Aberration of the Concave Grating. 147 



For the very centre oi: tlio iicld, therefore, the definition with 

 tho i)aral)olic oratino- is much better than with the spherical 

 oratino- for small angles of incidence, but interior to it for 

 ill roe "mgles of incidence. It is very nearly the same 

 throuohout as with a spherical grating on the Rowland 

 mounting. 



Substituting for a the value atang^i in (38), we find for 

 the limitintj; value of /S, 



V 2\ 



/^max^V Watang^r 



Hence, for the grating previously considered (W = 73 mm.), 



^max =0-100, fori= 5n 

 /3Lx=0-063, „ i = 10 

 /3' =0-049, „ z = 15° 



max. 



° I 



> . ■ (56) 

 I 



which are likewise very near the limiting values for ySmax. for 

 the Rowland mounting. 



When we consider the effect of the unsymmetrical aber- 

 ration in parts of the field away from the axis of the grating 

 the results are much less satisfactory. The general expression 

 for Zg for small values oi fc{ = 6) becomes 



1 ?/^ . . 



Zg = — -. -^ tang /c cos i (1 + cos i) 



+ 1-3 (^ + cos i) \ [cos'-^ ^-(l + 2 cos i^ tang^ k] sec^ /c- 1}. (57) 

 As in the case of (33), this may be put in the form 



Z;, = Z, \-^^.^ + sec2« (l-6tang2/c)--y 

 B ^ L cos z sm /S - ^ cos^ iJ 



= _^ (-V-I^\ (5^) 



tans^Acos^^ /' 



and therefore 



Z^ \cos^i / 



cot^i. . . . (59) 



In order to compare directly the definition of the parabolic 

 and spherical gratings at a given point in the field, it is also 

 desirable to find the ratio between the maximum values of R 

 and R^ Since the maximum value of Zb is always positive 

 (for positive values of «), and the maximum value of Z^ is 



L2 



