 2 cos 2 (< - 7) 
 for one sheet, and 
 
 2r [c 2 cos 2 + a 2 sin 2 0] 3 / 2 = - sin 26 (a I2 ' cos < + a 31 ' sin c/>) 
 
 + a' 2 sin 2 6 + cos 2 6 [b' 2 cos 2 ((/> - 7) + c' 2 sin 2 (</> - 7)] 
 
 for the other sheet. In the special case of complete symmetry 
 about the optic axis, these reduce to 2rc 3 = c 2 and 
 
 2r [c 2 cos 2 + a 2 sin 2 0] 3 / 2 = c' 2 cos 2 + a' 2 sin 2 6 
 
 respectively. Fig. 134 represents a quadrant of the section of this 
 surface by a plane through the optic axis, drawn to scale when 
 c = 0*611, a= 0'615, and (a'/ c/ ) 2 = 0*665, the constants being those 
 found experimentally for tourmaline. 
 
 Fig. 134. 
 
XlJ IN ABSORBING CRYSTALLINE MEDIA 315 
 
 In general the two sheets of the Absorptive Index Surface will 
 intersect, and the lines joining the origin to the points of inter- 
 section will form a cone, which may be called the cone of equal 
 absorption, from the fact that the absorption will be the same for 
 all waves whose normals are generators of this cone. We have 
 seen that for a wave whose normal coincides with the optic axis 
 the absorptive index cannot be derived from the equation 
 2/cv 2 [l 2 /(v 2 - a 2 ) 2 + m 2 /0 2 - b 2 ) 2 + n 2 /(v 2 - c 2 ) 2 ] 
 
 = a' 2 l 2 /(v 2 - a 2 ) 2 + V*m 2 /(v 2 - 6 2 ) 2 + c' 2 n 2 /(v 2 - c 2 ) 2 
 
 + 2a 23 / mn/(v 2 - b 2 ) (v 2 - c 2 ) + 2a 31 ' nl/(v* - c 2 ) (v- - a 2 ) 
 
 + 2a 12 ' lm/(v z - a 2 ) (v 2 - b 2 ), 
 
 but that when the direction of the displacement is known its 
 various values are determined by the equation 
 
 2/ti; 2 = a /2 X 2 + 6'> 2 + cV -f 2^23 V v + 2a 31 'i/X + 2a 12 'X/*. 
 From this it appears that the optic axes are not in general 
 generators of the cone of equal absorption. The equation of this 
 cone is obtained by giving equal roots to the equation of the 
 absorptive index surface, regarded as an equation in r, and by 
 putting I : m : n = x : y : z. In this way we obtain for a uniaxal 
 crystal the equation 
 
 [6 /2 (- y sin 7 + z cos 7)* + c 2 (y cos 7 -f z sin 7 ) 2 ] [c 2 x 2 + a z (y 2 + 2 2 )] 3/2 
 = c 3 (x 2 + f + 2 ) 1/2 [(y 2 + z z } {a 2 (y 2 - + z 2 ) - 2a^xy - Za^xz] 
 
 + [b' 2 (xy cos 7 4- ocz sin j) 2 4- c' 2 ( xy sin 7 4- xz cos 7) 2 )]. 
 In the case of complete symmetry the cone becomes a circular 
 one with its axis along the optic axis, and its semi-vertical angle 
 given by the equation 
 
 2q s sin 2 d =p 2 - 3q (p - q) 3 ' 2 (p + 3?) 1 / 2 , 
 where p - (a 2 - c' 2 )/a' 2 , and q = (a 2 - c 2 )/a 2 , 
 
 and, of course, unless be real the cone of equal absorption does 
 not exist. 
 
