[ 3^ 3 



IV. A Supplement to the Third Pari of the Paper en the Sum- 



matlon of hifinite Series, in the PJiilofophical Tranfadions 



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



communicated by Nevil MalkeJvne, D. D. F. R. S. and 



Aftronomer Royal, 



Read November 25, 1784. 



^"^HE reafoning in the third part of my paper on the Sum- 

 ^ mation of infinite Series having been midinderftood, 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 - * + 5 -^ hcz, fine 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 mufl 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 laft 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 mufl 

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

 — i + H. L. 2. If therefore in the folution of any problem, 

 the conclufion, whofe value 1 want, is exprefled by the above 

 feries, and which arofe from the necefhty of expanding fome 

 quantity in the preceding part of the operation, furely no one 

 can deny but that I may fubflitute for it - | + H. L. 2. For 

 whatever quantity it was, which by its expanfion produced at 



iirfl 



