ON

A GENERAL METHOD

OF EXPEESSINO

THE PATHS OF LIGHT, AND OF THE PLANETS, BY THE

COEFFICIENTS OF A CHARACTERISTIC FUNCTION.

BY WILLIAM R. HAMILTON,

ROYAL ASTRONOMER OF IRELAND.

FROM THE DUBLIN UNIVERSITY REVIEW FOR OCTOBER.

DUBLIN:

PRINTED BY P. DIXON HARDY, 9, CECILIA-STREET.

1833.

On a general Melhocl of expressing (he Paths of Lighl, and of (he Planets, bg (he Coe^cients of a Charac(eristic Funcdon. By William R. Hamilton, Royal Astronomer of Ireland.

The law of seeing in straight lines was known from the infancy of optics, being in a manner forced upon men’s notice by the most familiar and constant experience. It could not fail to be observed that when a man looked at any object, he bad it in his power to interrupt his vision of that object, and hide it at pleasure from his view, by interposing his hand between Ids eyes and it ; and that then, by withdrawing bis hand, he could see the object as before; and thus the notion of straight lines or rays of communication, between a visible object and a seeing eye, must very easily and early have arisen. This notion of straight lines of vision, was of course confirmed by the obvious remark that objects can usually be seen on looking througli a straiglit but not througli a bent tube ; and the most familiar facts of perspective supplied, we may suppose, new confirmations and new uses of the principle. A globe, for example, from whatever point it may be viewed, appears to have a circular outline ; while a plate, or a round table, seems oval wlien viewed obliquely: and these facts may have been explained, and reduced to mathematical reasoning, by shewing that the straight rays or lines of vision, which toucli any one given globe and pass through any one given point, are arranged in a hollow cone of a perfectly circular shape ; but that the straight rays, whicli connect an eye witti the round edge of a plate or table, compose, when they are oblique, an elliptical or oval cone. The same principle, of seeing in straight lines, must have been continually employed from tlie earliest times in the explanation of other familiar appearances, and in interpreting the testimony of sight respecting the places of visible bodies. It was, for example, an essential element in ancient as in modern astronomy.

The shapes and sizes of shadows, again, could not fail to ^yiggest the notion of straight illuminating rays : although opinions, now rejected, respecting the nature of light and vision, led some of the ancients to distinguish the lines of luminous from those of visual communication, and to regard the latter as a kind of feelers by whicli the eye became aware of the presence of visible objects. It appears, however, that many persons held, even in the infancy of Optics, the modern view of the subject, and attributed vision, as well as illumination.

B

On a General Method of jExpressing

to an influence proceeding from the visible or luminous body. But what finally established this view, and along with it the belief of a finite velocity of progress of the luminous influence, was the discovery made by Roemer, of the gradual propagation of liglit from objects to the eye, in the instance of the satellites of Jupiter ; of whicli we have good reason to believe, from astronomical observation, that the eclipses are never seen by us, till more than half an hour after they have happened ; the interval, besides, being found to be so much the greater, as Jupiter is more distant from the Earth. Galileo had indeed proposed terrestrial experiments to measure the velocity of light, whicli he believed to be finite ; and Des Cartes, wlio held that tlie communication of light was instantaneous, had perceived that astronomical consequences ouglit to follow, if tlie propagation of light were gradual : but experiments sucti as Galileo proposed, were not, and could not, be made on a scale sufficient for the purpose ; and the state of astronomical observation in the time of Des Cartes did not permit him to verify the consequences whicli he perceived, and seemed rather to justify the use that he made of their non-vérification, as an argument against the opinion witli which he had shown them to be logically connected. But when astronomers had actually observed appearances, wliicli seemed and still seem explicable only by this opinion of the gradual propagation of light from objects to the eye, the opinion itself became required, and was adopted, in the legitimate progress of induction.

By such steps, then, it has become an established theorem, fundamental in optical science, that the communication, whether between an illuminating body and a body illuminated, or between an object seen and a beholding eye, is effected by the gradual but very rapid passage of some thing, or influence, or state, called liglit, from the luminous or visible body, along mathematical or physical lines, called usually rags, and found to be, under the most common circumstances, exactly or nearly straiglit.

Again, it was very early perceived tliat in appearances connected witli mirrors, flat or curved, the luminous or visual communication is effected in bent lines. When we look into a flat mirror, and seem to see an object, sucli as a candle, behind it, we should err if we were to extend to this new case the rules of our more familiar experience. We should not now come to toucli the candle by continuing the straight line from the eye to a hand or other obstacle, so placed between the eye and the mirror as to hide the candle ; this line continued would meet the mirror in a certain place from which it would be ne-

tue Paths of Light ami of the Planets.

5

cessary to draw a new and different straight line, if we wished to reach the real or tangible candle : and the whole bent line, made up of these two straight parts, is found to be now the line of visual communication, and is to be regarded now as the linear patli of the light. An opaque obstacle, placed any where on either part of this bent line, is found to hide the reflected candle from the eye ; but an obstacle, placed any where else, produces no such interruption. And the law was very early discovered, that for every such bent line of luminous or visual communication, the angle between any two successive straiglit parts is bisected by the normal, or perpendicular, to the mirror at the point of bending.

Another early and important observation, was that of the broken or refracted lines of communication, between an object in water and an eye in air, and generally between a point in one ordinary medium and a point in another. A valuable series of experiments on sucli refraction was made and recorded by Ptolemy ; but it was not till long afterwards that the law was discovered by Snellius. He found that if two lengths, in a certain ratio or proportion determined by the natures of the two media, be measured, from tlie point of breaking, or of bending, on the refracted ray and on the incident ray prolonged, tliese lengths have one common projection on the refracting surface, or on its tangent plane. This law of ordinary refraction has since been improved by Newton’s discovery of the different refrangibility of the differently coloured rays ; and lias been applied to explain and to calculate the apparent elevation of the stars, produced by the atmosphere of the earth.

The phenomena presented by the passage of light througli crystals were not observed until more lately. Bartolinus seems to have been the first to notice the double refraction of iceland spar; and Huygens first discovered the laws of this refraction. The more complicated double refraction produced by liiaxal crystals was not observed until the present century ; and tlie discovery of conical refraction in such crystals is still more recent, the experiments of Professor Lloyd on arragonite (undertaken at my request) having been only made last year.

For the explanation of the laws of the linear propagation of light, two principal theories have been proposed, whicli still divide the suffrages of scientific men.

The theory ol Newton is well known. He compared the propagation of light to the motion of projectiles; and as, according to that First Law of Motion, of whicli he had himself established the trutli by so extensive and beautiful an induction, an ordinary projectile continues in rectilinear and uniform pro-

On a General Melkocl pf Expressing gress, except so far as its course is retarded or disturbed by the influence of some foreign body ; so, he thought, do luminous and visible objects shoot off little luminous or light-making projectiles, whicli then, until they are accelerated or retarded, or deflected one way or another, by the attractions or repulsions of some refracting or reflecting medium, continue to move uniformly in straight lines, either because they are not acted on at all by foreign bodies, or because the foreign actions are nearly equal on all sides, and thus destroy or neutralise each other. This theory was very generally received by mathematicians during the last century, and still has numerous supporters.

