178 



UNDULATORY THEORY OF LIGHT. 



sions of the wave front to be equal in extent, wliicli is sufficiently exact for the 

 purpose in view. The other dark stripes which form 

 Avithin the bright image of the opening aa' arc sub- 

 ject to fluctuations of intensity similar to those of the 

 central one. To understand this, let i be a point so 

 taken, that Z>«' — Z»a = >i. Join Z>R, and let the line 

 ha revolve round V\.h to the position he. Then Z»(?= 

 ha, and ha' — hc^^X. Divide ca' at d into parts, 

 such that ha' — hd=y., and bd—hc^^?-. Then, so 

 far as this portion of the wave is concerned, the 

 point h will be obscure in every position of h which 

 preserves this relation, whatever be the distance from 

 A. Also, at any distance of h for which the divisions 

 - Fig. 4-t. of the wave front ac, made as heretofore described, 



are an even number on each side of Rh, the whole effect of the wave at h will 

 be null, and the point h Avill be obscure. But if the number of these divisions 

 on each side of llh be an odd number, there will be a portion of the wave un- 

 neutralized, and h Avill be illuminated. 



The fringes exterior to the bright image of the opening are more beautiful 

 than those interior to it, being, especially when the aperture is very narrow, 

 richly colored. They are not subject to the fluctuations of brightness, as the 

 distance of 13 from the aperture varies, which attend the interior fringes ; since 

 the lines ha and ha', drawn to any point in any of those fringes from a and a', 

 the limits of the aperture, will be both on the same side of V\.h. 



The distances d from the central line B are all determined by the same 

 equation Avliich was found for the fringes formed by a narrow opjiquc object. 

 Indeed, the geometrical conditions in the present case are identically the same 

 as those in that. '^Fhe optical difference is, that the even values of n give the 

 loci of the dark stripes, and the odd those of the bright. The breadths yavj, 

 as before, inversely as c, which is the diameter or Avidth of the aperture. With 

 apparatus in A\'hich the opposed edges are moA^able, the expansion of the 

 fringes, as these edges are made gradually to approach each other, is A^ery 

 striking. When the aperture is a very slender isosceles triangle, they spread out 

 Avidely toAvard the A-ertex. The expression, 



2c 



also shows that the breadth A-aries directly as the length of the undulation. In 

 homogeneous light, therefore, the broadest fringes are obtained Avith red, and 

 tlic narroAvest Avith Aiolet. In such light, a dozen or tAventy may easily be 

 counted. When Avhite light is employed, the OA'erlapping of the colors, Avhile 

 it improves the beauty of the display, reduces A'ery much the number that can 

 be distinguished. When monochromatic light cannot conveniently be obtained, 

 the same effects may be substantially produced by A'iewing the fringes made by 

 white light through colored glasses. 



When, instead of a long and narrow aperture, a small circular opening in an 

 opaque plate is used, the fringes are, of course, circular. In this case, the cen- 

 tral dark stripe of the preceding experiment becomes a central dark round spot. 

 This spot disappears and reappears as the screen is brought nearer the plate, at 

 the same distances at Avhich this effect Avas observed in the -central stripe; in the 

 image of the oblong aperture. Referring to the last figure, and regarding aa' 

 as the diameter of the circular opeiiing, Avhen B« — BA=2x^A, there Avill be 

 eome point bctAveen a and A (suppose a") Avhich, if joined to B, Avill give 

 Ba" — BA=.y.. NoAv, it has been shown that B« — BA A\aries as y^; the radius 

 of the aperture (or of the part of it considered) being represented by ?/. Hence, 

 for the point supposed, a", we have Aa^=:A2a"^; or the circle of Avhich Aa is 



