On the Intensity of the Light reflected from a Pile of Plates. 481 



the plates be perfectly transparent, we may treat white light as a 

 whole, neglecting as insignificant the chromatic variations of reflecting 

 power. Let p be the fraction of the incident light reflected at the 

 first surface of a plate. Then 1 — p may be taken as the intensity of 

 the transmitted light*. Also, since we know that light is reflected 

 in the same proportion externally and internally at the two surfaces of 

 a plate bounded by parallel surfaces, the same expressions p and 1 — p 

 will serve to denote the fractions reflected and transmitted at the 

 second surface. We may calculate p in accordance with Fresnel's 

 formulae from the expressions 



sm/= , (1) 



sin a (t-i') tan 8 (i—i') f ~ 



P "s¥pj' torfrO' ' ' ' ' W 



according as the light is polarized in or perpendicularly to the plane 

 of incidence. 



In the case of perfect transparency, we may in imagination make 

 abstraction of the substance of the plates, and state the problem as 

 follows : — There are 2m parallel surfaces (m being the number of 

 plates) on which light is incident, and at each of which a given fraction 

 p of the light incident upon it is reflected, the remainder being trans- 

 mitted ; it is required to determine the intensity of the light reflected 

 from or transmitted through the system, taking account of the re- 

 flexions, infinite in number, which can occur in all possible ways. 



This problem, the solution of which is of a simpler form than that 

 of the general case of imperfect transparency, might be solved by 

 a particular method. As, however, the solution is comprised in 

 that of the problem which arises when the light is supposed to be 

 partially absorbed, I shall at once pass on to the latter. 



In consequence of absorption, let the intensity of light traversing 

 a plate be reduced in the proportion of 1 to 1 — qdx in passing over 

 the elementary distance dx within the plate. Let T be the thickness of 

 a plate, and therefore T sec i' the length of the path of the light 

 within it. Then, putting for shortness 



1 to g will be the proportion in which the intensity is reduced by 

 absorption in a single transit. The light reflected by a plate will be 

 made up of that which is reflected at the first surface, and that which 

 suffers 1, 3, 5, &c. internal reflexions. If the intensity of the inci- 

 dent light be taken as unity, the intensities of these various portions 

 will be 



and if r be the intensity of the reflected light, we have, by summing 



* In order that the intensity may be measured in this simple way, which saves 

 trouble in the problem before us, we must define the intensity of the light trans- 

 mitted across the first surface to mean what would be the intensity if the light 

 were to emerge again into air across the second surface without suffering loss by 

 absorption, or by reflexion at that surface. 



