Now turn B until the circle stops at 180°, and turn D until the image in the further face of the crystal coincides exactly with the line on the table. Then turn B in the other direction until the second image coincides, when the reading of the vernier will give the correct angle. Evidently the two faces are in turn brought into exactly the same position, and the angle between them equals 180° minus the amount through which the circle has been turned. It is sometimes more accurate, though a little more troublesome, to turn the crystal into any position by D, and bring first one image to coincide, and then the other. 180° minus the difference in the readings of the vernier give the required angle. Try this with different parts of the circle. Remove the crystal, attach it a second time to E, and see if the same result is attained as before. Repeat until readings are obtained differing from each other but a few minutes. Also measure some of the crystals less highly polished. An excellent test of the work is to measure the angles completely around a crystal, and see if their sum equals 180° (n - 2), in which n is their number.

72. ANGLE OF PRISMS.

Apparatus. One of the most valuable instruments in a Physical Laboratory is the Optical Circle, or Babinet's goniometer. This instrument may be used as a goniometer for measuring the angles of crystals, to find the index of refraction of liquids or solids, to study dispersion, or, as a spectrometer, to measure wavelengths. It is therefore often desirable to duplicate it, or perhaps better, to procure one large and very accurate instrument, and others of smaller size for work requiring less precision.

A

C

B

E

This instrument, Fig. 54, consists of a graduated circle on a stand, with two telescopes, A and B, attached to it. A is the collimator, or a telescope in which the eye-piece is replaced by a fine slit, whose width may be varied by a screw resting against a spring, and whose distance from the object-glass may be altered by a sliding tube with a rack and pinion. This telescope is attached permanently to the stand, while B, which is a common telescope with cross hairs in its focus, is fastened to an arm revolving around the centre of the graduated circle. It may be held in any desired position by a clamp D, moved slowly by a tangent screw E, and the angle

Fig. 54.

through which it has been turned, accurately measured by a vernier. To eliminate errors of eccentricity a second vernier is sometimes placed opposite the first, in which case the mean of their readings is always employed. For great accuracy a spiderline micrometer should be attached to B to measure small angles, as will be described more in detail in Experiment 77. It is often convenient to have both telescopes mounted on conical bearings so that they may be turned away from the centre of the circle when desired. They should also be supported in such a way that one end of each may be raised or lowered a little, so as to bring their axes perpendicular to that of the instrument. This is most readily accomplished by placing an adjusting screw under one of the Y's carrying them. C is a small circular stand on which prisms may be placed, and which may be turned around the centre of the circle and clamped in any position. Sometimes an arm and vernier is attached to measure its angular motion, but this is not absolutely necessary. Its principal use is to measure the angle of crystals, and by it the law of reflection may also be proved with great accuracy. The graduated circle is sometimes made to revolve, and the angle measured by one or more fixed verniers.

A

об

B

The whole is commonly mounted on a tripod with levelling screws, as shown in the figure. These are ornamental rather than useful, however, as in common experiments it makes no difference, except in appearance, if the circle is not properly levelled. In any case, except to raise or lower the instrument, only two screws are needed, and the third may be replaced by a fixed point. In this, as in all instruments mounted on three legs, the best form of support is that represented in Fig. 55. One leg rests in a conical hole A, a second in a wedge-shaped groove B, and the third on a plane surface C. A fixes the position of the tripod, which is prevented from turning by the groove B, while if the three legs change their relative positions, B can slide back and forth in its groove, and move freely over the plane surface. If instead, three conical holes were used, and these were not precisely in the right position, or the distance of the legs varied with changes of temperature, the whole instrument might be so strained as to introduce serious errors in the graduated circle. This instrument should be placed near the window so that sunlight may be reflected through it by means of a mirror, or if preferred, the light from an Argand or Bunsen burner employed. One or more flint glass prisms are also needed, all three of whose faces should be polished and inclined at angles of 60°.

Fig. 55.

Experiment. The following adjustment must always be made when the optical circle is used. Draw out the eye-piece of B,

Fig. 54, until the cross-hairs are seen with perfect distinctness. Then turn the telescope towards some distant object and focus it, moving both eye-piece and cross-hairs. Now both the object and cross-hairs should be perfectly distinct, and not change their relative positions as the eye is moved from side to side so as to look through different portions of the eye-lens. Sometimes the objective alone moves, and sometimes the distance is permanently fixed, so that it is in adjustment for parallel rays. Now turn the two telescopes towards each other and illuminate the slit either by placing an Argand burner behind it, or reflecting the light of the sky through it by means of a mirror. On looking through the observing telescope an image of the slit will now be visible. Focus it, moving it towards or from its objective, when its distance will equal the principal focal distance of the collimator, and the beam of light between the two telescopes will be parallel, or as if coming from a slit placed at a very great distance. Bring the image of the slit to coincide exactly with the vertical cross-hairs in B by the tangent screw, first clamping the telescope. If it is not vertical the slit may be turned, and if it is too high or too low it should be brought to the centre of the field by raising or lowering one end of one of the telescopes, as described above. Having rendered the coincidence exact, read the vernier and repeat the setting two or three times, as it gives the zero from which most of the following measurements are made.

