hitherto we have con fiued ourselves to the laws of reflection and refraction, which were in fact demonstrated by Huygens long before the principle of interference was known. Our subject naturally leads us, in the next place, to interference, to which a special article has already been devoted. [INTERFERENCE.] In that article the subject has been generally explained, and two fundamental experiments, due to Fresnel, have been mentioned, which show that interference is an essential property of light. We shall here, therefore, proceed to the formula which gives the intensity of the light resulting from the mixture of two interfering streams.
It will be necessary, in the first place, to express analytically the disturbance in a single stream of light. We shall suppose, for the sake of simplicity, that the waves are plane, and that the maximum accession of the particles of ether is the same at one point of space as another. This will be sufficient in any ease, provided we confine our attention to a small portion only of the fronts of the waves, and to variations of distance in a direction perpendicular to the front, such that the change of intensity due to convergence or divergenoe need not be taken into account. Let the ether be referred to the rectangular axes of x, y, z, s being measured in the direction of propagation ; let t be the time, and r the velocity of propagation. Since the waves are supposed plane, the disturbance will be independent of y and z, and therefore will depend only on x and t. Moreover, according to the fundamental notion of an undulation, whatever disturbance exists at the time t at the distance x from the origin will, after the lapse of the time 8 t, be found at a plane further in advance by 8 8 a being con nected with 8 t by the relation 8x=e8 t. Hence the disturbance remains the same, provided the difference between r t and .c remains unchanged, and therefore for one value of r t — .c the disturbance will have one value, for another value another, and so on. In other words, the disturbance will be some function sp (1, t — a.) of v t — x, the direc tion of the motion of the particle/ being left a perfectly open question. The character of the undulations will depend on the form of the func tion tio. Now we have seen mason to believe that in light we always have to deal with a succession of a great number of similar periodic disturbances, and, further, that the colour depends upon the periodic time. But in all optical phenomena the effects of lights of different colours are simply superposed, and therefore as regards the form of ty we are justified in restricting ourselves to the consideration of a single regularly periodic function. Among such functions there is one which claims our special attention, namely, that which is expressed by a sine or cosine, or a mixture of the two, that is, c sin m m + c' cos in w (so being written for t — .r), which again may be put under the form a sin (mw + a). For in the first place, any dynamical system in a position of stable equilibrium, and therefore capable, on being dis turbed, of performing small vibrations, has for its most general small motion a motion compounded of a finite or infinite number of motions, expressed, so far as the time is concerned, by a sine or cosine.
therefore, we suppose the vibration of the molecules of the self. luminous body in the first instance to have been of this character, the same would have been impressed on the ether to which these vibra• tions were communicated. In the second place, by a known theorem any periodic function of w, going through its period when w is altered by may be expressed by an infinite series of the form A,, + A, cos TA X + cos 2 in X - - - B, sin TA X + sin 2 in x + - - where the first term vanishes if the mean value of the function be zero. By virtue of this theorem, and of the principle of the superposition of small motions, we should have a right to resolve the function 4', sup posing it merely to be periodic, into such a series of circular functions and consider separately the disturbance due to each. Whether this be a judicious as well as a legitimate course to pursue, depends partly on whether any such resolution takes place physically, or whether, on the other hand, there are physical phenomena which we can refer to the form of the periodic function expressing the disturbance. In sound we have a sensible phenomenon, quality, which is referable to the form of the function expressing the disturbance, while vibrations of different periods are not, under ordinary circumstances, physically separated. But in light we have nothing answering to quality, and disturbances of different periods are physically separated, as in the phenomena of inter. ference and diffraction as well as in dispersion. That disturbances expressed by a sine or cosine, rather than by some other periodic func tion, should be those which are propagated within a refracting medium with a unique velocity, and should not consequently be separated by prismatic refraction, follows from the general laws regulating the small motions of a system slightly disturbed from a position of stable equilibrium. We are led, therefore, by the phenomena with which we have to deal, to express the function 4, by means of a simple sine or cosine. If A be the length of a wave, in x must change by 2 w when x changes by A, so that tu = 2 r A Hence we may take as the standard expression for the disturbance— Suppose, now, that the ether at the same part of space is simulta neously agitated by a second series of undulations, which came originally from the same source. Suppose the directions of propaga tion to be so nearly the same that In considering, as above, a small portion only of the ether, we may treat them as the same, or the wave fronts as parallel in the two series ; and suppose the directions of vibration to be likewise the same in the two series, except as to a small angle depending upon and comparable with the small angle between the directions of propagation. Suppose, however, that the amplitude of excursion of the particles is different in the second series from what it is in the first, and further, that from having had to describe a longer or shorter path, or from any other cause, the undulations in the second series are ahead of or behind those in the first. Then we may repre sent the disturbance in the second series by 2w b sin (ve — + n).