678 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
(5) Microscopical Optics and Manipulation. 
Abbe’s Method and Apparatus for the Determination of Focal 
Lengths.* — Dr. S. Czapsld gives a detailed account of the methods and 
apparatus recently devised by Prof. Abbe for the determination of focal 
lengths,| and explains the principles on which they depend. By the 
Abbe method, the fundamental conditions necessary for “ micrometric 
measurements by means of oj)tic images ” are more fully satisfied than 
by any other previously employed ; and accordingly the influence of the 
systematic as well as the accidental errors of observation on the result 
are considerably minimized. These fundamental conditions, as formu- 
lated by Prof. Abbe, are as follows : — 
(1) A determination of precision must not be made dependent upon 
the position of the optic image. 
For this position is always uncertain. The amount of play in an 
adjustment is directly proportional to the keenness of the observing eye 
and inversely to the angular aperture of the pencil forming the last 
image in front of the eye. For a normal eye the amount of play amounts 
to about 13 mm. For an eye armed with a lens this source of error is 
considerably increased. In the determination of the focal length of 
optical systems therefore, the distance of one image from another or 
from a fixed point must not enter in as a factor. 
(2) The measurement must not be indirectly affected by the adjust- 
ment. 
Suppose, for example, that the magnitude of the image P Q = y' 
formed by the system S of the object 0 M is to be measured, and that 
Fig. 76. 
an error in adjustment Q Q' = d x' has been made ; then the amount of 
error in the measurement of P Q is K P' = Q Q' tan P S Q, i.e. d y' = 
d x' tan (o. 
The error involved in calculating the magnitude of the object from 
that of the image is then 
d y d x r tan w 
V ~ Ny 
where N = y / y is the linear magnification. 
A means of reducing the error is to diminish the angle P S Q, which 
the axis of the pencil forming the image makes with the axis of the 
* Zeitselir. f. Instrumentenk., xii. (1892) pp. 185-97. f See ante , p. 427. 
