26 Transactions of the Society. 



by tlie fact that the amount of this difference in the colour amphfication 

 is very unequal in the central and peripheral parts of the objective- 

 opening, in the apochromatic objectives it is approximately constant 

 for all parts of the opening, and consequently allows of correction ly 

 means of the eye-piece. 



For this purpose it is only necessary to construct the eye-piece in 

 such a way that it may have an equal but opposite difference of 

 magnifying power (or focal length) for different colours, that is, to use 

 eye-pieces which are to a definite extent w^achromatic. To effect this 

 object it is necessary, if the eye-pieces are to be used with different 

 objectives — as must of course be the case — that this difference of 

 magnifying jjower for different colours should be made approximately 

 the same in all objectives. This renders it necessary that objectives of 

 small aperture (in which the difference of amplification can be easily 

 avoided and which are usually found free from any such defect) must 

 be purposely made defective in this respect to exactly the same extent as 

 is the case in the objectives of large aperture in which it is unavoidable. 



The result of this innovation is that now even objectives of relatively 

 large aperture will give images very free from colour over the whole field, 

 while their construction need not be more complicated for that purpose. 



As regards the production of a series of objectives corrected in the 

 above-described manner and satisfying the different practical require- 

 ments, the same considerations hold good which were formerly developed 

 by the author with reference to the conditions of construction then 

 existing. The altered conditions, as regards the degree of concentration 

 of the rays attainable, can be expressed by modifying the numbers there 

 given. 



As the starting-point for each construction, the numerical aperture 

 must be taken which finds its expression in the ratio of the free opening 

 of the objective to its focal length. The superior limit of this element is 

 in every class of objectives — dry, water-immersion, or homogeneous-im- 

 mersion — almost invariably determined by the theoretical maximum ; 

 within this limit the determination of a particular value for a definite 

 purpose remains a matter of free choice. When once the aperture is fixed 

 it determines the type of combination of the lens system in general, and 

 thence also the choice of the separate elements by which the require- 

 ments of the different conditions of correction are to be satisfied ; the 

 only point which remains open is the focal length with which the 

 aperture in question is to be associated, that is the scale of construction, 

 or the absolute dimensions of the system. 



The choice of the focal length or the scale for a rational construction 

 is based on the following considerations : — With the aperture are deter- 

 mined, on the basis of well-known and generally accepted laws, the 

 linear measure of the smallest detail of the object which can be delineated 

 by means of this aperture; the measure may be very approximately 

 expressed numerically for every aperture. 



It is then necessary that the smallest detail which can be reproduced 

 in the image should be presented to the eye under a visual angle which 

 is large enough for its clear perception, and the determination of this is 



