172 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
This completes the step. The proper way to check a sequence of such combinations is 
to proceed first from left to right and then from right to left; the final results will 
confirm one another and check the whole calculation, and the individual schemes 
arrived at will give in succession the steps across the surface (O), across the surface (O) 
and the space (l), across 0, I and the surface 2, and so on ; and in the reverse order 
across the surface 6, across 5 and G, across 4, 5, and 6, and so on. All these will be 
required after. I shall indicate them with the signs 0, 01, ..., G, 5G, .... 
For the individual surfaces the values of = — (l—7i)B are 
l\, = -(l-u.) B, = - '412848, = -(1-1/u) B^ = -1'585323, 
I'i = -(l-??i)B, = +1-145010, A, = -(l-l/m)Be = - '545089. 
In the following arrangement the separate schemes are written in the middle 
column, that corresponding to 3 being omitted as nugatory ; the calcidations forwards 
are written on the left and those backwards on the rinht. The latter beCTin at the 
O 
bottom and proceed upwards. Every figure used is recorded. 
0 
1-000000 
- -412848 
■653966 
} 
1-000712 
4019 
•996693 + -006366 
- -253990 
- - 629829 
- -883819 + -997672 
06 
01 i 
1-000000 
2477 
■997523 + -003924 | 
I 
•412848 + - 653966 J 
■997523 + -003924 
^ I - 
02 i _ 
1-581396 - 
-631298 + 
-006221 
■999999 t 
04 
[- 2-212694 + -993778J 
-997523 + -003924 
■V1-14217 4 + -004493 
-1-349927 + -606287 
- -207753 + -610780 
-> 
1 
1 - 000000 
* 
- 006000O 
1 - 000000 
I 
1-000000 * 
- 1-585323 +1-529130 
} 
1•000000 * 
+ 1-145010 + -610083 
+ -006004 
+ 3731 
+ 1-000712 + -009735 
- -001524 
+ 1-527096 
-253990 + 1-525572 
1 - 004580 
3868 
1-000712 + -003731 
+ 1-329225 
-1-583215 
16 
)■ 26 
- -253990 +1-527096 
1-000000 
+ 
4580 
1-004580 + -002440 
- -545089 
+ 1-874314 
r 46 
+ 1-329225 + -998670 
