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SUMMARY OF CURRENT RESEARCHES RELATING TO 
The formula) giving the focal length of a lens are 
/ = 
A A 
A ~ 9i -f- U 
and g = - / = 
- 9i 9-2 
/a “ 0i + * ’ 
where f x g l are the refractions for the first surface and f 2 g 2 those for the 
second, and t t is the thickness of the lens. These focal lengths are 
measured from the principal planes. The author designates the three 
segments into which the lens is divided by these planes by the names 
anteplane, interplane, postplane, and the interval between the second 
principal plane of one lens and the first principal plane of the following 
lens by the term transit. 
The formula) for determining the principal planes are 
a x (anteplane) = 
-At 
A-<h -H 
« 
a, 2 (postplane) = 
The above formulae give the focal lengths, &c., of a single lens, but 
by altering slightly the signification of the letters they also serve for 
the combination of two lenses to form a doublet. In this case f x f 2 , &c., 
denote the focal lengths of the individual lenses instead of those of the 
surface refractions, and t denotes the transit distance instead of the 
thickness of the lens. The same formulae then serve for combining 
the doublets into an objective, and finally for finding the dimensions 
and focal lengths of an imaginary lens equivalent to the whole 
Microscope. 
To illustrate the application of the formulae the author takes the 
case of a Microscope composed of an objective consisting of three 
doublets and the ordinary Huyghenian eye-piece. Each of the doublets 
C, B, A consists of a plano-convex flint lens (n = 1 • 6) combined with a 
double convex crown lens (n = 1 • 5). The thickness of the three flint 
lenses is 1/2, 2/3, and 3/4 mm., and that of the three crown lenses 1, 4/3, 
aud 3/2 mm. respectively, with radii of curvature 1, 4, and 10. The 
front vertex of the second and third doublets coincides in each case with 
the second principal plane of the underlying doublet. The eye-piece 
consists of two convex -plane crown lenses, with convex side downwards. 
The lower one is 3 mm. thick, and has radius of curvature 40, while 
the upper is 2 mm. thick, with radius of curvature 30. 
Taking as an example the case of the crown lens of the front doublet 
C, we have 
A = ~ 
n r 
n — 1 
= - 3. 
A = 
fl f 2 
A ~9i + t 
Anteplane = 2/5 and postplane = — 2/5. 
