MR R A. SAMPSON; A NEW TREATMENT OF OPTICAL ABERRATIONS. 
179 
We conclude from this calculation that the aberrational terms in the emergent 
ray are 
Sh' = &„.[+-20722 ( 6 / + C 0 ")--00526 (^,/3o + c,yo)--00188 {/3,^ + y,^)'] 
+ ^o[+‘31491 (6/ + c/)+- 00029 (6A + Coyo)--00188 (/3/ + y/)], 
= 6o[--21613(6/ + 0+-37086 (6,/8, + c,y,)--75647 (^o^ + yo^)] 
+ /8o[--09612 (V + e,/)--87730 (6,^o + Coyo)--0036l(/3,^ + yTO] • (27), 
with similar expressions for Sc', Sy if we replace h^^, outside the brackets Ijy c^, y^. 
We may without loss of generality put y^ = 0 , and in what follows this shall 
be done. 
The twelve coefficients above are not independent. We have seen that they must 
satisfy seven relations, and we shall now verify that they do so. 
We have, from p. 161, 
~ ^0 4” 4“ ^4 4" 
h 
- -41285 
- 1-58532 
+ 1-14501 
- -54509 
Term in QI. 
- -41285 
- 1-03675 
+ 1-14501 
- -33255 
and thus 
S 2 a-SiH= - -00526--62982 = - 
S 3 G-S 2 H= - -00376--00029 = - 
S 2 K- 81 L = + -37086+-19224 = + 
S 3 K- 82 L = - 1-51294+-87730 = - 
- -63714 
= *13 
63508 
4 IG = - -63503 
00405 
511H = - -00406 
56310 
qSK = + -56312 
63564 
il3L = - -63566 
Next for the relations (20); we have, noting that N = 1, 
NK2 = + -78113 
NKL = - -88176 
N(L2- 1) = - -00465 
The comparisons seem to point to a small accumulated error in 
If we transfer the origin for emergent rays to the focal plane, x' = +1-12771, we 
have finally 
Sh' = -03651 d'+-41296 d/3 cos 0--85496^'] 
+ /3 [+-20651 fP--98905 d/3 cos 0--00595/3'] 
Sc' = c [--03651 d'+- 4 1296 d^ cos 0--85496^'] 
+ * . (28), 
2-a 2 
s. 
+ L 84 .G. 
- KS,H. 
- H 8 ,K. 
+ GSjL. 
= 8 .N. 
1 
+ -41347 
+ -55665 
+-00275 
- -19160 
+-78127 
2 
- -00525 
+ -00026 
- -00236 
- -87440 
- -88175 
3 
- -00376 
- -00332 
+ -00963 
- -00720 
- -00465 