 The phenomena analogous to those discussed in Chapter vil. 
 when dealing with absorbing crystals may be investigated by the aid 
 of the formulae obtained in this chapter. We have to remember 
 that light incident on an absorbing plate gives rise to two waves 
 within the crystal, and that these waves have different velocities 
 and are differently absorbed. The difference in the velocities 
 
316 PROPAGATION OF LIGHT [CH. 
 
 leads to a difference of phase in the two waves on emergence, and 
 a,s the velocities are given, to our order of approximation, by the 
 same formulae as when the crystal is transparent, the change of 
 phase is the same as in the cases discussed in Chapter vn. The 
 absorption produces a change of amplitude, which can be calcu- 
 lated for each wave from the formulae of this chapter. On passing 
 through the crystalline plate the amplitude of a wave is diminished 
 in the ratio of e~* to unity, where <r = plv, p/ir being the frequency, 
 v the index of absorption, and I the length of the wave normal 
 within the crystal. Hence an incident wave represented by Ae ipt 
 will, on emergence, be represented by Ae~ cr e i(pt ~^\ where A is the 
 change of phase. Since <r = plv = 2?rZ:/X, where X is the wave 
 length, we see that a will be large and consequently e~ <T very 
 small, when the thickness of the plate is large, in all cases in which 
 K is appreciable. If, however, for any direction K be very small, 
 ~" may be appreciable even when the plate is not very thin. 
 
 Let us suppose, as in Chapter vn., that we have a polariser and 
 analyser whose principal planes OP and OA make angles cc and ft 
 respectively with Ox, the plane of polarisation of the quicker wave 
 within the crystalline plate. For brevity let c l} s lt C 2 , s 2 denote 
 cos a, sin a, cos /3, and sin respectively. On resolving an 
 incident vibration Ae ipt along Ox and Oy respectively, Oy being at 
 right angles to Ox, and omitting the factor e ipt , we get Ac^ and As lf 
 After passing through the plate these become Ac^e'^e^^ and 
 As^e~ ia ^e~ 1 ^ respectively, where o^ and o- 2 , A x and A 2 refer to the 
 two waves, the ellipticity of the polarisation within the crystal 
 being neglected, as this is very slight in cases to which the 
 formulae will be applied. On resolving along OA. we get 
 
 A [dWie-*^ + SiS. 2 e~' r *e i **\, 
 so that the intensity / is given by 
 
 / = A 2 [d 2 c 2 2 e~ 2<ri + sfsje-*** + 2s 1 s 2 c 1 c 2 e- (<r > + ^ cos A], 
 
 where A = A a - A 2 is the difference of phase between the two 
 waves on emergence. If the light used be not homogeneous we 
 must add the intensities due to each wave length, as was done in 
 Chapter vu. 
 
 Some interesting results are obtained with absorbing crystals 
 by using unpolarised light. In this way we may replace the 
 
XI] IN ABSORBING CRYSTALLINE MEDIA 317 
 
 incident light by two equal beams polarised at right angles, and 
 then the above equation for the intensity gives 
 
 If, moreover, there be no analyser, we get 
 
 We shall consider first the case of a biaxal crystal cut at right 
 angles to an optic axis, for which the formulae giving /^ and tc 2 
 have been explained on p. 310. The planes of polarisation of 
 the two waves within the crystal bisect the interior and exterior 
 angles A^NA* (Fig. 132), so that with the notation of p. 309 we 
 may take i/r/2 as the angle between the plane of polarisation of 
 one of the waves and the plane of the optic axes. If then the 
 polariser and the analyser be crossed the equation for / becomes 
 
 47 = A 2 sin 2 (2^ - i/r) [e~^ + e' 2 ** - 2e~^ + ^ cos A], 
 
 where x is the angle that the principal plane of the polariser makes 
 with the plane of the optic axes. Along the optic axis itself A/T 
 becomes indeterminate, and we must then proceed as on p. 311 by 
 resolving the incident light into two streams polarised at right 
 angles. These two streams traverse the crystal at the same speed, 
 so that there is no difference of phase between them, and the 
 intensity is /', where 
 
 4/' = sm 2 2a- <ri ' - e-** and <r' = 
 
 Ki and KZ having the values assigned to them on p. 311. Since 
 cr/ and cr./ are different, we see that /' does not vanish unless 
 sin 2! = 0, i.e. unless a x = or Tr/2. For any other azimuth of the 
 polariser, the extremity of the optic axis is not black. 
 
