598 LIGHT matical optics can only then attain a coordinate rank with mathe matical mechanics . . . , when it shall possess an appropriate method, and become the unfolding of a central idea. ... It ap pears 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 highest results of induction. . . . [This] 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 extraor dinary, along which light (whatever light may be) extends its influence successively in space and time : namely, that this linear path of light, from one point to another, is always found to be such that, if it be compared with the other infinitely various lines by which in thought and in geometry the same two points might 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. "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 which seems to offer naturally a method such 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 the first in the descending and deductive way. " The former of these two laws was discovered in the following manner. The elementary principle of straight rays showed that 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. These facts were thought by some to be instances and results of the simplicity and economy of nature ; and Ferrnat, 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 metaphy sical 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 with the law of refraction, discovered experimentally by Snellius, Format 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 refraction, 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 w r as 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 found, by his 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 per ceived that the supposition of a velocity inversely as the index reconciled his mathematical discovery of the minimum of the fore going sum with his cosmological principle of least time. Des cartes attacked Format s opinions respecting light, but Leibnitz zealously defended them ; and Huygens was led, by reasonings of a very different kind, to adopt Format 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 directly, not inversely, as the index, and that it was increased instead of being diminithed on entering a denser medium ; a result incompatible with the theorem of the shortest time in refraction. This theorem of shortest time was accordingly abandoned by many, and among the rest by Maupertuis, who, however, proposed in its stead, as a new cosmological principle, that celebrated law of least action which has since acquired so high 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 the 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 of action with Newton s estimate of the velocity of light as increased by a denser medium, and as proportional to the refracting index, and with Format s mathematical theorem 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, though not always to the least possible time. He proposed this view as reconciling physical and metaphysical principles which the 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 with 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 such curve be compared with 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) between the two near curves ; a theorem which I think may be advantageously expressed by saying that the action is stationary. Lagrange extended this theorem of Euler to the motion of a system of points or bodies which act in any manner on each other ; the action being in this case the sum of tlie 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." We give, first, a very brief indication of the nature of Hamilton s Varying method, as applicable directly to the corpuscular theory. Here the action. action of a corpuscle is the quantity which possesses the stationary Corpus- property. Let v be the velocity at any point x, y, z of the medium, cular ds an element of the path, o, ft, y the direction cosines of ds, which theory, are supposed to enter linearly and homogeneously into the expres sion for v. Then the action V is given by Y-fvds. Hence, for a path nowhere finitely distant from the first, But I dv dx dv^ -j- dy -j- dz dv a-r- the first three terms depending on the translation of the element ds, the others on its change of direction, and all the differential coefficients being partial. The homogeneity of v gives dv dv _ dp + 7 dy^ V Also dSx = Sdx = 8 .ads = 8a. ds + adds, with two similar equations in y and z. By the help of these, and a partial integration of the factors dSx, &c., we have ..,, dv^ dv^ dv _ ~da^df? J ^dy~ VT 5 *(^-4 y V . . . . 1 J L dx daj where the integrated part is to be taken between proper limits. If the initial and final points of the path be fixed, Sx, &c., vanish in the integrated part, and the stationary condition shows that we must have dv , , dv -rds - d = ax da 0, with other two similar conditions, only two of the three being in dependent because of the necessary relation a 2 + /3 2 + r = l. These may be regarded as the differential equations of the ray, or path of the corpuscle. But the essence of Hamilton s method of varying action depends upon a change of the terminal point of the ray, and leads at once to the three equations $V_dv 8V_*; 8V_rfv Sx~da Sy~dJ3 ~5z~dy which follow directly from the general value of 8V above, by taking account of the vanishing of the unintegrated part in consequence of the stationary condition. We may nor write d for 8 everywhere in these expressions.
