510 BECOKD OF CURRENT RESEARCHES RELATING TO 



images of which the whole image is composed be different, they may 

 still coincide in the axial point of the image-field, if the spherical 

 aberration is completely corrected, but at a distance from the axis, 

 and in proj)ortion to that distance, they will separate from one 

 another more and more. The image of a point at some distance from 

 the axis resolves itself into a circle of confusion, whose diameter 

 bears a finite— in some cases a large — ratio to its distance from the 

 axis, and consequently to the dimensions of the portion of surface 

 viewed, however small that may be. When, for example, the linear 

 amplification through the central portion of the aperture is 10 

 diameters, while the amplification through a marginal portion is 12 

 diameters, the overlapping of the image pi'oduced by the latter over 

 that produced by the former would introduce circles of confusion 

 whose diameter is one-fifth of their distance from the centre at every 

 part of the field. Hence a system, to be aplanatic, besides having 

 its spherical aberration corrected for a pair of conjugate points, must 

 satisfy the further condition of uniform amplification through all 

 parts of the available aperture, that is, for rays in every direction 

 which the angle of aperture embraces. 



By a geometrical analysis it may be demonstrated that the required 

 identity of amplification through different parts of the available aperture 

 only subsists when there is a definite ratio between the convergence 

 of the two conjugate pencils of rays whose centres are the axial 

 points of the object and of its image; the sines of the angles of 

 inclination of mutually corresponding rays towards the axis must 

 have a constant ratio throughout the whole range of both pencils. 

 By this proj)erty aplanatic points are contrasted with a second kind of 

 characteristic points which are important in, the formation of images 

 by rays of appreciable divergence, viz. those points on the axis in which 

 the tangents of the angles of inclination of conjugate rays are in a 

 constant ratio. These may very projierly be called orthoscopic points, 

 as on them depends the possibility of forming orthogonal, or similar, 

 images of extended (i. e. not infinitesimal) objects. 



The amended definition which Professor Abbe gives is therefore 

 as follows : — " Aplanatic points in a lens-system are those conjugate 

 points on the axis, the spherical aberration of which has been 

 corrected for a cone of rays of appreciable angular aperture, the sines 

 of the angles of inclination of conjugate rays being also proportional." 



The essential i)art of this definition was published by Professor Abbe 

 in 1873.* Professor Helmholtz independently established the same 

 principle, and showed f that the constant ratio of the sines was 

 the condition required if the quantity of light proceeding from the 

 object was to reach the image without loss or gain. Since " quantity 

 of light," according to the undulatory theory, is the energy of an 

 oscillatory movement, this mode of deriving the above theorem connects 

 the action of optical apparatus with the most universal principle in 

 modern physics. 



In microscopical objectives of large angular aperture the aplana- 



* 'Aicli. f. Mikr. Anat.' ix. (1873) p. 420. 

 t Toggcud. Auualt'ii, Jubelband,' p. 566. 