 In the more general case the equation for / shows that the 
 intensity vanishes when ^ = 2a 1} so that there is a black line 
 ^ = 2ai across the field of view, as in the case of transparent 
 crystals, with the difference just noted that this black line is 
 interrupted by a comparatively bright spot at the optic axis, 
 except in the special cases when a l = or ir/2. 
 
 I also vanishes with the factor e~ 2<Ti + e~ 2<rn ~ 2e~^ +0 " 2) cos A. 
 This cannot be zero unless a^ = o- 2 and A = 2?i?r. The condition 
 - 1 = <7 2 involves KI K%, which from the formulae on p. 311 leads 
 to tan ty (/ tc 2 '')/p', i.e. to -fy = ?r/2 in the special case when 
 
318 PROPAGATION OF LIGHT [CH. 
 
 the displacement and absorption axes coincide. In this azimuth 
 there will be black spots at distances from the optic axis given by 
 
 A = 2n7r, i.e. r = 2n\b*/eC = 2n\b*/e (a 2 - c 2 ) sin &>, 
 
 where e is the thickness of the crystalline plate (see p. 209). 
 
 To investigate the fluctuations of the intensity as r varies in 
 any given azimuth we have to consider the variation of the factor 
 
 e~ 2 " 2 2e~^ i+(T ^ cos A. 
 
 All the quantities cr^ <r 2 , and A are functions of r, ^ and o- 2 being 
 given by the formulae on p. 311, and A by the formula 
 
 A = (7re/X6 3 ) (a 2 - c 2 ) r sin &> = (ire/Xb' 2 ) Cr. 
 
 The intensity is stationary when dl/dr is zero, i.e. when dF/dr is 
 zero. This gives 
 
 B&-** + B 2 e- 2 "* - (B, + B 2 ) er^+'j cos A - #*-<* +*> sin A = 0, 
 
 i e B -*(*! sin * W+** cos 2 tfr/2 + ^ sin $) _j_ $ 2 g-2 cos 2 tfr/2 + <r a ' sin 2 ^/2 - q' sin ^) 
 
 + e -(<r/+<r 2 ') [(^ + 2 ) cos A + sin A] = 0, 
 
 where q = p . 27re/X. 
 
 This equation, combined with the formulae for B l} B 2 , and C 
 already obtained, gives r as a function of -v/r and so determines the 
 form of the rings. As either e~ Vt ' or er** will be small if there be 
 any appreciable absorption, the factor e~ (*i+< T *\ which is associated 
 with the terms sin A and cos A, will be small. Thus if there be 
 much absorption the terms involving A and therefore r in the 
 formula for dl/dr will be negligible, and the rings will consequently 
 be very indistinct. 
 
 The rings are not equally bright or dark throughout their 
 length, as the intensity varies with the azimuth. For a given value 
 of r the intensity is stationary when dl/dty vanishes, i.e. when 
 
 sin (2! - ^ ) [sin (2^ - ^)dF/d^ - ZFcos (2^ - ^)] = 0. 
 
 The factor sin (2^ i/r) equated to zero gives ^ = 2^, the 
 black line already referred to. Also we have 
 
 '+V) COS A. 
 
 Moreover 
 
 = (7re/X6 3 ) (a 2 - c 2 ) r 2 cos co sin -v/r, 
 
Xl] IN ABSORBING CRYSTALLINE MEDIA 319 
 
 and this is small owing to the factor r 2 . Hence as e-fo'+O is very 
 small we may neglect the variation of the factor g-fo'+O cos A 
 in finding dF/dty. Thus dl/d-ty vanishes when 
 
 sin (2! - i/r) [(0V - o- 2 ) sn ^ + 2# cos 
 
 + 2 cos (2! - ^) [e- 2 " + e- 2 " - 2er^+^ cos A] = 0. 
 
