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Theory of tides

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Theory of tides applies continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans (especially Earth's Ocean) under the gravitational loading of another astronomical body or bodies (especially the Moon).

Porto Covo, Portugal

Quotes

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  • [I]t was upon... inequality of motions in point of velocity that Galileo built his theory of flux and reflux of the sea; supposing that the earth revolved faster than the water could follow; and that the water was therefore first gathered in a heap and then fell down, as we see in a basin of water moved quickly. But this he devised upon an assumption which cannot be allowed, viz. that the earth moves; and also without being well informed as to the sexhorary motion of the tide.
    • Francis Bacon, Novum Organum (1620) as quoted in The Works of Francis Bacon: Translations of the Philosophical Works (1875) p. 212, Vol. IV of Translations of the Philosophical Works ed. James Spedding, Robert Leslie Ellis, Douglas Denon Heath.
  • When Gilbert of Colchester, in his “New Philosophy,” founded on his researches in magnetism, was dealing with tides, he did not suggest that the moon attracted the water, but that “subterranean spirits and humors, rising in sympathy with the moon, cause the sea also to rise and flow to the shores and up rivers”. It appears that an idea, presented in some such way as this, was more readily received than a plain statement. This so-called philosophical method was, in fact, very generally applied, and Kepler, who shared Galileo’s admiration for Gilbert’s work, adopted it in his own attempt to extend the idea of magnetic attraction to the planets.
    • Walter William Bryant, Kepler (1920), p. 35 Note reference: William Gilbert's New Philosophy about our Sublunary World or De Mundo Nostro Sublunari Philosophia Nova (1651)
  • A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed.
    • John L. Casti, "Correlations, Causes, and Chance," Searching for Certainty: How Scientists Predict the Future (1990).
  • Among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he nevertheless lent his ear and his assent to the moon's dominion over the waters, to occult properties, and to such puerilities.
  • J. Kepler was the first (that I know of) that discover'd the true cause of the Tide, and he explains it largely in his Introduction to the Physics of the Heavens, given in his Commentaries to the Motion of the Planet Mars, where after he has shewn the Gravity or Gravitation of all Bodies towards another, he thus writes: "The Orb of the attracting Power, which is in the Moon is extended as far as the Earth, and draws the Waters under the Torrid Zone, acting upon places where it is vertical, insensibly on included Seas, but sensibly on the Ocean, whose Beds are large, and the Waters have the liberty of reciprocation, that is, of rising and falling"; and in the 70th Page of his Lunar Astronomy,—"But the cause of the Tides of the Sea appear to be the Bodies of the Sun and Moon drawing the Waters of the Sea."
    • David Gregory, The Elements of Astronomy, Physical and Geometrical J. Nicholson (1715) Vol.2, p. 668.
  • Afterwards that incomparable Philosopher Sir Isaac Newton, improv'd the hint, and wrote so amply upon this Subject as to make the Theory of the Tides his own, by shewing that the Waters of the Sea rise under the Moon and the Place opposite to it: For Kepler believ'd "that the Impetus occasion'd by the presence of the Moon, by the absence of the Moon, occasions another Impetus; till the Moon returning, stops and moderates the Force of that Impetus, and carries it round with its motion." Therefore this Spheroidical Figure which stands out above the Sphere (like two Mountains, the one under the Moon and the other in the place opposite to it) together with the Moon (which it follows) is carried by the Diurnal Motion, (or rather, according to the truth of the matter, as the Earth turns towards the East it leaves those Eminencies of Water, which being carried by their own motion slowly towards the East, are as it were unmov'd) in its journey makes the Water swell twice and sink twice in the space of 25 Hours, in which time the Moon being gone from the Meridian of any Place, returns to it again.
  • Astronomy teaches the correct use of the sun and the planets. These may be put on a frame of little sticks and turned round. This causes the tides.
