506 Mr. T. Ohaundy: Method of Line-Coordinates for 



and Rm+ Sw to give the second approximation formulae for 

 m', u' , may conveniently be termed the "aberration functions 

 of the third order." Their coefficients A, B, . . . . K, L may 

 likewise be termed the " aberration coefficients of the third 

 ■order." They are optical constants of the system and in 

 terms of them all the third order aberrations can be stated. 



The aberration coefficients of the third order are connected 

 by the seven identities : 



SA-RB-QG + PH=(1-PV//*' or 1-P 3 if first and lariT 



media are the same . 

 SC-RD-QI+PJ=PQ/*//*'orPQ „ „ M 17 ) 



SE-RF-QK + PL = -QV/*' or -Q 2 „ „ J 



SA + RO-QG-PI = 0, "1 



SB + RD-QH-PJ=-X1//^/; the Petzval Sum, [ 



SC + RE-QI-PK = -2l//^/, the Petzval Sum, j (18 ^ 



SD + RF-QJ-PL = 0. J 



I omit the proof of these equalities from reasons of brevity. 

 The first three follow from the fact noted in § 3, that the 

 moment about the optic axis of any ray varies inversely as 

 the refractive index of the medium. 



(7) The Correction Conditions for the Second Approxi- 

 mation. 



Since the formulae (16) contain complete information 

 regarding the aberrations of the second approximation for 

 an optical instrument in which P, Q, R, S, A, B, . . . K, L 

 have been calculated, it is clear that we must be able to state 

 in terms of these quantities last-named, the conditions that 

 the optical instrument be perfect to the second approxi- 

 mation. The method is sufficiently illustrated, if we limit 

 ourselves to the simplest case, that of an incident parallel 

 beam. I shall suppose also the first and last media to be of 

 refractive index unity. 



Without loss of generality we may take the infinitely- 

 distant source generating the parallel beam to lie in the 

 plane zOx. Every ray of the beam will thus have the same 

 direction cosines Z, 0, n ; the other coordinates u, v will, 

 however, vary from ray to ray : in fact, they represent the 

 two degrees of freedom enjoyed by rays of the beam. To 

 visualise them we may remember that to a first approxi- 

 mation they are the Cartesian coordinates of the point of 



