150 On the Focal Length an ee 
variation, that comparison of these by means of their assumed focal 
lengths too often gives no true idea of their relative excellence. 
For example, if two quarter-inch objectives be compared, and one 
gives results much superior to that given by the other, we cannot 
be at all sure that the better lens is not really of shorter focus than 
its designation would indicate. 
The question immediately arises, what is the focal length of a 
compound objective? The focal length of a simple lens, or of a 
system of lenses in actual contact, is the distance from the optical 
centre of that lens or system to its principal focus. But as a 
system of lenses not in contact, like the triplet objective, has no 
optical centre, the term is only a general appellation serving to 
group together objectives of approximately the same magnifying 
power. If every system of lenses possessed an optical centre, or 
could be replaced by a single lens, we might define the focal length 
of such a system as being the focal length of a single lens equivalent 
to the system in magnifying power. But as this is not the case, 
a lens replacing the system when the conjugate foci are separated 
from each other by a distance, d, will not replace it if the foci are 
separated by any other distance, d'; and the difference in focal 
length of lenses replacing the system under these different cireum- 
stances varies with the internal arrangement of the system. But 
for any constant distance between the conjugate foci any system 
can be replaced by an equivalent single lens; and in order to 
attach a definite meaning to the term “ focal length,” as applied 
to triplet objectives, I would propose that the focal length of such 
an objective be understood as being the focal length of a lens 
replacing the system when the distance between the conjugate foci 
is 10 inches. 
‘The present series of experiments was begun upon the supposi- 
tion that a triplet microscope objective has an optical centre, and 
the formule applied to obtain the focal length were based on this 
supposition, which, on further investigation, proved to be incorrect. 
Observations recorded in the sequel showed, however, that though 
not exactly correct, the formulz offer a very close approximation to 
the truth for distances of about 10 inches between the conjugate 
foci; the variation in the focal length of the lens replacing the 
system corresponding to a difference of several inches in that dis- 
tance being very small. 
We may, then, deduce a formula by which to find an equivalent 
simple lens from the relations between the principal focus, the con- 
jugate foci, and the relative magnitude of object and image for given 
distances between them. 
The principal focus of a simple lens can be determined from the 
formule, é + —=-, in which p and p' are the conjugate foci 
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