Another theory however, proposed about the same time by another great philosopher, has appeared to derive some strong confirmations from modern inductive discoveries. This other is the theory of Huygens, who compared the gradual propagation of light, not to the motion of a projectile, but to the spreading of sound through air, or of waves through water. It was, according to him, no thing, in the ordinary sense, no hod)/ wliich moved from the sun to the earth, or from a visible object to the eye ; but a state, a motion, a disturbance, was first in one place, and afterwards in another. As, when we hear a cannon which has been fired at a distance, no bullet, no particle even of air, makes its way from the cannon to our ears ; but only the aerial motion spreads, the air near the cannon is disturbed first, then that whicli is a little farther, and last of all the air that touches us. Or like the waves that spread and grow upon some peaceful lake, when a pebble has stirred its surface; the floating water-lilies rise and fall, but scarcely quit their place, while the enlarging wave passes on and moves them in succession. So that great ocean of ether whicli bathes the farthest stars, is ever newly stirred, by waves that spread and grow, from every source of light, tilt they move and agitate the whole witli their minute vibrations : yet like sounds througli air, or waves on water, these multitudinous disturbances make no confusion, but freely mix and cross, while eacli retains its identity, and keeps the impress of its proper origin. Such is the view of Light whicli Huygens adopted, and which justly bears his name ; because, whatever kindred thoughts occurred to others before, he first shewed clearly how tins view conducted to the laws of optics, by combining it with that essential principle of the undulatory theory whicli was first discovered by himself, the principle of accumulated disturbance.

According to this principle, the minute vibrations of the

the Pathä of Light and of the Planets.

elastic luminous etlier cannot perceptibly affect our eyes, cannot produce any sensible light, unless tliey combine and concur in a great and, as it were, infinite multitude ; and on the otlier hand, sucli combination is possible, because particular or secondary waves are supposed in this tlieory to spread from every vibrating particle, as from a separate centre, with a rapidity of propagation determined by the nature of the medium. Anti lienee it comes, thought Huygens, that light in any one uniform medium diffuses itself only in straight lines, so as only to reacli those parts of space to whicli a straight patli lies open from its origin ; because an opaque obstacle, obstructing sucli straight progress, though it does not hinder tlie spreading of weak particular waves into the space behind it, yet prevents their accumulation within that space into one grand general wave, of strengtli enougli to generate light. This want of accumulation of separate vibrations behind an obstacle, was elegantly proved by Huygens : the mutual destruction of such vibrations by interference, is an important addition to the tlieory, which has been made by Young and by Fresnel. Analogous explanations have been offered for the laws of reflexion and refraction.

Whether we adopt the Newtonian or the Huygenian, or any other physical theory, for the explanation of the laws that regulate the lines of luminous or visual communication, we may regard these laws themselves, and the properties and relations of these linear paths of light, as an important separate study, and as constituting a separate science, called often mathematical optics. This science of the laws and relations of luminous rays, is, however, itself a brancli of another more general science, wliich may perhaps be called the Theory of Systems of Pays. I have published, in the XVth and XVlth volumes of the Transactions of the Royal Irisli Academy, a series of investigations in that theory ; and have attempted to introduce a new principle and method for the study of optical systems. Another supplementary memoir, whicli has been lately printed for the same Transactions, will appear in the XVIIth volume ; but having been requested to resume the subject here, and to offer briefly some new illustrations of my view, I shall make some preliminary remarks on the state of deductive optics, and on the importance of a general method.

The science of optics, like every other physical science, has two different directions of progress, which have been called tlie ascending and the descending scale, the inductive and the deductive method, the way of analysis and of synthesis. In every physical science, we must ascend from facts to laws, by the

On a General Method of E.vpressing

way of induction and analysis ; and must descend from laws to consequences, by tlie deductive and synthetic way. We must gather and groupe appearances, until the scientific imagination discerns their hidden law, and unity arises from variety : and then from unity must re-deduce variety, and force the discovered law to utter its revelations of the future.

It was witli such convictions that Newton, when approacli-ing to the close of his optical labours, and looking back on his own work, remarked, in the spirit of Bacon, that “ As in Mathematics, so in Natural Philosophy, the investigation of difficult things by the method of Analysis ought ever to precede the method of Composition. This analysis consists in making experiments and observations, and in drawing general conclusions from them by induction, and admitting of no objections against the conclusions but such as are drawn from experiments or other certain truths.” “ And although the arguing from experiments and observations by induction be no demonstration of general conclusions; yet it is the best way of arguing whicli the nature of things admits of, and may be looked upon as so mucli the stronger, by how mucli the induction is more general. And if no exception occur from plie-nomena, the conclusion may be pronounced generally. But if at any time afterwards, any exception shall occur from experiments, it may then begin to be pronounced witli such exceptions as occur. By this way of analysis, we may proceed from compounds to ingredients, and from motions to the forces producing them ; and, in general, from effects to their causes, and from particular causes to more general ones, till the argument end in the most general. This is the method of analysis : and the synthesis consists in assuming the causes discovered, and established as principles, and by them explaining the phenomena proceeding from them, and proving tlie explanations.” “ And if Natural Philosophy in all its parts, by pursuing this method, shall at length be perfected, the bounds of Moral Philosophy will also be enlarged. For, so far as we can know by Natural Philosophy, what is the First Cause, what power He has over us, and what benefits we receive from Him, so far our duty towards Him, as well as that towards one another, will appear to us by tlie light of nature.”

Tn the science of optics, whicli has engaged the attention of almost every mathematician for the last two thousand years, many great discoveries have been attained by botli these ways. It is, however, remarkable, that, while the laws of this science admit of being stated in at least as purely mathematical a form as any other physical results, their mathematical consequences have been far less fully traced than the consequences of many other

the Paths of Light and of the Planets.

laws ; and that while modern experiments have added so mucli to the inductive progress of optics, the deductive has profited so little in proportion from the power of the modern algebra.

It was known to Euclid and to Ptolemy, that the communication between visible objects and a beholding eye is usually effected in straight lines ; and that when the line of communication is bent, by reflexion, at any point of a plane or of a spheric mirror, tlie angle of bending at this point, between the two straight parts of tlie bent line, is bisected by the normal to the mirror. It was known also that this law extends to successive reflexions. Optical induction was therefore sufficiently advanced two thousand years ago, to have enabled a mathematician to understand, and, so far as depended on the knowledge of physical laws, to resolve the following problem: to determine the arrangement of the final straight rays, or lines of vision, along whicli a shifting eye should look, in order to see a given luminous point, reflected by a combination of two given spherical mirrors. Yet, of two capital deductions respecting tins arrangement, without whicli its theory must be regarded as very far from perfect—namely, that the final rays are in general tangents to a pair, and that they are perpendicular to a series of surfaces—the one is a theorem new and little known, and the other is still under dispute. For Malus, who first discovered tliat the rays of an ordinary reflected or refracted system are in general tangents to a pair of caustic surfaces, was led, by the complexity of his calculations, to deny the general existence (discovered by Huygens) of surfaces perpendicular to sucti rays ; and the objection of Malus has been lately revived by an eminent analyst of Italy, in a valuable memoir on caustics, which was published last year in the correspondence of the observatory of Brussels.

