(1).
In (1) α′ is the angle of refraction, and we see that, contrary to what might at first have been expected, the retardation is least when the obliquity is greatest, and reaches a maximum when the obliquity is zero or the incidence normal. If we represent all the vibrations by complex quantities, from which finally the imaginary parts are rejected, the retardation δ may be expressed by the introduction of the factor ε−iκδ, where i = √(−1), and κ = 2π/λ.
At each reflection or refraction the amplitude of the incident wave must be supposed to be altered by a certain factor which allows room for the reversal postulated by Young. When the light proceeds from the surrounding medium to the plate, the factor for reflection will be supposed to be b, and for refraction c; the corresponding quantities when the progress is from the plate to the surrounding medium will be denoted by e, f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave b, for the second cefε−iκδ, for the third ce3fε−2iκδ, and so on. Adding these together, and summing the geometric series, we find
| b + | cef ε−iκδ |
| 1 − e2 ε−iκδ |
(2).
In like manner for the wave transmitted through the plate we get
| cf |
| 1 − e2 ε−iκδ |
(3).
The quantities b, c, e, f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing δ = 0 in (2) and (3). This may be regarded as a development from Young’s point of view. A plate of vanishing thickness is ultimately no obstacle at all. In the nature of things a surface cannot reflect. Hence with a plate of vanishing thickness there must be a vanishing reflection and a total transmission, and accordingly
b + e = 0, cf = l − e2