Abbe's Theorems. By Prof. J. D. Everett. 387 



Iu the present case R 2 measures the luminous intensity at P; 

 and the expression for it, as due to the five spectra above included, 

 will consist of five coDstant terms, together with ten terms which 

 are simple-harmonic functions of x. 



If we include only one of the five spectra, R 2 is constant, and 

 the intensity at P does not vary as the grating moves. The field 

 is therefore uniform, with no trace of lines. 



If we include only C and A u we have 



R 2 



= c 2 + a x - + 2 c «j cos 2 7r (- + aA; 



showing that the intensity goes through one complete cycle of 

 values as x increases from to s. There is therefore one line in 

 the image for one line in the grating. 



If we include only A x and B 1? we have 



R2 = ai 2 + h 2 + 2 «! l x cos 2 ir ( 2 y + ai + &) ; 



showing that the intensity goes through its cycle while x increases 

 from to \ s, and goes through two cycles while x increases from 

 to s. There are therefore two lines in the image for one line in 

 the grating. 



If we include only Ai and B 2 (the two intervening spectra 

 C and B x being stopped out), we get 



H 2 = a x 2 + b, 2 + 2 «! b 2 cos 2 ir (— + a x + /3 2 ) ; 



showing that there are three lines in the image for one in the 

 grating. 



With any five consecutive spectra included, we shall have 

 xjs, 2 xjs, 3 x/s and 4 x/s in the arguments of the cosines ; and 

 the expression for the intensity will be reducible to a Fourier 

 series containing four periodic terms, of periods s, % s, | s, % s. 