To multiply sucli instances of the existing imperfection of mathematical or deductive optics would be an unpleasant task, and might appear an attempt to depreciate the merit of living mathematicians. It is better to ascend to the source of tlie imperfection, the want of a general method, a presiding idea, to guide and assist the deduction. For althougli the deductive, as opposed to the inductive process, may be called itself a method, yet so wide and varied is its range, that it needs the guidance of some one central principle, to give it continuity and power.

Those who have meditated on the beauty and utility, in theoretical mechanics, of the general method of Lagrange—who have felt the power and dignity of that central dynamical theorem which he deduced, in the Mécanique Analytique, from a

10 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;On a General Melhod of Ea-presatng combination of the principle of virtual velocities witli the principle of D’Alembert—and who liave appreciated the simplicity and harmony whicli he introduced into the researcli of the planetary perturbations, by the idea of the variation of parameters, and the differentials of the disturbing function, must feel that matliematical optics can only iken attain a coordinate rank with mathematical mechanics, or witli dynamical astronomy, in beauty, power, and harmony, when it shall possess an appropriate method, and become the unfolding of a central idea.

This fundamental want forced itself long ago on my attention ; and I have long been in possession of a method, by which it seems to me to be removed. But in thinking so, I am conscious of the danger of a bias. It may happen to me, as to others, that a meditation whicli has long been dwelt on shall assume an unreal importance ; and that a method whicli has for a long time been practised shall acquire an only seeming facility. It must remain for others to judge how far my attempts have been successful, and how far they require to be completed, or set aside, in the future progress of the science.

Meanwhile it appears that if a general method in deductive optics can be attained at all, it must How from some law or principle, itself of the highest generality, and among the higliest results of induction. What, then, may we consider as the highest and most general axiom, (in the Baconian sense,) to which optical induction has attained, respecting the rules and conditions of the lines of visual and luminous communication ? The answer, I tliink, must be, the principle or law, called usually the Law of Least Action ; suggested by questionable views, but established on the widest induction, and embracing every known combination of media, and every straight, or bent, or curved line, ordinary or extraordinary, along whicli light (whatever light may be) extends its influence successively in space and time : namely, that this linear patli of light, from one point to another, is always found to be such, that if it be compared witli the other infinitely various lines by which in thought and in geometry the same two points miglit be connected, a certain integral or sum, called often Action, and depending by fixed rules on the length, and shape, and position of the path, and on the media which are traversed by it, is less than all the similar integrals for the other neighbouring lines, or, at least, possesses, with respect to them, a certain stationary property. From this Law, then, which may, perhaps, be named the Law OF Stationary Action, it seems that we may most fitly and with best hope set out, in the synthetic or deductive process, and in the search of a mathematical method.

tAe Palhs of Ltgkl ami of the Plaitels.

Accordingly, from this known law of least or stationary action, I deduced (long since) another connected and coextensive principle, which may be called, by analogy, the Law of varying Action, and whidi seems to offer naturally a method sucli as we are seeking ; the one law being as it were the last step in the ascending scale of induction, respecting linear paths of light, while the other law may usefully be made tlie first in the descending and deductive way. And my chief purpose, in the present paper, is to offer a few illustrations and consequences of these two coordinate laws.

l'he former of these two laws was discovered in the following manner. The elementary principle of straight rays shewed, tliat light, under the most simple and usual circumstances, employs the direct, and, therefore, the shortest course to pass from one point to another. Again, it was a very early discovery, (attributed by Laplace to Ptolemy,) that in the case of a plane mirror, the bent line formed by the incident and reflected rays is shorter than any other bent line, having the same extremities, and having its point of bending on the mirror. 'I'hese facts were thought by some to be instances and results of the simplicity and economy of nature ; and Fermat, whose researches on maxima and minima are claimed by the continental mathematicians as the germ of the differential calculus, sought anxiously to trace some similar economy in the more complex case of refraction. He believed that by a metaphysical or cosmological necessity, arising from the simplicity of the universe, light always takes the course which it can traverse in the shortest time. To reconcile this metaphysical opinion witli the law of refraction, discovered experimentally by Snellius, Fermat was led to suppose that the two lengths, or indices, which Snellius had measured on the incident ray prolonged and on the refracted ray, and had observed to have one common projection on a refracting plane, are inversely proportional to the two successive velocities of the light before and after retraction, and therefore that the velocity of light is diminished on entering those denser media in which it is observed to approach the perpendicular : for Fermat believed that the time of propagation of light along a line bent by refraction was represented by the sum of the two products, of the incident portion multiplied by the index of the first medium, and of the refracted portion multiplied by the index of the second medium ; because he tound, by bis mathematical method, that this sum was less,' in the case of a plane refractor, than if light went by any other than its actual path from one given point to another ; and because he perceived that the supposition of a velocity inversely as the

C

On a General Method of E.rpressing

index, reconciled liis mathematical discovery of the minimum of the foregoing sum with his cosmological principle of least time. Des Cartes attacked Fermat’s opinions respecting light, but Leibnitz zealously defended them ; and Huygens was led, by reasonings of a very different kind, to adopt Fermat’s conclusions of a velocity inversely as the index, and of a minimum, time of propagation of light, in passing from one given point to another through an ordinary refracting plane. Newton, however, by his theory of emission and attraction, was led to conclude that the velocity of light was directb/, not inverseii/, as the index, and that it was increased instead of being diminished on entering a denser medium ; a result incompatible witli the theorem of shortest time in refraction. This theorem of shortest time was accordingly abandoned by many, and among tlie rest by Maupertuis, who, however, proposed in its stead, as a new cosmological principle, that celebrated law of least action whicli has since acquired so higli a rank in mathematical physics, by the improvements of Euler and Lagrange. Maupertuis gave the name of action to the product of space and velocity, or rather to the sum of all such products for the various elements of any motion ; conceiving that tlie more space has been traversed and the less time it has been traversed in, the more action may be considered to have been expended : and by combining this idea ot action with Newton’s estimate of tlie velocity of light, as increased by a denser medium, and as proportional to the refracting index, and with Fermat’s mathematical tlieorem of the minimum sum of the products of paths and indices in ordinary refraction at a plane, he concluded that the course chosen by light corresponded always to the least possible action, tliougli not always to the least possible time. He proposed tliis view as reconciling physical and metaphysical principles, whicli tlie results of Newton had seemed to put in opposition to each other ; and he soon proceeded to extend his law of least action to the phenomena of the shock of bodies. Euler, attached to Maupertuis, and pleased witli these novel results, employed his own great mathematical powers to prove that the law of least action extends to all the curves described by points under the influence of central forces ; or, to speak more precisely, that if any sucli curve be compared witli any other curve between the same extremities, which differs from it indefinitely little in shape and in position, and may be imagined to be described by a neighbouring point with the same law of velocity, and if we give the name of action to the integral of the product of the velocity and element of a curve, the difference of the two neighbouring values of this action will be indefinitely less than the greatest linear distance (itself indefinitely small) be-

ike Pai/is of Light and of the Planets.

