new Definition of Focal Length, 8fc. 139 



direction the light is passing. Let m be the magnification, or 

 the linear ratio of an image formed at this point to the object 

 producing it. 



Then it is easily proved that j— =f a constant ; and this 



constant is in fact the focal length. 



Hence v— v =f.m, 



where t' is the value of v when the magnification is zero, i. e. 



at the principal focus. 



dv 

 It follows from the constancy of -= — that dv and dm need 



not be infinitesimals ; so that if any two positions of A be 

 found for a clear image of an object moved in any way along 

 the axis, the distance between these positions bears to the 

 focal length a ratio which is simply the change in the 

 magnification. 



Similarly, if u is the position of the other focus but m 

 retains its first signification, 



- is constant =/', 



\m/ 



and ,, 1 



where u is the value of u when m is infinite. 

 Incidentally, 



U-UqV — v =/./', 



V — V f 2 



- ■=. •— ?n . 



u — u Q / 



If the end media are alike, /=/', which is the important case 

 u— u v— v =/ 2 , 



° = m\ 



u — u 



The former of these equations is a common one, but the 

 latter is not so well recognized. 



m 2 may be called the areal magnification. It clearly has 

 an importance in the question of photographic exposure. 



5 4 3 2 / A -I -2 -3 -4 



_J 1 ^J r-i U-i-J r-J i-J r- 1 H H 



In the sketch let the straight line represent the axis of a 



L2 



