20 THEORY OF THE MICROSCOPE. 



assumed refraction in the one case, and after it in the other. The 

 deviation caused by the latter is, therefore, the same as that which 

 would be brought about by an infinitely-thin leni bounded on both 

 sides by air (for which n = 1) and whose focal length = /. 



These ratios supply us with a ready means of combining any 

 two systems of optical cardinal points just as easily as two refract- 

 ing surfaces ; for, by virtue of the equations already given, it is 

 allowable to reduce all the refractions to the case of infinitely-thin 

 lenses, and to take the quantities U Q and u, which appear in the 

 formulae for ft and ft", as equal to the reciprocal focal lengths of 

 those lenses, taken as negative. 



If, for instance, E and 7 are the principal points of a lens, 

 whose (calculated) focal length = /, and E, I' those of another 

 lens, whose focal length is /', then, in accordance with the pro- 

 perties of principal points, a ray which passes through both will be 

 refracted as if in 7 and E' were situated infinitely -thin lenses with 

 the same focal lengths. The distance of the points 7 and E' has, 



j\T* _ N Q 

 therefore , the same signification as 7 = t' has in the 



case of a single lens. This also follows directly from the equations 

 for the ray before and after refraction, referred to the principal 

 points. For its direction before refraction we have 



y = (x - T ) + 1; 

 after the first refraction 



y = p (x - 7) + 6, 

 or, referred to E\ 



y = ff (x - E') + &*; 



after the second refraction 



?/ = 0* ( - 70 + ft* 



From the middle equations, making the expressions on the right- 

 hand side equal, results 



7,* = fc + (E' - 7) p, 



