196 Lord Rayleigh on the Manufacture and 



vanishes when — is an even multiple of — . The positions of the 



maxima (determined by tan — = — ) do not exactly bisect the 



angles between the vanishing directions ; but it will be sufficient 

 for our present purpose to note that the principal maximum 

 (R = 0) is unity, and that the others do not differ greatly from 



(2 \ 2 / 2 \ 2 / 2 \ 2 

 q- ) , \~~) > ( 7— ) , &c. It is evident that on either side of 



the principal direction the illumination falls off with great rapi- 

 dity. If A B is 1 (inch) and X = ^JL_ the angle C B P corre- 

 sponding to the first minimum is only about 5". 



The image of an infinitely narrow line of light (whose length 

 is perpendicular to the plane of the diagram), as formed by an 

 object-glass with aperture A B, is thus a series of parallel stripes, 

 composed of a central narrow band whose illumination varies 

 from a maximum in the middle to zero at the edges, enclosed by 

 parallel bands of rapidly decreasing illumination. We have now 

 to examine the effect of the ruling. 



For the sake of simplicity, we will take first the case of a gra- 

 ting composed of transparent bars of width a, alternating with 

 opaque bars of width d } and consider the central image or 

 spectrum of zero order. In the principal direction, B C, the 

 secondary waves are, as before, in complete agreement, but 

 the amplitude is diminished by the ruling in the ratio 

 a : a + d. In another direction, making a small angle with B C 

 such that the relative retardation of A and B amounts to a few 

 wave-lengths, it is easy to see that the mode of interference is the 

 same as if there were no ruling. For example, when the direc- 

 tion is such that the projection of A B upon it amounts to one 

 wave-length, the elementary components neutralize one another, 

 because their phases are on the whole distributed symmetrically, 

 though discontinuously, round the entire circumference. The 

 only effect of the ruling is to diminish the amplitude in the ratio 

 a : a + d; and except for the difference in illumination, the ap- 

 pearance of a line of light is exactly the same as if the aperture 

 were perfectly free. 



The lateral images occur in such directions that the projection 

 of the element a + d of the grating upon them is an exact mul- 

 tiple of \, The effect of each element of the grating is then the 

 same ; and unless this vanishes on account of a particular adjust- 

 ment of the ratio a : d, the resultant amplitude becomes com- 

 paratively very great. These directions, in which the retarda- 

 tion between A and B is exactly mn\, may be called the prin- 

 cipal directions. On either side of any one of them the illumi- 