 This equation is very nearly satisfied when ^ = cr 2 = (o-/ 4- <r 2 ')/2, 
 for then the first term vanishes and the second term becomes 
 gg-(<r,'+or s ') cos (2^ ^)sin 2 A/2, which is very small owing to the 
 presence of the factor e~ (<ri ' +0 A We have seen (p. 317) that 
 tan \|r = (AC/ - * 2 ')/p' when o-, = <r 2 , and that consequently ty = Tr/2 
 in the special case of symmetry when p f = 0. In this special case 
 we see also that another approximate solution of the equation 
 
 sn ! - - o-. sn + ^ cos r e~ <r ' - e~ <r 
 
 + 2 cos (2! - ^) [e~ 2 ^ + e~ 2 ^ - 2e-^i +0 ^) cos A] = 
 
 is i/r = or TT, for when ty has either of these values the first 
 term of the equation vanishes and the second reduces to 
 
 2 cos 2! [e- 2 ^ 1 + e- 8 '"' - 2e-^>'+*'> cos A], 
 
 which is small unless the plate is very thin or the absorption 
 negligible. It has been seen that the azimuths ^r= ?r/2 make 
 /a minimum, so that as maxima and minima must occur alternately 
 we should expect, and can easily verify, that the azimuths ty = 
 and TT correspond to maximum values of the intensity. Thus 
 there is a bright line in the plane of the optic axes and a dark 
 one at right angles thereto, in addition to the black line in the 
 azimuth ijr = 2^. 
 
 If the polariser and analyser be parallel, instead of being 
 crossed, the formula for / gives 
 
 / = A 2 [e~ 2(ri cos 4 (j - ^/2) + e~ 2(r * sin 4 (a, - ^/2) 
 
 + (1 /2) e -^+') cos A sin 2 (2a x - i/r)] 
 
 except in the direction of the optic axis itself when the intensity is 
 
 /' = A* [cos 2 ! e-** + sin 2 a x e~^}\ 
 The discussion of these formulae may be conducted on exactly the 
 
320 PROPAGATION OF LIGHT [CH. 
 
 same lines as before. If the plate be of sufficient thickness we 
 may neglect the term e~ (<Ti/+0 " 2/) , and we then have 
 
 = sn ft, - e-" cos a x - - e~ sn a, - 
 
 = sn ! - <r <r ' cos a a - ar - e~^ sn ^ - 
 
 - [g-2<r, _ e -2<7,] j^' _ ^ g j n ^ _j_ gg/ COS . 
 
 If ^ = o- 2 = (cr/ + cr 2 ')/2, the second term vanishes, and the first is 
 very small owing to the factor e-fo'+O. Also if q' = and ty = 
 or TT, the second term again vanishes and the first, which is always 
 small, vanishes completely when ] = 0. In the case where q' = 0, 
 we have c^ = cr, when ty = + Tr/2 and then, in the immediate 
 neighbourhood of the optic axis where A = 0, we have 
 
 / = A* [er*i cos 2 (a, - -f /2) + e~** sin 2 ( ai - ^/2)] 2 = A 2 erW +<r *\ 
 
 Moreover, if i = 0, the formula for /' gives I' A 2 e~ 2 " 2 ', so that 
 I IT = e~^'~ a ' 2 '\ and / is greater or less than I' according as e~" l< or 
 e~^' is negligible, i.e. according to the type of the crystal employed. 
 If Gtj = 7T/2, the relations between / and /' are reversed. 
 
 If unpolarised light be used, the intensity is given by the 
 formula 
 
 21 = A* [cos 2 ( 2 - i/r/2) e-^ + sin 2 ( 2 - ^/2) er^l 
 where a 2 is the angle that the principal plane of the analyser 
 makes with the plane of the optic axes ; whereas along an optic 
 axis itself we have 
 
 2/' = A* [cos 2 2 . e~^ -f- sin 2 2 . e~ 2 ^']. 
 When or 2 = 0, the formulae give 
 
 27 = A* [cos 2 yjf/2 . e~ 2 ^ + sin 2 A/r/2 . e~^\ 
 
 and 2/' = A 2 e~ 2<r *. 
 
 Hence 
 
 1 r)T 
 
 ~ = [cos 2 ^r/2 . e~^ - sin 2 ^r/2 . e~ 2 "*] 
 
 [(cr/ o-/) sin ^|r + 2^' cos ^] i sin ty (e~ 2<Ti - e~ 2(r *). 
 
