800 and thus OPTICS /x cos 2 <p If the refracting surface be curved, with curvature r~ l , we get by similar reasonin U. COS 2 COS 2 fJ. COS - COS ,n .
= - 1 ...... ) >
v 1 u r fJ, 1 _ fJt. COS - COS <f> ,^ . . 2 in which (1) and (2) are of course included as particular cases. When the incidence is direct, cos 1, cos In this case (3) and (4) become v u r To find the positions of the focal lines ofapcncil refracted obliquely through a plate of thickness t and index p. If be the angle of incidence (and emergence), the angle of refraction of the ray QAST (fig. 9), Sqi = v 1} Sq.^ v. 2> AQ = , we get by successive applications of (1) and (2) t cos 2 ,, t , 1 = tt + _ yZ., (6); If the incidence be direct, ^ COS" t /J. COS t .(7). .(8). Tlius, if we interpose a plate between the eye and an object, the effect is to bring the object apparently nearer by the amount On this result is founded a method for determining the refractive index of materials in the form of plates. A set of cross wires is observed through a magnifying glass. On interposition of the plate the glass must be drawn back through a distance given by (9) in order to recover the focus. If we measure this distance and the thickness of the plate we are in a position to de termine the refractive index. Prism. By a prism Fig. 9. is meant in optics a portion of transparent material limited by two plane faces which meet at a finite angle in a straight line called the edge of the prism. A section perpendicular to the edge is called a principal section. Parallel rays, refracted successively at the two faces, emerge from the prism as a system of parallel rays. The angle through which the rays are bent is called the deviation. The deviation depends upon the angles of incidence and emergence ; but, since the course of a ray may always be reversed, the deviation is necessarily a symmetrical func tion of these angles. The deviation is consequently a maximum or a minimum when a ray within the prism is equally inclined to the two faces, in which case the angles of incidence and emergence are equal. It is in fact a mini mum ; and this position of the prism is described as the position of minimum deviation, and is usually adopted for the purposes of measurement. The relation between the minimum deviation D, the angle of the prism i, and the refrac tive index /z is readily found. In fig. 10 the internal angles < , ^ are each equal to i. The external an gles <, ty are also equal, and are con nected with </> by the law of refrac- tion sin < = fj. sin </> . D Fig. 10. The deviation is 2 - </> ). Hence _ sin ^ ~ sin and this is the formula by which the refractive index is usually determined, since both D and i can be measured with great precision. The instrument now usually employed for this purpose is called a goniometer or spectrometer. Parallel rays are provided by a collimator, consisting of an object-glass and telescope-tube, by means of which the subject of examina tion, either a fine slit or a set of cross Avires, is seen as if it were at an infinite distance. The parallel rays from the collimator, after reflexion from a face or refraction through the body of the prism, are received by a telescope also provided with a set of cross wires at its focus. The table upon which the prism is supported, as well as the telescope, are capable of rotation about a vertical axis, and the position of either can be read off at any time by means of graduated circles and verniers. As a preliminary to taking an observation it is neces sary to focus the collimator and telescope. The first step is to adjust the eye-lens of the telescope until the cross wires are seen distinctly and without effort. The proper posi tion depends, of course, upon the eyesight of the observer, and is variable within certain limits in virtue of the power of accommodation. It is usually best to draw out the lens nearly to the maximum distance consistent with dis tinct vision. The telescope is now turned to a distant object and focused by a common motion of the cross wires and eye-lens, until both the object and the cross wires are seen distinctly at the same time. The final test of the adjustment is the absence of a relative motion when the eye is moved sideways across the eye-piece. The colli mator is now brought opposite to the telescope and adjusted until the cross wires in its focus behave precisely like the distant object. To measure the angle of a prism it may be placed with its edge vertical upon the table, in a symmetrical position with respect to the collima- tor (fig. 11). The telescope is then successively brought into such positions that the cross wires of the telescope coincide with the cross wires of the collimator when seen by reflexion in the two faces. The difference of the read ings is twice the angle of the prism. Another method is also often employed in which the telescope is held fixed and the prism is rotated. The angle between the two posi tions of the table found by use in succession of the two faces is the supplement of the angle of the prism. 11.
