so that, if we write
| 2πδ | = | π(a + b)s² | = | π | v² (2), |
| λ | abλ | 2 |
the effect at B is
| { | abλ | } | ½ | { cos | 2πt | ∫ cos ½πv²·dv + sin | 2πt | ∫ sin ½πv²·dv } (3) |
| 2(a + b) | τ | τ |
the limits of integration depending upon the disposition of the diffracting edges. When a, b, λ are regarded as constant, the first factor may be omitted,—as indeed should be done for consistency’s sake, inasmuch as other factors of the same nature have been omitted already.
The intensity I², the quantity with which we are principally concerned, may thus be expressed
I²= { ∫ cos ½πv²·dv}² + { ∫ sin ½πv²·dv }² (4).
These integrals, taken from v = 0, are known as Fresnel’s integrals; we will denote them by C and S, so that
| C = ∫ | v | cos ½πv²·dv, S = ∫ | v | sin ½πv²·dv (5). |
| 0 | 0 |
When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to ½, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v contribute but little to the result.