tween the two near curves ; a theorem which I think may be advantageously expressed by saying that the action is station-an/. Lagrange extended this theorem of Euler to the motion of a system of points or bodies which act in any manner on each otlier ; tlie action being in this case the sum of the masses by the foregoing integrals. Laplace has also extended the use of the principle in optics, by applying it to the refraction of crystals; and has pointed out an analogous principle in mechanics, for all imaginable connexions between force and velocity. But althougli the law of least action has thus attained a rank among the highest theorems of physics, yet its pretensions to a cosmological necessity, on the ground of economy in the universe, are now generally rejected. And the rejection appears just, for this, among other reasons, that the quantity pretended to be economised is in fact often lavishly expended. In optics, for example, tliougli the sum of the incident and reflected portions of the patli of light, in a single ordinary reflexion at a plane, is always the shortest of any, yet in reflexion at a curved mirror tliis economy is often violated. If an eye be placed in the interior but not at the centre of a reflecting hollow sphere, it may see itself reflected in two opposite points, of which one indeed is the nearest to it, but the other on the contrary is the furthest; so that of the two different paths of light, corresponding to these two opposite points, the one indeed is the shortest, but the otlier is the longest of any. In mathematical language, the integral called action, instead of being always a minimum, is often a maximum ; and often it is neither the one nor the other; tliougli it has always a certain stationary property, of a kind wliicli has been already alluded to, and which will soon be more fully explained. We cannot, therefore, suppose the economy of this quantity to liave been designed in tlie divine idea of the universe: though a simplicity of some high kind may be believed to be included in that idea. And though we may retain the name of action to denote the stationary integral to whicli it has become appropriated—which we may do without adopting either the metaphysical or (in optics) the physical opinions that first suggested the name—yet we ought not (I think) to retain tlie epithet least: but rather to adopt the alteration prososed above, and to speak, in mechanics and in optics, of tlie Law of Stationary j4ction.

To illustrate this great law, and that other general law, of varying action, whicli I have deduced from it, we may conveniently consider first the simple case of rectilinear paths of light. For the rectilinear course, which is evidently the shortest of any, is also distinguished from all others by a certain stationary

14 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;On u General Meihod of Expressing properly, and laugt;of varialion, which, being included in the general laws of stationary and varying action, may serve as preparatory examples.

'Fhe length F of any given line, straight or curved, may evidently be denoted by the following integral :

'^—f'dV^=J’'\/ dai‘ dÿ^ dz‘‘ - . nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(1)

If now we pass from this to another neiglibouring line, liav-ing the same extremities, and suppose that the several points ot the latter line are connected witli those of tlie former, by equations between their co-ordinates, of the form

X,-x e^, ^,=1/ er,, Ze=Z t^ C^) t being any small constant, and Ç, jj, ^, being any arbitrary functions of a:, ÿ, 2, whicli vanisli for the extreme values of those variables, that is, for the extreme points of the given line, and do not become infinite for any of the intermediate j)oints, nor for the value £—0, thougli they may in general involve the arbitrary constant £ ; the lengtli F, of the new line may be represented by the new integral,

^e—Jquot;Vdxe^ dge^ dz-^

~J^ ^ (rfa; elt;/f)“ (('/y c£Z9/)“ (£fe crf^)3, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(3) taken between tl)e same extreme values of a:, g, 2, as the former ; and this new lengtli F^ may be considered as a function of s, whicli tends to the old length F, when e tends to 0, the quotient

KK-F)

tending in general at tlie same time to a finite limit, which may be tlius expressed,

lim. y _y\= Pda^^ dy dy dz d^

tlie^ last of these forms being obtained from the preceding by integrating by parts, and by employing the condition already mentioned, that the functions Ç, ri, Z, vanish at the extremities of the integral. When the original line is such that the limit (4) vanishes, independently of the forms of the functions ^, ,,, 2, and therefore that the difference of lengths

the Paths of Light and of the Planets.

l^t — ^ bears ultimately an evanescent ratio to the small quantity e, (wliicli quantity determines the difference between tlie second line and the first, and bears itself a finite ratio to the greatest distance between these two lines,) we may say tliat the original line has a statiofmr^ length, F, as compared witli all the lines between the same extremities, whidi differ from it infinitely little in sliape and in position. And since it easily follows, from the last form of the limit (4), that this limit cannot vanisli independently of the forms of ?, ni Z, unless

tliat is, unless the ratios

da: dg ds

IrP* dr*

are constant throughout the original line, but that the limit vanishes wlien this condition is satisfied, we see that the property of stationary length belongs (in free space) to straigtit lines and to suclr only. The foregoing proof of this property of the straight line may, perliaps, be useful to those wlio are not familiar witli the Calculus of Variations.

To illustrate, by examples, this stationary property of the lengtli of a straight line, let us consider sucli a line as tlie common cliord of a series of circular arcs, and compare its lengtli witli tlieirs, and theirs witli one another. The lengtli of the straight line being called V, let 1 trbe the height or sagitta of the circular arch upon this chord ; so that At f”')^quot; shall be the diameter of tlie circle, and £ tlie trigonometric tangent of tlie quarter of an arc having the same number of degrees, to a radius equal to unity : we shall then have the following expression for the lengtli V^ of the circular arch upon tlie given chord r,

^,=y(.i t~^')ta.n.-\. (6)

Ou a Geueral Melfiod of E.rprest,iug

tlie arch Fquot;, increases continually witli its height at an increasing rate ; its differential coefficient being positive and increasing, when E is positive and increases, but vanishing with £, and sliowing, tlierefore, that in this series of circular arcs and chord the property of stationary lengtli belongs to the straight line only.

Again, we may imagine a series of semi-ellipses upon a given common axis 1^, the other axis conjugate to this being a variable quantity tf^. The lengtli of such a semi-elliptic arcli is

V^=V I -^(cos.fj)^ t^ sin.(jgt;‘‘')i dlt;jgt;, O

an expression whicli may be thus transformed,

sin. ^^ cos. ^^ dlt;jgt;

(cos. ^quot;^ E® sin. lt;p'‘)l’ thus the ratio of the elliptic arch E to its given base or axis P' is not only greater than unity, and continually increases with tlie height, but increases at an increasing rate, wliicli vanislies for an evanescent height ; so that in this series of semi-elliptic arcs and axis, the latter alone has the property of stationary length.

