14 THEORY OF THE MICROSCOPE. 



which the incident ray describes, and (, 77 its co-ordinates, we have 

 then, according to equation (1), 



^ 

 or, if we substitute for /3 and W their values from (11), 



77 = 

 consequently 



7* ~ 9 ~ 



If we put this value in the equation (7), which determines the 



direction of the ray after the second and last refraction, then the 



equation, if the terms multiplied by /3* are placed together, will be 



/P* 



</!/ """* I -i-V ^~ rt _ _ . i T-/\-v '^ 



* 

 4ind if we write by way of abbreviation 



** - ^-=4^ -iy. = r 



(13) 



= r,* 



the equation will then take the form 



3& i * / f*$fc\ 



Therefore, on the last path of the ray there must of necessity ^ie 

 ii point P*, whose co-ordinates are f * and 77* ; for, if in the above 

 equation we take x = (*, then y 77*. Since, then, f * and 77* are 

 determined solely through { and 77, in combination with the optical 

 constants entering into g, k, k, I, and are not dependent upon the 

 quantities /3 and &, it follows that every ray, which on its first 

 path passes through the point P, must on its last path pass through 

 the point P*. In other words, to the incident rays whose lines of 

 direction intersect in the point P there correspond emergent rays 

 which (if necessary produced) intersect at P*. The point P may, 

 therefore, be fegarded as the object, and P* as its optical image; the 

 object is, however, real only when P lies in the first medium, there- 

 fore if f N is negative. Similarly, the image can be real only 



