Aberrations of a Symmetrical Optical Instrument, 275 



For purposes of summation we shall also need the quantities 

 h u A 2 ,... hi,... which are the heights above the axis at which 

 a paraxial ray, starting from the axial point of the object 

 and passing through the system, cuts the successive surfaces. 

 It should be pointed out, however, that these are introduced 

 merely as a convenient device for summation, as used by 

 Seidel ; the quantities h must not be confused with the f 's 

 and ?/s which indicate the true order of approximation as 

 regards aperture. 



We have 



£i±! = */. *' = £±± ; and so on . . . (12) 



Hence the relation (6) may be written 



— = smnlarlv — = &c. . . . 



«t+l Si " Si-i 



= ^*^=X(snv), . . . (13) 



where X is a constant for the ray throughout its passage 

 through the instrument. 

 Similarly 



= <xc. . . = ' — = another constant //, (14) 



Summing the equations (10) and (11) we have 

 fijXi+igi+i _ /i .?-'ifi , *^' Pp-idpXpa p+ i t 



s i + l s l p=l s p$p + l 



s i+l s \ p=l SpSp+i 



which by (13) and (14) may be written compactly 



p =t d 



7 — — ~T +x Z — jH ; 7 ~ = T + a z 



th+i «i P =i fj-php'tp+i hi+i hi p=ipp'i P 'i P + 



(15) 



We can now effect the summation ol (9) for all the 

 refracting surfaces to the ith. For brevity we shall denote 



p %~ 1 d p t 1 1 A /.l\ 



2 — r-f — Dv ^ and ^ — by A ( — . 



p=i Pphphp+i " (*>iSi pi-is, V'V 



T 2 



