DETERMINATION OF THE CARDINAL POINTS. 185 



scribed by placing the lens with its anterior surface upon the stage, 

 and focusing the Microscope successively upon the surfaces. Where 

 the cell projects so much that it cannot be neglected, it is deter- 

 mined subsequently, and subtracted. 



From the data which these measurements furnish, the position 

 of the principal and focal planes can be calculated by simple 

 addition and subtraction. If we denote by JV , N lt N 2 , JV 3 , JV 4 , JV 5 , 

 the successive surfaces of the three double-lenses ; by < , ft, < 2 , 

 their focal lengths ; and e Q i, e i', e" i" t their pairs of principal 

 planes ; and further by (/) the focal lengths of the two posterior 

 lenses of the system ; and by (E}, (J)*, (F\ (F)* the principal 

 and focal planes of this system ; similarly, by / the focal length 

 of the whole objective ; and by E E*, F F*, its principal and focal 

 planes ; and finally, by d l and d 2 , in any given combination, the 

 distance of the corresponding focal points from the anterior and 

 posterior surfaces : the abscissae of the different principal planes, 

 reckoned from the front backwards, are given by the following 

 equations : 



e o = ^ + ft _ ^ P = Ni + dt 



e' = N 2 + ft d l i' = N 3 + d 2 ft 



e" = JV 4 + ft d 1 i" = N 5 + d 2 <f> 2 



and in like manner we obtain 



E* E = N, N Q + d 2 + d, 2 / 



N F=f (EXo). 



The relations, determined at the commencement, between the 

 cardinal points of two systems intended to be combined, and 

 those of the resulting systems, serve here as a check. We obtain, 

 by means of evident substitutions and transpositions, 



( f\ 01 02 



0i + 02 (0 ' 



^A fk 



in which (t) = e" i' = ft + ft 



and similarly, for the whole objective-system, 



J ~~j. i 



in which t = (E) H> = + (/) 



