160 MR.'R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
whence the condition 
H 4G-4H 
L 
{by-c/3) = 0, 
or 
— -- =: , ■ - = ‘4-h Sc 
H 
say. 
( 16 ) 
Also, this result must remain valid if we pass the emergent beam through any 
further optical system. This is a step that must frequently be taken, and it will be 
convenient to write down generally the formulae to which it gives rise. 
If w’e have 
g + Sg, h + Sh 
k + ^k, 1 + 81 
g' + kj, h' + kk 
k' + 8 k\ V + 8 V ^ 
> = 
G+dG, R + m 
K + (5K, L + (5L 
and if 
and, further. 
^9 = i{ ^i 9 (+ c") + 2 S 2 g (6^ + cy) + S^g {(S^ + y') 1, ..., 
V = i{W(&'^ + c'^)+ ... [, 
=^^{K9i{9b + hkiY + {gc + hyY~]+ ... },..., 
(^G = ^ {SiG {h'^+ c^)+ 2 S. 2 G {b^ + cy) + S^G: {l3‘^ + y~) f, ... 
then the following formulae result 
^iG = g'S,g + h'S,k+g {g'^g' + 2gkS,g' + k%g'} + k {g%h'+ 2gk<yi'+ k%h'}, 
8J1 = g'Sih + h'SJ,+h { ibid. }+^{ ibid. }, 
(?iK = k'S^g + l' 8Jc+g.{g^8Jd + 2gk8.Jc + Ti^8Ji'\ +k {g“8i V + 2gk82V +F4/'}, 
< 5 iL = k!8-Ji + V 8J, + h { ibid. }+M ihid. }, 
4 G- = g'82g + -^g {gb 8ig' + {gl + h k) + kU^g '} + ^) {gh 8ik' + {gl + hk) ' + kU^h '}, 
4 H = g'82h + h'821 + h { ibid. } + ^ { ibid. }, 
82K = k'8.2g + l'82k+g {gh8Jc'+ {gl + }ik) 82k!+ kl8Jc')+k {gh8,V +{gl + hk) V' +kl8,l'], 
= k!82h+V 8Jy+b { ibid. }+/ { ibid. }, 
^^G = g' 8 ^ + h' 8 Jc+g { 198 ^^ 2 ) 982 ^+ l%g')+k {)i\h!+ 2 ) 982 ) 1 '+ l%h'], 
= g' 8 ^)\ +)i) 8 ^l +)i { ib^d. }+/ { ibid. }, 
^ 3 K = k'8^ + l' 8Jc+g {)98^k' + 2)982k' + l%k'}+k {)98,V +2)989)' +l\l'], 
(^gL k'89^ +l' 89 +)i { ibid. }+^ { ibid. ]. 
These formulae with 
G = gg' + k)i', 
K = gk' + kV, 
H = )ig' + l}i', 
L = )ik'+ 11', 
