180 



UNDULATORY THEORY OF LIGHT. 



(r-i-s) 

 seen that Ba" — BA bus a value expressed by the formula h=— — -y^,yhGmgihe 



radius of the circle 



This may be resolved into the parts, 

 2r 2s 



of which the first is constant when y is constant, and the second varies inversely 

 as s, which is the distance B A. The less the value of k ( --=^7t X h^-), the greater 

 will be the distance at which the color corresponding to A will be suppressed. 

 And as the color which remains is the difierence between the color suppressed 

 and white, it folloAvs that, as the eye approaches A, in the line BA, the ring or 

 the aperture will assume successively all the tints of the spectrum from red up- 

 ward, and that this series may be several times repeated. Moreover, putting X 

 for the length of the red, and X' for the mean length of imdulation in the com- 

 pound color complementary to red, which will correspond nearly to the wave 

 length in the green, when Ba" — BA=(2?i-j-l) X^x' <(2»+l) x ^ X, a point b 

 may be found on any side of A, but very near it, at which ba" — bA-z^{2n+l)x^X. 

 A green circle will therefore appear surrounding A, Avhile A itself, whclher 

 it be an aperture or a ring, will be red. Also, at other distances, greater or less, 

 circles of other tints will appear; so that the ring or aperture will be encircled by 

 a corona displaying all the prismatic colors, from i-ed to violet, shading outward. 

 As the eye approaches A, the equation ba' — bA=i{2n+l)xhX will be true of 

 points nearer and nearer to B, until b and B coincide. The rings will therefore 

 appear to be successively absorbed into the aperture. In withdrawing the eye, they 

 will seem to be, in like manner, evolved out of it. In this experiment the aperture 

 should be very small or the ring very narrow, in order that the tints may be vivid. 

 It will readily be understood that the obscurity and the sharp edges of shadows 

 of bodies of considerable size are owing to the smallness of the values of X for 

 all the rays of light. On this account, if any point be taken within the line of 

 the geometrical shadow, and if the wave Iront, beginning at the edge of the 

 opaque body, be divided into portions whose extremes are remote from that 

 point by distances differing AX, these portions Avill neutralize each other'8 effects, 

 except for positions of the assumed point for which the divisions have (as they 

 may near the shadowing body) some slight inequality, and no material obliquity. 

 Such positions can only be found very near the line of the geometrical shadow. 



To the same cause it is owing that, in 

 refraction and reflection, tho beam re- 

 fracted or reflected is as sharply defined 

 as the incident beam. The demonstration 

 which we have given of these efi\'Cts, from 



Huyghens, contains an imperfection on 

 this point, which Fresnel has supplied. 

 Referring to the figure, suppose that an 

 undulation originating at B should take 

 the direction B/c, different from that of 

 the main reflected wave, BQ'. There 

 will always be found, to the left of B, a 

 point, as n, from which another undulation will follow in the parallel and nearly 

 coincident line nk, differing from the first by half an undulation. Draw 710 per- 

 pendicular to BB' and B^ perpendicular to nh; o?iB is the angle of incidcnce=j. 

 Putp for nBt. Then Bo=B« sint, and 7it:^Bn.s\np. 



Now, when the wave whose front is Bt starts from «, the movement which is 

 to produce the wave from B is at o. There will accordingly be interference, if 

 tU — Bo=^A; that is, if B/z.(sino — sinO^^'i- But since X is very small, if sin^ 

 sensibly exceed sin;, Bn will be very small; showing that interference will 

 take place from a point very near B. As sin/j approaches sinr, the distance of 

 the origin of the interfering wave will be greater; but there will be an interfering 



