On certain Properties of Numbers, by the REV. SAMUEL 

 VINCE, A M. F. R. S. and Plumian Professor of Astro- 

 nomy, in the University of Cambridge. An extract of a letter 

 to the Rev J. Brinkky, D. D. F. R. S. M. R. I. A. and An- 

 drews' Professor of Astronomy in the University of Dublin. 



Ramsgate, June 26, 1810. 



EuLER in his Introductio in Analysim Infnitorum, in the 

 chapter de partitione Numerorum, has shown, that bj a com- 

 bination of the numbers in each of the Geometric Series 1, 2, 

 4, 8, &c. and 1, 3, 9, 27, &c. all the natural numbers 1, 2, 3, 4, 

 &c. may be formed, as far as the sum of each series goes. This 

 he has proved, from assuming the products of an indefinite 



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number of factors(l+jr )(l + a' )(1+^ )^l + x ) &c. in the 



first instance, and( 0? +1+.T ){^ t + I +x j(^ x +1+0:' ^&c. 

 in the second; shewing that in each case, such products may 



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be represented by a series containing the terms i +x+x +x 



4 



+x + &c. the indices of which must necessarily arise from 

 the combination of the indices in the assumed factors. But 



