CHROMATIC ABERRATION. 51 



fore, in dispersive power, in the proportion of more than two 

 to one. 



Kemembering that concave and convex lenses act with oppo- 

 site effects, it is evident that the dispersion of colour produced 

 by a convex lens of crown may be completely neutralized by the 

 addition of a concave one of flint, without thereby cancelling the 

 deviation of the rays. For if the deviation were equal but opposite 

 in both lenses, the focal lengths therefore being equal, the flint 

 having dispersive power twice as strong, would not only neutralize 

 the colour-dispersion of the crown, but would produce an opposite 

 effect in about the same degree. A ratio between the focal 

 lengths may therefore be found, which would render the two 

 lenses achromatic for the extreme rays, whilst still possessing the 

 properties of a convex lens. 



If we denote the focal lengths of the flint-lens for red and violet 

 rays by F' r and F ', and the corresponding focal lengths of the 

 crown by F" r and F" v , the condition of achromatism is 



-L JL J l 



F' r F\ ~ F' v F\' 



If the flint is plano-concave and the crown bi-convex, and if the 

 radii of the three spherical surfaces are each = R, as we have 

 assumed, on substituting for the focal lengths their values, the 

 above equation becomes 



- (n'r- 1) -g + (n" r - 1) A = - (n',-1) ~ + , 1) ~ 



in which n' r , n' v , n" r , n" v are the refractive indices corresponding 

 to the focal lengths similarly denoted. By multiplication of all 

 the terms by R we obtain 



2 (n\ n" r 



that is, the dispersive power of the flint must be double that of the 

 crown if a double-lens constructed on the above assumption is to 

 be achromatic. As soon as 



2 (n\ - n" r ) > ri v - n' r) 



the influence of the crown, and in the converse case that of the 

 flint, preponderates. 



With a double-lens a perfect union of rays of different refrangi- 



E 2 



