When V = 0, viz. at the edge of the shadow, I² = ½; when V = ∞, I² = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing outwards from the edge of the shadow is, however, accompanied by fluctuations giving rise to bright and dark bands. The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = 0, and
½πV² = ¾π + nπ (24),
n being an integer. In terms of δ, we have from (2)
δ = (3⁄8 + ½n)λ (25).
The first maximum in fact occurs when δ = 3⁄8λ − .0046λ, and the first minimum when δ = 7⁄8λ − .0016λ, the corrections being readily obtainable from a table of G by substitution of the approximate value of V.
The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19)
| BQ = | a + b | AD = V √ { | bλ(a + b) | } (26). |
| a | 2a |
By means of this expression we may trace the locus of a band of given order as b varies. With sufficient approximation we may regard BQ and b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have
2ax² − V²λy² − V²aλy = 0,
which represents a hyperbola with vertices at O and A.