  • But to return to Kepler, his great sagacity, and continual meditation on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual betwixt bodies, and tells us that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds that the tides arise from the gravity of the waters towards the moon. But not having just enough notions of the laws of motion, he does not seem to have been able to make the best use of these thoughts; nor does he appear to have adhered to them steadily, since in his epitome of astronomy, published eleven years after, he proposes a physical account of the planetary motions, derived from different principles.
  • The allusion to the "puzzling" problem of [the orbit of] Mars shows that Galileo ought not to have been unaware of the great work of Kepler published in 1609: Astronomia nova... in which the first two of Kepler's laws were formulated. Yet he does not mention here at all Kepler's success in solving the problem, nor his laws, nor his name even, which is brought up... only to criticize his belief in the Moon's attraction [effect upon tides], which is quite reasonably presented in the Astronomia nova and founded on astronomical reasons and not on mystical speculations.
  • This world was once a fluid haze of light,
    Till toward the centre set the starry tides,
    And eddied into suns, that wheeling cast
    The planets: then the monster, then the man.
  • The Academy of Sciences at Paris proposed The Tides as the subject for a prize essay in 1740. Four essays were published in consequence at Paris. One essay was by a Jesuit named Cavallieri; this adopted the Cartesian system of vortices. The other essays were by Daniel Bernoulli, Maclaurin, and Euler; these are reprinted in the Jesuits' edition of the Principia, and it is stated that many errors in the original impression have been corrected. ...The second chapter of Daniel Bernoulli's essay contains some lemmas relating to the Attraction of Bodies. ...he determines the attraction at any superficial or internal point of an ellipsoid of revolution which is nearly spherical, neglecting powers of the ellipticity beyond the first. The method used consists in finding accurately the attraction of a sphere, and then approximately the attraction of the difference between the sphere and the ellipsoid on a particle at the pole or at the equator... this method had been previously used by Clairaut. But Daniel Bernoulli seems to claim the method as his own... Although Daniel Bernoulli employed attraction for the purpose of his essay, yet he seems to have had but a weak faith in the principle... Daniel Bernoulli added nothing to our subject; all his results respecting Attraction are included in the formulæ given by Clairaut in 1737. But his theory of the Tides is very important in the history of that subject...
    • Isaac Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth from the Time of Newton to that of Laplace, Vol. 1, pp. 124-125.
  • A Frenchman who arrives in London, finds a great alteration in philosophy, as in other things. He left the world full, he finds it empty. At Paris you see the universe composed of vortices of subtile matter, at London we see nothing of the kind. With you it is the pressure of the moon which causes the tides of the sea, in England it is the sea which gravitates towards the moon; so that when you think the moon ought to give us high water, these gentlemen believe that you ought to have low water; which unfortunately we cannot test by experience; for in order to do that, we should have examined the moon and the tides at the moment of the creation. You will observe also that the sun, which in France has nothing to do with the business, here comes in for a quarter of it. Among you Cartesians, all is done by an impulsion which one does not well understand; with the Newtonians, it is done by an attraction of which we know the cause no better. At Paris you fancy the earth shaped like a melon, at London it is flattened on the two sides.
  • Here lies
    Isaac Newton, Knight,
    Who, by a Vigour of Mind almost supernatural,
    First demonstrated
    The Motions and Figures of the Planets,
    The Paths of the Comets, and the Tides of the Ocean.
    He diligently investigated
    The different Refrangibilities of the Rays of Light,
    And the Properties of the Colours to which they give rise.
    An assiduous, sagacious, and faithful Interpreter
    Of Nature, Antiquity, and the Holy Scriptures,
    He asserted his Philosophy of the Majesty of God,
    And exhibited in his conduct the Simplicity of the Gospel.
    Let mortals rejoice
    That there has existed such and so great
    An Ornament of Human Nature.

