282 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 



is true accurately, and since when the objects are small, the denominators are 

 nearly 2.4,5, and 2A t B t , the proposition is proved approximately true. 



Using the expressions of Pi-op. III., this equation becomes 



1 xu 

 *- = *-/ 



c i C 



are 



Now by Pn^ IIL, when x and y are different, the focal lengths /, and f, 



f- J_ 

 ' J '~ C '-' 



x-y y-x 



therefore ^ = -^ = - by the present theorem. 



f, c t ^ 



So that in any instrument, not a telescope, the focal lengths are directly as 

 the indices of refraction of the media to which they belong. If, as in most 

 cases, these media are the same, then the two focal distances are equal. 



When x = y, the instrument becomes a telescope, and we have, by Prop. V., 



s* 



I = x, and = - ; and therefore by this theorem 



/"* 



We may find ? experimentally by measuring the actual diameter of the 

 image of a known near object, such as the aperture of the object glass. If O be 

 the diameter of the aperture and o that of the circle of light at the eye-hole 

 (which is its image), then 



From this we find the elongation and the angular magnifying power 



ii u, 1 



n = r, and m = - T . 

 /*, frl 



When /*, = /*,, as in ordinary cases, m = y= , which is Gauss' rule for deter- 



t 



mining the magnifying power of a telescope. 



