METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
is more serious in low-power than high-power objectives. Eye-pieces and 
pocket magnifiers should be corrected for astigmatism and for curvature of 
field as well. 
CURVATURE OP THE FIELD (PETZVAL CONDITION). 
After astigmatism has been corrected, the lens produces a point-shaped 
image of any point in the object field; these image points are not contained, 
however, in a plane normal to the principal axis, but on a curved surface 
generally concave toward the objective. In order that the image be flat, 
the radii of curvature and the refractive indices of the component elements 
must be such that the sum of their reciprocal products is zero (Petzval 
condition). The flattening of the curved surface of astigmatism is called 
astigmatic flattening of the image. An apparent lack of flatness of field 
in the microscope is usually due not to curvature but rather to imperfect 
correction of the other spherical aberrations (especially sine condition and 
chromatic differences of magnification). 
DISTORTION (TANGENT CONDITION). 
Even after the image is flat, it may not be similar to the object if there is 
variation in the magnification of the lens for points away from the optical 
The image of a rectilinear grating (Fig. 23, a) from a lens showing 

axis. 
\ 
/ 
\ 
1 
FIG. 23. 
distortion is no longer rectilinear but curved and cushion-shaped or barrel- 
shaped (Fig. 23, b, c). In microscope objectives the construction is such 
that distortion is not serious in the image. If distortion be present, it is 
usually due to the eye-piece. 
In a lens system it is not possible to correct all the above spherical aber- 
rations at once with equal perfection, and certain corrections are sacrificed 
for others, depending on the purpose for which the lens is to be used. In 
microscope objectives of high aperture, where wide pencils of light with 
small inclination of the chief ray to the axis are used, the spherical correc- 
tion of three zones of the aperture and the observance of the sine condition 
are generally sufficient; in the photographic lens, on the other hand, where 
the aperture is smaller and the field larger, astigmatism and distortion have 
to be corrected, while the spherical correction need not be carried so far as 
in the microscope or telescope where the image is highly magnified. From 
the eye-piece narrow pencils of light, highly inclined to the axis, enter the 
eye, the function of the eye-piece being that of a magnifier with large field. 
It is accordingly corrected, especially for astigmatism, curvature, and dis- 
tortion, but not necessarily for spherical aberration, which, in the axis, is 
not serious, since the diameter of the pencils is small equal to or less than 
the diameter of the pupil of the eye. 
