MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. I6I 
are of fundamental importance and cover all cases; they will he quoted as 
( 17 ) 
Apply them to the equation (16); we have 
(^gG — = g' — 4^) + h. —S.J,) + hn —^Ji') + In {S^g' — 
= k'{S^-S^) + r {S,Jc-SJ) + hn{S,k'-SJ') + ln{S,k'-S,r), 
n ~ gl — hk. 
where 
If we write in this 
S;^g — (^2^^_ 
we have 
I 
(^-^g' — SJi' SJc' — SJ,' f 
= + + {{^■^'-Kk')-Vg'], 
(^gK—4L = (l) + n^') L + nh {(4^— ^ik ')—I • 
In the same manner we find 
4G—= (iJ + nt*') G + 7jj \, 
Compare these with (16) and remember that the two systems {gh ...), {g'h'...) 
are arbitrary and independent of one another. Then we see that if for these systems 
S.^g —— S^h _(^ 2 ^—_ ^sk —^ 
g “ “ jr~ ~ I ~ 
then 
where 
<kjzM: = 
9 ' 
= P', 
(^2G-_ ^gG--^jli _ _ 0) 
G “ H “ ^ 
(18) 
Now if we examine the case of the single surface, for which 
= (1 -n) §.^ = 0, S.£ = 0, S,h = (l -n) B, S,h = 0, S^h = 0, 
^Jc = {l—n){—€ + n — n^)W, = — 71 ^ 4^ = —n(l—w^)B, 
8J, =-(l — w) ( —1+n—n^)B^, 4^ = —w^(l—n)B, 4^ = —w(l—n^), 
Qf = lj h = a, ^=—(l—n)B, I — a, 
the conditions are fulfilled and we have 
*) = -(i-m)B = M(VbB; 
\fj. n/ 
and 
VOL. ccxii.— A. 
V 