In more familiar words, if we construct on a base of a given length, suppose one hundred feet, a series of circular or of semi-elliptic arclies, having that base for chord or for axis, the lengths of those arches will not only increase witli their heights, but every additional foot or incli of height will augment the length more than the foregoing foot or inch had done ; and the lower or flatter any two such arches are made, the less will be the difference of their lengths as compared witli the difference of their heiglits, till the one difference becomes less tlian any fraction that can be named of tlie other. For example, if we construct, on the supposed base of one hundred feet, two circular arches, the first fifty feet high, and the second fifty-one feet high, of wliich the first will thus be a semicircle, and the second greater than a semicircle, the difference of lengths of these two arches will be a little more than double the difference of tlieir heights, that is, it will be about two feet ; but if on the same base we construct one circular arcli witli only one foot of heiglit, and another witli only two feet, the difference of lengths of these two low arches will not be quite an inch, thougli tlie difference of their heights remains a foot as before ; and if we imagine the two circular arclies, on the same base or common chord of one liundred feet, to have their heights re-

t/ie Palhs of Light and of (he Planels.

ducecl to one and two inches respectively, the difference of their lengths will thereby be reduced to less than the bundred-and-fiftieth part of an inch.

We see then that a straight ray, or rectilinear path of light, from one given point to another, has a sfatiotiar^ length, as compared witli all the lines which differ little from it in sliape and in position, and which are drawn between the same extremities. If, however, we suppose the extremities of the neighbouring line to differ from those of the ray, we shall then obtain in general a varying instead of stationary length. To investigate the law of this variation, whicli is the simplest case of the second general law above proposed to be illustrated, we may resume the foregoing comparison of the lengths V, K^ , of any two neighbouring lines; supposing now that these two lines have different extremities, or in other words, that the functions Ç, ^, Ç, do not vanisli at the limits of the integral. The integration by parts gives now, along with the last expression (4) for the limit of

KK-F),

the following additional terms.

diO dy dz , d'x'

which belong to the extremities of the given line, the accented being the initial quantities, and d' referring to the infinitesimal changes produced by a motion of the initial point along the initial element of the line, so that rf'Fis this initial element taken negatively,

lt;/'F=— y/ d'x''‘ d’y'^ d'z'^ ; (11)

when, therefore, the last integral (4) vanislies, by the original line being straiglit, and when we compare this line witli another infinitely near, the law of varying length is expressed by the following equation :

do ,d')/ ,.r«.^v.-r)^^- ,f, i- e— ,^f^ !:-,

On a General Method of Expressing

it may also be thus expressed,

sr= f/to-J/) ^(%r-%0 |,(8«-amp;'), (IS)

and shows that the length F STof any other line whidi differs infinitely little from the straight ray in shape and in position, may be considered as equal to its own projection on the ray.

It must be observed that in certain singular cases, the distance between two lines may be made less, throughout, than any quantity assigned, without causing thereby their lengths to tend to equality. For example, a given straight line may be subdivided into a great number of small parts, equal or unequal, and on eacli part a semicircle may be constructed ; and then the waving line composed of the small but numerous semicircumferences will every where be little distant from the given straight line, and may be made as little distant as we please, to any degree short of perfect coincidence ; while yet the lengtl) of the undulating line will not tend to become equal to the length of the straight line, but will bear to that length a constant ratio greater than unity, namely the ratio of tt to 2. But it is evident that sucli cases as these are excluded from the foregoing reasoning, whicli supposes an approacli of the one line to the other in shape, as well as a diminution of the linear distance between them.

From the law of varying length of a straight ray we may easily perceive (what is also otherwise evident) that the straight rays diverging from a given point x'y'z', or converging to a given point xyz, are cut perpendicularly by a series of concentric spheres, liaving for their common equation,

const.; (14)

and more generally, that if a set of straight rays be perpendicular to any one surface, they are also perpendicular to a series of surfaces, determined by the equation (11), that is, by tlie condition that the intercepted portion of a ray between any two given surfaces of the series shal1 have a constant length. Analogous consequences will be found to follow in general from the law of varying action.

It may be useful to dwell a little longer on tlie case of rectilinear paths, and on the consequences of the mathematical conception of luminous or visual communication as a motion from point to point along a mathematical straight line or ray, before we pass to the properties of other less simple paths.

t/m Paths of Light ami of the Planets.

It is an obvious consequence of this conception, that from any one point (^), considered as initial, we may imagine light, if unobstructed, as proceeding to any other point [B) considered as final, along one determined ray, or linear path; of whicli the shape, being straight, is tfie same whatever points its ends may be ; but of which the length and the position depend on the places of those ends, and admit of infinite variety, corresponding to the infinite variety that can be imagined of pairs of points to be connected. So that if we express by one set of numbers the places of the initial and of the final points, and by another set the length and position of the ray, the latter set of numbers must, in mathematical language, be functions of tlie former ; must admit of being deduced from them by some fixed mathematical rules. To make this deduction is an easy but a fundamental problem, which may be resolved in the following manner.

Let eacli of the two points ^4, B, be referred to one common set of three rectangular semiaxes OX, OY, 0X, diverging from any assumed origin 0 , • let the positive or negative co-ordinates of the final point B, to which the light comes, be denoted by .T, y, z, and let the corresponding co-ordinates of the initial point A, from which the light sets out, be denoted similarly by .Y, f, s' ; let P be the length of the straight ray, or line XB, and let a, ß, y, be the positive or negative cosines of the acute or obtuse angles whicli the direction of this ray makes with the positive semiaxes of co-ordinates : the problem is then to determine the laws of the functional dependence of tlie positive number P, and of the three positive or negative numbers a, ß, y, on the six positive or negative numbers a;, y, z, Y, g’, z'; and tills problem is resolved by tlie following evident formulae ;

f^= {x-x'y (ÿ -ÿ'f (z-zy, . (15)

x~x' nbsp;nbsp;nbsp;y-y

~ p^ ’ J7 ’ T^— JZ '

It is a simple but important corollary to this solution, that the laws of the three cosines of direction a, ß, y, expressed by the equations (16), are connected witli the law of the length P, expressed by the formula (15), in a manner whicli may be stated thus ;

ar ar _ sv ex’ ^ tÿ’ quot;^ Sz n

On a General Method of Ea^pressing

8 being here a characteristic of partial differentiation. We find, in like manner,

differentiating the function V with respect to the initial co-ordinates. And since the three cosines of direction, a, ß, y, are evidently connected by the relation

a’ ß’ y^ = 1, (19)

we see that the function E satisfies simultaneously the two following partial differential equations, of the first order and second degree.

The equations (17), (18), (20), will soon be greatly extended ; but it seemed well to notice them here, because tliey contain the germ of my general method for the investigation of the paths of light and of the planets, by the partial differential coefficients of one characteristic function. For the equations (17) and (18), which involve the coefficients of the first order of the function E, that is, in the present case, of the length, may be considered as equations of the straight ray which passes witli a given direction through a given initial or a given final point: and I have found analogous equations for all other paths of light, and even for the planetary orbits under the influence of their mutual attractions.

