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IV. A Supplement to the Third Part of the Paper on the Sum* 



mation of infinite Series^ in the Philofopliical Tranfa6tions 



for the Tear 1782. By the Rev, S. Vince, M ^. ; 



communicated by N^evil Maikelyne, D. D. F. R, S, and 



AJironomer RoyaL 



Head November 25, 1 784. 



1'^HE reafoning In the third part of my paper on the Sum- 

 . mation of infinite Series having been mifunderftood, I 

 have thought it proper to offer to the Royal Society the follow- 

 ing explanation. When I propofed, for example, to fum the 

 feries I - 4^+ I -" Sec, Jine fine, I wanted to find fome quantity 

 which, by its expanfion, would produce that feries, and that 

 quantity I called its fum ; not (as I conceived muft have been 

 evident to every one) in the common acceptation of that word, 

 that the more terms we take, the more nearly we fhould 

 approach to that quantity, and at lafl arrive nearer to it than. 

 by any affignable difference, for there manifeflly can be no fuch 

 quantity ; but as being a quantity from which the feries muft 

 have beeti deduced by expanfion, which quantity I found to be 

 — I +H. L. 2. If therefore in the folution of ^ny problem, 

 the conclufion, whofe value I want, is exprefled by the above 

 feries, and which arofe from the neceflity of expanding fome 

 <juantity in the preceding part of the operation, furely no one 

 can deny but that I may fubflitute for it - i -f H. L. 2. For 

 ^whatever quantity it was? which by its expanfion produced at 



firfl 



