SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
43 
and is the quantity whose vanishing gives the aplanatic points and must therefore be 
a factor of every coefficient after the first in the development of sin a 2 in powers of 
sin a 0 . 
If we write for shortness 
Q = ( 1 + njn^fx^l'ri + ( 1 -xjn) (l + n t fic 0 /n 2 r 1 ), 
and remember that 
1 + x 0 (n 2 —n 0 )/n 2 r 1 = l/M,, 
we find 
sin a 2 /sin a 0 = l/M, — |P* sin 2 a„/(l — iQ sin 2 a 0 ) 
= Mr 1 ! 1 — d-PMj + IQ) sin 2 a 0 }/(l —iQ sin 2 «o) 
= M, _1 { 1 +B sin 2 a t) /M, 2 }/{ 1+ C sin 2 ajM 2 } . 
correct as far as the second order inclusive, where 
B = -iPMd-IQM, 2 , C = -iQM, 2 , 
from which, after some reductions 
( 21 ) 
0 = —i (n 2 —n 0 ) 2 {(n 2 2 +n u n 2 +n 0 2 ) — 3(n 0 2 + n 2 )'M. 1 + 3(n 2 2 —n 0 n 2 +n 0 2 )M 1 2 }, (22) 
and 
B = \ (w 2 —n u )“ 2 (l —M x ) (n 2 —n$L x ) (n 0 —WaM^ + C 
= i (n 2 —n 0 )~ 2 { — (n 2 2 —n Q n 2 + n 2 ) + (n 2 —n ( tf M, — (n 2 + n 2 — 5n 0 n 2 ) M, 2 —2n 0 n 2 M, 3 }.(23) 
Keturning to equation (19) and using (21-) 
where 
A;r 2 = x 2 (C —B) M,- 2 sin 2 uj{ 1 + BM,- 2 sin 2 
= na/iAMj -2 sin 2 a 0 /{l +BM 1 _2 sin 2 a 0 }, . 
(24) 
A = x 2 (C-B)/n 2 f 
= —£ ( n 2 -n 0 Y 2 (njn 2 ) (1 -M ,) 2 (n 2 -n 0 M,) (w 0 -n 2 M,) 
- n 0 n 2 + (n 2 + n 0 ) 2 M, — 2 (n 2 + n 0 n 2 + n 2 ) M, ; 
+ {n 2 + w 0 ) 2 M, 2 — 
= ? {n 2 -n 0 )~ 2 (n 0 /n 2 ) 
• (25) 
All the above formulae are correct to the second order of aberrations inclusive. We 
note that A, B and C are polynomials of degree 4, 3 and 2 in the magnification 
respectively. 
If we express the aberration in terms of tangents instead of sines we have at 
once 
Ax 2 = nj\ AMr 2 tan 2 *J{ 1 + (B + M, 2 ) Mr 2 tan 2 a 0 } 
= n 2 j\A tan 2 fi.J {1 + B tan 2 fi 2 ] 
(26) 
