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arrangements are fixed at the ends of a brass trough, of semi-circular section, D, resting on the stand P. The size of the trough should be such as easily to take tubes 3 decimetres long with their water jackets.

$ 58. The peculiar feature in Laurent's instrument is the thin plate of quartz PQ, cut parallel to the axis, Fig. 39. Let the polarizer

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be first so adjusted that the plane of polarization of the transmitted pencil of light is parallel to the axis of the plate-that is, lies in the direction A Bthe two halves of the field of vision will then appear equally dark or equally bright in every position of the analyzer. But if the polarizer be inclined to A B, at an angle a, the plane of polarization of the rays passing through the quartz plate will undergo deviation through an equal angle, a', in the opposite direction. Therefore, when in the uncovered half the plane of polarization has the direction AC, in the covered half it will have the direction AC'. If now we turn the analyzer, then, according as its plane of polarization lies in the direction cc or c'é, so will either the rays polarized parallel to AC or to A C'be extinguished, and the corresponding half of the field of vision will appear completely dark, while the other half merely suffers a partial decrease of brightness (Figs. 40, 41). In the middle position, b b, Fig. 42, there is a uniform shading over the two halves, but a very slight movement to and fro of the analyzer will at once destroy the equality. These phenomena repeat themselves when the analyzer has been moved through an angle of 180°.1

The degree of uniform shade obtained by bringing the analyzer into the middle position will be greater, the smaller the angle a (Fig. 39), which the plane of the polarizer makes with the axis of the quartz plate. The parts are set by the instrument-maker so that these two

1 For the theory of the phenomena produced by polarized light with plates cut parallel to the axes of uni-axial crystals, see Willner's Lehrbuch der Physik, 3 Aufl. Bd. II. S. 568. Laurent, in his earlier instruments, employed a thin plate of gypsum instead of the quartz, which gave the same results (Comptes Rend. 78, 349).

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directions are parallel to one another, but, as before observed, § 57, adjustment is arranged for by allowing the polarizer a slight amount of rotation, by means of the ring B, in the slot B, Fig. 38, whereby the field of vision is brightened. In this way the sensitiveness of the instrument can be altered. This is always greater the smaller the departure from the paraliel position, when, consequently, the less will be the amount of movement of the analyzer requisite to produce perfect obscurity of one or other half of the field of vision. The deepest shading suitable should therefore be chosen.

$ 59. In setting up the instrument it is directed towards a sodium flame, and the telescopic eye-piece so adjusted that the edge of the quartz plate appears to divide the diaphragm by a sharply defined vertical line. The analyzing prism must then be turned until both halves of the field of vision appear equally dark, and the polarizer adjusted to that position where the least displacement of the analyzer is required to produce an appreciable change in the appearance of the field.

In determining the zero-point, the analyzer is brought into the middle position, where the partition-line becomes invisible. Moreover, it is better to fill the experimental tube with water, so as to equalize the conditions in respect to absorption of light with those holding in observations of active liquids. One can then make the actual zero-position correspond as nearly as possible with the zeromark by means of the screw L (Fig. 38). Then, introducing the

. liquid, a rotation of the analyzer with its disc to the right will be necessary to restore the reference position if the substance be dextrorotatory, and to the left if it be lævo-rotatory.

If now, owing to colouring or any slight turbidity, the field of vision is too dark, greater brightness can be obtained by a slight movement of the polarizer on its axis (see § 58), but this entails the disadvantage that larger movements of the analyzer are then required before any alteration on the uniformity of shade is apparent, and the readings accordingly are more divergent. In clear solutions these do not differ by more than single minutes. For determining the direction of rotation in active substances of high rotatory power, the procedure given under the head of Mitscherlich's instrument (8 46) is equally appropriate here.

A bright sodium flame is necessary, which is best obtained by using the lamp shown, $ 46, Fig. 22. If tubes provided with waterjackets are employed, the complete arrangement of the instrument corresponds exactly with the description given § 50, Fig. 29. To eliminate errors in the Nicols, the observations should be made at two positions 180° apart, and the mean taken. The subjoined table contains, as an example, a series of observations with one of Hofmann's instruments, graduated from 0° both ways to. 180°. The zero-point was approximately at 90° in each half circle. The active liquid was a solution of cane-sugar :

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(d.) Comparison of Mitscherlich's, Wild's, and Laurent's Instruments.

$ 60. To determine the degree of concordance possible between the above instruments, observations were made with the same tubes on two suitable solutions of sugar. In each case, forty readings were taken, one-half of which were for the determination of the zeropoint. The following were the results :

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From this it will be seen that the results obtained with these several instruments agreed to the tenth of a degree, the variations not appearing till the second, and in some cases not before the third decimal place. It is therefore immaterial which instrument is used, any superiority consisting merely in the comparative facility with which observations can be made. An idea of the amount of difference that may be expected between the results obtained by different observers is shown by the following table, in which are recorded the angles of rotation obtained with the same sugar-solution by two observers of almost equal experience in the use of the three several instruments :

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Here we see the observations of the two observers varying within limits of hundredths or even only thousandths of a degree.

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8 + 8 + 1 Mean error for a single determination +

when n

1 0,0,...8. represent the differences between individual observations and their arithmetical mean, and n the number of observations taken.

The variations that occur in observations with different tubes are similar in amount, provided indeed that, as in the above instances was the case, their lengths have been measured correctly to within 0.05 millimetre. (See further $75.)

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(e.) Determination of the Angle of Rotation for different Rays.

[METHOD OF BROCH.] § 61. The instruments thus far described serve only to measure the rotation of the yellow sodium ray D. It is possible, indeed, at least with Mitscherlich's and Wild's instruments, by introducing into the flame, instead of common salt, some other substance, such as lithium or thallium compounds, to produce monochromatic light of another colour. The red lithium flame has the disadvantage of being too weak in illuminating power to admit of exact observations; besides which, it retains an admixture of yellow rays, which, however, can be absorbed by placing a red glass slide in front of the flame. The volatility of thallium compounds, on the other hand, renders it difficult to maintain the green-coloured light with sufficient intensity for any length of time.

§ 62. A method which admits of the determination of the rotation for a whole series of rays of known wave-length was proposed by Broch in 18461, and about the same time by Fizeau and Foucault,2 and has since been adopted by various other observers, as HoppeSeyler, Wiedemann, &c. For this purpose, solar light is employed, reflected horizontally by means of a heliostat into a darkened chamber. The beam passes in succession through-(a) a polarizing Nicol; (b) the layer of active substance; (c) the analyzing Nicol; (d) a spectroscope, consisting of a collimator, prism, and telescope, which last must be furnished with cross-threads. If, leaving out the liquid, the analyzer is first adjusted to the position of greatest darkness—the zero-point-and the active liquid then introduced, the spectrum with Fraunhofer's lines will at once appear in the telescope. If the analyzer be then rotated, a position will be found at which a vertical black line makes its appearance, and can be made to move across the field of vision by continuing the rotation. This is caused by the Nicol as it revolves, extinguishing in succession the rays whose planes of polarization are perpendicular to its own.

If the cross1 Broch: Dove's Repertorium d. Physik. 7, 113. ? Fizcau and Foucault : Comptes Rend. 21, 1155.

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