123 





the two first, for example, becoming 



0=a(^-^ + ^); (H'") 



which contain under a convenient form, the known theorems for the 

 change of the central focus, and of the principal coefficient of aber- 

 ration, by refraction of a spheric surface ; r being the radius of this 

 surface, and being considered as positive or negative, according as the 

 convexity or concavity is turned towards the incident rays. 



If, for instance, we consider an infinitely thin lens in vacuo, hav- 

 ing |M. for its refractive index, and having r, r', for the radii of its two 

 spheric surfaces, (positive when those surfaces are convex towards 

 the incident rays,^ we may take the point of central incidence for 

 origin, and the equation (H'") will become, 



~ A' A + r ' ^ A" A' ^ r- ' ^/^ ^ 



— A, — A', — A", being the ordinates of the central focus in the 

 three successive states of the system ; and similarly, (I'") will give 



- ^t£ _ ^ (JL. J_\ (^ i_^ 4. ^—^ • 1 



A'* A* ~~ \A'^ A' J \rA" rA'J "^ »^ ' ^ 



(L'") 



2i^' 2^' /_1_ j^\ _ /_2_ _2f^\ 1_-^ ( 



L, L' , L" , being the three successive values of the principal coeffi- 



