18 BELL SYSTEM TECHNICAL JOURNAL 



The two principal focal distances measured from the principal planes 

 are, in a similar manner, designated as T^i and F^; then, from equation 

 58, 



F\ = fi — Oil = kifi, 



Fi = fi — Oi2 = kifi, 



(60) 



and, from equation 55, 



^i/^2=-J^- (61) 



2v: 



Substitution of the new focal distances in the lens equation 56 now 

 gives 



{Fi - D2){F, - DO = FxFi (62) 



or 



This is the general equation relating the conjugate focal distances in 

 any lens. With the aid of equation 61, it may be written in the more 

 familiar form • 



^2v2 ^2^1 _ 1 



~^~~d7 ~f' 



where 



1 V2iJ2 ^|2vl 



Fy 



(64) 

 (65) 



The Principal Points of a Lens 



The points locating the two principal planes on the axis of a lens 

 and its two principal focal points are called the cardinal points of the 

 lens. The preceding theory of a thick lens shows that its performance 

 is completely determined by the locations of these four points. ^^ 

 The theory does not furnish a general method for calculating their 

 locations, but it does show that they can be determined from a knowl- 

 edge of two so-called principal rays. The first is a ray leaving the 

 lens parallel to the axis. If its entrance and exit paths are projected, 

 they intersect as shown in Fig. 11, and the intersection locates the 

 first principal plane. The projected incident ray also intersects the 

 axis, and this intersection locates the first principal focal point. The 

 second principal plane and the second principal focal point may be 

 located in a similar manner from the entrance and exit paths of a ray 

 entering the lens parallel to the axis. 



1^ We are here speaking only of rays near the axis. 



