186 TESTING THE MICROSCOPE. 



The magnitudes N F and E* E are determined by the 

 equations 



^ 



which are generally applicable for any two systems of cardinal 

 points by substituting the corresponding values. 



As an example of a determination of cardinal points, we give 

 the following numerical data, which we obtained for a No. 9 

 objective of Beneche. The focal lengths, focal distances, and 

 surface-distances, expressed in millimetres, were as follows : 



(1.) Focal Lengths. 



= 2-6 ^ = 12 2 = 114 



(/) = 6-24 

 / = 315 



(2.) Focal Distances. 



For the first lens ...... d 1 = 1-75; d 2 = 2-8 



second,, ...... d l = 9*5 ; d 2 = 13'5 



third ...... d l = 94 ; d 2 = 114 



second and third lenses d l = 2'86 ; d 2 = 5'8 

 whole objective ... d l = 43; d 2 = 1-05. 



(3.) Surface-Distances. 



JVi - J\T = 1-37; JV 3 - N 2 = 2; N 5 - JV 4 = 2; 

 ffi t - N 2 = 5-ro; X 5 - N = 7; 

 and therefore 



JV 2 - A\ = 48 ; JV 4 - JV 3 = 1-15. 



From these quantities we obtain, in the first place, for the 

 principal planes 



c = N + '85 V = JVi + -1 



e' = JV 2 + 2-5 *' = JV 3 + 1'5 



e" = N 4 + 1-9 T = JV 5 4- 



(^) = N 2 + 3-38 (.)* = # 5 + -44 



^ = ^o + 272 ^* = JV + 2-8 



i" - e' = 2-65 

 00)* - e = 5-71. 

 The further deductions are contained in the following table 



