DETERMINATION OF THE PATH OF RAYS. 17 



f* .V* = oc. Therefore, evidently, 77* = oc, and, as we 

 assume that ?; > has a finite value, 



OC . 



Since, in this expression, the numerator is a finite quantity, the 

 denominator must = 0, and, therefore, /,; (f N) = nl, from 

 which we get 



f = N + *. (16) 



111 like manner for the analogous third case, in which the object- 

 point lies at an infinite distance, we get from equation (13) the 

 corresponding value of f*. If, for instance, we consider that 

 the quantities n h and n I, which occur in the numerator and 

 denominator, disappear in comparison with the infinite quantity 

 (f N), it is readily seen that 



* = #*- ?Ll. (17) 



rC 



These values of f and f* evidently correspond to the abscissa 

 of the two focal points F and F*, and the planes drawn through 

 them at right angles to the axis are the focal planes. As soon as 

 one of the two points lies in the corresponding focal plane, the 

 other moves to an infinite distance. 



The distances of the respective principal and focal points may 

 be determined from their abscissae, given in equations (14) to (17), 

 by simple subtraction of the corresponding values of the abscissae. 

 We get 



F* - E* = - 



71 n* 

 The quantities r - and -,or the distances of the principal 



K K 



planes from the corresponding focal planes, are called the focal 

 frn.(//hs of the system. They have under all circumstances since 

 ?i and n* are, from their nature, positive numbers the same, and 

 k the opposite, sign. If they are positive, and if, therefore, E lies 

 behind F, and F* behind E* t the system is called a convergent 

 one ; it acts as a convergent lens and produces real images. In the 



c 



