180 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
where the subscript (O) is dropped, y is taken as zero, and h = d cos (f), c — d sin <p, 
as on p. 165. 
These give the aberrations of the lens at its principal focus. 
The corresponding normal scheme is 
h'= ^ +ri31453^, 
= -•883818?>+ -997671/3.(29) 
In order to fix ideas, compare (28) with the case of a parabolic reflector of the 
same focal length, given on p. 167, for which we get 
= * +-441916/3+ * ] + /3[ + -22095d"-l-000006^+ * ]; 
we see that there is a close resemblance, except for the value of 4Gr, so that the two 
hardly differ in any sensible way, except in the curvatures of the fields. It will then 
cause some surprise that Seidel concluded that the Fraunhofer glass was free from 
coma which is so marked in the reflector. It was, in fact, a misapprehension, as the 
diagrams given by Steinheil and by Finsterwalder sufficiently demonstrate. 
Seidel’s argument presents an interesting feature.! He puts together the four 
components of his sum S (2), 
+ 0,412 
-12,672 
+ 13,454 
- 1,662 
S(2) = - 0,468 
and draws his conclusion from the approximate balance, within one-thirtieth, of the 
large positive and negative members. It is evident, however, that this amounts to no 
more than saying that the two internal surfaces nearly annul one another. But the 
point I wish to make is that these numbers are in fact the same as those found 
on p. 178 above. If we transfer to the principal focus by addingto (^G, we have 
for 
10 . 
'A- 
82 G. 
S^G xf. 
- -36445 
- -36445 
+ -4125 
--12406 
+ 11-32385 
+ 11-19979 
-12-6721 
+-13396 
-12-02515 
-11-89119 
+ 13-4543 
--00718 
+ 1-47599 
+ 1-46^8 81 
- 1-6619 
+ -41296 
+ 0-4672 
The connection does not appear to be so close in the case of others of Seidel’s sums, 
but it is interesting to notice this common ground. 
Let us now compare my calculations with those of Steinheil. First as to 
t ‘ Ast. Nach.,’ No. 1029, 326. 
