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Rh distance between two planes, with an error less than one in two millions, and probably one in ten millions.

But the distance corresponding to 400,000 wave lengths is roughly a decimeter, and this cannot be determined or reproduced more accurately than, say $$\tfrac{1}{500000}$$, so that it would be necessary to increase this distance. This can be done by using the same instrument, together with a comparator.

The intermediate standard decimeter a b is put in place of the mirror b; a b is a piece of cast iron, c and d are plane glass mirrors, c rests on the points of three studs, d rests on the points of three screws, c and d are made exactly parallel and the distance c d is made conveniently near to some aliquot part of the meter.

The end c is now adjusted till colored fringes appear in white light. These can be measured to within one twentieth of a wave length (probably within one-fiftieth).

The piece a b is then moved forward till the fringes appear again at d; then the refractometer is moved in the same direction till the fringes appear again at c, 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 of the meter would be one in two million.

By repetition, this would of course be reduced.

A microscope attached to the carriage holding the piece a b 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 will be needed.

Probably there would be considerable difficulty in counting 400.000 wave lengths, but this can be avoided by first counting the number of waves and fractions in a length of one millimeter and using this to 'step off' a centimeter. This will give the nearest whole number of waves, and the fractions may be observed directly. The centimeter is then used in the same way to 'step off' the decimeter, which again determines the nearest whole number, the fractions being determined directly as before.

These 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 a b, a c 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 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 $$x=\tfrac{y^{2}}{d^{2}}$$.

There is a slight difficulty to be noted in consequence of there being two series of waves in sodium light. The result of the superposition of