ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
783 
II. — The plane of the circle of least confusion of the brightest rays 
is reckoned as the focal plane of the system. Its position could then 
be determined, by the method of Aug. Kramer,* from the expression for 
the distance of union. But Kramer foresaw that the least section 
through the light-particles may be bounded by two zone-rays of opposite 
Fig. 86. 
\0-55 
0^55 
h=o / 1-5 2'0 25 
inclination, and not always by a marginal and a zone-ray. Since, how- 
ever, nothing definite can be known on this point, the author contents 
himself with determining the distance of the focal plane (£') from the 
simple relation z = a — bh 2 -j- ch^. 
The latter was deduced from the four-termed relation by the method 
of least squares. But z is to be so determined that the error of the 
transversal aberration 
d(T = (z — z )- = («-a)^-(&-b)f + (c- c) h 5 + py 
shall be kept within the narrowest possible limits. This will be the 
case if the three maxima and the positive limit of that error determined 
by the marginal ray are equal to one another. 
Now, in the function 
y = m x -f- n oc 3 -f- p x 5 -f- q as 7 , 
its positive limit (for x = «'), and its three maxima are equal to each 
other, for 
7 7 7 
m = - Gl q X '*’ U = + 8 q X '*’ P = "“.4 9- x ’ 2 ' 
This introduced into d <r leads to the determination 
a- A= b-B=-~Dh'\ c_C= 
or, since 
A = 169*16, B = 4-779, C = 0-9593, D = 0-05218, h' = 2-5: 
a = 167-77, b = 2-995, c = 0-3886, 
so that 
(2) z 0 . 65 = 167-77 - 2-995T* 2 + 0-3886 A 4 . 
* ‘ Allg. Theorie der Fernrohr-Objektive,* § 31. 
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