4 Transactions of the Society. 



a point and its image. The second is a corollary, easily deducible 

 from the first, which establishes the equality of all optical paths 

 between a flat object and its flat image formed according to the 

 sine law.* 



Both may be comprised in the one proposition, namely, that in 

 a fully corrected system all paths between the aperture and the 

 focal plane are equal. Now it is obvious that this proposition can- 

 not lie true for two discon jugate focal planes. The following 

 diagram will make this clear. Here let S x S 2 S 3 be the aperture, 

 and let P x P 2 be central points in two disconjugate focal planes. 

 Then if the system be aplanatic for the point P 2 , all optical dis- 

 tances between the aperture S x . . S 3 and the point P 2 will be equal 

 to one another. 



In like manner, if we assume that the system is aplanatic also 



Fig. 1. 



for the point P 1} we shall have all paths between the aperture 

 Si . . S 3 and the point Pi equal to one another, and therefore the 

 path S 3 . .Pi is optically equal to the path S 2 . .Pi. Let S 3 ..P X 

 be extended to P 3 and make P x . . P 3 = P x . . P 2 . Wherefore the 

 path S 3 . . . P 3 = the path S 2 . . . P 2 , and therefore P 3 is a point in 

 the image formed by the aperture S a . . S 3 of a plane surface con- 

 jugate to the surface in which P 2 and P 3 lie. But by construction 

 this last-named surface is not a plane but a sphere having its 

 centre at P^ and the optical system occupying the aperture Si. .S 3 

 is not corrected to give a flat image in this region. If we make 

 the necessary correction to yield a flat field it is clear that we 

 shall incidentally render the point Pi non-aplanatic, and it follows 

 therefore that no optical system can be fully corrected so as to be 



* This second proposition does not appear ; to be so generally understood as the 

 first. A regular proof of it is given in a note — Note I. — in the appendix to a p.iper 

 on the Helmholtz Tlieorv of the Microscope, which I had the honour of laying before 

 the Society in 1903 (Journ. E.M.S., 1903, p. 420). 



