SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
49 
Further, if the factor (1 + 4w 0 2 sin 2 aJn 2 H) is introduced into the denominator 
of Ax 2 , we no longer obtain expressions of the simple type (24) and (26), and 
endless complications are introduced when we come to consider a compound 
system. 
Can we make our expression B given by (23) give a tolerable approximation to 
(An^M^/n /R) for those regions where the denominator factor is really needed, namely 
for M, large, positively or negatively ? 
To get the answer to this question we develop (An 0 sr M. 1 2 /n/R) in descending powers 
of M,. 
This is found to be (the most rapid method is to break up first into partial 
fractions) 
l{n 2 -n 0 )- 
— 2 n Q n 2 M, 3 — (n 0 2 — 5n 0 n 2 + n 2 2 ) M : 2 
- [(n 2 —n 0 ) 4 + n*nf\ M,/2 n 0 n 2 -[_(n 2 —n 0 ) 3 ( n 2 3 -n 0 3 ) + n£nf\l^n*n? }>. (40) 
+ terms in l/Mj, &c. 
If we now compare (40) with (23) we find that the most important terms when M, 
is large, namely those in M, 3 and Mj 2 agree in the two expressions. 
We may, therefore, take it that the approximations (21) and (24) which we have 
seen hold good to the second order when expansion in series is convergent, will 
probably not be numerically very far out when M, has a large value, in which case 
the normal method of development cannot be used. 
It is important, at this stage, and to justify the above assertion, to consider a few 
numerical examples. 
Tables I. and II. give the values of Ax 2 and sin a 2 for a single refracting surface, 
calculated for a number of values of M, and two inclinations in each case. The 
inclinations are fixed from the perpendicidar distance vs of Aj from the incident ray. 
This, for moderate inclinations, is sensibly the same as the intercept made by the 
incident ray on the principal plane, vs has been given the two values 0'5 and 0‘25 
in every case, except for Mj = 2 where vs — 0'5 leads to a physically impossible value. 
In this case vs — 0‘25 and vs — 0'125 have been used to define the ray. 
In each case four values have been computed (l) the correct one, from trigono¬ 
metrical calculation ; (2) the values given by formulae (21) and (24)—-these are shown 
in the column headed “fractional formula” ;'(3) the values obtained by expansion in 
series, up to the optician’s first order of aberrations inclusive, that is including sin 3 a 0 
in the development of Ax 2 and sina 2 —these are shown in the column headed “first 
order” ; (4) the same series carried to the second order of aberrations inclusive, i.e., 
to the terms involving sin 5 a 0 —these are shown in the column headed “second order.” 
It should be noted that these first and second order approximations are the most 
accurate that can be obtained, much more so than more usual ones, proceeding in 
powers of sin a 2 or tan a 2 . 
