160 



Transactions of the Society. 



many written during the aperture controversy, and its object was to 

 prove that the aperture of 180° in air could be exceeded by an oil- 

 immersion lens. All who remember that controversy and the bitter- 

 ness engendered by it, will be thankful that it has passed away never 

 to return ; it was, however, not without its use, for it was the means 

 of disseminating a large amount of optical literature, which un- 

 doubtedly enlarged the views of microscopists at that time, and 

 enabled them to form correct ideas about the nature of the instru- 

 ment they were using. 



The aplanatism of a single refracting surface was first investi- 

 gated by Descartes, who described the various curves of a surface 

 that would refract light aplanatically ; but the proof of the proposition 

 appears to have been first published by Newton, in his ' Frincipia ' ; 

 for he says, " The invention of which, since Descartes concealed it, 

 I have thought proper to lay open in this proposition." 



These refracting surfaces were curves consisting of ellipses, hyper- 

 bolas, parabolas, and Cartesian ovals ; but all these have disappeared 

 from practical optics, with the exception of the parabola, which is still 

 used in Newtonian reflectors ; it is, however, with glass lenses and not 

 with reflectors that we are at this moment concerned. Under certain 

 varying conditions of the distances of the point and its image from the 

 vertex of the curve of the glass refracting surface, the Cartesian oval 

 becomes more and more like a spherical curve, until a definite position 



of the point and its image is 

 arrived at, when the curve be- 

 comes truly spherical. This 

 position is, when the distance 

 of the virtual image from the 

 vertex of the refracting surface 

 is /n times that of the object 

 from the vertex. In fig. 40, let 

 A be the vertex, O the object, 

 and V the virtual image, then 

 the condition for aplanatism by 

 a spherical surface is that A V 

 must be equal to /n A ; for simplicity, call A 0, p, and A V, q, 

 then 



Q = F- V- 



(*•) 



Call the semi-angle of aperture, viz. C H, 6, and the angle CVH, 

 <f>, and the radii C A, C H, r. 



Now p, q, and r are related by the common formula for refraction 

 at a spherical surface, viz. 



1 

 </ 



V 



h 



- 1 



(ii.) 



