The end, l, is now adjusted so that colored fringes appear in white light. These can be measured to within one-twentieth of a wave length, and probably to within one-fiftieth. The piece, lm, is then moved forward till the fringes again appear at m. Then the refractometer is moved in the same direction till the fringes appear again at l, and so on till the whole meter has been stepped off. Supposing that in this operation the error in the setting of the fringes is always in the same direction, the whole error in stepping off the meter would be one part in two millions. By repetition this could of course be reduced. A microscope rigidly attached to the carriage holding the piece, lm, would serve to compare, and a diamond attached to the same piece would be used to produce copies. All measurements would be made with the apparatus surrounded by melting ice, so that no temperature corrections would be required. Probably there would be considerable difficulty in actually counting 400,000 wave lengths, but this can be avoided by first counting the wave lengths and fractions in a length of one millimeter and using this to step off a centimeter. This will give the nearest whole number of wave lengths, and the fractions may be observed directly. The centimeter is then used in the same way to step off a decimeter, which again determines the nearest whole number, the fraction being observed directly as before. The fractions are determined as follows: The fringes observed in the refractometer under the conditions above mentioned can readily be shown to be concentric circles. The center has the minimum intensity when the difference in the distances, ab, ac, is an exact number of wave lengths. The diameters of the consecutive circles vary as the square roots of the corresponding number of waves. Therefore, if x is the fraction of a wave length to be determined, and y the diameter of the first dark ring, d being the diameter of the ring corresponding to one wave length, then x = y squared/d squared. [Illustration: