80 



Proceedings of the Royal Jris/t Academy. 



9. The theory has been built up by considering the successive refractions 

 and their combined effect. For some purposes the intermediate steps are 

 unnecessary, and a general qualitative theory of the symmetrical instrument 

 can be based on the consideration of the first and final ray alone. The 

 equation (1). now written in the form 



is general. The equations, corresponding to (7) 



Kj'Lj" (y/ - yj) 



„ y [L, - Kj) ' 



Kj'LJ' (sj - zj) 

 prf (Lj - Kj) ' 



m i = 



)ii = 



in,. = 



n,. = 



AW 

 ,,,,/yA- A',' 



K(L( (s ' - z ) 

 ^(L t - KJ 



represent a general transformation. Here (y ' t z '), (i/f, zj) are not strictly 

 the intersections of the ray with the planes of the entrance and exit pupils, 



but coincide with them t" the iir>t (Gaussian ler. The result for the 



, y is, in the notation of $ 6, 



K /.,",, 



K'L/% 



, , , Lj 11, A -^i 2 \ 



= -'/") >i "i ■ J ' K ■ /J- "■' g n ■ '/o I 



= - •,.' >> (<, q ct) 



I II l-,'ll, , ■ ,j 



' ' " K II '' K II ' ''* ' " ''" T 

 Therefore, if 



the variation ol S depends on (»j„ Z t ),(nf, %f) only, and 



>o 



"> 



y y 

 V» = - ~7pt tj - So = 



3 



y l y i 



;,s' 

 - 



(ii) 



to 



N .* can contain uo terms of the second ordei se iy = ij„, . . 



th<- first order. And, owing t<j the Bymmetry of the instrument, S can be 

 developed in terms of the three axial invariants p , pj, and r v : The meaning 

 ertain classes oi terms in S is now i en. 



