146 Prof. F. L. 0. Wiulsworth on the 



For these we have from (48) and (51) 



„ , 1 y^ cos 6 + cos i . ^ 



Zi'= ^ "^ TT^ sin ^ cos I 



2 p^ cos- 6 



1 y^ cos ^ + cos i r . o . . . J g /I / /I . -v -T / ^ ^x 



— o q TT^ ^sln2^ + 4tallo;^^(cos^ + cos^) cosz . (o2) 



8 p^ cos^ 6 ^ » V / J \ / 



Comparing this with (50) or (8)^ we see that the unsym- 

 metrical aberration is just the same in amount and sign for 

 the two surfaces (spherical and parabolic). The symmetrical 

 aberration is, however, less for the parabolic surface. This 

 may be more readily seen by putting the second term of (52) 

 in the form 



1 ?^ cos^+c osj rcos2i_(cos6> + 2cosi)2tang^6>--cos*^6']. (53) 

 8 p^ cos^ 6 ^ ^ / & J V / 



The parabolic surface is therefore in general better than a 

 spherical surface for a concave grating when the 0. S. type 

 of mounting is used. The degree of improvement effected 

 by the change in the form of surface depends on the type of 

 mounting used. For all types for which B has any appreciable 

 value, the first term of (52) will, as we have already seen, be 

 large compared with the second term, and the improvement 

 in definition will be slight. On the other hand, for those 

 images for which 6 is zero or very small, the improvement 

 may be very considerable. 



As before, the type of mounting which will be most 

 advantageous will be the type A-B, for which ^ = for the 

 centre of the field. For these types we have for the parabolic 

 grating 



Z^=g^3(l + cosi)(cos2i-l) 



= K[a-(l + cosz)] = -Katang''^z. . . (54) 

 Comparing this with (10) and (10 a), we see that the altera- 

 tion in the form of the surface has not only decreased the 

 aberration but has altered its sign. The relative values of 



h: and ^^'for the different fields, So, So^ So'^ So'^ and So^ 

 used in previous comparisons, run as follows :• — 

 For So So' So" So''' So- 



?^ =2-00 -i-1-98 +1-83 +1-40 + •38 = fl 

 K 



^ =0-00 - -015 -0-13 -~0-47 -l-17 = -atanA 

 K ^ ' 



also 5^=0*00 - -01 - '07 - '29 -^l'b = a,. 



