Ahevration of the Concave Grating. 141 



that the "^ definition " (sic) begins to deteriorate when the 

 aberration exceeds one fourth of the wave-length of light. 

 Definition is itself rather of an indefinite term in a physical 

 sense, although it has a well-founded physiological basis. 

 Of this point it will be necessary to speak later. Assuming 

 for the present the limit assumed above, we-can determine on 

 that basis the maximum angular aperture /9max. and the 

 maximum field /<:max. that can be utilized in any given case by 

 the equation 



Z<iX (37) 



For the case A of the O.S. spectroscope this equation 

 becomes 



psm /3= — 

 a 



or sin3/3=2^=4^^ .... (38) 



v/a a 



where W is the semi-ruled surface of the grating, and uq the 

 angular resolving-power of a telescope of an aperture equal 

 to that of the grating. 



Equation (38) determines the maximum permissible semi- 

 angular aperture /3max. which can be used for any given value 

 of /. This varies, as will be seen, inversely as the cube root 

 of the linear aperture, 2W, of the grating, and also inversely 

 as the coefficient a. For a grating of the usual linear 

 aperture W^72 mm. in the position i = 15° we have for 

 y^max. in the visual region of the spectrum (\=5500 tenth- 

 metres) 



3 / . 



/-'max. ^^ \/ -1 .Q 



0011 _.ooo 



For the same grating used in the position z = 60° 



A,ax = -034, (39) 



or since 60° is about the maximum permissible angle of 

 diffraction, (39) is about the maximum permissible semi- 

 angular aperture of a spherical grating of 15 cm. aperture 

 used as an objective spectroscope. 



Equation (37) also enables us to determine for any given 

 grating (whose semi-angular aperture is less than /5max.), the 

 maximum permissible field of good definition Km* To find 

 this we put 



■^B(max.) — i" J- 



since the maximum aberration is always positive. 