The equations (16) when put under the form

X—x'=aP'^ y—y'=ßj^, Z—z'=yV, (21) give evidently by differentiation

d.rzzadK, d^=ßdp’', dz=-fdV', nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(22)

and therefore

dP'^zzdaif dÿ^ dz^t nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(23)

the symbol d referring here to an infinitesimal change of tlie

the Puths of Light and of the Planets.

final point B, by a motion along the ray prolonged at its extremity ; in such a manner that the equations (22) may be regarded as differential equations of that ray. They give the expressions

dx - nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;dz

~ nbsp;nbsp;nbsp;nbsp;nbsp;~ dV’

which may, by (23), be put under the form

8 implying still a partial differentiation, and dV being treated here as a function of dx, dÿ, dz. And comparing the expressions (25) and (17), we obtain the following results, which we shall soon find to be very general, and to extend with analogous meanings to all linear paths of light.

«F _ MT ^ _ êdV Sjf _^SdV

Sx nbsp;nbsp;nbsp;nbsp;idx ’ ^ My ’ Sz Sdz

It must not be supposed that these equations are identical ; for the quantities in the first members are the partial differential coefficients of one function, F, while those in the second members are the coefficients of another function dV.

In lilie manner, if we employ (as before) the characteristic d' to denote the infinitesimal changes arising from a change of the initial point .A, by a motion along the initial element of the ray, we have the differential equations

df = —af V, flf z z — ßd/ P’',df= ~yfl'K, (27)

d’ V being as before the initial element taken negatively, so that

d'P'^ = d'x' (f^'^ d'z^;

we have therefore

a= - ^^ nbsp;nbsp;_^^ nbsp;- _^^

Sd'x' ’^d'y ’ ^ ~ nbsp;nbsp;nbsp;^d'z'

and consequently, by (18), ^ _ 8£r ^ _ Sd'P Sjf _ SdT

22 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;On a General Melhocl of E.epressing

The same remarks apply to these last results, as to the equations (26).

The general law of stationary action, in optics, may now be tlms stated.

Tlie optical quantity called action, for any luminous patli liaving i points of sudden bending by reflexion or refraction, and having therefore f l separate branches, is the sum ofi l separate integrals,

Action =r=^fdy^'' = V''- K- V^^^

- r^quot;^ •• rO b, (31) of wliicli each is determined by an equation of the form

= J'dV^’'^ = J'v^'^ 's/ dx^'i*■i-dÿ‘''gt;* ds^r)»(32) the coefficient v^'^ of the element of the path, in the r® medium, depending, in the most general case, on the optical properties of that medium, and on the position, direction, and colour of the element, according to rules discovered by experience, and such, for example, that if the r** *medium be ordinary, v^'^ is the index of that medium ; so that dr^’')is always a homogeneous function of the first dimension of the differentials (lx'') , dp'’'gt; , (Iz^’quot;’, which may also involve the undifferentiated co-ordinates x^')p‘')z^’'^ themselves, and has in general a variation of the form

g(/r^’'^=a^’'^ W^ t'’'gt; Sdÿ'”’ vquot;’Sdz quot;-gt;

if we put for abridgment

err ^dr^')

~ Sdx''» ’ ’■ quot;nbsp;Sdÿ'’'gt; ’quot; ~ Sdz^v gt;nbsp;nbsp;nbsp;nbsp;(31)

and

ds^'^ = vlt;amp;ƒ»•;* «yrj* 4.f^; nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(.g^^

we have also, by the homogeneity of df^^’'gt;,

If now we change the co-ordinates x^'^i/'^z^'')of (be luminous patli to any near connected co-ordinates

Jr:”’ = x'''gt; ,^'rgt; yCrj _yfrj ^^^(n , ^W - ^cn 4, ^^,gt;; (gy)

Ike Pulks of Ligkl and of ike Planets.

£ being any small constant, and ^‘'’'if^il^'’'gt; any functions of £ and of the co-ordinates xW y^’'^ ^W , which do not become infinite for t—o, nor for any point of the r“ ' portion of the path, and which satisfy at the meeting of two such portions the equation of the corresponding reflecting or refracting surface, and vanish at the ends of the whole patli ; we shall pass hereby to a near line having the same extremities as the luminous path, and having its points of bending on the same reflecting or refracting surfaces ; and the law of stationary action is, that if we compare the integral or sum, r=s/6Zrw, for the luminous path, witli the corresponding integral P, for this near line, the difference of these two integrals or actions bears an indefinitely small ratio to the quantity £, (which makes the one line differ from the other,) when this quantity £ becomes itself indefinitely small : so that we have the limiting equation,

lim.l(r.-D=0, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(38) that is,

lim. SX rw - F« ) = 2/lira, 'fdVf^ -dF^gt;-^)=0, (39) or finally

(40)

To develop this last equation, we have, by (33) and (37),

„W

6 nnmg ot the 'portion of the path, J*(h )^'quot;’ -yW^'W ^(r)^(r) _/(r)^q,) nbsp;„(r)^W

—Vlt;*’) j('(r)

Vquot; (s« nbsp;nbsp;nbsp;nbsp;nbsp;)

On a General Method of Expressing

Jquot;' (Sw '^^'’■^- ^quot;‘’■’)•

And since the extreme values, and values for the points of arbitrary functions ?nÇ, are subject to the following conditions :

^'''^ = 0, V‘*’ = 0, ^(’) = 0, ^G i) = 0, n(H-i) = 0

^^^ ’)=0, and

$(’•) = ^(* 1), ,(»•) = ^Xr-H), ^(r) - ^Xr l),

r varying from 1 to z; and finally, for every value of r within the same range, to the condition

0,

M^ ) being either serainormal to the ’reflecting or refracting surface at the r*» point of incidence, and n^;^ nlt;r) ,r^1 being tlie cosines of the angles which nW makes with the three rectangular positive semiaxes of co-ordinates xpz ; the law of stationary action (40) resolves itself into the following equations :

1A- =|gt;gt;. (46)

and

ct'('' 1) — aW = XW fj^{r) . /(r l) _ ^w _ XW^W.

wW)-vW^xWn«'-); nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(47)

in which XW is an indeterminate multiplier. The three equations (46), whicli may by the condition (36) be shown to be consistent with each other, express the gradual changes, if a^iyj of a ray, between its points of sudden bending; and the equations (47) contain tlie rules of ordinary and extraordinary

^”4 refraction. All these results of that known law, which I have called the law of stationary action, are fully confirmed by experience, when suitable forms are assigned to the functions denoted by uW.

For example, in the case of an uniform medium, ordinary or extraordinary, the function uW is to be considered as independent of the undifferentiated co-ordinates a;Wygt;-) ^(’') , and the differential equations (46) of the r**quot; portion of the luminous path become simply

(he Paths of Light and of the P/anets.

d'r^'^ = 0, d-'’'- = 0, tZv'quot;’ = 0, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(48)

and give, by integration,

«r'’^’ = const; T^''’ = const; v'’’ ^ = const- j (49) they express, therefore, the known fact of the rectilinear propagation of light in a uniform medium, because in such a medium, ctWtWuW depend only on the colour and direction, but not on the co-ordinates of the path, and are functions of a^’')ßt''’ yl’’^ not including x^’'^y^'' gt;z^'^ , if we put for abridgment

„(0 = nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;ß(r)

so that a{^} ß(’') yf) represent the cosines of the inclination (in this case constant) of any element of the r*» portion of the patli to the positive semiaxes of co-ordinates. The formulæ (46) give also the known differential equations for a ray in the earth’s atmosphere.

