184 ME. E. A. SAMPSOF: A NEW TEEATMENT OF OPTICAL ABEERATIONS. 
In considering what discrepancies may be expected, we have to recall that the 
method developed in the preceding pages omits terms of the fifth order, which may 
amount to, say, 
coejf. xdx '000001. 
Taking d = 35^'1, for unity as coefficient, we should have an error of 4 units in 
the last place retained above. We have no means of saying what the coefficient may 
be, but it is clear that it may affect the last digit. Yet I believe that these 
calcidations are not only very much easier, but also more correct than the trigono¬ 
metrical ones, for though the formulae for the latter are exact, the number of 
operations they require is very large. Thus, for each ray which meets the axis, there 
are fully 50 operations of which at least one-half consist in taking out a logarithm 
or an antilogarithm with seven decimal places ; for each ray which does not meet the 
axis the work is rather more than four times as great. 
Steinheil has calculated seven of the former rays and nine of the latter. 
The controls that exist are of the most meagre description and give little help in 
locating an error. But, even if the whole is done in strictest accordance with the 
tables, at any step an error may be introduced which falls only short of haff a unit 
in the last place. Thus, in the rays which do not meet the axis, an irremovable 
accumulated error of 10 or more units could cause no surprise, and for this reason the 
trigonometrical method loses any advantage over the formulse given above which it 
might claim from resting upon exact formulae. The differences under discussion are, 
however, minimal, since 550 units in the last decimal place only amount to 1 second 
of arc. 
But pursuing the question a little further I believe, in spite of the evident care 
with which the whole of Steinheil’s calculations have been carried through, that the 
comparison above shows that a small error has crept in in respect to ray (5). 
If we take the general agreement as showing that the trigonometrical calculation 
does in fact bring in no terms of the aberrations beyond the 3rd order, we can readily 
analyse Steinheil’s numbers in more than one way so as to derive the coefficients 
•^jG, ... from them. 
Take the formulae (24); on the outer ring 0 = 0, 45°, 90°, 135°, 180°, correspond 
respectively to the rays 2, 3, 4, 5, 6 ; thus we have 
d^jSS^Gr = Sc'^ — Sc\ = ^ 
KdSf+id\G+id/3%G = 
d^l3 (<5iH + 4G)+= Sb's -t Sh', 
= -^{Sc's+lc's) = Sc't 
1 
{Sb',-Sb',) = Wb',-Sh’,) 
x/2 
i" {^b'2 -t + Sb', 
/3%R = Sb\ 
(34) 
