40 MEMOIES OF THE NATIONAL ACADEMY OF SCIENCES. 



The product of these equations is, remembering that the product of all the p's is p^: 



The value of the ftictor //'^ I is easily changed by substituting successively the value of 

 /j^ and of Ck given in (a), to the form 



,H=;.-I 1 ..>-. 1 



[1— Ca U !ffA_i+/ix-i /^''-^Ca-i, 



in which (Va_i and (h-i are constants depending solely upon the physical constants of the system 

 and not at all upon the value of any of the wave curvatures. If we make another substitution of 

 the values of yu^-i and of Ca_i in this last equation we have, 



A I A 2I 1 



I 1 'oa_2+/3a-2 /'^-'Ca-2 



If we repeat this pi'ocess of substitution A times we shall have, finally 



1 = 1 

 ^1 « + /ic' 



in which a and /i are determinable functions of the ;/'.«, the fs and the p's — that is, of the physical 

 constants of the system, and the ratio of the ultimate image to the object, or the magnification, 

 becomes 



°y=^- ..,.,;+,.) <"' 



There is, however, another very important expression for the magnification which may be de- 

 rived as follows: 



Suppose we have a diaphragm at the first surface with the small diameter 2«. Let the semi- 

 angular diameters of the wave surfaces, incident and refracted, thus limited be designated by iJ 

 and coi, and so on for the successive refracting surfaces; then 



('•) 