History of the Inductive Sciences (1837)

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from the Earliest Times to the Present, by William Whewell.
  • The application of the general doctrines of mechanics to fluids was a natural and inevitable step, when the principles of the science had been generalised. It was easily seen that a fluid is, for this purpose, nothing more than a body of which the parts are moveable amongst each other with entire facility; and that the mathematician must trace the consequences of this condition upon his equations. This accordingly was done, by the founders of mechanics, both for the cases of the equilibrium and of motion. ...
    The explanation of the Tides, in the way in which Newton attempted it in the third book of the Principia, is another example of a hydrostatical investigation: for he considered only the form that the ocean would have if it were at rest. The memoirs of Maclaurin, Daniel Bernoulli, and Euler, on the question of the tides, which shared among them the prize of the Academy of Sciences in 1740, went upon the same views.
    The Treatise of the Figure of the Earth by Clairaut, in 1743, extended Newton's solution of the same problem, by supposing a solid nucleus covered with a fluid of different density. No peculiar novelty has been introduced into this subject, except a method employed by Laplace for determining the attractions of spheroids of small eccentricity, which is, as Professor Airy has said, "a calculus the most singular in its nature, and the most powerful in its effects, of any which has yet appeared."
    • Vol. 2, Book VI., Ch VI.
  • Laplace... took up the subject of waves propagated along the surface of water; and deduced a very celebrated theory of the tides, in which he considered the ocean to be, not in equilibrium, as preceding writers had supposed, but agitated by a constant series of undulations, produced by the solar and lunar forces. The difficulty of such an investigation may be judged of from this, that Laplace, in order to carry it on, is obliged to assume a mechanical proposition, unproved, and only conjectured to be true; namely, that "in a system of bodies acted upon by forces which are periodical, the state of the system is periodical like the forces." Even with this assumption, various other arbitrary processes are requisite; and it appears still very doubtful whether Laplace's theory is either a better mechanical solution of the problem, or a nearer approximation to the laws of the phenomena, than that obtained by D. Bernoulli, following the views of Newton.
    • Vol. 2, Book VI., Ch VI.
  • In most cases, the solutions of problems of hydrodynamics are not satisfactorily confirmed by the results of observation. Poisson and Cauchy have prosecuted the subject of waves, and have deduced very curious conclusions by a very recondite and profound analysis. The assumptions of the mathematician here do not represent the conditions of nature; the rules of theory, therefore, are not a good standard to which we may refer the aberrations of particular cases; and the laws which we obtain from experiment are very imperfectly illustrated by à priori calculation. The case of this department of knowledge, hydrodynamics, is very peculiar... we want, in addition to what we have, true and useful principles, intermediate between the highest and the lowest;—between the extreme and almost barren generality of the laws of motion, and the endless varieties and inextricable complexity of fluid motions in special cases. The reason of this peculiarity in the science of hydrodynamics appears to be, that its general principles were not discovered with reference to the science itself, but by extension from the sister science of the mechanics of solids...by a perception that the parts of fluids are included in that range of generality which we are entitled to give to the supreme laws of motion of solids. ...[S]olid and fluid dynamics resemble two edifices which have their highest apartment in common, and though we can explore every part of the former building, we have not yet succeeded in traversing the staircase of the latter, either from the top or from the bottom. If we had lived in a world in which there were no solid bodies, we should probably not yet have discovered the laws of motion; if we had lived in a world in which there were no fluids, we should have no idea how insufficient a complete possession of the laws of motion may be, to give us a true knowledge of particular results.
