76 Proceedings of the Royal Irish Academy. 



Aj /•, z,\, ' h{ /•, /r,' n ' 

 W = {Li - /r,)- 1 , 



H in !Hi = Ki+JK", Jf;' n H[ = Li' n /Li", 

 H { E; = rfiLi-Ki), 



For the beginning it is possible to write 



.;/,- = „/. //,' = /;;. //,= ^(z,- a\vA'. 

 /,, = ,,'/;/• & t ' = A , /r 1 (z 1 -/r 1 ). 



I lose Formulae completely determine the Gaussian approximation 



l| = . . . = 1); = . . . = ){/, 



' - / 



i; =... = »(, = . . . = >ij , 



which can be used t" find the terms of the third order. 



Tlie two Bysterus of conjugate planes are -juite arbitrary. With 



schild the fin lentified with the object plane, the second C ' 



with the entrance pupil. The lasl plane Cj of the first set is then the image 



plain- in the ' last Cj of the second set coincides with 



the exit pupil. Thus the limits of i;,' or iy define the limits of the effective 



1 which forms the im 



Id remarks "ii the analogy of the method with the theory of 



planet iry perturbations. But a more Bpecial analogy in the variation of 



constants will be Been in the 'method of Delaunay's Lunar Theory. The 



essential point consists in the treatment "!' the Becond-order terms in the 



iracteristic function, whereby the principal part is removed and only 



Inal terms of .1 bighei (sixth) order are left. Thus the analogy is very 



7. The formulae of is -1 are easily adapted to a refl system. It is 



only necessary t" v. 



1 /. 1. A ' K r 1 K" = AV, ¥ 1 



/ /. ■, - 1, Z," = Z.r,- i 1. 





J /.,- - Jd? + (2A7 + L? - IKjLj) L?r> 



n{KM - i) 



B { A 



/. . 



A = bjrf + A", ■/.. - K 

 E K>\> + Z, 2Z. - A', I 

 A. ; . • K.I.Jn. 



