Plummkk — On the Symmetrical Optical Instrument. 73 



It is now necessary to calculate TV/" according to (4), and here it is 

 clearly possible to use the lowest approximation (6) so as to express the 

 result in terms of y Uv >/-, z H , z{. Thus (7) give 



_ Lj" W'?// - Kyi-,) _ K{ i L/'/n - Zj>i) 



and therefore, by (2), 



Y Wul^Kl - K(') - K-i/n 'L- - Lj") 

 n ' rf {Li - IQ (, H - &J 



Y- = Wy{-K i ' Vi Jlu{L l -Ki). 



Let 



Bi = L{" tot* + zr), R^ = K{> { yi \ x + Zi \), 1\ = Li'Ki {yfy Ul + z-z^. 



Then 



Y* vP = (i2/ + i^ - 2 /',) / »v ( £. - K t f 

 m-- + % a = (£/1R/ + Z/'^ - 2K i ' / L i "T i )/^r i ' (J, - K t f 

 mc. x f nr_ L = WR; + Z/^ - 2K{L{T i )l t i*_ 1 r i *(L i - Ktf. 



Hence 



W \ + ^ + (7 2 +^)/(ai-c i )r i 



_ £/"£/ + V^ - IKl'U'Tj + nib" (Bi' + i^ - 2fl) 

 tfrfiLt-W 



= TiKiKfRj' - ZrjLtKCTj + nR^ (Jfo - 2^ + ^A,) 

 rfrfiLt-KF 



and finally by (4) 



8 (Z £ - *d« W? = -j±= „ {KiK/'Bi' - 2L i K{'T i + I?;., {Lfr t - 2^L { + „,•£,);* 



2 \ y , (^i/^'A-' - 2Z t AVr f + i^, r#r, - 2^Li + ^A',),- 



-%( JH -^ i . 1 )r^{M/-2T i + B i .^ 



= - 27V 1 (-W-! + JW + 4W + 2PiRi-iBi - 4^,i?,-. 1 r l 



- 4iW*V) 



where 



, = , ( &<(w- /ii-i) (ZjV,- - 2/iA + jUj/ C) 2 (Z,"/V - 2^/,, + ^gj) 8 

 ' ' \ rr t&iKi" HiJriKi' 



B { = \{h Cm - ^•. 1 )r < - 3 - A7 {Kflittn - A7 /«,-,->.»; 



C\ = I 1 6,- Oi,- - £) r«* - Z/ 2 (K/'/^ri - K{ /> K «r ( ; 



