274 Mr. E. Howard Smart on the Third-Order 



So the last equation takes the final form 



This is the recurrence formula for the longitudinal 

 aberration of the successive images P z , P»+i. 



To sum this throughout the system we require first relations 

 between successive £ ? s and 77's, the coordinates of the points 

 at which the ray strikes the successive surfaces. 



The condition that the refracted ray passes through the 

 point (f i+1 , r}i+.v ^i+?;+i)> where d i is the distance between 

 the ith and (z + l)th surfaces, is 



fc+i-R _ Vi+i-Vi _< + S+i-S 



In approximating to this we need only retain first-order 

 quantities ; so remembering that 



Bjs.., &/-,! As/ £ , £.,, 



are all of the second order, we get from the equality of the 

 first and third 



f- t 1 1 ^A &i+idi 



Making use of (6) this may be written symmetrically 



(10) 



*»+l 



— ~ j 



since 



*i=8i+i + di 



Similarly 





^ i+ iV i+1 



^-Wi Pi-AViVi+i 



S i+l S t $#j +] 



