Aberration of the Concave Grating. 139 



As before there are two values of R for each value of Zb, 

 oorrespondino- to the two sides, right and left, of the grating- 

 surface. These are designated as before by Rj and Ro. For 

 positive values of k, Ri is negative and Ro is positive. For 

 negative values of k these quantities are simply interchanged, 

 i. e. 



Ri (for -Ac)=Ro (for +/c) 



R2 ( ,, J, ) = Rl ( ?, M )• 



In general, in considering the prejudical effect of aber- 

 ration w^hen the whole surface of the grating is used we must 

 take the maximum numerical values of Z and R. But 

 inspection of equation (33) and of the values in Table III. 

 shows that for each half of the gratmg-surface there is a 

 point y^o ^^ ^tie field for which the aberration for that half is 

 zero. In the case of the right-hand half (for which the 

 aberration is Zi) of grating /3i this point lies between a:= + 10^ 

 and /c=+5^ for values of i from 0^ to 30°_, and between 

 /c= +5\and a: = for /=60°. For gratings ^2 ^i^d /5;3 the 

 point K^) is farther from the centre of the field. 



The exact location of the point Kq for any grating and any 

 value of i is easily found by solving the equation 



Zb = 0, 

 or from (16a) or (33) 



4 1 — 5tang2^ 



+ -; — 5 .taug«:= ^r^^— . . . ^34) 



~ sm (3 cos I ^ cos- k ' 



Put -^^ — 75 . =/. Then since k is by assumption always 



sm p cos I "^ -^ ^ - 



small, we may readily obtain from (34) 



tang/.= +/72^(l + \/l+— 2) 



= ±isin/3cos/(l- A). . . (35) 



For grating /5i = -01 we find for k^ 



^o=±(8'4O^0 for 1 = 0°, 

 /^= + (7^3O'0 „ /=30°, 

 /co=±(4^20^') „ 2 = 60°. 



For /33 = *05 we similarly find 



A^o = (38UO^0 for2' = 0°, 

 ,., = 37^4- ,, i = 30°, 



and /fo = 2r ?y^" „ /=60°. 



