1903.] Tlie Theory of Symmetrical Optical Objectives. 267 



"The Theory of Symmetrical Optical Objectives." By S. D. 

 CHALMERS, B.A. (Cantab.), M.A. (Sydney), St. John's College, 

 Cambridge. Communicated by Professor LARMOR, Sec. E.S. 

 Received and read June 18, 1903. 



This paper deals with the relations between the aberrations of a 

 lens system, used with a front stop, and those of the compound system 

 formed by two such systems disposed symmetrically with respect 

 to the stop. The method of Hamilton's characteristic function is 

 used as adapted by Maxwell, the notation employed and the expres- 

 sions for aberrations of any system being those given by Thiessen.* 



The optical length between the points (#] y\ ,?i), (# 2 y^ 2 ) in the 

 medium n 12 is n^ 2 {(x\-x^f + (y\-y^f-\-(z\-z^fY' ) which is ex- 

 pressible in terms of x\, x^ yi, 3/2 when the points lie on the surfaces 

 s\ =/i (x, y) and z 2 =/ 2 (x, y). 



If these be surfaces of revolution about the axis of z, their equations 

 may be written z = 0,1 + bipi 2 + Cipi 4 + . . ., where pi 2 = x^ + ?h 2 , and 

 the characteristic, omitting the constant and writing xi-2 for - x.x* - 

 is given by 



1 12 Aj2/l" "I" -012/02" ~t~ 2C*i2Xl2 "J" J-)l2/l ~t~ ^12/2 "^ 4.E 12Xl'- - 



to terms of the 4th order in a;, y, where, 



^12 representing a 2 - #1, 



B/ 1 , i \ pi 

 TO ^ Tlio I -t- Oo I . \j'\-~> = 



\ 2/ / 

 + CT ^ , GI^ = - n u ( -3 + -I^-J 



The characteristic of the media between (1) and (3) is 

 where x% ?/ 2 are to be eliminated by the relations 



' = 0, 



C/^2 Uy2 



4 Sitzungsberichte der K. Preuss. Akad.,' Berlin, 1890. 



