It was soon found, however, that the variations recurred with great irregularity: in less than two years the phases differed by as much as seven hours from the predicted times. At length the subject was taken up by Dunér at Upsala. A series of observations made with the 14-inch refractor at Upsala in 1891 and 1892 convinced him in the latter year that two eclipses take place in the course of one revolution: one star occults the other. Dunér showed that the intervals between minima were thus—1 day 9 hours; 1 day 15 hours; 1 day 9 hours, and so on. Thus, the first, third, fifth, and seventh sets of minima obeyed a different law from the second, fourth, sixth, and eighth. Dunér proved that two stars revolve round their centre of gravity in less than three days, alternately occulting each other, while the ellipticity of the orbit explains the irregularity of the light changes. In April 1900 Dunér gave his final conclusions as follows: “The variable star Y Cygni consists of two stars of equal size and equal brightness, which move about their common centre of gravity in an elliptical orbit, whose major axis is eight times the radius of the stars.” He also stated the exact period of revolution and the eccentricity of the orbit. In the case of the short-period variables, such as β Lyræ, δ Cephei, ζ Geminorum, and η Aquilæ, the variations do not seem to be due to eclipse. It was discovered by Professor Pickering that β Lyræ is a spectroscopic binary, but Vogel and Keeler showed that the supposed orbit is incompatible with the eclipse theory. Vogel says: “I am convinced that β Lyræ represents a binary or multiple system, the fundamental revolutions of which in 12 days 22 hours in some way control the light change.” The eclipse theory, however, is still maintained by Bélopolsky, who has framed a hypothesis according to which the chief minimum of the star’s light corresponds with the obscuration of the lesser star, the lesser minimum with that of the primary, implying that the primary is much less luminous in proportion to its light than its satellite,—a state of affairs which Miss Clerke concludes to be improbable. The variable stars, δ Cephei and η Aquilæ, were both found in 1894 by Bélopolsky to be binaries; but as the times of minimum light do not correspond with those of eclipses in the hypothetical orbits, he concludes that the variations cannot be explained on the eclipsing satellite theory. Miss Clerke is inclined to the theory that the increase of luminosity in short-period variables is due to tidal action, so that while the revolutions of the stars control their variability, they are inherently unstable in light. A large number of these stars are known, and it is a remarkable fact that the majority of these variables lie on or near the Galaxy, so that their variations have probably something to do with their vicinity.