a » . 



i . Mr. A.aams uu w«c [^: 



no intention to appropriate to himself the credit of my invention, 

 but I sincerely regret that, before promulgating the modifica- 

 tion of my method, he had not consulted me on the subject. 

 This would have prevented all chance of misunderstanding 

 between me and Dr. Henry, whose accomplishments as a gentle- 

 man and a chemist, I have been accustomed to admire. The 

 readers of the Dictionary will perceive, under the articles Cal- 

 culi, Coal Gas, Gas, Salt, &c. that I have not suffered temper to 

 influence my judgment, but have done merited honour to the 

 Doctor's researches on every scientific occasion. 

 I have the honour to be, Sir, 



Your most obedient servant, 



Andrew Ure. 



Article IV. 



On the Finite Values of Circulating Decimals. 

 By Mr. James Adams. 



(To the Editor of the Annals of Philosophy.) 



SIR, Stonchousc, near Plymouth, May 8, 1821. 



Problem 1 . — To find the sum of n terms of a geometrical pro- 

 gression. 



Let A, B, C, D, E, be any series in continued proportion, 

 then will A : B :: B : C :: C : D :: D : E ; where A, B, C, D, or all 

 the terms, except the last, are antecedents ; and B, C, D, E, or 

 all the terms, except the first, are consequents. Put S equal to 

 the sum of all the terms, then will S — E = sum of all the 

 antecedents, and S — A = sum of all the consequents, whence 

 (12 . e . 5) A : B :: S - E : S - A ; therefore, A S - A* = 



B S —BE, from hence S = _ . 



Corollaru. — In a decreasing series continued in infinitum, the 

 last term E vanishes, and the sum of the whole series is S = 



A-B* 



Problem 2. — To find the finite value of any pure circulate. 



•3 — JL j. _£ + — -I - t- &c Or 



* ~ io + ioo ^ iooo + ioooo + °"" W > 



** =w.j'+(^)'+(^r+(iV)'+M 