With respect to the rules of reflexion or refraction of light, expressed by the equations (47), they may in general be thus summed up ;

zr^ffW ^^△’■^quot;^ Z?gt;AvW= 0: nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(51)

in which A refers to the sudden clianges produced by reflexion or refraction, and t^ft^ft^f are the cosines of the inclinations to the positive semiaxes of co-ordinates, of any arbitrary line t^’''', whicli touches the r'quot; reflecting or refracting surface, at the r''’ point of incidence, so that

t^fn^p t^'^n^;^ ffn^f = 0.

In the case of ordinary media, for example, we have aW =vW«W, rW =üWßW, „W ^^^y^; and the equation (51) may be put under the form A.v^’'gt;«W=0, in which

= «'quot;’e^ ß'quot;gt; ZW yW^-), (55)

so that the unchanged quantity «fr; «^n« the projection of the index «(»•)on the arbitrary tangent t(r), eadi index being measured from the point of incidence in the direction of the Torres-

On a General Method of Expressing

ponding ray : which agrees with the law of Snellius. In general, if we put

’ = ^/ ffW«4- t(’')’-)-vW* , nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(56)

and

a'quot;’ =pWpW, r'*-’ =vWvW , v'quot;’ = v'’'’v^ , nbsp;nbsp;nbsp;nbsp;nbsp;(57)

we may consider akOTG) wW as the projections, on the axes of co-ordinates, of a certain straight line vCO, of which the lengtli and direction depend (according to rules expressed by the foregoing equations) on the form of the function «(»■)or dE(.r}^ and on the direction and colour of the element of tlie luminous path, before or after incidence ; and if we put

vW = pW^W v^’/W v«ZW, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(58)

tlie equation (51) will take the form

A.p^rW^O, nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(59)

which expresses that the projection of this straight line rf) on any arbitrary tangent tCO to the reflecting or refracting surface, at the point of incidence, is not changed by reflection or refraction, ordinary or extraordinary : whicli is a convenient general form for all the known rules of sudden change of direction of a patli of liglit. In the undulatory theory, I have found that the line vG) is the reciprocal of the normal velocity of propagation of the wave ; and its projections may therefore be called components of normal slowness: so that the foregoing property of unchanged projection of the line pW, may be expressed, in the language of this theory, by saying that the component of normal slowness in the direction of any line which touches any ordinary or extraordinary reflecting or refracting surface at any point of incidence is not clianged by reflection or refraction. It was, however, by a different method that I originally deduced this general enunciation of the rules of optical reflexion and refraction, namely, by employing my principle of the characteristic function, and that other general law, of which it is now time to speak.

This other general law, the law of varying action, results from the known law above explained, by considering the extreme points of a luminous path as variable : that is, by not supposing the six extreme functions (43) to vanisli. Denoting, for abridgment, the three final functions of this set by g ut, and the three initial functions by K'nZ'j and writing similarly

and d'Vbeing, according to the same analogy of notation, the infinitesimal change of the whole integral V, arising from the infinitesimal changes d'ad, d'f, d'z', of the initial co-ordinates, E

28 nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;^gt;i ft General Method of Expressing that is from a motion of the initial point .r' y'z along the initial element of the luminous patli ; so that lt;/'k'1s the initial element of tlie integral taken negatively,

dV—— V -^(l'x'^-j-d'ÿ'^ d'z'^. nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(6'1')

If tlien we consider the integral or action F as a function (wliidi I have called the characteristic function) of tlie six extreme co-ordinates, and if we differentiate this function witli rcispect to these co-ordinates, we see tliat its six partial differential coefficients of the first order may be represented generally by the equations (26) and (30), whicli were already proved to be true for the simple case of rectilinear paths of liglit. And as, in that simple case, those equations, being then equivalent to the formulæ (17) and (18), were seen to determine the course of the straight ray, which passed with a given direction througli a given initial or a given final point ; so, generally, wlien we know tlie initial co-ordinates, direction, and colour of a luminous path, and the optical properties of the initial medium, we can determine, or at least restrict (in general) to a finite variety, the values of the initial coefficients

g£F sd'r ^d'y

Sd'x' ’ ^d'y' ’ Sd'z'’

wliicli form the second members of the equations (30) ; and therefore we may regard as known the first members of the same equations, namely the partial differential coefficients

ck_ gk gt£

of the characteristic function V, taken with respect to the known initial co-ordinates : so that if the form of the function V be known, we have between the final co-ordinates, x, y, z, considered as variable, the three following equations of the path, or at least of its final branch,

Sfquot; , sr gr

= const., = const.,— = const. (65)

These three equations are compatible witli eacli other, and are equivalent only to two distinct relations between the variable co-ordinates xyz, because in general F must satisfy a partial differential equation of the form

0 = fi' (a', f,v,x,^’,ff nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(66)

lAe Pal/n of Li^/it and of l/tc Planefs.

in which, by what has been shown,

z _ nbsp;1^ - _ 8F , 8r

quot; nbsp;nbsp;S./’- gy'’* '- - gZ’

and which is therefore analogous to the second formula (20) ; this equation (66) being obtained by eliminating the ratios of d'x',d'y', d'z, between the general formulae (30). In like manner the formulae (26) give generally a partial differential equation of the form

_ z87 gr grX

Z/’ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;’ nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;(68)

analogous to the first of those marked (20), and the tliree following compatible equations between the variable initial co-ordinates xfz' of a patli of light whidi is obliged to pass with a given direction througli a given final point,

gF nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;gr nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;nbsp;gr

— = const., — = const. , — = const. (69)

But for the integration and use of these partial differential equations, the limits of the present communication oblige me to refer to the volumes, already mentioned, of the Transactions of the Royal Irish Academy.

I may, however, mention here, that my employment of tlie characteristic function P, in all questions of reflexion and refraction, is founded on an equation in finite differences, which, by the integral nature of this function P, is evidently satisfied, namely.

f^P’'—0=\u ;

A referring, as before, to the sudden changes produced at any reflecting or refracting surface having for its equation

M=0;

and A being an indeterminate multiplier, employed for the purpose of being able to treat the co-ordinates of incidence as three independent variables. For example, the formulae (47), for a sudden change of direction, result immediately from (70), under the form

, gF nbsp;nbsp;nbsp;gw . 8F „ gw gF g« ,,

△ g.r“ 8.T ’ △ gyquot;gy’ △ g^ “ '^^ ’ ^'^^

Oft a General Melhod of E-cpressing

by differentiating with respect to the co-ordinates of incidence,

as three independent variables, and then reducing by the equation (71) of the ordinary or extraordinary rellecting or refracting surface. These results respecting the change of of a luminous patli may be put under the form

\So: Sy nbsp;nbsp;Sÿ Sx/ nbsp;nbsp;nbsp;nbsp;’

