INTRODUCTION. 7 



of the single lens. A ray proceeding from b through the first 

 principal point E undergoes, in a sense, no deviation, since the 

 directions of transmission before and after refraction form the 

 same angle ; but the emergent ray, produced backwards, cuts the 

 axis at the second principal point E* instead of E. It may be 

 treated, therefore, as an unrefracted ray which has been displaced 

 along the axis to a distance equal to E E*. A similar displace- 

 ment must be taken into consideration in the case of every other 

 ray, whatever its direction may be. The emergent rays always 



FIG. 4. 



behave as if the refraction had taken place in the first principal 

 plane, and then a displacement of the refracted ray had taken 

 place along the axis to the distance E E*, or, what amounts to 

 the same thing, as if the displacement of the incident rays had 

 taken place first, and then the refraction in the second principal 

 plane had followed. We can, accordingly, determine without 

 difficulty the direction of a ray which is incident parallel to the 

 axis, it must be produced to the second principal plane and then 

 carried on through the focal point F*. Conversely, a ray which 

 cuts the axis in the first focal point F must be produced from the 

 first principal plane in a direction parallel to the axis. 



Consequently, from a given point I of the object (Fig. 4), not 

 being on the axis, there may be drawn three distinctive rays, for 

 which the corresponding emergent rays may be constructed just 

 as readily as with the single lens. Two of them are sufficient to 

 determine the point I* of the image, for the others will, obviously, 

 be refracted to this point also, as indicated in the figure. 



It is evident, therefore, that the combined action of a refracting 

 system is in every respect analogous, though not entirely equiva- 

 lent, to the refraction of an infinitely-thin lens. A lens of equal 



