168 UNDULATORY THEORY OF LIGHT. 



back than the wave which our construction makes i\ie_ following wave. This 

 construction, therefore, embraces all possible cases. 



But if" the rays which are thus brought together are polarized in planes at 

 right angles to each other, then it will be manifest that the movements of neither 

 can interfere with those of the other; but, as in the case of the vibrating solid 

 again, they may produce a resultant of which the character may vary from a 

 plane vibration through every form of ellipse to a circle. Equation [1] ex- 

 presses the circumstances of this case. 



Thus, if, in that equation, we assume the difference of phase represented by 

 to be 90°, cos 6 becomes =0, sin = 1, and the equation is 



which is the equation of the ellipse when the axes of figure coincide with the ases 

 of co-ordinates. If we make a = a', then we have 



?/2 -I- 2^2 __ ^2^ 



which is the equation of the circle. 



If the difference of phase is 0°, then 



(I'^IJ^ — 2(3a'r y -f- a?x^ = 6, or a'y — ax = 0, 

 which is the equation of a straight line ; and if a once more be put = a', ar = y. 

 or the straight line makes equal angles with the directions of the original 

 molecular movements. 



If the planes of the two undulations are neither normal to each other, nor co- 

 incident, there will be an interference which will be more or less complete, as 

 the inclination of the planes is less. 



Rays of common light, if the difference of their paths be not very gi-eat, will 

 interfere, notwithstanding the fact that their undulations are confined to no 

 determinate azimuth. This fact proves, what has been above assumed, that the 

 changes of azimuth in common bght cannot be incessant. But there is one con- 

 dition absolutely indispensable to produce interference in any case; it is that 

 the rays shall have a common origin. 



If the light subjected to experiment be unpolarized, the necessity of the con- 

 dition is easily explamed. The changes of the azimuth of vibration in two 

 such rays could not, except upon a supposition Avhich has an infinity of chances 

 against it, take place at the same intervals and in the same order; and if they 

 did, the chances would be equally great against the coincidence of those planes. 

 It appears, however, to be true, as well of polarized rays as of common light, 

 that they will not interfere unless from the same origin. We are obliged, there- 

 fore, to resort to the supposition, which has a priori, moreover, strong 

 probability in its favor, that there are irregularities at the very origin of the 

 undulations, or at the surface of the luminous body, which are propagated 

 with the undulations, and which will prevent the permanent coincidence or 

 conflict of two sets of undulations, unless both are equally affected by the same 

 irregularities. Thus, if we observe the flame of a candle, we shall sec that its 

 wavering motion will make the point of departure of the undulations it gene- 

 rates unsteady. But a diffei-ence of a single one hundred-thousandth part of 

 an inch in the position of the origin of two successive sets of undulations, 

 would put them into entirely opposite phases. Considering the activity and 

 the energy of the forces at work at the surfaces of incandescent bodies, it is 

 impossible to believe that the luminous waves which they generate can have 

 their origins absolutely invariable in position. 



These things premised, we are prepared to apply the theory of undulation 

 to the explanation of all the phenomena of diffraction, polarization, and the 

 colors of thin or thick plates, in regard to which we have heretofore stated only 

 the facts. It is worth while, however, in the first place, to give a moment's at- 

 tention to an ex]ieriment first suggested by Fresnel upon purely theoretic 

 grounds, and afterward made by him with complete success; in which the cir- 

 cumstances preclude the application of any of the special hypotheses which 

 had been previously conceived, for the purpose of accounting for the phenomena. 



