( 147 ) 



APPENDICES 



APPENDIX I 



TO ASCERTAIN THE N. A. OF AN ORDINARY PHOTOGRAPHIC LENS* 



F F 



This may be deduced from the usual values stamped ou the diaphragms (such ^ ^ etc. etc.) 

 as follows : 



F 



If a diaphragm is marked - (n being any one of the usual figures 8.1 1.16.22 etc. etc.) the mean- 

 ing is that the effective diameter D, to which it reduces the working aperture of the lens, is equal 

 to that amount algebraically. 



(I.) D = ^ or transposed § = i . 



On the other hand, according to Abbe, the N. A. of any lens, provided it is aplanatic, is obtained 

 by dividing half its working diameter by its focus. 



- I D D 

 (II.) N. A. = 2 = - Introducing the value of ^ as given in equation (I.) into (II.) we 



obtain 



cm.) K A. = - X - = - . This wouU de the true N. A. of the lens, if the object was at the 

 ^ ^ 2 n 2X1 



principal focus, producing an infinitely magnified image at an infinite distance behind the lens. 



Really, however, the lens will be producing a moderately magnified image at a reasonable distance, 



and this demands that the object should be at a greater distance from the lens than the principal 



focus. At this greater distance the working diameter of the lens will subtend a smaller angle than 



at the principal focus, hence the N. A. will be smaller than that found by Equation (III.)- The 



reduced N. A, may be found with sufficient accuracy as follows : 



Let the inaguification be in, then the image will be formed at a distance (m + i) F behind the 



HI "4~ I 



optical centre of the lens, and the object will be at a distance — — F in front of the optical centre 



of the lens. The distance from the object to the lens is therefore increased over that assumed in 



m + I. 

 m 



fore we get the final equation 



m I 



Equation (III.) in the proportion — ^ ' and the N. A. is diminished in inverse proportion. There- 

 3 we get the final equation 



(IV.) For magnification M, the actual N. A. = 



m + I 2 n 

 F 



Example : Diaphragm Introducing n = 16 into (III.) we get 



the N. A. at the principal focus ^ — 7: = — = "03125. 



^ 2 X 16 32 



* I am indebted for this lucid description to the pen of Mr. Conrady. 



