DIFFRACTION OF GRATINGS. 



181 



wave, (if the surface AB is milimited,) iu every case except that in which 

 sin^=siu c ■ that is, in Avhich Bk coincides in direction with the regularly re- 

 flected wave. 



In like manner, in the case of refraction, if we suppose a wave to diverge in 

 the direction Br, draw nq parallel to Br and Bp perpendicular to it. Call the 

 angle nBp, p, as before. Then Bo^Bn.s'mt, and n2:i^^Bn.i<'mf>. But np being 

 the path of a wave in th(! denser medium, it must be multiplied by the. index of 

 refraction, iu order to obtain the equivaleiit distance, or dhtance which the tcave 

 would have moved in the same time, in the rarer. Let n be the index of re- 

 fraction, and we have, for the condition of interference, B«.(7?.sin,o — sin£)=.^/l. 

 If n^i\\<> is sensibly greater than siut, Bn must be very small. And for any 

 value of n^mp — siuf, there will be a distance Bn furnishing a wave of inter- 

 ference, if the surfece AB is unlimited; except only for the value n^mp — sin.'=0, 

 when the ray Br ceases to diverge from the direction of the main refracted wave. 

 These reasonings assume that the forces of the elementary derivative waves 

 are the same in all directions. But it is probable that these forces are less in 

 lines oblique to the direction of progress of the primitive wave than in that di- 

 rection. How far this is true could be easily investigated experimentally, by 

 employing apertures less than the length of a half undulation in diameter, were 

 it not that the extreme minuteness of such apertures (the mean length of a half 

 undulation not exceeding one one-hundred-thousaiidth of an inch) would render 

 the light too feeble for the purpose. 



Some material for the formation of an opinion on this subject may, however, 

 be gathered from certain phenomena of diffraction first observed by Fraunhofer, 

 more remarkable and more brilliant than any which have been thus far men- 

 tioned. If a single very minute aperture will not furnish light enough for 

 experiment, an assemblage of very many very minute apertures, closely grouped, 

 may do so; and if these be so arranged that, for any determinate point in the 

 shadow, they feihall allow only such portions of the wave front to pass as conspire 

 in their effects at that point, while the intervals between them obstruct those 

 portions which conflict, we shall possibly find that the tendency of a wave 

 originating in a single molecular impulse to expand equally in all directions, is 

 much more decided than had been supposed. Fraunhofer's original experiments 

 were made with gratings formed by stretching an exceedingly fine wire across 

 two parallel screws of a great number of threads to the inch — the threads 

 serving to keep the wires equidistant. He subsequently employed gratings 

 formed by cementing leaves of gold to glass and. cutting them through in very 

 fine parallel lines ruled with a sharp instrument. Instead of these, also, he 

 employed similar lines ruled with a diamond on glass itself. 



The results of such an arrangement may easily bo predicted. The image of 

 an aperture closed by such a grating will appear bright, as though the obstruction 

 were not interposed. But toward either side, in the direction perpendicular to the 

 lines of the grating, will be found several points for which the part of the wave 

 which the grating obstructs v,^ould if allowed to pass be more or less in conflict with 

 those which it transmits ; and which, therefore, arc bright when the grating is pres- 

 ent, and dark when it is absent. Suppose, for simplicity, that the open spaces and 



tlie opaque bars are equal in breadth. Let 

 a, a, a, represent several of these open spaces, 

 and b, b, b, &c., the intermediate bars. A 

 point, P, may be found from which lines being 

 drawn as in the iigure, and perpendiculars let; 

 fall upon them from the edges of the aper- 

 tures, as at c, c, d, d, will give cd =^ 1,1, 

 db = U, and therefore cb = ?.. The distances 

 from P to the corresponding parts of the sev- 

 eral openings Avill thus differ by an entire un- 

 waves which reach P through them will be in harmony. 



