It is sufficient to inspect the figure, in order to be convinced of the necessity of having the two mirrors nearly in the same plane, if we wish to obtain fringes of tolerably large dimensions; for in the little triangle _bni_, the side _bi_, which represents the length of a semiundulation, being little more than the hundred thousandth of an inch for the yellow rays, for example, the side _bn_, which measures the half breadth of a fringe, can only become sensible when _bn_ is very little inclined to _in_, so that their intersection may be remote from _ib_; and the inclination of _bn_ to _in_ depends on the distance AB, which is the measure of the inclination of the mirrors.
If A and B, instead of being the images of the luminous point, were the projections of two very fine slits cut in a screen RN, through which the rays of light were admitted from a luminous point placed behind the screen in the continuation of the line _b_DC, the two paths described between the point and the slits A and B being equal, it would be sufficient to compute the paths described by the rays, beginning from A and B, in order to have the differences of their lengths; and it is obvious in this case, that the calculations which we have been making of the breadth of the fringes, produced by the two mirrors, would remain equally applicable, at least as long as each slit remained narrow enough to be considered as a single centre of undulation, relatively to the inflected rays which it transmits. It may therefore be said that the breadth of the fringes, produced by two very fine slits, is equal to the length of an undulation supposed [p135] to be multiplied by the interval between the two slits, and divided by the distance of the screen from the wires of the micrometer employed for measuring the fringes.
This formula is also applicable to the dark and bright stripes which are observed in the shadow of a narrow substance, substituting the breadth of this substance for the interval which separates the two slits, as long as the stripes are far enough from the edges of the shadow: for when they approach very near to the edges, it is shown, both by theory and by experiment, that this calculation does not represent the facts with sufficient accuracy; and it is not perfectly correct in all cases, either for the fringes within the shadow, or for those of the two slits, but only for the fringes produced by the mirrors, which exhibit the simplest case of the interference of rays slightly inclined to each other. In order to obtain from the theory, a rigorous determination of the situation of the dark and light stripes in the two former cases, it is not sufficient to calculate the effect of two systems of undulations, but those of an infinite number of similar groups must be combined, according to a principle which will shortly be explained, in treating of the general theory of diffraction.