On Bull's-eyes for the Microscope. By E. M. Nelson. 311 
C to represent the aperture, by which means the diameter of the lens 
may be found. This being determined from the centre C, we can 
draw the curve D, and so the meniscus is constructed. If it is found 
that the proper amount has not been allowed for A D, the thickness 
of the lens, it can now be measured, and the curve computed with the 
new value of A 0 or P. To complete the doublet it is necessary to 
place a converging lens of minimum aberration in front of A to 
parallelize the rays which have their focus at B. 
Some care is necessary in doing this, for it is important that the 
focus for the marginal rays be accurately determined, leaving the 
aberration to act on the central pencils, as they are of less importance. 
First, we must decide the form of the lens. In the case of a glass 
of low refractive index, it would be better, perhaps, to have a crossed 
lens, but with flint of 1*62 I find that the difference in the co- 
efficients for aberration in the piano and in the crossed lens amounts 
to only ‘006, whilst in glass, whose refractive index is 1*516, it is 
•075. Crossing a flint lens is therefore a work of supererogation. 
Let us, in the first instance, investigate the procedure with a plano- 
convex flint lens where //, = 1 * 62. The plane side will face A. 
Its focal length will obviously be B A + the distance between the 
lenses + the distance of the nodal point from the plane side -f the 
spherical aberration for the semidiameter of the lens. B A or P' we 
already know from the formula P' = //, P. The distance between the 
lenses may he made small, say 1 /20 in. ; n the distance of the nodal 
point from the plane side can be found by the formula n =- 
where d = the thickness of the lens. 
To determine the spherical aberration is a longer business, and as 
this paper is intended to be entirely practical and not theoretical 
I have no intention of giving the formulae at length, especially as I 
have done so previously (R. M. J., 1887, p. 928.) It will be sufficient 
to point out that it consists of the product of two quantities which we 
will call a and h. The first of these, viz. a, varies principally with the 
refractive index of the glass used. When /x = 1 *516, for a crossed 
lens, a = l* 025, and for a plano-convex, a = 1*1. When [i = 1 * 62, 
for a plano-convex lens, a = * 804. 
The quantity h is the square of the semidiameter of the lens 
divided by the focus : thus if y is the semidiameter, h = - . The 
total spherical aberration is a h. 
If we put in for the value of / the sum of the values already 
obtained, viz. the distance B A or P' -f the distance between the 
lenses, say 1/20 in. + the distance of the nodal point from the plane 
side of the lens, sufficient accuracy will be obtained; if, however, 
extreme accuracy be required, determine the spherical aberration 
by this value of f and by adding it to the above quantities 
obtain a new value for / by which a true value may be obtained of h. 