 Thus when q' = 0, dl/difr vanishes when ^ = or TT, and is very 
 small when c^ = er 2 , i.e. when i/r = + Tr/2. In the former case 
 
XI] IN ABSORBING CRYSTALLINE MEDIA 321 
 
 (>jr = or TT) we have / = I', and in the latter case (-\Jr = + 7r/2) 
 we have I/I' = e'^ 1 ''"^. Hence with crystals belonging to the 
 type for which cr/ > cr 2 ' there is a dark line across the field of view 
 perpendicular to the plane of the optic axes, this line being inter- 
 rupted by a bright spot at the centre ; while with crystals of the 
 other type for which cr/ < cr./ the dark line lies in the plane of the 
 optic axes and is continuous. 
 When 2 = 7T/2 we have 
 
 27 = A 2 [sin 2 ^/2 . e~ 2 ^ 4- cos 2 ^/2 .e'**], 
 
 and 21' A 2 e~ 2<Ti '. In this case, in the azimuths ^r = or TT we 
 have / = 7', and in the azimuths ^ = + ?r/2 we have 
 
 1 1 1' = e-W-n'l 
 
 Hence, if cr/ > cr 2 ', a continuous dark line crosses the field in the 
 plane of .the optic axes, while with crystals of the other type the 
 dark line is perpendicular to that plane and is interrupted by a 
 bright spot at the centre. 
 
 Lastly, if there be no analyser we have 27 = A 2 [e~ 2 * 1 + e~ 2 " 2 ], 
 and 27' = A* [e~ 2 ^' + e~ 2 ^]. Hence 
 
 87/3f = A 2 [e~ 2 "* - e~ 2 ^] [(cr/ - o- 2 ') sin ^ + 2q cos ^r], 
 so that 7 is stationary when o^ = <r 2 , and also when 
 tan ^ = 2 ? 7(cr 2 ' - a/) = 2p7(* 2 / - */). 
 
 In the case of symmetry when p 0, the former corresponds to 
 -vjr = + 7T/2 and the latter to -v/r = or TT, so that we have a dark 
 line perpendicular to the plane of the optic axes with a bright spot 
 at the centre. 
 
 As the final application of these formulae we shall take the 
 case of a uniaxal plate cut at right angles to the optic axis. The 
 displacements for the two waves are in the plane of incidence and 
 at right angles thereto, the former corresponding to the ordinary 
 and the latter to the extraordinary wave. If the polariser and 
 analyser be crossed the formula for the intensity gives 
 
 47 = sin 2 2 e~ 2 ^ + e~ 2(r * - 2 cos A . e -K+^), 
 
 where </> is the angle that the principal plane of the polariser makes 
 with the plane of polarisation of the quicker wave. Along the 
 optic axis, we have c^ = o- 2 and A = 0, so that I' = 0. The intensity 
 vanishes when sin 2<f> = 0, i.e. when </> = or ir/2, so that there is a 
 M. L. 21 
 
322 PROPAGATION OF LIGHT [CH, 
 
 black cross in the field of view with its arms parallel to the 
 principal planes of the polariser and analyser. 
 We have 
 
 2 dr dr 
 
 cos A FP + ^- 2 +sinA^ L 
 ( \dr dr) dr}] 
 
 On putting d = I in the formula of p. 209, we have 
 
 A = (7re/Ac)(a 2 -c 2 )r 2 . 
 
 Also from the formulae of p. 312 we get o- 2 = (ire/X) (c'/c) 2 , and 
 ai = (7re/X) (c' 2 cos 2 6 + a' 2 sin 2 0)/(c 2 cos 2 4- a 2 sin 2 0) 
 = (jre/X) [c /2 + (a /2 - c /2 ) r 2 ]/c 2 
 = (7 2 [l+r 2 (a /2 -c' 2 )/c /2 ]. 
 Hence 
 
 [c (a 2 - c 2 ) sin A - (o /a - c /2 ) cos A - e~ r2<r * <**- c/2 )/ c ' 2 ] . 
 
