If R = ½λ, I vanishes at ξ= 0; but the whole illumination, represented by ∫+∞−∞ I dξ, is independent of the value of R. If R = 0, I = (1/ξ²) sin² (2πξh/λƒ), in agreement with § 3, where a has the meaning here attached to 2h.
The expression (5) gives the illumination at ξ due to that part of the complete image whose geometrical focus is at ξ = 0, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point O due to all the components which have their foci in its neighbourhood, we may conveniently regard O as origin. ξ is then the co-ordinate relatively to O of any focal point O′ for which the retardation is R; and the required result is obtained by simply integrating (5) with respect to ξ from −∞ to +∞. To each value of ξ corresponds a different value of λ, and (in consequence of the dispersing power of the plate) of R. The variation of λ may, however, be neglected in the integration, except in 2πR/λ, where a small variation of λ entails a comparatively large alteration of phase. If we write
ρ = 2πR/λ (6),
we must regard ρ as a function of ξ, and we may take with sufficient approximation under any ordinary circumstances
ρ = ρ′ + ωξ (7),
where ρ′ denotes the value of ρ at O, and ω is a constant, which is positive when the retarding plate is held at the side on which the lue of the spectrum is seen. The possibility of dark bands depends upon ω being positive. Only in this case can
cos {ρ′ + (ω − 2πh/λƒ) ξ}
retain the constant value -1 throughout the integration, and then only when
ω = 2πh / λƒ (8)
and