Telescopes and Spectroscopes for Lines of Finite Width. 319 



for a train of N prisms of refracting angle <f> through which 

 the rays pass at minimum deviation, and 



r = tti n-o = mn (3) 



//'cost? v ' 



for a grating of n lines. 



In the case of the grating the expression for the resolving 

 power may be put into a form which will bring out more 

 clearly one fact that is not generally emphasized in the theory 

 of the gratings, i.e., that for a given position of the grating, 

 the resolving power is independent of the number of lines n 

 and is determined, as in any optical instrument, simply by the 

 linear aperture b. This proposition may be very simply 

 proved from the fundamental equation of the diffraction- 

 grating, 



?7iX=s(sin2 -f sin 6). 



Multiplying both sides by n we get 



mw\=7is(sin2-fsin 6) (4) 



But mn = r and ns = b the linear aperture of the grating. 

 Hence 



r=- (sin2 + sin<9), (5) 



an expression which is independent of n and depends only on 

 b and on the position into which the grating is turned. 



The maximum value of r is that for which 2 = = 90°. 

 Then we have 



?W. = 2- , 



which shows that the resolving power of a grating is an 

 expression of the same form as the corresponding expression 

 for a microscope, telescope, and reflecting mirror. The 

 maximum resolving power is the same (though expressed in 

 different units) as that for a mirror of the same horizontal 

 aperture. 



This theoretical maximum, however, can never be realized, 

 because for large angles of incidence and diffraction the 

 angular aperture of the grating becomes very small, and the 

 light consequently excessively faint. In practice the angle 

 of incidence i never exceeds 60° for an angle of diffraction 

 = 0, nor more than 45°-50° when the angles of incidence 

 and diffraction are equal (Littrow type). Hence maximum 

 practical resolving power, which we will call r , varies from 



lb . 3 6 



ffl= u tor ° = n"' 



2C 2 



