150 Prof. F. L. 0. Wadsworth on the 



Since the second term of the quantity within the brackets 

 is always small compared to unity for any practicable values of 

 yS, it may in general be neglected. The only difference there- 

 fore between the expression ((32) for a parabolic grating and 

 the expression (40) for a spherical grating is in the presence of 

 the factor tang^ i as coefficient of the term in sin /S. This 

 will increase the value of k for values of i less than 45°, and 

 decrease it for values above 45^^. Thus, for the same grating 

 before considered (W^73 mm., ^ = *011), we find for Km 



Km = +54', for i = 1 



Km= ±56', ,,2-154 . . . (C4) 



Km= +2M0^ „ /=60°J 



Comparing these results with those of (40) _, we see that the 

 effect of parabolizing the surface of this given grating is to 

 increase the field by only about 15 per cent, for small values 

 of ?, and decrease it about the same amount for large values 

 of i. 



For larger angular apertures the field of good definition 

 is reduced, as we have already seen^ in the inverse ratio of 

 the square of the aperture. Thus for a grating of angular 

 aperture ^^ — '06 the field of good definition is only a little 

 over 5' (2^ 35^^ each side of the axis) for normal incidence, 

 and it would become zero for a value of i a little less than 

 15° [see (55)]. 



For any given angular aperture /3 the value of Km varies 

 also very nearly inversely as the linear aperture 2W. Thus, 

 if we decrease the size (aperture and focal length) of the 

 grating one half, we shall double the field. Or, keeping the 

 field the same, we may increase the maximum permissible 

 value of ySmax. This will vary for any given value of i in- 

 versely as the cube root of the linear aperture [see (55)], the 

 field being confined to the image on the axis of the grating. 

 For any smaller values of i the field will be increased, and 

 since Km decreases only as the square of the angular aperture, 

 it follows that by decreasing the linear aperture we may 

 simultaneously increase the value of ySmax. for large values of 

 i, and the values of K,n for small values of i. Thus, if we 

 decrease the linear aperture of the grating considered in (54) 

 and (64) to, say, 2 cm. (W = 10 mm.), we may increase 

 permissible angular aperture 2/3 to nearly 0"10 (yS^^'OS) 

 for 2 = 60°, and increase the angular field to about 0^*6 



(Km=l^' to K„^=n') for i = tO 2 = 30°. 



It will also be instructive to compare the preceding results 

 for the limiting values of Km and /3„i from another point of 



