REFRACTION 



619 



medium, have been at BRD. It has not, however, 

 got so far as R in the time ; the central part of 

 the wave-front has only got as far as R', where 

 AR : AR' : : /t : 1. Any non-axial ray, such as 

 XP, which would have reached Q, can only have 

 originated a disturbance at P, which would have 

 travelled from P in some direction to a distance 

 not equal to PQ, but to PQ reduced in the same 



X' 



Fig. 3. 



ratio of /t : 1. We mijjht then, knowing p, the 

 relative index of refraction of the denser medium, 

 draw, with centre P and radius = PQ -j- n, an arc 

 of a circle ; the disturbance will have got to some 

 point on that circle. Doing the same for all the 

 P's, we have a series of circular arcs which may be 

 connected by a line drawn so as to touch them all. 

 This line will be a curve ; and it will, for some 

 distance from the axis, coincide very nearly with 

 the arc of a circle whose centre is at X', so that the 

 wave-front will travel in the denser medium 

 approximately as if it had originally come from 

 X'. The relation between the distances AX, AX', 

 and AC is given by the formula n^/ AX.' - Ha/AX. 

 = (j*i - fv)JAC, where HQ is the refractive index of 

 the original, and /*, that of the refracting medium. 

 For example, let li, = 1 (air) and /i, = 1'5 (crown- 

 glass ) ; AC = 2 inches ; AX = - 1 inch ( Le. the 

 source of light is one inch to the left of A) ; then 

 \\l AX' + 1/1 = V2 ; whence AX' = - 2, or the 

 light travels in the denser medium as if it had 

 come from a point 2 inches to the left of A. If the 

 wave-front be plane as it approaches A, that is 

 equivalent to AX = - infinity or Mo/AX = ; 

 wlience AX' is equal to + 6, or the light converges 

 on a point in the denser medium 6 inches to the 

 right of A. If, however, a plane wave -front 

 approach A in the denser medium, that is equiva- 

 lent to AX = + infinity ; but, as the original 

 medium is now the denser one, ^ = $ and ^ = 1 ; 

 whence, by the formula, AX' = - 4, and the con- 

 vergence is on a point 4 inches to the left of A. 

 These distances of the points of convergence for 

 plane waves, at - 4( = /) and + 6( =/*) from A, 

 are the Principal Focal Distances for the curved 

 surface and the media in question ; and they bear 

 numerically the same ratio to one another as the 

 refractive indices do ; from which, together with 

 the previous equation, we get - f/ AX. + f / AX.' 

 = 1 ; which shows, still keeping to our numerical 

 example, that when the object lies at a greater 

 distance than 4 inches to the left or 6 inches to the 

 right of A, the image is a real one on the opposite 

 aide of A ; whereas when it is at a less distance 

 from A, X and X' are on the same side of A, and 

 the image is virtual. X and X', thus determinate 

 when one of them is known, are conjugate foci ; 

 and they are interchangeable, so that an object at 

 either will produce an image, real or virtual as the 

 case may be, at the other. 



The refracting medium may not be of indefinite 

 extent, but may be bounded in the path of the 

 lii.'lil hy another surface. If this be symmetrical 

 with respect to the first spherical surface we have 

 a li-tit : and then, by repeating our calculations of 



Fig. 4. 



the refraction at the second surface as if the injage 

 produced by the first were itself an object, we 

 arrive at the formulae given in the article on 

 LENSES. 



If a parallel beam of light enter one plane surface 

 and be there refracted and emerge by another which 

 is not parallel to the first, we have the essentials of 

 a Prism. Assume the incident light to be mono- 

 chromatic ; then fig. 4 shows the incident beam SP 

 taking the course SPQR. 

 The elements of the pro- 

 blem are, ft being the rela- 

 tive index of refraction of 

 the prism : ( 1 ) /* sin QP' 

 = sin SPn ; ( 2 ) ^ sin PQ' 

 = sin RQm ; (3) angles 

 QPn' + PQ' = angle A, 

 hy the geometry of the 

 figure ; and (4) angles SP/i 

 + RQro = angles A + 

 inn' n, this last being the 

 Deviation produced by the 

 prism. These four equations contain seven terms ; 

 and it is sufficient to measure three of these, say the 

 angles A, SI '//, and mn'n, in order to ascertain the 

 rest, including ft, the relative refractive index of the 

 prism for the particular monochromatic light em- 

 ployed. If, however, the light employed be not 

 monochromatic but mixed, as ordinary daylight, 

 we find that the prism sends each wave-length 

 each colour-sensation-producing component of the 

 daylight (see COLOUR) to a different place, and 

 thus produces a Spectrum (q.v.). Each wave- 

 length has its own , and its own deviation ; the 

 more rapid, shorter waves being the more refran- 

 gible by a given piece of glass. 



If in fig. 4 the prism be turned so that S and R 

 lie symmetrically with reference to the angle A, 

 the deviation is then a minimum ; and in that 

 position of minimum deviation a monochromatic 

 beam, divergent from S, will come to focus at R. 

 In examining the spectrum of light from a source 

 S it is necessary to turn the prism so as to ensure 

 sharpness by producing this minimum deviation for 

 each part of the spectrum in succession. When 

 the deviation is a minimum everything is sym- 

 metrical ; SPn = RQt ; QP' = PQ' : whence, 

 by equations above, SPw = J( A + mn'n), and 

 QP'=}A; whence / = {sin 4 ( A + mn'n ) -j- sin 

 ^A}, which determines n, when A (the angle of the 

 prism ) and mn'n ( the deviation ) have l>een meas- 

 ured. The refractive indices of liquids and of 

 gases are determined by enclosing them in hollow 

 prisms of glass whose walls are made of truly 

 parallel glass ; the parallel glass produces no devi- 

 ation. In liquids the angle of total reflection or 

 ' critical angle ' may also be readily measured ; 

 then the sine of this angle = I//*. The refractive 

 index varies with changes of density, p - 1 being 

 approximately proportional to the density : and it 

 bears certain intimate relations with the molecular 

 constitution of the refracting matter. 



Why ether-disturbances of differing wave-lengths 

 are differently refracted in such a medium as glass 

 is not yet perfectly clear. The fact that ether- 

 disturbances of greater frequencies are propagated 

 more slowly through optically denser matter may 

 be fairly inferred to arise from a mutual interaction 

 of the ether, periodically stressed and released, and 

 the matter amid whose molecules the disturbance 

 is propagated. The question is complicated by the 

 downright absorption or non-transmission of many 

 particular wave-lengths, and by the peculiar be- 

 haviour of some particular transparent substances 

 which produce anomalous dispersion : for example, 

 iodine vapour refracts' red light more than blue, and 

 blue more than violet ; an<T fuchsine refracts blue 

 and violet light less than it does red, orange, and 



