ME. R A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 171 
trigonometrically.'^ Later it was used as an illustration by Seidel,! who named his 
second condition the Fraunhofer condition, under the misapprehension that coma was 
elfectively corrected in it. 
In 1889, Dr. A. Steinheil gave a particularly thorough and instructive calculation 
of its field, and assigned first, the modifications of its curves necessary to remove a 
remaining trace of spherical aberration, and next, to correct the coma.| Finally, 
Finsterwalder recalculated Seidel’s sums for it, using the first corrected curves of 
Steinheil. § In the following pages I shall first of all show how the calculations will 
run with my formulm, and shall return to compare them with Steinheil’s results. 
The data given by Bessel are in “ lines,” of which 144 = 1 Bavarian foot. To 
render the arithmetical work more compact I have increased the unit to 1000 lines, 
which brings the measures of the radii of the surfaces and the focal length of the 
whole to the neighbourhood of a unit. 
The radii and curvatures of the surfaces, with the spaces between them, are the 
following;— 
Pd — + 838164, 
P 2 = ~ ‘333768, 
- ‘340536, 
Pe = -1‘172508, 
Bd = +1-193084, d, = -006, 
B 2 = -2-996093, ^3 = ‘000, 
B, = -2:936547, d, = ‘004. 
Bg - - ‘852873, 
The semi-aperture is 0‘0351. 
The refractive indices he takes as 
n = = ‘653966, l/n = 1‘529130, 
m = P 3 //X 5 = ‘610083, l/w = 1‘639121. 
We first form the normal scheme for the whole combination by writing down and 
combining the schemes that represent each surface and each space between two 
surfaces. To perform the step of combination 
yg + }]Ic., yh + rjl 
kQ + X/.’, kK + X/ 
—as to which it must be remembered that light passes through to reach 
{y ...} — we set up the number y as multiplier upon the multiplying machine, multiply 
it into g and h, and place the products as above, then set up and multiply into h 
and I, placing the products as shown; this gives the top line of the combination 
completely ; then set up /c, multiply into g and /? , set up X and multiply it into k and 1. 
* ‘ Untersuchungen,’ Bd. L, p. 101. 
t ‘ Astronomische Nachrichten,’ No. 1029, p. 325. 
I K. Bayer. Akad. d. Wiss., ‘ Sitzungberichte d. math.-phys. Classe,’ Bd. XIX., Heft III., 1889. 
§ K. Bayer. Akad.. d,. Wjss., ‘ .Abhandlungen,’ Bd. XVIL, Abth. III. 
z 2 
