IMPORTANCE OF THE ANGLE OF APERTURE. 95 



refer once more to the construction of the path of the rays in Fig. 

 42. If t f o s' is the pencil proceeding from the object, then its 

 marginal rays, and consequently all the others, meet the peripheral 

 portion of the objective, and the centre of the objective receives 

 nothing. If we turn the figure round, and regard s o t as the 

 pencil coining from the object, all inclinations from to 15 

 are represented ; it will therefore occupy half the aperture of 

 the objective. If, then, an objective with a large angle of 

 aperture is less aplanatic for central rays up to 15, and up- 

 wards, than for outer rays of 45 60, which is actually the 

 case, then the inclination of the incident pencil essentially in- 

 fluences the clearness of the image ; and if, in practice, the 

 optician can eliminate the aberrations of pencils incident obliquely, 

 which meet only the margin of the objective more easily than 

 the direct ones, then a large angular aperture is of decided 

 advantage. 



There is, however, as has been shown by Abbe, a second con- 

 sideration of far greater weight and more important signification, 

 in respect to which the magnitude of the angle of aperture 

 exercises an influence which may be mathematically proved and 

 experimentally fixed. The point in question has not only 

 reference to a corrective against accidental incompleteness of 

 construction, but also to a specific function of the angular aperture 

 in relation to the rays deflected in the object-plane, and inter- 

 fering in the upper focal plane, or in its neighbourhood, which 

 may be proved by the fact that they alone show in the micro- 

 scopic image the fine structure of the object, such as the stride of 

 diatoms, &c. 



The theory of this image-formation by interference will be 

 subsequently explained when treating of the optical action of 

 objects. We will here limit ourselves to the elucidation of a few 

 facts which prove beyond doubt the signification of the angle of 

 aperture for the delineation of fine structure. The facts are the 

 results of careful comparisons instituted by Abbe with a series of 

 objectives, as perfect as possible, of different focal lengths and 

 angles of aperture, testing their capacity by various kinds of test 

 objects, among which were coarse and fine powders, and also fine 

 systems of lines ruled in extremely thin layers of silver. He 

 thereby proved: 



(1.) As long as the free aperture of the objective remains so 