Sy Sz nbsp;nbsp;S~ Sÿ' nbsp;nbsp;nbsp;nbsp;’

Sw ST SwST\

Sz Sx Sx Sz'

and in general, all tlieorems respecting the changes produced by reflexion or refraction in the properties of an optical system, may be expressed, by the help of the formula (70), as permanences of certain other properties. Tlie remarkable permanence, already stated, of the components of normal slowness of propagation of a luminous wave, was suggested to me by observing that my function V is (in the undulatory theory) the time of propagation of light from the initial to the final point, and tlierefore that the waves (in the same tlieory) are represented by the general equation

F = const., (^5)

and the components of normal slowness by the partial differential coefficients of T of the first order. The properties of the function r, on whicli my whole optical method depends, supplied me also, long since, with a simple proof of the contested theorem of Huygens already mentioned, namely, that the rays of any ordinary homogeneous system, which after issuing originally from any luminous point, or being (in an initial and ordinary state) perpendicular to any common surface, have undergone any number of reflexions or refractions ordinary or extraordinary, before arriving at their final state, are

the Paths of Light ami of the Planets.

in that state perpendicular to a series of surfaces, namely, to the series (75), whicli are waves in the theory of Huygens : because, by the properties of my function, the differential equation of tliat series is

aSjT ß^^ 782=0, (76)

a,ß, 7, being the cosines whicli determine the final direction of a ray. It was also by combining the properties of the same cliaracteristic function Fquot; with the physical principles of Fresnel, that I was first led, (from perceiving an indeterminateness in two particular cases in the relations between the coefficients

SdV MV MV

Mx’ My’ Mz’

and the ratios of dx, dy, dz,) to form that theoretical expectation of two kinds of conical refraction whicli I communicated in last October (1832) to the Royal Irisli Academy and to Professor Lloyd, and whicli the latter has since verified experimentally. Mr. Mac Cullagli has lately informeel™® that the same two indeterminate cases in Fresnel’s theory had occurred to him from geometrical considerations, some years ago, and that he had intended to try to what geometrical and physical consequences they would lead.

The method of the characteristic function has conducted me to many other consequences, besides those whicli I have already publislied in the Transactions of the Royal Irisli Academy: and 1 think that it will hereafter acquire, in the liands of other mathematicians, a rank in deductive optics, of the same kind as that whicli the method of co-ordinates has attained in algebraical geometry. For as, by the last-mentioned me-tliod. Des Cartes reduced the study of a plane curve, or of a curved surface, to the study of that one function wliicli expresses the law of the ordinate, and made it possible tliereby to discover general formulae for the tangents, curvatures, and all other geometrical properties of the curve or surface, and to regard them as included all in that one law, that central algebraical relation : so I believe that mathematicians will find it possible to deduce all properties of optical systems from the study of that one central relation wliich connects, for eacli particular system, the optical function V witli the extreme co-ordinates and the colour, and which has its partial differential coefficients connected witli the extreme directions of a ray, by the law of varying action, or by the formulae (26) and (30).

Chi a General Melhod of Expressing

It only now remains, in order to conclude tlie present re-marks, that [ should brielly explain tlie allusions already made to my view of an analogous function and method in the research of the planetary and cometary orbits under the influence of tlieir mutual perturbations. I'he view itself occurred to me many years ago, and I gave a short notice or announcement of it in the XVtli volume (page 80) of the Transactions of the Royal Irish Academy ; but I have only lately resumed the idea, and have not hitherto published any definite statement on the subject.

To begin witli a simple instance, let us attend first to the case of a comet, considered as sensibly devoid of mass, and as moving in an undisturbed parabola about the sun, whicli latter body we shall regard as fixed at the origin of co-ordinates, and as having an attracting mass equal to unity. Let r be the comet’s radius vector at any moment ( considered as final, and r' the radius vector of the same comet at any other moment ^' considered as in itial ; let alsorquot; be the chord joining the ends of r and r', and let us put for abridgment

r-Q^r P-i-rquot; T 2 fr r'-rquot; ; '(77)

then, I find, that the final and initial components of velocity of the comet, parallel to any three rectangular semiaxes of co-ordinates, may be expressed as follows by the cofiicients of the function V,

dx SV dg gr gz ^v dl ~ Sx ’ dt - Sg’^quot;^^^

dC Sx'' dC Sg'’ ft' ’

and that this function E satisfies the two following partial differential equations,

whicli reconcile the expressions (78) with the known law of a comet’s velocity. I find also that all the other properties of

Ike Polks of Light and of Ike Planets.

a comet’s parabolic motion agree with and are included in the formulae (78), when the form (77) is assigned to the function J^. They give, for example, by an easy combination, the theorem discovered by Euler for the dependence of the time (t—t') on the parabolic chord (rquot;) and on the sum (r'-f-r') of the radii drawn to its extremities.

More generally, in any system of points or bodies which attract or repel one another according to any function of the distance, for example, in the solar system, I have found that the final and initial components of momentum may be expressed in a similar manner, by the partial differential coefficients of the first order of some one central or ekaracteristic function P of the final and initial co-ordinates ; so that we have generally, by a suitable choice of P,

dzi SV

dxi SV

and

w,, m,, amp;c., being the masses of the system, and the function P being obliged to satisty two partial differential equations of the first order and second degree, whicli are analogous to (79), and may be thus denoted

tlie function F involving the final co-ordinates, and tlie function F' involving similarly the initial co-ordinates, and the common form of these two functions depending on the law of attraction or repulsion. In the solar system

On a General Method of Expressing, ^c.

U being a certain constant ; and in general the partial differential equations (82) contain the law of living forces, while the other known general laws or integrals of the equations ofmotion are expressed by other general and simple properties of the same characteristic function E: the coefficients of whicli function, when combined with the relations (80) and (81), are sufficient to determine all circumstances of the motion of a system. By this view the research of the most complicated orbits, in lunar, planetary, and sidereal astronomy, is reduced to the study of the properties of a single function E; which is analogous to my optical function, and represents the action of the system from one position to another. If we knew, for example, the form of tins one function E for a system of three bodies attracting according to Newton’s law, (suppose the system of Sun, Earth, and Moon, or of the Sun, Jupiter, and Saturn,) we should need no further integration in order to determine the separate paths and the successive configurations of these three bodies ; the eight relations, independent of the time, between their nine variable co-ordinates, would be given at once by differentiating the one function F, and employing the nine initial equations of the form (81), which in consequence of the second equation (82) are only equivalent to eight distinct relations, the positions and velocities being given for some one initial epoch ; and the variable time t of arriving at any one of tlie subsequent states of the system would be given by a single integration of any combination of these relations with the equations (80). The development of this view, including its extension to other analogous questions, appears to me to open in mechanics and astronomy an entirely new field of research. I shall only add, that the view was suggested by a general law of varying action in dynamics, which I had deduced from the known dynamical law of least or stationary action, by a process analogous to that general reasoning in optics which I have already endeavoured to illustrate.

Observator^ of Trinity College Dublin, September, 1833.