420 Mr. R. T. Glazebrook on 



and taking the negative sign, so that sini|r= — > 



* , (o sec ylr x n\ /11N 



and Q 1 Q , = W / S^=!^. 



6 



Thus, carrying our approximation as far as terms in © 3 in 

 equation (4), we find that the position of the image formed, 

 considering only two of the lines as producing diffraction 



effects, is not at Q 1 but Q', where QiQ'= — w- . Hence, if we 



consider the whole grating, using the same notation as before, 

 the breadth in a direction normal to AQ X of the image formed 



will be comparable with ^ — , © being the whole semi- 

 aperture. Expressing this in terms of the radius of the gra- 

 ting a and the distance between the lines <7, we find the value 



\{k + k r ) 2 — . Thus the breadth of the image will depend on 



the square of the number of lines. In the grating first con- 

 sidered this quantity, \(k + k f )\(o, is about ^Jq of a centimetre 

 for yellow light, while the distance between the D lines is about 

 jfe centim., or ten times as much; while in the second grating 

 this lateral aberration is -£$ centim., the distance between the 

 D lines being about seven times as great. If the size of this 

 last grating be reduced to f of what it actually is, the extreme 

 lateral aberration will be reduced to ■£% or about \ of its 

 actual value, thus becoming about -§\~5 °f a centimetre, and 

 the extreme difference of phase in the light of a given wave- 

 length X reaching any point of the diffracted spectrum will never 

 exceed A/4, the dispersion will remain unaltered, the defini- 

 tion and the brightness of the spectrum will both be increased. 

 It is clear that in both cases the outer portion of the grating 

 not merely impairs the definition, but actually renders it less 

 bright than before. For Fig. 4 



consider two points P ls P 2 

 (fig. 4) equidistant from 

 A, such that the difference 

 in phase in the waves 

 coming from P x and P 2 to 



Qi is ^ (since the differ- 

 ence of phase for the ex- 

 treme rays is in both cases 



greater than ^,these points 



can be found). Then the light reaching Q x from above P 2 is 