    • Vol. 2, Book VI., Ch VI.
  • That all the parts of the universe are drawn and held together by love, or harmony, or some affection to which, among other names, that of attraction may have been given, is an assertion which may very possibly have been made at various times, by speculators writing at random, and taking their chance of meaning and truth. The authors of such casual dogmas have generally nothing accurate or substantial, either in their conception of the general proposition, or in their reference to examples of it... But among those who were really the first to think of the mutual attraction of matter, we cannot help noticing Francis Bacon; for his notions were so far from being chargeable with the looseness and indistinctness to which we have alluded, that he proposed an experiment which was to decide whether the facts were so or not;—whether the gravity of bodies to the earth arose from an attraction of the parts of matter towards each other, or was a tendency towards the centre of the earth. And this experiment is, even to this day, one of the best which can be devised, in order to exhibit the universal gravitation of matter: it consists in the comparison of the rate of going of a clock in a deep mine, and on a high place. Huyghens, in his book "De Causâ Gravitatis," published in 1690, showed that the earth would have an oblate form, in consequence of the action of the centrifugal force; but his reasoning does not suppose gravity to arise from the mutual attraction of the parts of the earth. The influence of the moon upon the tides had long been remarked; but no one had made any progress in truly explaining the mechanism of this influence; and all the analogies to which reference had been made, on this and similar subjects, as magnetic and other attractions, were rather delusive than illustrative, since they represented the attraction as something peculiar in particular bodies, depending upon the nature of each body.
    That all such forces, cosmical and terrestrial, were the same single force, and that this was nothing more than the insensible attraction which subsists between one stone and another, was a conception equally bold and grand; and would have been an incomprehensible thought, if the views which we have already explained had not prepared the mind for it.
    • Vol. 2, Book VII., Ch II.
  • Newton, in the Principia, had inserted a series of propositions, the object of which was to prove, that the machinery of vortices could not be accommodated to one part of the celestial phenomena, without contradicting another part. A more obvious difficulty was the case of gravity of the earth; if this force arose, as Descartes asserted, from the rotation of the earth's vortex about its axis, it ought to tend directly to the axis, and not to the centre. The asserters of vortices often tried their skill in remedying this vice in the hypothesis, but never with much success. ...The mathematical prize-questions proposed by the French Academy, naturally brought the two sets of opinions into conflict. The Cartesian Memoir of John Bernoulli... was the one which gained the prize in 1730. ...The last act of homage of this kind to the Cartesian system was performed in 1740, when the prize on the question of the tides was distributed between Daniel Bernoulli, Euler, Maclaurin, and Cavallieri; the last of whom had tried to amend and patch up the Cartesian hypothesis on this subject.
    • Vol. 2, Book VII., Ch III.

