Professor Abbe's Paper on the Microscope. 193 
where it may be examined with a lens. But the image of the 
object, so far as it is produced by the objective alone, lies in or 
close to the upper focal plane of the objective, where also it may be 
seen by looking down the tube of the microscope with the naked 
eye. These two sets of images are interconnected by common 
relations, the determination of which affords a key to the solution 
of questions scarcely to be approached by any other means. All 
the characteristics of the object images hang together with certain 
characteristics of the aperture images, and vice versa. 
The principle on which is founded the study of these aperture 
images leads to various results, depending for their full develop- 
ment upon a principle which constitutes at the same time a law of 
fruitful application throughout the whole theory of the microscope, 
and which may be thus formularized. 
When an objective is perfectly aplanatic for one of its focal 
planes, every ray proceeding from this focus strikes the plane of 
the conjugate focus at a point, whose lineal distance from the axis 
is equal to the sum of the equivalent focal length of the objective 
X the sum of the angle which that ray forms with the axis. 
Now as this condition must be fulfilled in every correct in- 
strument, both for the objective and for the whole optical part of 
the microscope, the formula above given establishes a relation of 
quantity between the angle of aperture of the microscope and the 
lineal diameter of the aperture images above the objective and ocular. 
Moreover, it is thus possible to determine, by micrometric 
measurement of the position in the upper focal plane of the objec- 
tive which the track of any ray occupies, the direction which it 
took before entering the microscope. Consequently the aperture 
images formed above the objective, when examined with a suitable 
micrometer eye-piece, can be used for measurement of the diver- 
gence which the rays coming from the object undergo. 
In the next place, we need a more characteristic exposition of 
the optical functions which, in the case of images formed under 
larger angles, by rays having a great inclination to the axis, differ 
greatly from the abstraction by which theory represents the action 
of a set of lenses in forming an image. And such an exposition 
offers itself when we can define by axioms of general validity the 
mode in which an image is focussed and spread out on the focal 
plane of an optical system, and distinguish th e focussing function 
and the extension of image over a surface as the two principal 
factors of the image-forming process, alike independent in their 
abstract idea, and distinct in actual specific function. Apart from 
the fact that no exhaustive analysis of a faulty image nor any 
means of perfect correction are possible until such characteristic 
distinction can be laid down, we have no other means of determining 
