ON “OPTICAL TUBE-LENGTH.” 15 
to their focal lengths: thus a 1-2 in. will not necessarily give double 
the power of a 1 in. 
Conversely, two eye-pieces will not magnify in proportion to their 
focal lengths, though used with the same tube and objective. 
Indeed, the true magnifying powers may differ from the powers 
which would be obtained on the ordinary assumptions by more 
than 100 per cent., and Prof. Abbe records the existence of objec- 
tives (of somewhat exceptional construction it is true) which 
exhibit this paradoxical behaviour : that one of longer focal length 
amplifies much more than one of shorter focal length; that one 
gives the same amplification with a long and a short tube, and that 
one gives a higher amplification with a short tube than with a 
long one. 
What then is the explanation of these paradoxes? 
The explanation is not to be found in any question of the length 
of the objective or eye-piece, or the character of their respective 
settings, but depends upon the fact that hitherto microscopists have 
regarded the outside only of the tube and have left out of consider- 
ation the optical action which goes on within it. 
To properly understand the matter it will be necessary to consider 
the principles on which the action of the Microscope in regard to 
magnifying power is founded. 
The magnifying power of a lens depends of course upon its 
focal length and varies inversely with it; the ordinary mode of 
obtaining the power being to divide the distance of distinct vision Z 
(assumed as ro in.) by the focal length, or expressing it by a formula 
Z : Ps irl ge : 
N mee Thus if the focal length f of an objective is 1-8th in. 
10+1=80, The same applies to the action of the Microscope 
as a whole, that is with eye-piece and objective combined ; when 
we have determined its focal length we similarly obtain its mag- 
nifying power.* 
We have therefore to ascertain the proper mode of determining 
the focal length 7 of the entire Microscope, having given the focal 
length /* of the objective and the focal length /* of the eye- 
piece. 
The usual assumption hitherto has been that / is determined by 
multiplying 77 and /? together and dividing by the length of the 
tube ro in., or 
Io 
w 
* The quotient obtained by dividing Io in. by the focal length gives the linear 
amplification of an image—real or virtual—which is projected by an objective to 
a distance of 10 in. from its posterior focus, and not from the objective, as has 
been so commonly assumed. 
