Ahcrrathn of the Concave Grating. 127 



has not yet been used, to my knowleiloe^ except in the sniull 

 trial instrument there iUustrated. 



In this case we have for the aberration in the central 

 imaoe 



Zc= ±psin-'yesin/+^ sm'/3(l-l) tang^i) cos i. . (18) 



The values of the function 



/^(;)=(l-9tano-2/)cosi = cosi-9cto = c . . (19) 



are tabulated in column 6, Table I. 



(D) When the grating is mounted as just considered, and 

 is used for photographic work, we have to consider not only 

 the line at the centre of the field 6=—i, but also lines at a 

 distance k each side of the centre. For any given value of i 

 we have for 6j) 



0^z=-i±K (20) 



The aberration in this case is most easily computed directly 

 from the general equation (8) by substituting for 6 and i the 

 values determined by the relation (20). 



(E) Let the grating be mounted so that the incident light 

 is always normal to the surface. This case has been investi- 

 gated for the Rowland mounting^, but not for the O.S. 

 mounting. In the latter we have from (7) for i = 



_ P_ 



1 4- cos (9 



w= .— .cos26>, (21) 



which is the equation of the curve on which all focal images 

 lie, i. e. the focal curves for both central and lateral images 

 coincide. 



The aberration for any imaoe on the focal curve is 



l + cos6>^ 



p . . Vl + cost^\ 



+ |sin^^ii^|-^[l-(2 + co.tfftang-^^]. . (22) 



The last term of {'1'2) may be put in the form 



u . . ^ 1 f COS ^ . ^ rcos-^ 1 + cos ^ . ^-| 

 8 cos-^ L^'ii ^ cos''^ J 



* " Fixed Arm Concave Grating Spectroscopes," Astropliysical Journal, 

 vol. ii. p. 370. Forms of mounting fultilliug this condition are shown 

 on plates xiii. (for i=0) and xv. 



