DETERMINATION OF THE PATH OF KAYS. 21 



whilst we found above [equation (4)] 



/v* v 

 ft * - J + -T,'--^' 



The analogy is, therefore, apparent. As, however, besides the re- 

 fraction upon which b* is alone dependent, there also comes into 

 account the displacement of the rays from one principal plane to 

 the other, then, with respect to the position of the emergent to 

 the incident ray, the first principal point J corresponds to the 

 anterior lens-vertex, and the last /' to the posterior. If, therefore, 

 in the preceding, E and E* t in their position to N and N*, are 

 determined by the equations 



the principal points of the resulting system, denoted by E and 

 will be given by 



E = E - l " l , E* = /' + l ~- ff . 



/J K 



The same holds good also for the focal points. We have 

 F = E + , F* = /' - . 



SUMMARY OF THE RESULTS. 



It is probable that many who are not able to follow readily the 

 preceding analytical explanation will wish us to summarize the 

 most important of its results. 



The quantities which enter into the formulae for the determina- 

 tion of the principal and focal points, and which evidently remain 

 constant for a given system, are #, /, fc ; these are, in their turn, 

 dependent upon two series of other quantities, viz., w, u', u" . . . . 



and ', r . . . . , in which denotes the focal length (taken as 



negative) of an infinitely -thin lens, which would produce a devia- 

 tion equivalent to the first refraction, and u' t u" have the same 

 signification for the second and third refraction, while t' t t" . . . 

 denote the distances of the imaginary equivalent lenses. Hence it 



