MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 163 
Now consider the substitution 
h' - (G + ^G)6+(H + ffi)/3, 
/3' = {K + SK)h+{L + SL)f3. 
Then if we shift the origin in the emergent system to d' the first will read 
b' = l(G + ^G) +d' (K + SK)} 6+ {(H + ,^H) +c^'(L + (^L)} /3. 
Choose d' 60 as to make the coefficient of /3 zero; then 
}/ = |(G + ^G)-(K + ,^K)(H + ^H)/(L + (^L)}6, 
and the coefficient 
G + dG- (K + ^K) {R + SB.)I{L + SL) 
is the linear magnification for narrow pencils emerging in the general direction (/3) 
from the point (O, b, O) in the meridianal plane. Again from 
the angular magnification for the same is L + (5L. 
Hence (G + dG) (L + (5L) — (H + (5H) (K + (5K) is equal to the ratio of the effective 
refractive indices. But we have seen on p. 157 that the change of ray effected by an 
aberrational system is equivalent to the use of refractive indices 
iul' ... throughout. So that the expression above is equal to 
where /3' may be taken as the final value of ^ after any number of transformations; 
or equal to 
N{l+i(K6 + L/3)^-i/3^}. 
Identifying term by term with the expression above we have the relations 
()\N = GS,L + U,G-m,K-KSJl = K^N, 
4N = G.^2L + U2G-m2K-K4H = KLN, . . . . 
4N = G^L + L^G-H^K-K^L = (L2-l)N. 
( 20 ) 
The relations (20) may also be proved from a sequence formula out of the equation 
f17); thus 
(^jN _ 
N n 
^.>N ^^71 
I 2 ^'^77, " 7 , 7 ■> do/?' 1 
— + \g' ^ ^•2gk^ + t , 
' n 71 71 ' 
^2^0 I 7 SiTZ / 7 , 1 7 \ do// 7 7 do// 
71 
n 
71 
71 J 
( 21 ) 
