322 



SUMMAEY OF CURRENT RESEARCHES RELATING TO 



in air, and u* the semi-angle of the emergent pencil which depicts 

 the image. According to the law of aplanatic convergence "j" — 



or, as n* = 1 for air, 



t It may be useful to give tlie history of this law. It has its origin in a 

 publication of Lagrange (" Sur une loi ge'ne'rale d'optique," ' Me'moires de I'Acad. 

 de Berlin,' 1803), where he first showed that there is a fixed relation between 

 the amplification and the divergence or convergence of the rays at any pair of 

 conjugate foci, provided (1) both foci are in the same medium, (2) the system is 

 composed of infinitely thin lenses, and (.3) the pencils are infinitely narrow, i. e. 

 the angles of convergence very small. 



The formula then being — = N for every system of iufiuitely thin lenses 



u* 



(;i and n* being equal). 



In the famous reproduction (or rather reformation) of the Gaussian theory in 

 the ' Physiologische Optik ' (1866), Helmholtz showed that tliis formula holds 

 good for every composition of an optical system, and for different media, n and n*, 

 on the general supposition, however, of the Gaussian theory — infinitely narrow 

 pencils. The generalized law therefore became 



Instead of u and u* Helmholtz took the tangents of these angles (which is 

 the same thing as long as the angles are very small), and in this shape the 

 proposition first obtained its characteristic feature, sliowing the existence of a 

 general fixed relation between amplification and divergence or convergence at 

 conjugate foci entirely independent of the elements of the systems, and indicating 

 a difi"erent equivalent of equal angles in different media. 



The next stop was to apply this formula to systems with wide-angled pencils ; 

 and in 1873 Prof. Abbe signalized the feet that in the case of aplanatic foci the 

 convergence or divergence of the rays does not vary with the angles or with the 

 tangents, but with the sines. The same result was proved independently by 

 Prof. Helmholtz by a different method, and was published by him six months after 

 that of Prof. Abbe. 



At a later period Prof. Abbe has expressly called attention to the bearing of 

 the law of the sines to the practical performance of wide-angled systems, and 



