The ring micrometer due, like so much other valuable apparatus, to Fraunhofer, is convenient and widely used for determining positions. It consists, as shown in Fig. 130, of an accurately turned opaque ring, generally of thin steel, cemented to a plane parallel glass or otherwise suspended in the center of the eyepiece field. The whole ring is generally half to two thirds the width of the field and has a moderate radial width so that both the ingress and the egress of a star can be conveniently timed. It depends wholly on the measurement of time as the stars to be compared drift across the ring while the telescope is fixed, and while a clock or chronometer operating a sounder is a desirable adjunct one can do pretty well with a couple of stop watches since only differential times are required. For full directions as to its use consult Loomis’ Practical Astronomy, a book which should be in the library of every one who has the least interest in celestial observations. Suffice it to say here that the ring micrometer is very simple in use, and the computation of the results is quite easy. In Fig. 130 F is the edge of the field, R the ring, and _a b_, _a′b′_, the paths of the stars _s_ and _s′_, the former well into the field, the latter just within the ring. The necessary data comprise the time taken by each star to transverse the ring, and the radius of the ring in angular measure, whence the difference in R. A. or Dec, can be obtained.[20] [20] r the radius of the ring, is given by, r = (15/2)(t′-t) cos Dec., t′-t being the seconds taken for transit. Difference of R. A. = ½ (t′-t)½ (T′-T) where (T′-T) is the time taken for transit of second star. To obtain differences of declination one declination should be known at least approximately, and the second estimated from its relative position in the ring or otherwise. Then with these tentative values proceed as follows. Put x = angle _aob_ and _x_′ = angle _a′o′b′_ Also let d = approximate declination of _s_ and d′ = approximate declination of _s′_ Then sin x = (15/2r) cos d (T′-T) sin x′ = (15/2r) cos d′ (t′-t) and finally Difference of Dec. = r (cos x′-cos x), when both arcs are on the same side of center of ring. If on opposite sides, Diff. = r (cos x′ + cos x). [Illustration: _Chamber’s “Astronomy”_ (_Clarendon Press_). FIG. 131.—Double Image Micrometer. (_Courtesy of The Clarendon Press._)]