476 Transactions of the Society. 



microscopic limit occurs when the grating-interval is equal to half 

 the wave-length of the light in operation. 



The method employed in 1896 depends upon the use of 

 Fourier's theorem. The critical case, where the structure has 

 just disappeared, may be treated in a somewhat more elementary 

 manner as follows. It is required to prove that 



sin 2 u sin 2 (u + ir) sin 2 (u — tt) 

 ~W + {u + ir) 2 >(u - tt) 2 



, sin 2 (w + 2 7r) , sin 2 (« — 2 if) n( >\ 



(u -j- 2 Try (u — 2 try 



obtained by writing ir for v in (22) above, is the same for all 

 values of u. In (76) the (sine) 2 have all the same value, so that 

 what has to be proved may be written 



—=—0 = ~9~"J~ 7 1 \o "f" - / V> "1 / \ o \2 T • • • • (J l) 



sm 2 u u 2 {u + 7r) 2 (u — -rry (u + 2 iry 



This follows readily from the expression for the sine in factors. 

 If we write 



sin u = C u {u + 7r) (w — ir) (u + 2 w) . . -. . , 



or 



log sin u = log C + log u + log (u + tt) + , 



we get on differentiation 



and again 



d log sin w 1,1 ,1 , 



(III U U + 7T U — TT 



d- log sin % _ 1 , _1 , 1 . 



du 2 ' u 2 (u -f 7r) 2 (% — 7r) 



2 



In these equations 



d log sin u = . u _ d 2 log sin u 



du du 2 sin 2 u 



from which (77) follows. 



We infer that a grating of the degree of closeness in question 

 presents to the eye a uniform field of light and no structure, but 

 it is not proved by this method that structure might not reappear 

 at a greater degree of closeness. If however we take v — \nr, 

 that is, suppose the lines to be exactly twice as close as above, 

 a similar method applies. The illumination at the point is now 

 expressed by 