To measure the angle of a prism, stand it on the centre-plate with its edges vertical, and with the faces whose angle is to be determined about equally inclined to the axis of the collimator. To eliminate parallax in case the telescopes have not been accurately focussed for parallel rays, it is better to place the edge of the prism over the centre of the graduated

circle. To prevent motion of the prism when B is turned, C should be clamped. Let ABC, Fig. 56, represent the prism whose angle A is to be measured, and DD' the axis of the collimator prolonged. Turn the observing telescope into the position AF, when an image of the slit will be seen on looking through. Bring it to coincide

B

'D'

Fig. 56.

with the cross-hairs by the tangent-screw, first clamping the telescope, and read the vernier. Then turn the telescope into the position AE and set again. The difference in the readings divided by two, equals the angle of the prism. For D'AC equals 90° - the angle of incidence, and FAC, 90° the angle of reflection; hence they are equal. In the same way, EAB = D'AB, or FACEAB = BAC, and FAE=2BAC. Move the prism a little, repeat the measurement, and see if the same result is obtained as before. Determine in the same way the three angles of the prism, and their sum should equal 180°. If either of the reflected images of the slit is too high or too low, the base of the prism is not perpendicular to the edges. In this case it must be adjusted by placing pieces of paper, or tinfoil, under one or two of its corners, until both images are in the centre. If either is out of focus when the telescopes have been adjusted for parallel rays the reflecting surface is curved instead of plane, while a distortion of the image shows that the surface is irregular. In either case, an accurate measurement is impossible, since the angle will vary for different parts of each face.

By the plan just described, the angle of a prism may be found if, as is often the case, the centre plate has no vernier attached to it. With such a vernier, however, the angle may be determined more readily, as follows. Set the telescopes nearly at right angles, and stand the prism on the centre-plate, as before, with its faces vertical, and the edge to be measured over the axis of the instrument. Turn the centre-plate until one of the faces is equally inclined to the axes of both telescopes, when the image of the slit reflected in this face will be seen in the field on looking through the observing telescope. Bring it to coincide with the cross-hairs by the clamp and tangent-screw, and read the vernier. Turn the centre-plate, taking great care not to disturb the position of the prism on it, until the image reflected in the other face coincides with the crosshairs. 180° minus the difference in the readings of the vernier gives the angle of the prism. Repeat as before, and also measure. the three angles and see if their sum equals 180°.

When a vernier is attached to the centre-plate this instrument serves to prove the law of reflection with great exactness. For this purpose it is only necessary to turn the centre-plate into vari

ous positions, bring the reflection of the slit to coincide with the cross-hairs of the observing telescope, and read the verniers attached to each; or in fact, to repeat Experiment 70, replacing the sight-hole by the slit and collimator, and the needle by the observing telescope.

73. LAW OF REFRACTION. I.

Apparatus. In Fig. 57, DBCG is a tank, like that of an aquarium, with the side BD of glass. Two horizontal scales are attached to CG, one over the other, and the tank filled so that one shall be seen above, the other below, the liquid. A is a plate of brass with a vertical slit in it, larger above, and tapering to a point. It is used as a sight, and is placed at a distance AB equal to BC. A small plumb-line may be hung in front of DB to serve as an index. A tank without glass sides may be employed instead, by regarding Fig. 57 as a vertical instead of a horizontal section, and placing one scale at the surface of the water, BD, the other at the bottom, CG. The divisions of the upper scale should then be one half those of the lower.

Experiment. On looking through A the lower scale will be seen through the water, the upper through the air only. The divisions of the former will therefore,

D

E

B

Fig. 57.

by refraction, appear larger than those of the other, and from the amount of this increase the law of refraction may be deduced. Placing the plumb-line at D, and looking through A, it will be seen projected on the upper scale at G, but on the lower, owing to the bend- 4. ing of the ray at the surface BD, at F. If placed at B, however, the reading on both scales will be the same, since the incidence being normal there is no bending of the ray. To find this point, read both scales, and if the reading of the upper scale is the greatest, move B to the right, otherwise to the left, until both read alike. The object of the varying width of the slit is to read approximately through the upper part, and then lowering the eye to eliminate parallax, and read more exactly by the lower portion. Move the plumb-line a short distance, read both scales again, and thus take ten or fifteen readings between B and D.