and 



Aherration of the Concave Grating. 



123 



h = 2p sin sill li— IjA 



= p sill yS sin / — ^ siii-/3 cos i — ^ sin'^yS cos i. (2) 



Expanding (1) by the binomial theorem in terms containing 

 successive powers of /3, and neglecting as before those powers 

 hiiiher than the fourth, we obtain 



ho' = u—p sin /3 sin 6 



— ^ sin-/6 cos 6[u ■ p cos 6) 

 2u 



1 



P 



J—^ sin^/S sin ^cos 6[u—p cos ^) 



2i/ 



— ^y^sin'/3 sin-^ cos ^(i/ — /9Cos 6) 

 + -^sin*/8(p — ?i cos ^) 



— -^ sin*/S cos^^ (?/ — p cos 0) '^ 



(3) 



; 



Combining this value with that for hh from (2), and sub- 

 tracting the same from ii^ we find for Zj 



Zi=zp sin /S (sin ^ — sin i) = M "i 



4-^, sin2;8(cos^4-cos/-^cos26>) 

 2 u 



\ (4) 

 I 



= N 

 + ^i.sirr'^lsme cos d{u-p (-030)] =0 



+ |-2sin4yS( 2/3 sin^^ cos O-^p cos 6'- |- sin^i? 



- ^ sin26> cos26> + /' cos^l9\ = P 



+ § sin^y8(cos ^ + cos i) = Q j 



The expression for Zi can also be derived at once from 

 Glazebrook's paper by putting ii in equation (3) equal to co , 

 and substituting for c^, -v/^, and (o the corresponding values 

 i, 6, and y8. 



For the left-hand half of the surface we have similarly 



Z^-M + N-O-hP + Q, .... (5) 



