170 UNDULATORY THEORY OF LIGHT. 



By means rf a movable eyepiece, pi-ovided with a microitctrical apparatus, 

 Frcsnel accurately measured the distances of these stripes from each other, 

 and thus deduced the lengths of the undulations by which they are produced. 

 In fact, as the locus of the central bright stripe is in the line of intersections, of 

 which o is one, and that of the adjacent bright stripe is in the line of intersec- 

 tions, of which h is one, we have a small triangle ahc, whose sides arc severally 

 perpendicular to those of the triangle «SS' ; and accordingly, 



cS : SS' :: he : ac; or ac= pr— . [7.1 



aii *■ ■* 



But ac is the length of the undulation, whence it appears that this length is 

 equal to the distance between two adjacent similar stripes multiplied by the 

 distance between the two radiant centres, and divided by the distance of either 

 centre from the screen. As the radiants in this experiment arc merely virtual 

 and not actu.il, the values of SS' and aS cannot be conveniently measured. But 

 it may be observed that the fraction 

 SS' 



-^=2sin JSaS'=2sin BAA'. 

 ao 



Hence, ac=bc.2s'm ^SaS'—bc.2sm BAA'. [8.] 



The angle S«S' may be directly measured by an instrument placed at a, or 

 the angle BAA', which is the inclination of the mirrors, may be otherwise deter- 

 mined. 



Putting A for the length of the undulation, <f for the angle iS«S', and S for 

 the distance between the stripes, the I'oregoing equation gives 



;.=2sincr X ; or 0=—, ; f 9.1 



Whence it appears that the distance between the stripes will be greater as 

 ^ is less, or as the radiant centres are nearer together. The same process 

 applied to the distance from the middle stripe to the second one on either side 

 will give — 



0'= — ; — : and for the third, S"^= — ; — , &:c. 

 2smc' 2sm^ 



So that the successive stripes .are equidistant from each other. 



Grimaldi's case of diftVaction, in which the radiant centres were two minute 

 apertures very near to each other through which light was introduced into a 

 dark room, was manifestly analogous in principle to this. To that case the first 

 of the formulfie just given may be directly applied. 



The mirrors in the experiment of Fresnel require very careful adjustment. 

 If, at the edges where they meet, one or the other projects, however slightly, 

 the effect will be sensibly impaired. A prism of glass having two adjacent faces 

 very slightly inclined to each other might be used to produce the interferences 

 by total reflection from these inclined surfoces, without being subject to this ob- 

 jection. The other faces of the prism would require to be so adjusted that the 

 light might enter and emerge through them sensibly at right angles. The dis- 

 advantage would be that the angle of the reflecting faces would be invariable. 

 The experiment admits of being performed, and has been perfoimcd, by the 

 help of a single mirror, placed almost but not quite parallel to the original rays, 

 so as to cause a portion of the wave very slightly to deviate, and thus to inter- 

 fere with the portion which is not reflected. In this case it is obvious that the 

 system of fringes produced can embrace only one-half of those which are seen 

 in the experiment of Fresnel. 



In place of Fresnel's mirrors Mr. Arago employed a glass prism to produce 

 interference by refraction instead of by reflection. Arago's prism has a cross 

 section of the form of a very obtuse angled isosceles triangle ; the light being 

 received in the experiment perpendicularly upon the base, and emerging at ilu'. 



