distance of the objects. On this account the lines D and G' are united for ordinary photographic objectives; the optical as well as the actinic image is chromatically inferior, but both lie in the same place; and consequently the best correction lies in F (this is known as the actinic correction or freedom from chemical focus). Should there be in two lenses in contact the same focal lengths for three colours a, b, and c, i.e. , then the relative partial dispersion must be equal for the two kinds of glass employed. This follows by considering equation (4) for the two pairs of colors ac and bc. Until recently no glasses were known with a proportional degree of absorption; but R. Blair (Trans. Edin. Soc., 1791, 3, p. 3), P. Barlow, and F. S. Archer overcame the difficulty by constructing fluid lenses between glass walls. Fraunhofer prepared glasses which reduced the secondary spectrum; but permanent success was only assured on the introduction of the Jena glasses by E. Abbe and O. Schott. In using glasses not having proportional dispersion, the deviation of a third colour can be eliminated by two lenses, if an interval be allowed between them; or by three lenses in contact, which may not all consist of the old glasses. In uniting three colors an achromatism of a higher order is derived; there is yet a residual tertiary spectrum, but it can always be neglected. The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colors; and should they be compensated for one color, the image of another color would prove disturbing. The most important is the chromatic difference of aberration of the axis point, which is still present to disturb the image, after par-axial rays of different colors are united by an appropriate combination of glasses. If a collective system be corrected for the axis point for a definite wavelength, then, on account of the greater dispersion in the negative components — the flint glasses, — overcorrection will arise for the shorter wavelengths (this being the error of the negative components), and under-correction for the longer wavelengths (the error of crown glass lenses preponderating in the red). This error was treated by Jean le Rond d'Alembert, and, in special detail, by C. F. Gauss. It increases rapidly with the aperture, and is more important with medium apertures