"Tides and Waves" (1845)

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George Biddell Airy, Encyclopædia Metropolitana; Or, Universal Dictionary of Knowledge (1845) Vol. 5, pp. 241-396. Also see Tides and Waves: Extracted from the Encyclopaedia Metropolitana (1845) Tom. v, pp. 241 - 396.
  • We propose... to enter at some length into the mathematical theories, and the experimental observations, applying to the two subjects of Tides and Waves of water. But we do not intend to treat them with the same extension. We shall give the various theories of Tides in detail sufficient to enable the reader to understand the present state of the science... and we shall advert to the principal observations which throw light either on the ordinary phænomena of tides, or on the extraordinary deviations that occur in peculiar circumstances. In thus treating the Tides, it will be necessary for us to enter largely into the theory of Waves. We shall take advantage of this circumstance for the introduction several propositions, not applying to the theory Tides, but elucidating some of the ordinary observations upon small Waves. But these investigations will be limited to that class which is most closely connected with tides, namely, that in which similar waves follow each other in a continuous series, or in which the same mathematical process may be used as when similar waves follow each other. In this class will be included nearly all the phænomena of waves produced by natural causes, and therefore possessing general interest. But it will not include the waves of discontinuous nature produced by the sudden action of arbitrary causes, which have been the subject of several remarkable mathematical memoirs, but which possess no interest for the general reader.
    • Introduction
  • We shall describe cursorily the ordinary phænomena of tides.
    • Introduction
  • We shall explain the Equilibrium-Theory of Tides, including the first tidal theory given by Newton, and the more detailed theory of his successors, especially Daniel Bernoulli.
    • Introduction
  • We shall give a sketch of Laplace's investigations, (founded essentially on the theory of the motion of water,) in the general form in which he first attempted the theory, as well as with the arbitrary limitations which he found it necessary to use for practical application.
    • Introduction
  • We shall give an extended Theory of Waves on water, applying principally to the motion of water in canals of small breadth, but with some indications of the process to be followed for the investigation of the motion of Waves in extended surfaces of water.
    • Introduction
  • The results of a few Experiments on Waves will be given, in comparison with the preceding theory.
    • Introduction
  • We shall investigate the mathematical expressions for the Disturbing Forces of the Sun and Moon which produce the Tides, and shall use them in combination with the theory of Waves to predict some of the laws of Tides.
    • Introduction
  • We shall advert to the methods which been used, or which may advantageously be used, for Observation of Tides, and for the Reduction of the Observations.
    • Introduction
  • We shall give the results of extensive observations of the Tides, as well with regard to the change of the phænomena of tides at different times in the same place, as with respect to the relation which the time and height of tide at one place bear to the time and height at other places, and shall compare these with the results of the preceding theories, as far as possible.
    • Introduction
  • And as Conclusion, we shall point out what we consider to be the present Desiderata in the Theory and Observations of Tides.
    • Introduction
  • Caesar, in his account of the invasion of Britain, (De Bella Gallico, lib. iv.) alludes to the nature of spring tides as perfectly well understood in connection with the moon’s age. Some of the peculiarities of river tides, however, were not published in scientific works till the beginning of the last century; and some of the properties of the tides in the English and other channels were not known till the end of that century. ...In the present century, the elaborate discussions of immense collections of accurate tide-observations by M. Laplace, Sir John W. Lubbock, and Professor Whewell, have brought to light and reduced to law many irregularities which were before that time unknown.
  • Before entering upon either of the theories explaining the Tides, we must allude to their inadequacy, perhaps not to the explanation of the facts already observed, but certainly to the prediction of new ones. This inadequacy does not appear to arise from any defect in the principles upon which the theory is based,... but from the extreme difficulty of investigating mathematically the motions of fluids under all the various circumstances in which the waters of the sea and of rivers are found. For the problem of the Tides, it is evident, is essentially one of the motion of fluids. Yet so difficult are the investigations of motion that, till the time of Laplace, no good attempt was made to determine, by theory, the laws of the Tides, except on the supposition that the water was at -rest. Since that time theories of motion have been applied...
  • Indeed, throughout the whole of this subject, the selection of the proper theoretical ground of explanation is a matter of judgment. In some cases we may conceive that we are justified in using the Equilibrium theory; in others the Wave-theory will apply, completely or partially... as a last resource, in almost every case, we shall be driven to the same arbitrary suppositions which Laplace introduced. ...In the instances which it does not master completely, it will show that there are ample grounds for the arbitrary alterations of constants introduced by Laplace in his suppositions...
  • [W]e are precluded from further advance, partly by our almost necessary ignorance of the forms of the bottom in deep seas, and partly by the imperfection of our mathematics. ...the first principles of our explanation are correct.
  • The popular explanation of the Equilibrium-theory is very simple. If we conceive the earth to be wholly or in a great degree with water, and consider that the attraction of the moon upon different particles (according to the law of gravitation) is inversely as the square of their distance, and is therefore greatest for those particles which are nearest to it; then it will be obvious that the moon attracts the water on that side which is next to her, more than she attracts the great mass of the earth, and therefore tends to raise the water from the earth on the side next to her; but she also attracts the great mass of the earth more than she attracts the water upon the side most distant from her, and therefore tends to draw the earth from the water on the side most distant from her; which will produce exactly the same effect as if a force tended to draw the water away from the earth on that side. Thus the moon’s action tends to raise the water on two opposite sides of the earth; and similarly the sun’s action tends to raise the water on two opposite sides. The close relation, however, which the times of high water bear to the times of the moon’s passage, shows that the moon’s influence in raising the tides must be much greater than the sun's. If the sun and moon are together, as seen from the earth, the elevations produced by these two bodies will coincide in place, and will therefore be added together. Thus Spring Tides will be produced. In other relative positions of the sun and moon, it may happen that the elevation produced by the sun will occur at a place where the moon causes depression: the action of the sun there tends to counteract that of the moon, and Neap Tides will be produced.

