Objectives for the Compound Microscope. 
67 
on the other side of the lens, which may be received on a screen. 
This image will be larger, and formed at a point more distant from 
the lens, the nearer the object approaches to the focus for parallel 
rays ; and two equations are given which express the relationship 
of the distances to each other, and to the magnifying power, viz. 
Ill «' 
= — I — ; , and — = m, in which / is the length of the focus 
f p p V 
for parallel rays, p the distance of the lens from the object, p its 
distance from the image, and m the true magnifying power, that is, 
the size of the image divided by the size of the object ; p and p 
are termed the conjugate foci, and are variable quantities ; / is 
termed the principal focus, and has an unchangeable value for each 
single lens. 
If now we combine the above equations, representing p -f- p or 
the sum of the conjugate foci by l, we may deduce the formula 
f = ^ ^ ^ , which represents in the case of any single convex 
lens the relationship existing between the length of the principal 
focus, the magnifying power, and the distance from the object to 
the screen. This formula, which I think rather more convenient 
than that of Mr. Cross, differs from it only in using m = the 
magnifying power, instead of n = the reciprocal of the magnifying 
power ; it may be deduced from his by substituting for n its value 
— and reducing, or it may be derived directly from the primitive 
equations. In either shape the formula yields the same numerical 
results, and if for any single convex lens m and l are given, the 
accuracy of the value of / resulting will be limited only by 
the degree of precision with which m and l have been measured. 
If now there were any such actual equivalence between achro- 
matic objectives and single lenses as the nomenclature assumes, it 
would only be necessary to set up the objective to be rated, in such 
a manner that the image of a micrometer should be focussed upon 
a white screen, using of course no eye-piece, to measure the distance 
from the micrometer to the screen, to determine the magnifying 
power by measuring the image of the micrometer, and substituting 
these values of l and m in the working formula to calculate the 
value of /. Unfortunately, however, if with any compound 
objective we repeat this operation several times, merely varying 
the distances, we obtain as many different values for f as there are 
distances used, instead of obtaining hut one value for all distances 
as we do with a single lens. 
Mr. Cross* has already pointed out this circumstance, which 
results from the fact that the modern achromatic objective has 
* Loc. cit., p. 100. 
