PtiUMMEB — On the Symmetrical Optical Instrument. S3 



Hence 



If the terms found in $ 3 are restored to Wi\ omitting the corresponding 

 terms in ;., 



Si = W t - W? + \{L['yl - L( y ;_ x ) + {Kl'Vi - K(y^ L(U '\KIK(' 

 - 2(Z ; V - Li !,{.,) (JQ'yi - KrVi-,) 1{'W\ KIKl'r&fa - wJW 

 = Wi - W« + KIKl'n [fa' - n [^f + [tU - Zt-tfWfa - „,.,) H? 



+ Li'IfnKni - n^f + (Zt - Zi.^/2 fa - m^HP 

 - KlLl'n { fa - nj fa' - m '. x ) + (Zi - Zi-d {Zi - Zt'-J I / 



[fH - fx;.\)HiEi' 



Thus S !i = 0, S iv = W iv , and for the sixth order, giving the errors of the 

 fifth order, 



st* = y 



?W/ V 7) /'-i \ 7)W- iv 7) /'-i 



wj h , d Jp- . JL 2 wA + d -p- . A (y Wi 



Wo o'lJ \*=i J 01,0 olj \k=i 



irT »,. \° w i v W" dWfidWfi \j 



' Ki Li ' ' ' 1 dW ~W ~W W I / ( '" " " m) HiHi . 



since it is possible here to use the first approximation 



'Jo =••• = '»» =••• = »#) £o = . . . = Zi = • • ■ = Zj '., 

 77,, =... = »),- = . . . = jy , 4„ = . . . = d = . . . = L,j. 



In calculating Wi H the expressions (2) will no longer suffice to represent 

 the law of refraction. To the third order the expression 



->((" F+ ^) + i(^ + ^)-^ + ^ + , ') + w( 7, + ' , J , |' 

 is a minimum. Therefore 



. Y( \ Y(Y° + Z*){ 1 \f 



= mm; - tH . 1 m i . 1 + — f & - !x ; A + -^ ^1 + btj f ,„ - ,<,-., 



with a similar equation obtained by interchanging (y,m)and {Z, n). The 



