ACADEMY OF SCIENCES. 279 



in terms of those diameters, the consecutive diameters of the circles 

 ■which will subdivide the ring into a given number of circular rings 

 of equal area with each other. 



Let d' represent the outer, and d the inner diameters of the 

 given circular ring ; w, the number of circular rings, x the first 

 diameter interior to d', y the second, and (iv-V) and iv the last 

 two ; then, 



x^ = 1— i '. . yi = L_i '. ; etc. 



11 n 



, .^ (»-2) (/2 + 2fr2 . (n-l) (12 + d'2 



n n 



III. Having subdivided the circular ring as in problem II, de- 

 termine, in terms of the two given diameters, the consecutive diam- 

 eters of the required circular rings inside the given ring and 

 having equal areas Avith the prescribed subdivisions. 



Let d\ d, x,y, etc., represent quantities as before, i', i'\ i"\ etc., 

 the consecutive diameters of the inner circular rings, and suppose 

 the given circular ring is divided into p circular rings ; and there 

 are required n inner circular rings, then, 



,2 ^ {n + p) fZ2 _ nJ'2 



p 



IV. Having subdivided the circular ring as in problem II, de- 

 termine, in terms of the given diameters, the consecutive diameters 

 of the circular rings outside the given ring, and having equal areas 

 with the prescribed subdivisions. 



Let d', d, X, ?/, etc., represent quantities as before, o', o'\ o"\ 

 etc., the consecutive diameters of the outer circular rings, and 

 suppose the given circular ring is divided into -p circular rings ; 

 and there are required n outer circular rings, then, 



2 _ (" + /') f/'S — «f/2 

 n '7, 



