Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/866

802 OPTICS displaced laterally from the position occupied by the first, a second spectrum overlapping the former will be thrown upon the screen, and a second kind of light will be admitted to the eye. In this way we may obtain a field of view lighted with a mixture of two or more spectrum colours, and we may control the relative proportions by varying the widths of the slits. For instance, by mixing almost any kind of red with any kind of green not inclining to blue we may match the brightest yellows, proving what so many find it difficult to believe, that yellow is a com pound colour. In Maxwell s systematic examination of the spectrum, mixtures of three colours were used, and the proportions were adjusted so as to match the original white light incident upon the apparatus. A similar arrangement (with one original slit) was em ployed by He!rnholtz in his examination of a fundamental question raised by Brewster. The latter physicist main tained that there was abundant evidence to show that light of definite refrangibility was susceptible of further analysis by absorption, so that the colour of light (even of given brightness) could not be defined in terms of refrangibility or wave-length alone. The appearances which misled Brewster have since been explained as the effect of contrast or of insufficient purity. It is obvious that light, e.g., from the red end of the spectrum, may be contaminated Avith light from some other part, say the yellow, in such propor tion that though originally entirely preponderant it may fall into the second place under the action of a medium very much more transparent to yellow than to red. To obtain light of sufficient purity for these experiments Helmholtz found it advisable to employ a double prismatic analysis. A spectrum is first thrown upon a screen per forated by a slit in the manner already described. The light which penetrates the second slit, already nearly pure, is caused to pass a second prism by the action of which any stray light is thrown aside. Using such doubly piiri- jied light, Helmholtz found the colour preserved, whatever absorbing agents were brought into play. Light of given refrangibility may produce a variety of e/ect-s, visual, thermal, or chemical, but (apart from polarization) it is not itself divisible into parts of different kinds. If yellow light produces the compound sensation of yellow, we are to seek the explanation in the constitution of the retina, and not in the divisibility of the light. In all accurate work with the prism the use of a collimating lens to render the incident light parallel is a matter of necessity. If the incident rays diverge from a point at a finite distance, the pencil after emergence will be of a highly complicated character. There are, however, cases in which a collimator is dispensed with, and thus it is a problem of interest to find the foci of a thin pencil origin ally diverging from a point at a moderate distance. Even when a collimator is employed, the same problem presents itself whenever the focusing is imperfect. For the sake of simplicity the pencil is supposed to pass so near the edge of the prism that the length of path within the glass may be neglected in comparison with the dis tances of the foci. We denote as usual the angles of incidence and emergence by 0, s/ , and the corresponding angles within the glass by <f> , f/ . Tlie distance AQ from the edge of the prism to original source is denoted by u ; the corresponding distances for the primary and secondary foci q-i, q* by v^, v. 2 . By successive applications of the results already proved for a single refraction, we get so that (1) (2). In order that the primary and secondary foci may coincide we must have ^ = ; that is to say, the ray must pass with minimum deviation. This is sometimes given as a reason why this arrange ment should be adopted in spectroscopes ; but in reality, since the slit is parallel to the edge of the prism, a slight elongation in this direction of the image of a point is without detriment to the defini tion. Hence a good image will be .seen when the telescope is adjusted for the primary focus ; and it is not clear that any improve ment would arise from coincidence of the two foci, the question being in fact one of aberration. The position of minimum devia tion is, however, usually adopted for the sake of dcfiniteness, and sometimes it is convenient that the fixed lines and the extremities of the slit (or the markings produced by dust) should be in focus together. The deviation is a symmetrical function of </> and i//, and therefore is not altered >y an interchange of these angles. The correspond ing values of v are thus by (1) reciprocals, and their product is equal to u 2 . This principle has been ingeniously applied by Schuster a to the adjustment for focus of the telescope and collimator of a spectroscope. The telescope is so placed that the deviation necessary to bring the object upon the cross wires is greater than the minimum, and the prism is adjusted in azimuth until the effect is produced, that position being chosen for which the angle of incidence is greater than the angle of emergence, so that f x is greater than u. After focusing the telescope the prism is turned into the other position which gives the same deviation, and the collimator is focused, the telescope remaining untouched. The prism is next brought back to the first position, and the telescope is again focused. A few repetitions of this operation, always focusing the telescope in the first position of the prism and the collimator in the second, will bring both into perfect adjustment for parallel rays. Lenses. The usual formula for the focal length of lenses (vol. xiv. p. 593), ignores the fact that the various parts of a lens bounded by spherical surfaces have not the same focus, and is appli cable in strictness only when the aperture is small. It is not necessary here to repeat the process by which (1) is usually obtained, but before passing on to give the form ulae for the aberration of lenses it may be well to exhibit the significance of (1) from the point of view of the wave- theory. Taking the case of a convex lens of glass, let us suppose that parallel rays DA, EC, GB (fig. 13) fall upon the lens ACB, and are collected by it to a focus at F. The points D, G_ E, G, equally distant from ACB, lie upon a front of the wave before it impinges upon the lens. The focus is a point at which the different parts of the wave arrive at the same time, and that such a point can exist depends upon the fact that the propagation is slower in glass than in air. The ray ECF is retarded from having to pass through the thickness (i) of glass by the amount (p - 1). The ray DAF, which traverses only the extreme edge of the lens, is retarded merely on account of the crookedness of its path, and the amount of the re tardation is measured by AF - CF. If F is a focus these retardations must be equal, or AF-CF = (M-1)<. Now if y be the semi-aperture AC of the lens, and / be the focal length CF, AF - CF = V{/ 2 + y-} ~/=4 - approximately, 13. In the case of plate -glass /* 1 = -| nearly, and then the rule (2) may be thus stated : the semi-aperture is a mean proportional betu>een the focal length and the thickness. The form (2) is in general the more significant, as well as the more practically useful, but we may of course express the thickness in terms of the curvatures and semi-aperture by means of 1 Phil. Mag., February 1379.