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



always negative (for the samc^ values), we have 



R"=^^=___5^' (60) 



By the aid of the preceding Table III., which gives the values 

 of R for a spherical grating, we can readily obtain the values 

 of W for a parabolic grating, and the values of R^^ gi^ hig the 

 ratio of the aberrations of the two. These values of R^ and 

 R" have been computed for the fields S, S^, and S''' for a 

 grating ^83 = '05, and for fields ^"' and S^^ for a grating 

 ^2 = '025. They are tabulated in Table IV. Comparing the 

 values of R' of this Table with those of R of the preceding- 

 Table, we see that for values of i less than 45^ the variation 

 in the amount of the aberration at different parts of the field 

 is greater with the parabolic than with the spherical grating, 

 but that the absolute amount, as shown by R", is everywhere 

 less. Above 45° the reverse of both propositions is true. 

 Hence, for large values of i the parabolic grating is inferior 

 in definition to the spherical grating, although the field is 

 more uniform ; for small values of i the definition of the 

 parabolic grating is much the better at the centre of the 

 field, but the field is far less uniform. 



A further inspection of the values of R^' for these small 

 values of i shows that for values of k greater than 1° the 

 parabolic grating is not much superior to the spherical 

 grating in definition. The smaller the value of /3, the less 

 pronounced is this difi*erence. Hence, for points any distance 

 from the axis of the grating, there is not much to choose 

 between the two. 



The maximum field of good definition that can be obtained 

 with any given grating is found as before by putting 



^B(max.)=~4 (^^ 



Since in this case the maximum aberration is always negative, 

 this gives for the value of /Cmax., 



, 1 / X asin^, g A f-i lahdnB 



tang «±» = ± 2^ (wsi^ - ~T- *=^"8 V L^ " 8 ~1^ 



For ? = (field S) this becomes simply 



