We shall now investigate some simple cases of refraction. 1. And first of refraction nt a plane surface. Let DIMN (fig. 1) be any medium bounded by a plane DI, and let R be a radiant point, and RD and RI two incident rays of a divergent pencil proceeding from R to the surface of the medium; then RD being perpendicular to the surface, suffers no refraction, but proceeds along DM within the medium; hut RI is refracted in tne direction IN, which, produced out wards, meets the normal DF in F. Therefore, a small pencil of rays proceeding from R, and having RD, perpendicular to the surface, for axis, will be refracted into another pencil diverging from the imaginary focus F; for all the rays inter mediate 'between RD and RI will converge very near F when the pencil is small. An eye within the medium, and between N and M, would thus, the pencil being small, sec the luminous point R, as if it were at F, or further off than it really is. In the opposite case, in which the luminous point is within the refracting medium, similar reasoning shows that after the rays emerge from the plane surface into the air, they will, if the pencil be small, appear to proceed from an imaginary focus nearer to the surface than the luminous point.
2. The case of refraction through a prism, which we are next to consider, is, in fact, the case of refraction through a medium bounded by plane surfaces which are not parallel. Conceive two planes at right angles to 'the plane of the paper,.and making on that plane the figure BAC (fig. 2). The question is as to the laws of transmission of a ray, SPQR, of homogeneous light through the prism. mil and n'n perpendicular to the sides. Then n'PQ and n'QP are respectively the angels of refraction at the first, and of incidence at the second surface. Now, as n'QA and n'PA are each of them right angles, and as all the angles in the figure 71,QAP are equal to four right angles, it follows that the angles at n' and at A together are equal to two right angles.
But the angle at n', together with the angles n'PQ and n'QP, are equal to two right angles; therefore must the angles n'PQ and n'QP together be equal to the angle at A. In other words, in refraction through a prism: The sum, of the angles of refrac tion at the first surface, and of incidence at the second, is equal to the angle contained between the plane sides of the prism. From this it might be shown, that the deviation of a ray causqd by passing through a prism is always towards the thicker part of the prism, if the medium be denser than the surrounding atmosphere. It is a geometrical proposition which the student may solve
for himself, that if i be the angle of incidence at the first surface, and e that of emerg ence at the second, and if a be the angle of the prism, then 6, or the change of direc tion of the ray in its passage, is obtained from the formula 6 = I+ e—cr.
3. We now take up the case of refraction at a single spherical surface of a medium denser than the surrounding air. And first, of parallel rays refracted at a convex spherical surface. Let ABP (fig. 3) be the refracting medium, whose terminating convex surface is spherical, C being the center of the surface, and V its vertex. Let XV be the axis of a pencil of parallel rays, of which any ray, RI, is incident at I. Then, if CIN be a normal, the angle of refraction, CIF, will be less than the angle of incidence, RIN, and the refracted ray will thus turn towards the axis, and meet it at some point, F. the pencil is small, or the aperture, AVB, of only a few degrees, the rays will clearly nearly all converge to the same point, F. To find the position of F, we have, in the triangle ICF, the angle CIF= r, the angle of refraction, and ICF, the ,supplement of ICY or NIR (by paral ) lel lines), i.e., of or the angle of inci dence. Therefore, IF is to CF : : sin is to sin r. And as for a very small pencil, IF may be taken = VF, we have FV : Fe : : sin i : sin r, or : : A : 1. And putting FV = F, the principal focal distance, and VC = R, we have F = R. If the medium be for p — which the value of p is I, we have F = 1 R, or F = 3 R; i.e., the principal focal I — distance is equal to three times the radius of-the sphere. The student may, by similar reasoning, ascertain for himself the focus of parallel rays incident on a concave spher ical refracting surface, as also the focus in the case of a pencil of parallel rays within the medium and emerring'from it: The case of a divergent pencil is incapable of such elementary treatment as to justify its insertion here. For branches of the subject treated under separate heads, the reader should refer to the articles CAUSTIC, LENS, and REFRACTION. tinder REFRACTION, he will find a table of the values of is—the refract ive index for various media and kinds of light. See also the articles SPECTRUM and