 Owing to the smallness of the factor a' 2 c' 2 , this yields very 
 approximately 
 
 \c - = - 7T6^1 2 (a 2 - c 2 ) re- (-' +(r 2) sin 2 2(/> sin A, 
 
 so that I is a maximum or minimum when A = mr. There will 
 thus be a series of bright and dark concentric circles in the field of 
 view, their radii being given by the same law as that which holds 
 for transparent crystals. Owing, however, to the presence of the 
 factor e~ (ori+ " 2) in the expression for 9 1/dr, the variation of the intensity 
 will be very slight if the absorption be at all large, so that the rings 
 will be very indistinct. When the absorption or the thickness of 
 the plate is considerable, we have / very nearly proportional to 
 (g-an + e -2r 2 ) S i n 2 2 ^ i. e . to e~** [1 + e -^(' 2 -^)/ c ' 2 ] sin 2 2</>. The 
 term e -r*<r*(a"z-c'*)/c'* w jn b e ver y sma ll for crystals belonging to 
 the type in which a is considerable and c' very small, while it will 
 be large for crystals of the other type in which of is very small and 
 c considerable. Hence with crystals of the first class the field 
 away from the black cross will be dark, and with those of the other 
 class it will be bright. 
 
Xl] IN ABSORBING CRYSTALLINE MEDIA 323 
 
 If the polariser and analyser be parallel, the formula for the 
 intensity gives 
 
 I = A 2 [cos 4 $ . e~ 2 ^ + sin 4 <j> . e~ 2 ^ + (1/2) sin 2 2< cos A . e- 
 and 
 
 Thus / is a maximum or a minimum when <f> = or Tr/2. In the 
 former case (< = 0) we have ^ = ^r 2 ' 1 = ^-a^Li + rMa' 2 -^)/^ 
 and in the latter case (</> = 7r/2) we have I 2 = A 2 e~ 2<T2 , so that 
 
 At the end of the optic axis we have I = A^e' 2 "* = / 2 . Hence with 
 crystals for which a' is large compared with c', we have a bright 
 line perpendicular to the principal plane of the polariser and a dark 
 line at right angles thereto, this dark line being interrupted by a 
 bright spot at the end of the optic axis. On passing to crystals of 
 the opposite type for which c is large compared with a, we must 
 interchange bright and dark in the description of these lines. 
 
 When unpolarised light is employed, we have 
 
 27 = A 2 [cos 2 (f> . e~ 2<r * + sin 2 </> . e~ 2 ^] , 
 
 where (/> is the angle that the analyser makes with the plane of 
 polarisation of the quicker wave, and 2/' = A^e' 2 "*. From these 
 equations we get 2d//3</> = J. 2 sin 2< [e~ 2 " 2 e~ 2<ri ], so that / is 
 stationary when <f> = or Tr/2, and the bright and dark lines are 
 the same as those in the case last discussed. 
 
 Finally, if there be neither polariser nor analyser we have 
 
 2I = A*e- 2<r i + e- 2 < r * and I' 
 
 Everything is independent of 0, so that there are no bright or dark 
 lines crossing the field. The intensity varies slightly with r ; but 
 if the absorption be appreciable the rings are very indistinct, and 
 the appearance presented is that of a bright or a dark field 
 according to the type of the crystal, the bright field having a 
 dark spot and the dark field a bright spot at the centre. 
 
 All these conclusions relative to the behaviour of thin crystal- 
 line plates that have slight absorptive power are in thorough 
 agreement with the results of experiment. 
 
INDEX. 
 
 Absorbing media 235, 298, 301, 312 
 Absorption, axis of 298 
 
 index of 302, 313 
 
 weak 304 
 
 cone of equal 315 
 Achromatic lines 207 
 Action, principle of 14, 32 
 Aim of science 1 
 Amplitude 16, 44, 72 
 Angle between ray and wave normal 
 
 148 
 Axis, optic 134, 137 
 
 of single ray velocity 151 
 
 ray 151 
 
 of displacement 298 
 
 of absorption 298 
 Azimuth, principal 242, 245, 262, 295 
 
 of axis of orbit 27, 84 
 
 of disappearance of rings 98 
 
 Bacon, method of induction 2 
 
 Baltimore lectures 12 
 
 Biaxal crystals 137 
 
 Biot 142, 205 
 
 Boundary conditions 33, 131, 159, 190 
 
 Brewster's law 42 
 
 experiments on reflection 44 
 refraction 47 
 Newton's rings 98 
 polarising angle 172, 174 
 
