58 



PR R. A. SAMPSON ON A 



the solutions of which are 



4, = +2-390547, q, ( = + 4'936038, 



e u = +'164675. 



. (33A) 



1C with these values of q 4 , q* we calculate the curvatures of the two lenses from the 

 t'>rmul>u (24) of p. 49, we find that the completed approximation directs us to replace 

 the numbers of p. 54 from which we set out by 



B 4 = -'693009, B' 4 = + 2'048193, 



B, = -'024163, B' = -2-612025, (34) 



together with the value of e just written down. 



Turning back to p. 49, where these data from a previous approximation are set 

 down, we see that the chief effect of the step is to restore B' g to the value given on 

 p. 49, throwing the change in focal length which is demanded for achromatism, in 

 accordance with p. 53, almost exclusively upon B 6 , which is a surface that contributes 

 very little to these aljerrations. The changes are thus in reality smaller than they 

 appear. Following now strictly the plan given on p. 54, the next step is to take 

 the new system as a whole and calculate exactly its numerical aberrations at its 

 principal focus; it is unnecessary to give the details of this step, which contains 

 nothing new ; the following numbers show first the normal schemes from the surface 

 O a up to each other point, and then the contribution of each surface to each of the 

 coefficients <5,G.. .<S 3 H at the principal focus F' 7 . 



Surface O 



O a to O 2 

 O tf to O a and surface O 2 



to O' a 

 O 9 to O', and surface 0' 2 



to O", 

 B to O", and surface 0", 



O to O 4 

 O, to O 4 and surface O 4 



O, to 0' 4 

 O, to O' 4 and surface O' 4 



O toO 8 

 O to () and surface O, 



to O', 

 O. to O 7 . and surface 0'. 



to F- 



