ME. E. A. SAMPSON: A NEW TEEATMENT OE OPTICAL ABEEEATIONS. 188 
We derive these as follows. We have throughout ^ unchanged :—■ 
For the rays (4), (12), (20), 
„ (3), (11), (19), 
„ (5), (13), (21), 
h = 0, c = d; 
h = c ^ d sin 45° = dx 70711, 
-h ^ c = dsin 45° = dx 70711. 
Hence the additional calculations required run as follows. 
In the calculation of the coefficients replace the columns 4^6/3, dgH^/S by the 
following :— 
Ray. 
SiGbfS. 
S2mi3. 
3 
+ 1431,1 
-3427,6 
11 
+ 954,1 
-2285,0 
19 
+ 477,0 
- 1142,5 
21 
- 477,0 
+ 1142,5 
13 
- 954,1 
+ 2285,0 
5 
- 1431,1 
+ 3427,6 
In the rays (4), (12), (20) replace these by zero. 
Therefore 
Eay. 
Coefficient 
b or c. 
Coefficient 
= W. 
Sc'. 
3 
- 332,4 
- 895,0 
- 82- 125 = - 
00207 
-'•00082 
11 
- 559,5 
-1165,8 
- 93- 163 = - 
•00256 
- -00093 
19 
- 886,7 
- 871,4 
- 73 - 122 = - 
•00195 
- -00073 
21 
- 1840,7 
+ 1413,6 
+ 152 + 197 = + 
•00349 
- -00152 
13 
- 2467,7 
+ 3404,2 
+ 408 + 475 = + 
•00883 
- -00408 
5 
-3194,6 
+ 5960,2 
+ 793 + 832 = + 
•01625 
- -00793 
4 
- 1763,5 
+ 2532,6 
+ * +354 = + 
•00354 
- -00618 
12 
- 1513,6 
+ 1119,2 
+ * +156 = + 
•00156 
- -00354! 
; 20 
1 
- 1363,7 
+ 271,1 
+ ^ +038 = + 
•00038 
- -00160 
Steinheil. 
Sb'. 
Sc'. 
-'•00204 
! 
-'•000801 
- -00251 
- -00092! 
- -00191 
- -00070! 
+ -00351 
- -00151 
+ -00879 
- -00410 
+ -01605 
- -00758* 
+ -00357 
- -00627 
+ -00160 
- -00353 
+ -00041 
- -00162 
There is a slight discrepancy in the ray (5), 
