196 Mr. Blackburn's Analjjtical Theorems 



time been the object of my experiments, and for an account 

 of some of the results of which I may at a future day beg a 

 place in this Journal. 



The second part of this paper, containing the application 

 of the above principles to the analysis of minerals containing 

 manganese and iron, and an examination of those methods 

 now used, will appear in a future Number. 



28, Golden Square, London, December 12, 1834. 



XXXIII. Analytical Them-ems relating to Geometrical Series. 

 By Charles Blackburn, A.B, 



To the Editors of the Philosophical Magazine and JourJial. 

 Gentlemen, 



T^HE following properties of numbers have not been noticed 

 -■- in any work of Algebra that I am aware of. If on inspec- 

 tion you find them to be new, you may think them sufficiently 

 curious for insertion in your Philosophical Journal, 



It seems not impossible that they may admit of some useful 

 application in analysis; but at any rate they may be of service 

 to persons entering on the study of algebra, or even to per- 

 sons engaged in tuition, from the inexhaustible fund of ex- 

 amples the formulae will supply. They are exhibited in the 

 forms they assume when r is negative as well as positive ; and 

 when the number of terms is even as well as odd. Three of 

 them have already appeared before the public, but without any 

 demonstration; they are here inserted with the investigations. 

 Kensington, Feb. 23, 1835. 



Theorem I. 



1 . S'*^ 2.2'* (W-1)2" 



^'■^ l+r + r^ r»-' =0-^ + ^^ »• ' 



x{l-»- + r* +/"-')n 



X {i_y+X~.'... rC"-')*""'} "" beinsrodd. 



Let l+r^%'-^-^" + rC«-')2''=S, 



't> 



1+ r + r^ + r = s; 



then, since each of these is a geometric series, we have 



n 

 ?h2 



S _ ('•'"-!) (>•-■) (a). 

 's (r"'-l){r'"-X) 



