cos ρ′ = −1 (9).
The first of these equations is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.
The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that
cos {ρ′ +(ω − 2πh/λƒ) ξ} = cos {ρ′ − g1ξ}
= cos ρ′ cos g1ξ + sin ρ′ sin g1ξ,
we see that the second part vanishes when integrated, and that the remaining integral is of the form
| w = ∫ | +∞ | sin² h1ξ cos g1ξ | dξ | , |
| −∞ | ξ² |
where
h1 = πh/λƒ, g1 = ω − 2πh/λƒ (10).
By differentiation with respect to g1 it may be proved that
| w = 0 | from g1 = −∞ | to g1 = −2h1, |
| w = ½π(2h1 + g1) | from g1 = −2h1 | to g1 = 0, |
| w = ½π(2h1 − g1) | from g1 = 0 | to g1 = 2h1, |
| w = 0 | from g1 = 2h1 | to g1 = ∞. |