SPHERICAL ABERRATION. 
3 
siderably nearer to the lens than the point F, where the central 
parts of the pencil converge. The distance F f, is called the 
longitudinal spherical aberration, and a the smallest diam- 
eter of the converging pencil, is called the lateral spherical 
aberration, 
6. It is easily seen that if the piano convex lens is turned 
with its convex surface towards parallel rays, the points where 
the lateral rays undergo refraction would then be in advance 
of the positions where the rays, passing near the centre, are 
refracted, and consequently the spherical aberration would be 
diminished by the difference between the thickness of the lens 
at the centre and the thickness at the border. The difference is 
really much greater than this, for if the plane side of the lens 
is turned towards parallel rays, as shown in Fig. 2, the longi- 
tudinal spherical aberration is about times the thickness of 
the lens, but if the opposite face is towards parallel rays, the 
aberration is only about lyVo times its thickness.* 
A double convex lens has the least spherical aberration 
when the two faces have their radii of curvature in the propor- 
tions of about one to six. Such a lens with its flatter surface 
* The spherical aberration of a lens varies with the refractive power of the 
material of which the lens is made. 
The proportions here given are taken from Lardner’s Optics. Other writers state 
them somewhat differently. 
According to Prechtl, if w=:index of refraction, r=the radius of curvature of the 
anterior surface of the lens, and R=radius of posterior surface, then for parallel 
rays the form of least aberration will be expressed by the equation 
r 4-}-^ — 
Then if the index of refraction equals 1^, the form of least aberration will be ob- 
tained if the two surfaces have their radii as 1 to 6, the side of deeper curvature 
being towards the parallel rays. If the spherical aberration of such a lens in its 
best position be reckoned as unity, the aberrations of other lenses will be as 
follows : 
Plano convex with plane surface towards distant objects, 4.2. 
“ “ “ convex surface towards distant objects, 1.081. 
Plano concave the same as piano convex. 
Double convex or double concave with both faces of the same curvature, the 
aberration will be 1.567. 
CATALOGUE OF ACHROMATIC MICROSCOPES. 
