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Colorado College Studies. 



Euler), is noteworthy as containing only the symbols B, r 

 and a with no auxiliary quantities. The data of this and the 

 preceding paragraph afford, of course, the material for writing 

 similar equations for polygons of 2" or 3 X 2" sides, if, instead of 

 introducing as known quantities in the equations the elements 

 of previously solved problems, we should eliminate all such 

 quantities. For polygons not belonging to these groups an in- 

 dependent investigation would be necessary for every polygon 



of J: sides (where A- is an odd num- 

 ber) in order to apply the preced- 

 ing method to extend the result to 

 k X 2" sides. Equations between 

 B, r and a had been obtained before 

 the date of Jacobi's memoir, cited 

 at the beginning of the present 

 paper, by Nicolaus Fuss and by J. 

 Steiner, the former of whom ob- 

 tained formulae applicable to each 

 of the polygons of sides not greater 

 in number than eight, and the general method indicated by 

 Jacobi has since been applied to polygons of still higher num- 

 bers of sides by F. J. Kichelot and others; but, so far as I know, 

 without appending methods of geometrical construction. * 



It is especially to be noticed, in the problems of the two pre- 

 ceeding paragraphs, that when an initial vertex has been 

 chosen, the remaining vertices are thereby fixed, not only for 

 that polygon, but for all that are successively derived from it by 

 doubling the number of sides; and so that if, in passing in a 

 determinate direction around the polygon, the number of sides 

 between the two vertices is to the whole number as m to n, this 

 holds true for the same two points in all the polygons subse- 

 quently derived. These two points, then, always have between 



* The memoirs of Ricbelot are to be found in Crelle's Journal vols. 5 and 38; there 

 is also an article in vol. 81 of the same Journal (1875), by Max Simon ; the latter 

 paper succeeding a Latin dissertation by the same author which appeared in 1867. 

 These writers have made extensions of the problem in several directions. 



