182 
MR. R. A. SAMPSON; A NEW TREATMENT OP OPTICAL ABERRATIONS. 
We take next the oblique rays in a plane through the axis, that is, we take c = 0. 
The rays considered are taken at an angle of 48' with the axis, following Bessel 
and Steinheil ; hence ^8 = tan 48'='01396353, and we take in succession 
6 = d=+'03510, +'02340, +'01170, -'01170, -'02340, -'03510. The final 
numbers given below have been multiplied by 1000 in order to compare with 
Steinheil, 
The centi'al ray meets the chosen plane at a distance from the axis 
(H + L^/')^ = ( + l'131453-'000040) x/3.= '01579852. 
Giving as before the calculations in full, the formulae (28), supplemented by the 
term K^'. 6, give the following :— 
Sh'~ 
h. 
+ 
+ ^SiG#. 
+ mbji. + |S3G)S2. 
CoefT. b. 
+ 
4- 
Coeff. ft. 
+-03510 
+ 353,5 
- 449,8 
+ 2023,9 - 1667,2 
+ 260,4 
+ 2544,2 
-4847,3 
-11,6 
-2314,7 
■02340 
ibid. 
- 199,9 
+ 1349,3 ibid. 
- 164,3 
+ 1130,8 
- 3231,5 
ibid. 
-2112,3 
+ -01170 
ibid. 
- 50,0 
+ 674,6 ibid. 
- 689,1 
+ 282,7 
- 1615,8 
ibid. 
- 1344,7 
- -01170 
ibid. 
- 50,0 
— 674,6 ibid. 
-2038,3 
+ 282,7 
+ 1615,8 
ibid. 
+ 1886,9 
■02340 
ibid. 
- 199,9 
- 1349,3 ' ibid. 
-2862,9 
+ 1130,8 
+ 3231,5 
ibid. 
+ 4350,7 
- -03510 
ibid. 
- 449,8 
- 2023,9 ibid. 
-3787,4 
+ 2544,2 
+ 4847,3 
ibid. 
+ 7379,9 
The comma is placed between the 7‘^ and 8*^'" decimals. 
Hence we have the following, with unit 1 line :— 
b. 
8b'. 
b’ + 8b'. 
Steinheil, p. 419. 
+ 35^-1 
+ 
91 - 
323 = 
■00232 
15^-79628 
15^-79622 
23 -4 
— 
38- 
295 
— 
333 
■79519 
■79528 
+ 11 -7 
- 
81 - 
188 
— 
269 
■79583 
■79587 
0 -0 
0- 
2 
— 
2 
■79850 
•79852 
- 11 -7 
+ 
238 + 
263 
+ 
501 
■80353 
■80349 
23 -4 
+ 
669 + 
608 
+ 
1277 
•81129 
•81130 i 
-35-1 
+ 1329+ 1031 
+ 
2360 
■82216 
•82212 
1 
(32) 
There is a slight discrepancy for h = +23'4. 
Steinheil now considers the rays which do not meet the axis. Fig. 6 is 
taken from his memoir and shows the object-glass on reduced linear scale. He 
divides the object-glass into three rings, and computes all the rays which Impinge 
upon it at an angle of 48' with the axis, at the points indicated in the figure. The 
rays 2, 10, 18, 1, 22, 14, 6 are those just given; of the remainder, those upon the left 
may be written down from symmetry from those upon the right, so that he computes 
in all nine Independent rays which do not meet the axis. 
