473 



UXDULATORY THEORY OF LIGHT. 



UNDULATORY THEORY OF LIGHT. 



471 



or cosine, or a mixture of the two, that is, c sin m w + c' cos m w, 

 (w being written for v t .;), which again may be put under the form 

 a sin (m to + a). For in the first place, any dynamiual 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. If, 

 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 2 IT m-', may be expressed by an infinite series of the form A,, + 

 A, cos m x + A 2 cos 2 m x + + B, sin m x + B 2 sin 2 m 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 </*, 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 

 iuiti as well as a leyitimaie 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 differenl 

 periods are not, under ordinary circumstances, physically separated. Bui 

 in light we have nothing answering to quality, and disturbances 01 

 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 ij/ by means of a simple sine 

 or cosine. If A be the length of a wave, mx must change by 2 IT when 

 x changes by A, so that m = 2 ir A '. Hence we may take as the 

 standard expression for the disturbance 



2ir -I 



(rt - x) + A |- 



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 



[2-r -. 



i sin ^ (i't x) + B y 



And compounding the disturbances belonging to the two series, we 

 shall have for the result 



2r 2r 



(a cog A + 6 COB B) sin -^ (rt x) + (a sin A + 6 aiu B) cos (rt r), 



which may be transformed into 



/ 2ir i 



csin ^ (vt x) +'v V 



c and c being given by the equations 



M c 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 pkane 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 (1), that according to 

 the value of A B, 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, according 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 It 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 

 j 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 c 3 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 the mixture of independent streams the mean 

 value of a 2 *- i s + 2a6 cos (A B), or a 2 + 4- ; or the intensity will be the 

 sum of the intensities of the separate streams, aa 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 2 a, and those of the brightness o and 4 a-, that is absolute dark- 

 ness, and four times the brightness of either stream alone. 



We shall apply these formula; to express the intensity at any point 

 of the field of view in the case of the mixture of two streams of light 

 coming originally from a luminous point, and afterwards reflected from 

 two slightly inclined mirrors. [INTERFERENCE.] 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 leus 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 r, 

 i', and the point of the focal plane at which the brightness is sought by 



M, and if we take c sin (rt - x) to denote the disturbance coming 

 from I, that coming from i must be denoted by c sin [ v I x 



(I'M m)j. Let o be tho middle point of the linen'. 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 1 1'. 

 Let p, q be the co-ordinates of M, 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 I' perpendicular to the mirrors. We 

 shall suppose, in conformity with the experimental circumstances of 

 the case, that :/, ]>, q are small compared with a and 4. Let 

 a' = ^(a 2 \d 2 ) be the distance of o from the intersection of the 

 mirrors. Then I M 2 = (o' + b)"- + (p - J df + <f, i' M 2 = (a' + 4) 3 -r (p + 



q-, and I'M-IM = (i'v 3 -iM ! )-Hi'M + iM) = 2/>ci-:-(i'M + iM) = 



a 

 nearly. We have therefore for the intensity (L) of the mixture 



Zxpd 



= 4C 2 COS 2 : 



Since L does not involve q, 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 --- ~ .'-, L is a maximum, and equal to i c 2 . At points 



a 



lidway between these, where therefore p is an odd multiple of half 

 that quantity, t, vanishes altogether. Hence we have a series of 

 alternately bright and dark bars, extending on each side of the central 

 plane, or that bisecting 1 1' 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 we infer 

 that the wave length also decreases from the red to the violet. The 

 obliteration of the bars at a moderate distance from the centre when 

 white light ia used has already been explained. [INTERFERENCE.] 



