14 



HANDBOOK OF PHOTOGRAPHY 



Focal Length of a Lens. — The above process is a very laborious one, and to simphfy 

 it Gauss, about 1841, devised his system of principal and focal points. These are 

 defined as follows: In Fig. 5 is shown a general lens system of any internal construc- 

 tion, such as a photographic objective, and a series of rays from a very distant axial 

 object point is shown entering the left-hand end of the system. These will all 

 emerge from the other end of the lens as a converging beam, as shown, and each ray 

 will evidently possess an "equivalent refracting point" where the entering and emerg- 

 ing portions of the ray intersect. The surface which joins vip all these equivalent 



Fig. 5. — The equivalent refracting surface of a lens. 



refracting points is called the "equivalent refracting surface" of the lens, and where 

 this crosses the axis is the "second principal point" Pi of the lens. Also, the point 

 at which the innermost rays Ijang close to the axis cross the axis is called the "second 

 focal point" Fi, the distance from Pa to F-i being the focal length of the lens. If rays 

 from a distant object point enter the lens at the right-hand end, there will be another 

 principal point P\ and another focal point F\, the distance between them being another 

 focal length. It can be proved that, in any lens whatever, these two focal lengths 

 are equal. ^ 



Combination of Two Lenses. — If two lenses of focal lengths /i and Ji are used 

 together in succession, the focal length F of the combination will be given by 



i = 1 4-i _ _1 



(4) 



where d is the distance between the second principal point of the first lens and the 

 first principal point of the second lens (Fig. 6). 



Fig. 6. — A system composed of two separated lenses. 



It should be noted that if /i and fi are both positive, i.e., convex lenses, then 

 increasing their separation will lengthen the combined focal length. On the other 

 hand, if a positive and a negative lens are separated, the focal length of the combina- 

 tion is shortened. In the case when d is equal to /i, the combined focal length is also 

 /i, and hence the presence of a second lens in the focal plane of the first lens does not 

 affect the focal length of the first lens. In the limiting case when d = fi + /a, the 



1 This statement is incorrect if the image and object happen to be situated in different mediums. 

 The eye and the oil-immersion microscope objective are the only important systems in which this 

 occurs. 



