ME. R. A. SAMPSON; A NEW TREATMENT OF OPTICAL ABERRATIONS. 181 
spherical aberration. Take rays parallel to the axis, that is, /3 = 0, and consider the 
focus where they unite for impact upon the original plane at fractions 0, l/3, 2/3, 
and 1 of the semiaperture, ^.e., for 
d = 0-00000, 0-01170, 0-02340, 0-03510. 
For any of these the distance of the point from the last surface (6) is 
- (G + 
or 
-G/K[l+id^{^iG/G-WK)],.(30) 
and 
G = +-996692, K = --883818, = +'20722, = -'21613. 
.-. ^[S,G/G-SJL/K] = +-20791-'24454 = -'03663. 
Hence the rays meet at the following points along the axis :— 
Steinheil, p. 417. 
Axial. 
. 1127^-712 
1127^-712 
l/3 semiaperture . 
'706 
'706 
2/3 
-689 
-687 
1 
-662 
•659. 
In these and the following comparisons the unit of length has been brought back 
to 1 line by multiplying by 1000, to preserve Steinheil’s numbers unchanged. 
In consequence of this residue of spherical aberration the best setting for focus at 
the middle of the field is not the axial focus but a point within it. Steinheil takes 
this point at 1127'670, following presumably the theory of Bessel, which gives a 
position for the greatest apparent concentration of light that is slightly within the 
least circle of aberration (1127'672).* 
Adopting the corresponding point, which allows for the slightly smaller aberration 
shown by my numbers, and multiplying by 10“^ to bring the units into agreement 
with formula (28), we see, in accordance with p. 165, we must include with Sb' of 
p. 179 the term 
+ KSf' .h = -'8838 X -'0000400 x& = +'000035356, 
with a corresponding term for Sc' in terms of c. 
The diameters of the image-disc in the focal plane and at this setting are 
respectively the corresponding extreme values of Sb', doubled, or 
Steinheil. 
0^-00316 — 
0^-00067 0^-00071. 
* Bessel , loc. dt., p . 104 . 
