Aberration of the Concave (jrvatinr/. 143 



First, by determining the value of the differential coeffi- 

 cient iVLjdr as a function of /3. 



Thus, in tig-. 2, let aJ> be the spherical surface which we 



Fig 2. 



O' 



have already considered, and let ag be the new surface for 

 which we wish to determine the aberration. If the central 

 elements, «, of the two surfaces coincide, the variation in the 

 difference of path between the central and outer rays will 

 evidently be 



AZ = (hh-{-bo')--(hg-[-go') = kg+gI 

 =. hg(cos I cgk + cos I cgl) 



dr cosO-^)+cos/tang-i ^ ^+/^\ • (^2) 



I 



u cos 6 — '2p sin^ ~ 



This expression (42) can, after some transformations, be 

 developed in terms of successive powers of like (3). If 

 then we add the corresponding terms of these two develop- 

 ments together, we shall obtain the expression for the 

 aberration Z^ of any surface whose equation, referred to the 

 centre of curvature of the central element, is 



7' = p-\-dr 



= const.+/(/5, 6*, (43) 



By making the term in sin'^ /3 in the general equation for Z^ 

 equal to zero we can determine the form of the function 

 fl/Sj 6, i), I. e. the form of the grating-surface which will 

 eliminate the unsymmetrical aberration ; by a similar 

 operation with the term in sin^ ^ we can likewise eliminate 

 the symmetrical aberration : it will not, however, be possible 

 in general to satisfy the general conditions for eliminating 

 both kinds of aberration simultaneously. 



The method above indicated of determining the form of 



