Correction of Telescopic Objectives. 407 



departure from the sine condition are 



«-4/3Mh-(2^ + 3)M 2 , (1) 



*-/3(3M + S)+wM(M + S) + M(M + 2S), - (2) 



and «-/3(3M-l) + ^M(M-l) + M(M-2) . . (3) 



respectively, where 



1— m 1—5 



m and s being the magnifications for the object and for the 

 aperture stop respectively. It will be noted that (3) may 

 be derived from (2) by putting S= — 1, i. e. s — go , showing 

 that apart from the satisfaction of an aberrational condition 

 the coma and the departure from the sine condition are only 

 measured by the same expression when the centre of the 

 aperture stop is situated at the first principal focus *. 



If — and — are substituted for m and s, M and S are 

 m s 



changed in sign but not in magnitude. This is sufficient to 



indicate that a, /3, and -ar are symmetrical | functions of the 



curvatures and refractive indices of the system. It is easy 



to show that if the system is reversed, thus changing the 



sign of the curvature of every surface, a and -sr are unaltered 



and (3 is only changed in sign. 



When the form of the lens is varied by making the same 



change in the curvature of each surface, -st, which is the 



Petzval sum. remains unchanged, but the other two quantities 



are altered. By choosing a suitable zero conformation to 



which such deformations may be referred, the change in a 



and /3 due to the impression of the additional curvature r on 



each surface of the system may be expressed in the form | 



a =* +4r 2 (2*r+l), (4) 



£=&+ 2r(tr+l), ( 5 ) 



which involve no new constants. Although it does not 

 appear in these equations there is in effect one additional 

 constant involved, inasmuch as a standard conformation for 

 the system has been introduced by imposing the condition 



* A detailed discussion of the relation between the spherical aber- 

 ration, the coma, and the sine condition is given in Proc. Phys. Soc. 

 vol. xxix. p. 293. 



t In Mr. Allen's expressions nine un symmetrical coefficients occur. 



% Proc. Phys. Soc. vol. xxvii. p. 485. 



