COLOURS OF THICK PLATES. 



[173] 



Section VI. 



Investigation of the angles of diffraction. 



34. Something yet remains to be done in order to complete the theory of these rings 

 and bands, namely, to compare the two diffractions which a wave of light experiences at its 

 entrance into the glass and on its return, respectively. For the phase and intensity of a ray 

 diffracted in a given direction depend altogether on the circumstances under which the dif- 

 fraction takes place ; and were these circumstances materially different in the case of the two 

 diffractions abovementioned, the rings might be modified, or might even disappear altogether. 



Let us consider first the case of a conca,ve mirror when the luminous point and its image 

 are in the same plane perpendicular to the axis. In this case, if we consider any point P on 

 the dimmed surface, and any point M in the plane of the rings, the angle of diffraction for the 

 ray diffracted at emergence will be L 3 PM*. For the ray diffracted at entrance, the angle of 

 diffraction measured in air will be LPM 3 , that is to say, M 3 P is the course of a ray in air 

 which by regular refraction into glass would be brought into the direction of the ray diffracted 

 at P. If C be the intersection of the axis and the plane of the rings, C will be the centre of 

 the system, and the middle point of both the lines LL t and MM 3 , and therefore LM 3 will be 

 equal and parallel to ML 3 . Hence, on account of the smallness of the obliquities, the angles 

 of diffraction LPM 3 , L :i PM are sensibly equal, and their planes sensibly coincident, but the 

 deviations take place in opposite directions. But between the two diffractions the light under- 

 goes reflexion ; and since the mutual inclination of two rays is reversed by reflexion, we must 

 conceive the direction of deviation reversed in the first diffraction, in order to compare the 

 circumstances of the two diffractions. Allowing for this reversion, we see that not only are 

 the angles of diffraction sensibly coincident, but the directions of deviation are the same. 



Accordingly, the interference connected with diffraction, and the interference which gives 

 rise to the colours of thick plates, take place independently of each other. For, let I, I' 

 denote the vibrations at M due to two streams of light diffracted by any particle of dust P on 

 entering the glass, and passing on opposite sides of P ; let J, J' denote the vibrations due to 

 two streams diffracted at emergence, and passing on the same sides of P as I, /', respectively ; 

 and let I + I' denote the resultant of I and /', and similarly in other cases. Let ^ be the 

 difference of phase corresponding to the retardation R, and w the difference of phase of 7, V, 

 and therefore also of J, «/', on account of the similarity of the two diffractions. We may 

 represent the phases of the four vibrations by 9 + ^ + co, + ^, 6 + w, 6, respectively . 

 Writing down for greater clearness the phases along with the symbols of the vibrations, we 

 may express the resultant of the whole four vibrations by 



Moreover, on account of the similarity of the two diffractions, the coefficients of the two 



* In speaking of angles of diffraction, such as L 3 PM, I 

 shall distinguish between L 3 PM and MPL 3 , using the former 

 notation to denote that the deviation takes place from PL 3 to 



PM, and the latter to denote that it takes place from PM to 

 PL.. 