Life of Thomas Young (1855)

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by George Peacock, source.
  • Newton pointed out and assigned generally, not only the nature and the magnitude of the periodical forces which are concerned in producing the tides, but likewise indicated their true character as undulations, in one very remarkable proposition, as well as in a special explanation of... the tides of the Port of Batsha. The equilibrium theory of Daniel Bernoulli adopted the first part of Newton's views but altogether neglected the second.
  • It had been shown that if the earth was a spherical body covered with water, and if both the earth and moon were at rest, the water would assume the form of a spheroid of equilibrium, of extremely small eccentricity, such as would be due to the disturbing action of the moon's forces. A similar but less eccentric spheroid would be formed beneath the sun. Under such circumstances the joint effect of the elevations or depressions of the two spheroids would produce the elevation or depression of the water, or the tide. The theory further assumes that the same effects would follow if the earth revolved round her axis and the earth and moon in their orbits, and that no effect was produced by the spontaneous oscillations of the sea. Totally false as are the principal assumptions upon which this theory is founded, it is extremely remarkable that it not only sufficiently separates from each other the principal movements of the tides, but represents generally the law and order of succession of the periodical phenomena which they present. "The greatest mathematicians and the most laborious observers of the present day," says Professor Airy, "including Sir John Lubbock and Dr. Whewell... have agreed equally in rejecting the foundation of this theory, and comparing all their observations with its results."
  • The same eminent authority [Professor Airy] has pronounced the theory proposed by La Place in the Mécanique Céleste,—if viewed with reference to the boldness and comprehensive character of its design rather than to the success of its execution—"as one of the most splendid works of the greatest mathematician of the past age." The problem, however, was not considered by him [La Place] in the most general form which it is capable of receiving. He assumed the earth to be entirely covered by water, and its depth to be uniform, at least throughout the same parallel of latitude, and he neglected the resistance both of the particles of the fluid amongst each other, and of that which arises from the irregular surfaces in the channels over which the tide is transmitted. He was consequently obliged to omit the consideration of the tides in canals, rivers, and narrow seas, which constitute some of the most interesting, and by no means the most unmanageable, of the problems which later, and even in some respects more simple, investigations of the oscillations of the sea have brought within the control of analysis. Imperfect, however, as the results of this theory were as it came from the hand of its author, their importance cannot easily be estimated too highly. Dr. Young adopted the general principles which they involved, though he has subjected them to a totally different treatment; and Professor Airy, who has materially simplified the investigations which it contains, by rejecting some conditions which they included, such as the density of the sea, by which they were made needlessly difficult and complicated, has not only verified the more remarkable of the conclusions at which La Place arrived, but has also made important use of his methods in his own theory of waves and tides, which is by far the most complete and comprehensive that has ever yet appeared.
  • There is one result of a very unexpected kind, which La Place regarded as one of the happiest of his discoveries,—it is the entire evanescence, if the sea be of uniform depth, of the diurnal tide in elevation, but not in horizontal motion. At the equator, under such circumstances, the water moves north and south, resting for a moment at the change of motion. At the poles the motion is transverse to the meridian passing through the luminary. At all other points on the earth's surface it is perpetually changing. Few persons have attempted to follow the mazes of the difficult analysis by which this great mathematician has arrived at this conclusion, which has been verified by the Astronomer Royal. Its correctness, however, has been disputed by Dr. Young, who contends that the diurnal tide will not disappear, unless the depth of the sea be not merely uniform, but evanescent.
  • Though Dr. Young was not disposed to give his assent to the results of an extremely difficult analysis,—which few persons of his age could venture to follow, and which might appear to those who could not trace them through the long train of consequences... to be either paradoxical or contradictory to the first principles of mechanics—he was sufficiently prepared to seize the general purport of other parts of this comprehensive theory; and by divesting it of the unnecessary generalizations by which it was encumbered, not only to bring its principles to bear immediately upon the ordinary phenomena of the tides, but to apply it to cases which it was otherwise incompetent to reach. Such were the tides of narrow seas and rivers, and the modifications which those tides undergo from the effects of the resistance of the particles of water upon each other, or upon the channels through which they are propagated. The same questions have been made the principal subject of the investigations of the Astronomer Royal, in his Article on Tides and Waves, in the Encyclopædia Metropolitana, where they have been treated with that rare combination of mathematical skill and clearness and completeness of exposition for which all his writings are so remarkable. It will be found, however, that there are not many of his results which Young had not already attained, though in a much less definite form, by methods which are, it is true, much less regular and systematic, but which are not less distinguished for the sagacity and philosophical power which they display.

See also

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