 Bright or dark centred rings 99 
 
 Brightness of Newton's rings 99, 101, 
 103 
 
 Cause 5 
 
 Change of phase on reflection 51, 191, 
 
 285, 287 
 
 Circular polarisation 25, 28, 227 
 Colour term 206 
 
 of Newton's rings 106 
 Compensated plates 211 
 Complex coordinates 300 
 Composition of coordinates 26, 28 
 Compression 33 
 Cone of equal absorption 315 
 Conical refraction 149, 150 
 
 Conjugate vibrations 26, 29 
 
 curves 220 
 Conroy on intensity of reflection 53 
 
 on polarising angle in crystals 172 
 
 on principal incidence and azimuth 
 245, 262, 293 
 
 on reflection from steel 267 
 Continuum 37 
 Critical angle 55 
 Crystals, propagation in 130 
 
 positive and negative 134, 137 
 
 uniaxal 134, 167 
 
 biaxal 137 
 
 reflection and refraction 157, 189 
 
 boundary conditions 159 
 
 twin plate 195 
 
 plates 203 
 
 absorbing 298 
 Curl of the displacement 32, 147 
 
 d'Alembert's principle 13 
 de Senarmont 248, 259 
 Descartes on hypotheses 7 
 
 law 39 
 Deviation 165, 171, 176 
 
 by metallic prism 240 
 Displacement 32 
 
 direction of 48, 140 
 
 electric 237 
 
 Dissipation function 235 
 Drude on optical constants of metals- 
 
 249 
 Dynamical principle 14 
 
 equations 33, 131 
 
 Electric displacement 237 
 
 Electrons 237 
 
 Elliptic polarisation 25, 27, 83, 188,. 
 229, 230, 302 
 
 End of science 1 
 
 Energy, conception of 10 
 
 kinetic 11, 32, 35, 36, 39, 133 
 potential 11, 32, 35, 36, 39, 133 
 lines of flow 22, 155 
 
 Equations (dynamical) 33, 131 
 
 Ether 10, 16, 32, 37 
 
INDEX 
 
 325 
 
 Explanation 1 
 Extraordinary wave 134 
 
 Fourier's theorem 17 
 
 Frequency 16 
 
 Fresnel's formulae 40, 51 
 
 rhomb 61 
 
 equation 306 
 
 Gauss' principle of least constraint 14 
 Glazebrook's experiments on index sur- 
 face 135, 138 
 on reflection 169, 170 
 
 Hamilton's construction 24 
 Helmholtz on relativity 4 
 
 aim of science 5 
 
 concealed masses in motion 14 
 
 principle of action 14, 15 
 Hertz, principles of mechanics 14 
 
 criticism of action principle 14 
 Huyghens' principle 20 
 
 construction 22, 50, 146, 154 
 Hypotheses, place of, in science 7, 8, 10, 
 13 
 
 Incidence (principal) 60, 242, 245, 261, 
 
 293 
 Index of absorption 302 
 
 surface 134, 135, 138, 314 
 Induction 2 
 Intensity 18, 35, 53, 205, 207, 250, 
 
 284, 289 
 
 Interference, principle of 20 
 Isochromatic lines 207, 211, 212, 215 
 
 surfaces 222 
 Isogyric lines 207 
 Isotropic media 20, 94 
 
 Jamin, experiments on reflection 73 
 
 Kelvin on models 12 
 
 concealed masses in motion 14 
 Kepler, preface to his books 7 
 
 laws 8 
 Kurz, experiments on reflection 72 
 
 Larmor on the principle of action 14 
 
 Layer of transition 61, 72, 117, 178, 
 250, 393 
 
 Least action 14, 15, 32 
 
 Lloyd's experiments on conical refrac- 
 tion 150, 155 
 
 Loops 36 
 
 MacCullagh, theorem of polar plane 161 
 
 Malus 42 
 
 Mascart 293 
 
 Matter and ether 37 
 
 Maupertuis, principle of action 15 
 
 Meslin on reflection 26, 294, 297 
 
 Metallic prism 239 
 
 film 283, 288 
 Models 12 
 Monochromatic light 18 
 
 Natural light, features of 29 
 Necessity in causation 6 
 Neumann on reflection 169 
 Neutral lines 206, 219 
 Newton on hypotheses 10, 13 
 
