154 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
therefore 
and 
= —(/X— fj.') 06\ 
00 ' = {i^-.q){i 3 '-q) + {y-r){y'-r). 
Hence the right-hand inenil)er of this equation reads 
(m —mO ^ +!■ (/d;d' + yy')+i- (^'+ —^-yr —|-yV] 
+ i- (m —/) {/d,d' + yy') —i(/d + d') (m/ 3 —/yd') —i (y + y') (My —mVO] 
= Q [(m—+' i'(/'^~ m 0 (id^ + y^)+i-M‘^(/d^^+y^^)]5 
or, since 
and 
therefore 
q ^ _B/>„(l-lr/-lr=^)-lC6o(?>o^ + Co^) 
?>o = + = h 0-^13 {cf+ r^)l^, 
» 
and the equation becomes 
(3 [m—I'M (/3“ + y'*) + i' (m—mO (^^+^'^)]—[m^—(. d^"^ + y^*^)] 
= - [(m-m') B + i (m-m') C (^^+e^) + (/3^+/) + Im'B (d'^ + y'^)], 
or dividing by the coefficients of and writing 
m/m' = >1, 
d' = />[-(l-;i)B--i(l-M,)C(/d-t-c^) + lM,B(d'^ + y'^-d^-/)] 
+ d [n+In (d'^ + y'^ - d^ - /) - i (1 - «■) (g^ + ^)]. 
= h-\-ci^^j3 — 1 / (1x^(3'. 
jy = — j 
d'= —(l—n)B6 + nd; 
1/ = &[l+i-(l-n)(g^+r^)]4-d[i(l->^)(g‘+'^’^)/B]; 
« = 1(1-;,) (,/+,.2) = 1 (!_„,) B^(/;^ + c^), 
V'= (d'^’+y'^-d^-r), 
/>' = 6 [l + w]-td [tf>/B], 
Also 
Therefore 
but approximately 
therefore 
or if we write 
we may put 
d' = ^ 
— ( 1 — ?l) B + Bij/y — ^ OJ 
+d “t V'—^] • 
( 6 ) 
(J') 
