298 Transactions of the [^Tm^^i^ST^m} 
change in the aplanatism of the final image viewed by the eye- 
piece ; it not merely makes a convenient adjustment for the errors 
caused by using different thicknesses of covering-glass, which varies 
from the xio th to the xo^oths of an inch, but the index on the screw- 
collar may be used as a measure of spherical aberration occurring 
at the final focal image. The aberration sensibly changes for every 
different distance of the final focal image from the object, and con- 
sequently with the same object and covering-glass for differently- 
constructed eye-pieces. 
The form of the caustic or curve whose successive tangents 
represent the aberrating rays passing through the last or back lens 
is exceedingly acute, almost approaching a straight line. If F be 
the focus for parallel rays passing through the lens L, and inter- 
secting the axis at F ; let be the course of an aberrating ray 
intersecting the axis at q : and let it touch the caustic curve EPF 
at P ; let FN = X and PN = Y. Then the square of PN is pro- 
portional to the cube of FN or varies as X-^=:NX^. And the 
aberration consists of two distinct dimensions. The lateral aberra- 
tion is, in this case, represented by PN, and the longitudinal by F^ ; 
it has therefore breadth and length. But lenses may be so com- 
bined that for a certain distance of q from L these aberrations can 
be reduced almost to nothing. 
And it is possible to compensate practically one aberration by 
introducing another equal and opposite. 
On these principles the investigation of the circumstances requi- 
site for enhancing, clearing, or sharpening high-power definition 
may possibly be successfully carried out. 
For a given distance of the object from the object-glass, the 
aberration caused by refraction through a plate of glass of thickness 
t is doubled or trebled, &c., by making the covering- glass twice or 
thrice, &c., as thick : but it also varies in a high ratio as the angle 
of aperture of the objective increases. In other words, the con- 
fusion of the final image is represented by multiplying the aperture 
by the thickness. 
To define some of the most minute lines of diatoms fine defi- 
nition is often sacrificed to enlarged aperture, which however gives 
the additional advantage of increased light. 
It was only by very oblique and most skilful illumination that 
black-lined shadows could be obtained in the finest specimens and 
also by an aperture large enough to admit such oblique rays, that 
the lines could be seen at all: without this aperture, definition was 
of no avail with the power employed. Perhaps until a more exqui- 