 Eegulae Philosophandi 11 
 
 laws of motion 13 
 
 on size of rings 104 
 
 on colour of rings 106 
 Newton's rings 94, 263 
 
 azimuth of disappearance 98, 101, 
 114 
 
 Brewster on 98 
 
 bright or dark centres 99, 114 
 
 brightness of 99, 101, 103, 267, 273 
 
 size of 104, 268, 274, 278 
 
 colour of 106, 270, 272, 281 
 
 beyond critical angle 109, 114 
 
 viewed through analyser 111 
 
 effect of layer on 117 
 Nodes 36 
 
 Optic axis 134, 137 
 
 Optical constants of metals 249, 256, 
 
 259 
 
 Ordinary wave in crystals 134 
 Oscillation, sense of 27 
 Osiander on hypotheses 7 
 
 Period 16 
 Phase 16 
 
 change of, 011 reflection 51, 191, 285, 
 287 
 
 difference of 52, 61, 71, 73, 82, 248, 
 
 296 
 
 Pile of plates 124 
 Plane waves 18 
 
 polarisation 24 
 Plates, reflection from 94 
 
 pile of 124 
 
 thick 123 
 
 thin crystalline 203, 315 
 
 thin metallic plates 283, 284 
 
 compensated 211 
 Polarisation 24 
 
 plane of, in crystals 140 
 
 circular 25, 28, 228 
 
 elliptic 24, 25, 27, 83, 188, 229, 230, 
 302 
 
 plane 24 
 Polariscope 215 
 
 Polarising angle 41, 42, 50, 79, 164, 
 171-175, 178, 185, 187, 244, 295 
 Polar plane 50, 161 
 Principal azimuth 242, 245, 262, 295 
 
 Incidence 60, 242, 244, 245, 261, 293 
 
326 
 
 INDEX 
 
 Prism, metallic 239 
 Quasi-polarising angle 244, 295 
 
 Ray 21, 35, 156 
 
 velocity 22, 151 
 
 axis 151 
 Bayleigh on intensity of reflection 53 
 
 reflection at twin-plane 202 
 
 dissipation function 235 
 Reality 4 
 
 Reflection, positive and negative 52 
 Reflective power 75, 249 
 Refracted ray, construction of 146 
 Rood on reflection 53 
 Rotational ether 32 
 
 Savart's polariscope 215 
 
 Schmidt on uniradial polarising angle 
 177, 187 
 
 Secondary waves 21 
 
 Seebeck on polarising angle 173 
 
 Sense of oscillation 27 
 
 Simple harmonic motion 16 
 
 Snell's law 39 
 
 Stationary waves 36, 250 
 
 Stokes on index surface 135 
 reflection at twin-plane 202 
 Newton's rings formed by metallic 
 reflection 263 
 
 Superposition, principle of 20, 26 
 
 Thing-in-itself 4 
 
 Total reflection 55, 61, 80, 190, 191 
 
 Transition layer 63, 71, 117, 178, 250, 
 
 293 
 
 Transmission through a plate 95, 287 
 Twin-plane 195 
 
 Uniaxal crystals 134, 167, 311 
 Uniradial directions 161 
 
 polarising angle 176, 185, 187 
 
 Vectors 32 
 
 Velocity of propagation of wave 17, 
 18, 134 
 
 ray 22, 151 
 Vibrations, composition of 26, 28 
 
 conjugate 26, 29 
 
 Wave, length of 17 
 
 ordinary and extraordinary 134 
 
 plane 18 
 
 progressive 34 
 
 secondary 21 
 
 stationary 36, 250 
 
 surface 20, 143, 145, 146 
 
 theory 16 
 
 velocity 17, 18 
 Wiener on stationary waves 250 
 
 CAMBRIDGE : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS. 
 
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