176 



UNDULATORY THEORY OF LIGHT. 



the shadow, hut to some distance into the light on each side. Should any evi- 

 dence seem to he needed to confirm the theory on which the formation of these 

 fringes has been explained, it may be found in the fact that if, by an interposed 

 card, the light from one side of the opaque object be arrested, all the fringes 

 will instantly disappear from the shadoAv. 



As for the form of the loci of these fringes in space, since each is determined 

 by the intersection of radii from A and C, having a constant difference, they are 

 necessarily hyperbolas, having A and C for their foci. But in this case it is the 

 principal axis which is small, while the conjugate axis is comparatively very 

 great ; so that the curves are widely open, having but slight curvature even at 

 their vertices. 



Let B'A — li'C^ii X tp', n being any integral number, even or odd. It is 

 evident, from the law of the hyperbola, that hd is the principal a xis of th e 

 trajectory of B'., And, putting CA=c, the conjugate axis will be ^/ d^ — ^w^/^. 

 Making this the axis of y, and the former the axis of x, the equation of the 

 curve gives us — 



„ c^—hi^)? {nix 



y 



Suppressing the minute term \n'^f?' 

 equation with respect to x, we obtain- 



(nlx „\ 



from the numerator, and reducing the 



4 



iy'' + ^)n'A^ 



[14.] 



According to the notation heretofore used, y, which is the distance of B from 

 the object AC, may be replaced by s. Also c^, in the numerator under the 

 radical, may be dropped without appreciable error, except when B is quite near 

 to the object. The simplified expression will then be — 



snl 



d=—, which is the equation of a straight line. 



2c 



[15.j 



At any considerable distance from AC, therefore, as compared with AC, the 

 hyperbolic trajectory sensibly coincides with the asymptote to the curve. In 

 fact, the ecjuation of the asymptote being — 



, A , UX 



"B 



3/ — 



VrP. 



[16.] 



n"/. 



by rejecting the minute term under the radical, we obtain — 



^ nh/' snl 1 • 1 . . n . , -11 P 

 x=^o^^ — ^=r-— ■ : which IS identical with the lormer. 

 2c 2c 



By substituting different numerical values for n, this equation serves for all 

 the fringes, light or dark. The even numbers give the loci of the bright stripes, 

 and the' odd those of the dark. The distance <5 is in all cases measured from 

 the middle of the central bright stripe. 



The expression for the value of o indicates, at sight, that the fringes Avill 

 increase iu breadth, as the opaque intercepting object diminishes in diameter. 



In fact, '5 is inversely as c, and to double the 

 breadth of the fringes, wo have only to r(!duce 

 the diameter of the object one-half. Accord- 

 ingly, if a tapering object, as a sewing needle, 

 be employed, the fringes will spread out toward 

 the top with a beautiful plumose appearance. 

 This becomes still more striking when the taper 

 is more rapid, as when we use an acute-angled 

 ^''g- ^2. or even a right-angled plate of thin metal. The 



fringes, which in this case are very remarkable, have been called Grimaldi's 

 crests. 



