CASSEORAIN KKFLKCTOR WITH CORRECTED FIELD. 35 



we have 



**', V* = J.G = 0, 

 J,H = Sfi + Sfi = -)>->' = -#, A.H = J,H = 0, 



where E is the sum of terms in e, e' for each of the lenses ; 



^,L = 4,K-K$ = K'-K'P-i.G, 



J,L = ^K-9 = K, ^ 3 L = ............. (8) 



Thus, to form the coefficients cS,G, ... for any system of thin lenses in contact, we 

 require to know only the forms for ^,G and <5,K. I add the forms of these for three 

 lenses, 



-k") tf + l(2k+2V + U')(k + V)tf'. . (9) 



From these, if necessary, the general case may be written down by analogy without 

 much difficulty, e.g., in (5,K the coefficient of Jj>'</' ' 8 three times the k of the 

 preceding system minus the X: of the following system ; but I shall not require more 

 than three. 



We may employ these equations where we require to obtain algebraically rough 

 but reliable indications of the properties of a given actual system. Thus, consider 

 the aberrations of any set of thin lenses in contact, at their principal focus, that is, at a 

 distance K" 1 beyond their common surfaces. We must form S { T = <5,G K~M,K, ... 

 where ^G, ... are the quantities just found which refer to the surfaces of the lenses as 

 origins. Hence for example, referring to p. 30, we see that the radius of the focal 



F 2 



