Ahernition of the Concave Grating. 125 



by the conditions of the mountino-. It will also be noted tkit 

 the sions of the terms in sin'yS are opposite in the two cases. 

 This is a point of special significance, the importance of 

 which will be develo[)ed later. 



The general equation (8) reduces to a simpler form in 

 several special cases corresponding to dift'erent forms of 

 mounting. These will now be considered. 



(A) Let the grating be mounted so that the eyepiece or 

 the centre of the photograi)hic plate lies on the axis of the 

 grating, /. e. on the normal to the centre of the ruled surface. 

 This is the form of mounting described and illustrated on 

 pp. 54-57, and on Plates VII. and IX. of my previous paper, 

 and is the one that has been most commonly used in sub- 

 sequent work. In this case ^ = at the centre of the field, 

 and we have for the aberration 



Z/ = Z2' = Z^= + I sin^^ Gos2i(l-f cos i). . (10) 



For the usual Rowland mounting in which this condition 

 is also fulfilled we have similarly from (9) 



p . 4^sin2/ 



Zo=-^. «in^P (11) 



b cos 2 



Equations (10) and (11) show that with this form of 

 objective grating mounting the aberration decreases with 

 increasing angles of incidence, and becomes vanishingly small 

 for values of i near 90°. With the Bowland mounting the 

 reverse is the case. 



Since the factor -^ sin^/S is common to both (10) and (11) 

 o 



the relative amount of aberration for any given grating with 



the two mountings will be given by the two trigonometrical 



functions 



/j(i) =cos^?(l + cos2) =a .... (12) 

 ,.... sin^i ,_,. 



^^«=i^r"° (1^) 



For convenience the values of these coefficients, a and ao, 

 have been computed for values of i from 0° to 90°, and are 

 tabulated in Table I. columns 2 and o. 



(B) In using the grating photographically we cannot 

 satisfy the condition ^ = for all parts of the field simul- 

 taneously. If we bend the plate to conform to the focal 



3026^ 



