ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 695 



equally well corrected) tbey must always give the same image of the 

 same object. With the notation indicated above, the equivalent focal 

 length F of the total system (S + 2) is 



F ~ <p ' d + / " 5 ' 

 the linear amplification N (of the ultimate image at P), 



5 // 



'-Hi) 



and the aperture angle u of the total system (resulting from the linear 

 aperture a of the objective 2), 



M = 



therefore 



= . _ . o (where v = - ) 

 d <p v 5/ 



M = - ■ - ■ 

 d <p 



To take an example : let S be an \ inch and % a T L objective , 

 d = 400 mm., 8 = 200 mm., a — 3 mm.,/ = 3 mm., = 2 mm. — then 

 we have 



1^_300 1 200 _1___ 3 1 17 



F~20l)'4W + 36o'200 _ 800 + 300 _ 2400 



F = -— = 141 mm. (= 5| inches approximately) 



200 300 _ 3 

 ~ 400 ' 200 " I 



The ultimate image at P is therefore a slightly (3:4) diminished 

 image of 0. 



_ 3_ 300_ 9 



U ~ 400 ' 200 ~ 880 



which is an aperture angle of about § °. 



Thus the simple matter of fact is that if the miniature of O is 

 observed at P we observe the real object O by means of a very low- 

 power objective (5^ inches) of very low aperture (|°) under a very low 

 linear amplification, and nothing more is shown therefore by the 

 observation but this, that spider-lines and similar things can be seen 

 through very low-power objectives, which nobody will doubt. 



The formulae for F, N, and u show that the focal length of the 

 actually effective system, the ultimate amplification, and the aperture 

 angle do not depend on any other elements except (1) the distances 

 d and 8 of the object and the ultimate image, (2) the ratio of the focal 

 lengths of the objectives S and %, and the latter in addition (3) on 

 the linear aperture of the objective. Now F, N, and u, as has been 

 said, comprise all elements of the effective system (S -f- 2) which can 

 possibly have any influence on its performance (spherical correction of 



