174         ON THE COLOURS OF THICK PLATES.
The numerical quantity n_0 may conveniently be called the central order, since when it lies between i — -J- and i + ^, where i is any integer, the colour at the centre belongs to the bright 'ring of the i^th order. If v be the radius of the central fringe, v will be equal to the semi-difference of the quantities ah (h + c)"^1 and ah (h - c)"^1,
whence
acA
Having found the centre of the system of circles and the projection of the image, or the point where the line joining the eye and the image cuts the mirror, describe a circle passing through this projection. This will be the central line of the bright fringe of the order 0, and its radius will be equal to v. Now describe a pair of circles whose radii are to v as V(^o ± 1) to V?V These will be the central lines of the two bright fringes of the first order, for the particular colour to which the assumed value of X relates. The central lines of the two bright fringes of the second order will be a pair of circles with radii proportional to V(^o ± 2), and so on. The fringes will be broader on the concave than on the convex side of the central white fringe. When the fringes become straight, n_Q becomes infinite, and the system becomes symmetrical with respect to the central fringe. This agrees with observation.
13. When the luminous point is situated in a line drawn through the eye perpendicular to the mirror a = 0, and we have simply
In this case the achromatic line of the system is reduced to a point, and the rings are analogous in every respect to the transmitted system of Newton's rings. For the bright ring of the first order R= ± X, and therefore the radius of the ring is equal to
. c/i
which becomes infinite when c=h. Hence if the luminous point be at first situated in front of the eye, and be then conceived to move backwards through the eye till it passes behind it, the rings will expand indefinitely, and so disappear, and will reappear again when the luminous point has passed the eye.