Part I. Method of displacement.—This method permits the determination of glancing-angles in terms of linear measurements only. It consists in letting a properly limited beam of x-rays, after reflection, fall on a photographic plate placed successively in two parallel positions which differ in distance from the crystal by a known amount. The theory of the method and its practical application to the present case are briefly discussed with references to Uhler's paper.

Part II. Apparatus and Adjustments.—A new high precision spectrometer, including a specially designed bearing on which the crystal rotates and a unique form of plate holder using air pressure to ensure the flatness of the photographic plate, is fully described. A special form of hot filament cathode which gives an approximately linear source of x-rays has been originated. Precision methods for adjusting and testing the reliability of the instrument are discussed at length. It is believed that the instrument is free from any defects that might cause a constant error of as much as 0.04″ in the glancing-angle.

Part III. Experimental prodedure.—The cleavage planes of a nearly perfect specimen of Iceland spar were used as grating. The bearing of its perfection on the widths of the spectral images is discussed. The coefficient of expansion of calcite normal to the cleavage planes was computed from the most reliable data obtainable. A mean value which is sufficiently precise, over a sixteen degree range, to reduce values of glancing-angles to 18°C within 0.008″ is ${1.02}_3$${\mathrm{}\left(10\right)\mathrm{}}^{-5\mathrm{}}$ per °C. Attention is called to the fact that the commonly used value ${1.04\left(10\right)\mathrm{}}^{-5\mathrm{}}$ is not a sufficient approximation.

Distances between spectral images on each negative were measured on a Gaertner measuring engine, the errors of which were carefully investigated. The "displacements" of the photographic plate were equal to fixed intervals marked on a measuring bar and calibrated against the screw of the same measuring engine. The accuracy of these measurements and the effect of their errors on the glancing-angle are discussed.

The experimental procedure and precautions used in taking and measuring spectrograms are fully described. A complete protocol of observations is given for one plate. Evidence for the accuracy of the measurements is presented and discussed.

Part IV. Results.—The weighted mean value of the glancing-angle for the $K{\alpha }_1$-line of silver from the cleavage planes of calcite at 18°C is 5°17′ 13.81″ ± 0.06″ giving a wave-length, if $log 2d_{18}=0.7823350\mathrm{}$, of 0.558238A. This result is compared with those of other observers. It differs from Kellström's value quoted by the Int. Crit. Tables by more than ten times the combined probable errors. The only probable source of an adequate constant error in either result seems to lie in the crystal. Compton's measurements of the density of calcite are invoked to prove that different specimens of Iceland spar do differ enough in density to account for more than the discrepancy, if changes in density and grating space are interdependent.

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