| cos | 2π | {Vt − x cos α − y sin α}, |
| λ |
and
| cos | 2π | {Vt − x cos α + y sin α}, |
| λ |
so that the resultant is expressed by
| 2 cos | 2πy sin α | cos | 2π | {Vt − x cos α}, |
| λ | λ |
from which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity V/cos α. Considered as depending on y, the vibration is a maximum when y sin α is equal to O, λ, 2λ, 3λ, &c., corresponding to the centres of the bright bands, while for intermediate values ½λ, 3⁄2λ, &c., there is no vibration.
From (1) we see that the linear width Λ of the bands, reckoned from bright to bright or dark to dark, is
Λ = λD / b
(2).
The degree of homogeneity necessary for the approximate perfection of the nth Fresnel’s band may be found at once from (1) and (2). For if du be the change in u corresponding to the change dλ, then