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of which shows the rotation for a length of 50 millimetres, the latter for 100 millimetres.

§ 54. Wherever exactness is required, the observations should be repeated in each of the four quadrants of the circle. It will be found that appreciable differences exist between the angles of rotation thus obtained, differences which van de Sande Bakhuyzen1 has shown originate in defective construction of the Nicol, as well as in improper placing of the two calc-spar plates of the Savart. These errors, however, disappear altogether when the mean of the four values for the angle of rotation is taken. When the observations are repeated in two opposite quadrants only, and the mean of the readings is taken, the compensation of errors is not indeed complete, but the degree of accuracy attained is usually enough for all ordinary purposes. The deviations from the true value do not exceed 0.03° at the most, and are generally less than 0.01°. To take observations in two adjoining quadrants has not, of course, the same compensatory

effect.

To obtain very precise results, it is obviously requisite that a still larger number of observations should be taken. As a rule, five observations in each quadrant will be enough, so that allowing for the verification of the zero-points, which should be repeated at least once each day on taking observations, the angle of rotation finally obtained will be the result of forty readings. Where considerable differences are found in the readings, the number of observations must be increased. This will occur when the solutions are not absolutely clear; slight colorations, on the other hand, do not materially affect the observations.

The degree of accuracy attainable is shown in the two series of observations appended. These were taken with an instrument of Hermann and Pfister's manufacture, graduated to divisions of 5 minutes, and allowing of approximate reading to single minutes. The liquid employed was an aqueous solution of cane-sugar containing 19.45 grammes in 100 cubic centimetres. The length of tube was 219.79 millimetres.

1 van de Sande Bakhuyzen: Pogg. Ann. 145, 259.

2 Differences of 20 minutes in the readings may easily occur with unpractised observers, but with care these can soon be much reduced in amount, so that a hasty opinion should not be formed of a newly-purchased Wild polariscope. When the observer has become accustomed to the instrument, and the latter is properly constructed, the difference in the readings will seldom exceed 5 minutes.

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0° 20.0' 28° 43.8′ 90° 19.0' 118° 40.0' 180° 13.8' 208° 38.6' 270° 17.8′ 298° 46'8'

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0° 20-4' 28° 44.6′

90° 18.0' 118°38.8' 180° 14.4' 208° 38-4' 270° 18.6' 298° 47.2'

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(c.) Half-Shade Instruments (Polarimètres à Pénombre) of Jellett, Cornu, and Laurent.

In these instruments the mechanism for sensitiveness is arranged to produce a circular field of vision divided into halves, which in certain positions of the analyzing Nicol are unequally illuminated, but in one particular position exhibit a uniformly faint shade. This position, which can be fixed with great accuracy, is taken as the point of reference. The use of monochromatic sodium light is pre-supposed.

§ 55. The earliest instrument of this kind was constructed by Jellett in 1860.1 In this, between the polarizing and analyzing Nicols, and close behind the former, is placed a prism of peculiar form. An elongated rhombohedron of calc-spar which, by grinding the ends, has been converted into a right prism, is divided longitudinally into halves by a plane nearly, but not quite, perpendicular to its principal section, and the two halves then reunited, but in reversed positions. The prism is mounted in a case, furnished at the extremities with diaphragms having circular apertures. The circular field so obtained appears divided diametrically by the section into equal halves, in which the planes of polarization are slightly inclined to each other. A plane polarized ray passing through can, by turning the analyzer, be extinguished by either half of the prism, these points of extinction lying very close together, whilst between them lies the position of uniform shade. The appearance of uniform shade can also be made to vanish by the introduction of an active liquid, and, to bring it once more into view, the analyzer must be turned on its axis through a certain angle, which can be taken as measure of the deviation of the ray produced by the active substance.

§ 56. Cornu's instrument consists of an ordinary Nicol as analyzer, with a polarizer of peculiar construction. The latter is formed out of a Nicol prism, by bisecting it in the direction of the plane passing through the two shorter longitudinal diagonals, cutting down the sectional faces 210 and reuniting the halves. In this way we have a double Nicol prism, having its two principal sections forming an angle of 5° with each other. When, therefore, by turning 1 Jellett Reports of the British Association, 1860, 2, 13. 2 Cornu: Bull. Soc. Chim. [2], 14, 140.

the analyzer, we bring its principal section exactly perpendicular to one of the two principal sections of the polarizer, perfect obscuration follows in the corresponding half of the field of vision, the other half remaining illumined. A rotation of 5° reverses these conditions, the dark half then becoming bright and vice versa, while midway between these two positions lies a point where the halves exhibit equal degrees of incipient shadow.

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§ 57. The half-shade instrument, however, which has come into most general use is that of Laurent, of which a representation is given in Figs. 37, 38.

Laurent: Dingler's Polyt. Journ. 223, 608.

In this, the light from a sodium flame passes through the following optical apparatus:

1. A thin plate, a (Fig. 37), cut from a crystal of bichromate of potash, serving to free the yellow ray from intermixture of green, blue, and violet light. This is enclosed between a couple of glass plates and fixed in a movable diaphragm.

2. A double refracting calc-spar prism, b, as polarizer.

These two pieces are placed one at each end of the tube AB, Fig. 38, which is inserted in the fixed portion C C', of the instrument, within which it is capable of rotation through a small angle. The amount of this movement is regulated by means of the screw-stop ß, passing through the slot at C.

3. A circular diaphragm c, containing a glass plate, to which is affixed a thin plate of quartz, cut parallel to the axis, and just large enough to cover exactly one half of the circle. The thickness of the quartz plate must be so regulated that the yellow rays polarized parallel and perpendicularly to the axis may in their transmission undergo a retardation of half a wave-length. (In an instrument manufactured by Dr. Hofmann, of Paris, the thickness of this quartz plate is 0.11 millimetre.)

4. The solution-tube d.

5. An analyzing Nicol e, furnished with rotatory movement.
6. The lenses ƒ and g, forming a small Galilean telescope.

The analyzer rotates in a piece with the divided disc E, within the stout ring, M. For this purpose, the back of the disc is furnished with a bevelled toothed wheel, driven by a small pinion worked by the milled-head F. The vernier is screwed firmly to the arm G, hanging down over the graduated edge of the disc. In reading, a magnifier, H, is used, which has a motion round the point 0, and is provided at the top with a metal reflector, J. The latter can be made to reflect light on the divisions either from the sodium flame or some other convenient source. The Nicol can be turned in its case slightly by means of the screw L, so as to alter the zero-point. The telescopic lenses are mounted in tubes, KN, the latter of which has a draw motion. The graduated circle, a front view of which, with the parts pertaining thereto, is given in the figure, has a diameter of 250 millimetres, and the vernier reads to single minutes. The optical

1 May be obtained of Dr. Hofmann, 29, Rue Bertrand, Paris; Schmidt and Haensch, Berlin; J. Duboscq, 21, Rue de l'Odéon, Paris; Bartels and Diederichs, mechanicians, Göttingen. Fig. 38 is drawn from one of Hofmann's instruments.

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