38 



MEMOmS OF THE NATIONAL ACADEMY OF SCIENCES. 



I.— GENERAL EQUATIONS. 



P. 



FiCfl 



Let p, Pi and p' be I'espectively tiie geometrical centers of the lens surface, the incident wave 

 surface, and the refracted wave surface; also let ;/, c' andc'i be the curvatures of these surfaces in 

 order, a positive curvature corresponding to the case where the center of curvature is in the direc- 

 tion of propagation of light from the surface. Let x, x' and ,r, be the sagittas of these surfaces in 

 order named, corresponding to the common semicord y; then for small values of y we have 



itr. 



x' = ii/^C', 



iy^o.. 



From the laws of wave propagation we have, if p represents the ratio of the velocity in the 

 medium to the right of the surface ;/ to that in the medium to the left, 



x-x, = p {.r-x'). 



Substituting the above values of x, x' x, in this equation, eliminating the common factor hf, 

 and solving for c, we have 



ci = r (i-p) +pc'. 



This wave surface will be propagated with uniform velocity and uniformly decreasing radius 

 until it reaches the second refracting' surface at a distance t, from the first, when its radius is re^ 

 duced fi'om — to — t, and therefore its curvature to the reciprocal of this, or to 



if we define /t' by this equation. After refraction at this second surtace, the curvature will be 

 according to exactly the same reasoning as that applied to the first surface. 



c-2 = y' (1— P') + /''/•>''■/• 

 This completes the general solution, and it can be extended to any numb 



C;l 



r (1— p) + /'/Jt'' 

 r" (i-p") + /'"p" 



/'' = (1- 



l" =: (1- 



)f refractions, thus 



(«) 



-1 (l-/.^->) 



.= (1- 





