And compounding the disturbances belonging to the two series, we shall have for the result w w (a cos A + b cos n) sin (rt — + (a sin A + b sin n) cos (rt x), which may be transformed into w c sin { +:0 A c and c being given by the equations a sin A + b sin n (1.) +2ab cos (A—B) ; (2.) tan c= a toe We learn, therefore, that the resultant of the two series is an undu lation of the same character as the component undulations, differing from them only in the magnitude of the coefficient of vibration, c, and in the state or phase of vibration at a given point of space and at a given instant, as determined by the value of c compared with those of A and B.
It is with c that we are chiefly concerned ; the value of c does not enter into account unless when we have a third stream interfering with the two former. Now we see from the formula (I), that according to the value of A —n, c varies between two extreme limits, which are the sum and the difference of a and b. These are, the one greater, the other less, than the greater of the two a, b. Hence we see that two streams of light from the same source reinforce each other, or else give an effect less than that of the stronger stream alone, to their difference of phase.
The question now arises, what precise function of the coefficient of vibration ought we to take as a measure of the intensity ? Various considerations tend independently of each other to the same conclusion, —that we must take the square of the coefficient of vibration as a measure of the intensity. It will be sufficient here to mention one or two.
Suppose that we have two streams of light just as before, only that in this case they come from independent sources. The theoretical difference between the present case and the former consists in this, that in the former, whatever may affect the constant A equally affects B, and therefore leaves the difference A—B unchanged ; and whatever affects a affects b in the same proportion, which is not the case when the streams are independent. In the case of streams from the same source, we may, for example, suppose that the actual disturbance consists of a series of regular periodic disturbances followed by a dis tinct series, and that by another, and so on, there being a great number of such changes in one second. The mode of interference will not thus be affected. But in the case of independent streams, A— B, though constant, it may be, during a great number of successive undulations, will go through all sorts of values a great number of times in one second, and the mean value of the term + 2ab cos. (A —B) in the expression for will be zero. If now we take the square of the coefficient of vibration for the measure of the intensity, we shall have for the intensity of tho mixture of independent streams the mean value of + + 2ab cos (A— B), or + V; or the intensity will be the sum of the intensities of the separate streams, as it ought to be; whereas if we were to take some different function as a measure of the intensity that would not be the case. Again, all mathematical investi gations relative to the propagation of small vibrations in a medium disturbed at one place show that at a great distance from the centre of disturbance the coefficient of vibration varies inversely as the distance. But experiment shows that the intensity of light varies inversely as the square of the distance, and we are thus led in a perfectly inde pendent manner to the same conclusion.
If the interfering streams are of equal brightness, b=a, and the limits of the coefficient of vibration c of the resultant stream are o and 2a, and those of the brightness o and that is absolute dark ness, and four times the brightness of either stream alone.
We shall apply these formulae to express the intensity at any point of the field of view in the case of the mixture of two streams of light coining originally from a luminous point, and afterwards reflected from two slightly inclined mirrors. [INTERFEBENCli.] Supposing, for sim plicity, the light to be reflected in a plane perpendicular to the line of intersection of the planes- of the mirrors, let a be the distance of the luminous point, and therefore that of either virtual image, from that line,*b the distance from the same line to the focus of the lens with which the fringes are viewed, d the distance of the two images. Since the length of path of either stream is the same as if it had started from the virtual instead of the actual image, if we denote these images by I, s', and the point of the focal plane at which the brightness is sought by 2 w 31, and if we take c sin — (r t —x) to denote the disturbance coming from x, that coming from f must be denoted by c sin y e — x A (f 11-1101 . Let o be the middle point of the line If. It will be readily seen that the plane drawn through o and through the line of intersection of the planes of the mirrors will be perpendicular to i e. Let p, q be the co-ordinates of 1st, measured in the focal plane of the eye-lens, the first perpendicular, the second parallel, to the plane through o just mentioned, the origin being at the intersection of that plane with a plane through I o x' perpendicular to the mirrors. We shall suppose, in conformity with the experimental circumstances of the case, that d, p, q are small compared with a and b. Let a = —#d') be the distance of o from the intersection of the mirrors. Then a (a'+ (p + (a'+ + (p + + and = m+Int)=2pd in+130= pd is +b nearly. We have therefore for the intensity (L) of the miaturo 2srpd mrpd cos = A b) A (d+Since L does not involve 7, the illumination will be arranged in bars parallel to the axis of q, and it will be sufficient to discuss its variation along the axis of p. At a series of equidistant points, for which p=o ' + or a multiple of A(a b) is a maximum, and equal to 4 At points midway between these, where therefore p is an odd multiple of half that quantity, L vanishes altogether. Hence we have a series of alternately bright and dark bars, extending on each side of the central plane, or that bisecting I e at right angles, which is in the middle of a bright bar. The scale of the system is found to depend upon the colour, decreasing from the red to the violet, from whence wo infer that the wave length also decreases from the red to the violet. The obliteration of the bare at a moderate distance from the centre when white light is used has already been explained. [INTERFERENCE.] From the expression for t. we see that the mesa illumination, or the value of Ldp for a large value of p' +//', 20, which • ft would be the uniform illumination, if the streams mixed without interfering. This is a particular example of a general principle from which inferences have already been drawn [ransom-nos], that when two or more distinct streams of light interfere, no light is destroyed by Interference, which merely causes a different distribution of the illumination.