dulum. The distance of a cycloidal pendulum from its place of rest is expressed by a sin {t in {ε √(2) + 0 }, or a sin (nt + C). The required function then for the disturbance of a particle is a sin { (ut - a) +1}. For while we consider the motion of a single particle only, x is constant, and the expression is пх a sin (nt + C), where CA- v At the same time it is a function of vt-x, and therefore satisfies the condition requisite for an undulation. We will therefore assume as the expression for the disturbance But it is plain that, without any loss of generality, we may get rid of A by altering the origin of time from which t is reckoned, or the origin of linear measure from which x is reckoned. We may therefore take as the expression for the disturbance when one undulation only (consisting of an indefinite number of similar waves) is considered. 8. A form somewhat more convenient may be given thus. The expression goes through all its periodical values while nt increases by when a single undulation only is considered. It is to be observed that a is the maximum vibration of any particle. PROP. 5. To explain the interference of undulations. 9. By interference is meant the co-existence of two undulations in which the length of a wave is the same. The conception of interference is not in any circumstances easy+, and it is more particularly difficult with regard to Physical Optics, from our ignorance of the physical causes to which the undulation is due. This is the form of the function tacitly assumed by Newton for the disturbance of particles of air, in his investigation of the velocity of sound. (Principia, Lib. II. Prop. 47). The simplest illustration is perhaps to be found in the crossing of two waves on the surface of water, each of which affects the surface in the same manner as if the other were not there. If we conceive two series of waves, produced by agitating the surface at two points, to spread in circular forms with equal and uniform velocities, and if one agitation was created a little before the other, so that the wave proceeding from one has proceeded as far (in a given direction) as the hollow between two waves proceeding from the other, then it may be imagined that at every point where this holds, the elevation of one wave may exactly fill up the hollow of the other, and the surface will be, in fact, undisturbed. 10. *If we investigate, from the known properties of air, the motion of the particles (supposed parallel to a fixed line), we find this differential equation for the disturbance of a particle d2X d2X v2 :0 dx2 (v being a constant, = mgH in the common notation, and X being the disturbance from the state of rest), and the solution is where the form of the functions is to be determined by the initial circumstances. Or if we suppose the wave of air to move only in one direction, the expression for the disturbance will be (vtx). And this may be divided into several different expressions, ¥ (vt − x) +x (vt − x) + v (vt − x) +&c. where the form of each is to be determined by the initial circumstances, or by the cause of the undulation. If there was only a single original cause for the undulations, there would be only a single term (vtx) to be preserved. But if there were several distinct original causes for the undulations, there would be a single term corresponding to each of these to be preserved, and the whole disturbance would be the sum of all these terms. And it is particularly to be remarked, that the whole disturbance thus found as the effect of all the original causes together, is precisely the sum (with their proper signs) of a number of disturbances, each of which would have been produced by one of the original causes acting separately. 11. Now if we examine to what this property of the solution of the differential equation (namely that it can be broken up into several parts all similar to each other and to the whole) is due, we find it is owing to the circumstance that the differential coefficients of u were raised only to the first power in The reader who is not familiar with the investigation of the problem of Sound may omit the next three articles. the equation, or (to express it in other words) that the equation was linear. For the differential coefficient of the sum of a number of functions is the same as the sum of the differential coefficients: but the square of the differential coefficient of a sum of functions is not the same as the sum of their squares, &c. If then the differential coefficients (and the unknown quantity itself if it enters in the equation) be all of the first dimension, the substitution of a sum of functions is the same as the sum of their substitutions separately, and therefore if each of those functions satisfies the equation, their sum will satisfy the equation. But if they are raised to a higher power, the substitution of the sum is not the same as the sum of the substitutions, and therefore if each function satisfies the equation, their sum will not. 12. If now we retrace the steps of the investigation for air, it will be seen that the linearity of the differential equation depends upon this physical fact, that upon altering by a small quantity the relative position of particles, the forces which they exert undergo variations very nearly proportional to that small quantity. And in any other case where this holds, the equations will be linear; and the wave-disturbance of any particle, produced by a number of agitating causes, will be the sum of all the wave-disturbances which these causes would singly have produced. We can hardly conceive any law of constitution of a medium in which undulations are propagated, where this does not hold, and we shall therefore suppose it to be true for light. 13. Taking it then as a fact that the disturbance of every particle produced by two co-existent undulations will be the sum of the disturbances which they would produce separately, we will consider the nature of the disturbance produced by the superposition of two such undulations as those treated of in (7) and (8), each of which is represented geometrically by fig. 1, if the vibrations are in the direction of the wave's transmission, and by fig. 2, if they are perpendicular to that direction. For convenience of figure, we will suppose them of the latter class: but all that we say will apply as well to the former. We will suppose the length of a wave the same in both undulations. In fig. 3, let the Italic letters of (a) represent the state of an undulation, at the time T, where the law of vibration is and let (B) represent the state of another undulation at the same time where the law of vibration is If from any point in (a) we measure upwards a distance equal to the elevation of the corresponding point of (B), or measure downwards a distance equal to the depression of the corresponding point of (B), we shall determine the position of the Roman letters. Their distances from the straight line represent the effect of the superposition of the two undulations. This is evidently an undulation of the same kind, and with waves of the same length, as either of the others. But in the instance as we have supposed it, the addition of the undulation (B) to (a) has diminished the maximum vibration of the latter, and has made the maximum to exist at a different part of the line. Thus we see that the magnitude of vibration in an undulation may be diminished by the addition of another undulation transmitted in the same direction. This is a point of great importance, and deserves the reader's attentive consideration. 14. The geometrical figures which we have given are merely illustrations: the conclusion that we have arrived at will be more readily obtained from the algebraic expressions. Adding together the two disturbances |