42 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



paper on the limit of power iu microscopes, and was proved very indirectly by Helmlioltz in his 

 pai^er on the same subject as a consequence of the second law of therinodyuamics. I am not aware 

 that any proof purely optical has heretofore been given. 



Ill — SIMPLIFICATION OF GENEEAL EQUATIONS! («). 



In the discussion to this point we have made c' the curvature of the incident waves at the first 

 vertex of the system and CA+i,the curvature of the finally refracted waves at the last vertex. We 

 will now seek the relation of the curvatures for incident, and finally refracted wave surfaces at 

 points situated, respectively, at a distance.ro from the first vertex and.Ci from the last. 



Let us suppose that the incident wave surfaces are bounded by a circular diaphram of radius 



a° at Xq, then the finally refracted wave surfaces will be also bounded and will have a definite 



semi-diameter at X\, which we shall designate by a°i; then, if C and C\ represent the wave 



curvatures at the points Xq and x^ for incident and finally refracted waves, respectively, we have 



the following relations: 



6" 

 sin od' = a° C c'= = — ^, — 



1+G'Xo 



Substituting these values in (c) we have 



= Po 



By replacing the left member of this equation by its value given in (b) and in the latter the values 

 above for & and Ca^,, we have 



whence the value of Ci is given by the equation 



«o _-■ 



^~" ax°+ft aP 



If 

 since a°i is simply the image of a°, and the expression for d becomes greatly simplified. We will 



call the value of *^for any such case k. .This valuecan be found by substituting jind — fore' 



rtPj Xq Xi 



and (^^A+i in equation {b), whence, remembering that 0° corresponds to o', we have 



a°i_l_ Xi 



ao~k~'^°aXo+P' 

 We have also the following relations: When C'=o c'==o also, that is, if the incident waves are flat 

 when they reach Xq, they are also flat at the first vertex. Call c°a+] the xaluc of the finally 

 refracted waves as computed from the equations {a) under the supposition that e'—o; then 



substituting this and the value , the corresponding value of Ci,in the above 'equation cou- 



necting ^a+i and Ci we find 



fc«— 1 , 



Xi +' 



Substituting these expressions in the general equation we have this remarkably sini])l(' expression 

 to replace equations (a) — 



Ci=A-«'cOa+i + PoFC:,". ((7) 



Before discussing the methods for finding tlie contents Jc, a, and c°a+i in this general equation we 

 will first establish expressions for the magnification and for the change in direction of the wave 

 surface in ]irogressing from Jq to Xi. 



