22 THEORY OF THE MICROSCOPE. 



follows, that if, with any combination, two refractions only come 

 into account, whether at surfaces or through imaginary lenses, the 

 first series of these quantities are reduced to u and u ', and the 

 second to t'. In the preceding mathematical consideration we have 

 confined ourselves to this simple case. 



If we denote by r and r the radii jof curvature of the anterior 

 and posterior lens-surfaces, and by / and f the focal lengths of 

 lenses whose action is equivalent to the refraction at those surfaces, 

 in the sense that to equal angles of incidence correspond equal 

 angles of refraction, and if, further, the index of refraction of the 

 lens-substance is represented by n, then, if the surrounding medium 

 is air, 



1 . n ~ 1 



yo " r o 



1 1 - n 5_"_1 . 



/' r ' r 



ft must also be observed that the radii of curvature r and r are to 

 be taken as positive or negative according as the incident ray meets 

 the convex or the concave side of the surface. If the surface is 

 plane, then r oc and, consequently, the corresponding u = 0. 

 Similarly, the focal lengths, where these are brought into account, 

 are to be taken as positive or negative according as the refrac- 

 tion causes the rays to converge or diverge, that is, positive 

 for converging and negative for diverging lenses and systems 

 of lenses. 



For the quantity t' we get, in the case of a single lens whose 

 thickness = d, 



d t 



n ' 



and, in the combination of two systems, if the principal points of 

 the first are denoted by their abscissre J3 and /, and those of the 

 second by E' and T, 



t' = E' - 7, 



that is, t' is equal to the distance of the principal points which are 

 turned towards each other. 



The operations which are necessary for the determination of the 

 optical cardinal points can now since u, u, and t' may be regarded 

 -be easily collected in a tabular form. 



