U 8r ] 
IV. On Infinite Series . By Edward Waring, M. D, JF\ R* S« 
Lucafian Profeffor of Mathematics at Cambridge. 
Read December 15, 1785. 
i.TN the Paper, which the Royal Society did me the honour 
JL to print, on Summation of Series, is given a method of 
finding the fum of a feries, whofe general term ^ (where 
p 
^ is a fra&ion reduced to its lowed: terms) is a determinate al- 
gebraical function of the quantity (%) the diftance from thefirft 
term of the feries, which always terminates when the fum of 
the feries can be expreffed in finite terms. 
2. The terms of every infinite feries mufl neceffarily be 
given by a function of s, or by quantities which can be re- 
duced to a function of s. 
3. Let Q = AxA 7 x A /z x . . A /fl x B x B' x B /2 x . . .B /w xC 
C'xC /2 x . . C /r x&c. where A 7 , A /2 , A /3 . . . h'% are fuc- 
ceflive values of A ; that is, refult from A by writing in it for 
% refpe£tively % + 1, z + 2, 2 + 3, . . %-) r n ; and B', B / • % B' * 3 , 
B lw , refult from B, by writing in it for % refpe&ively % + 1 , 
ss 4* 2, 3,. . . ,z + m ; but Bis not a fucceffive value of A ; &c. 
Let the numerator P = E . E' . E /2 . . E 1 ^ 1 . F . F / . F /z . . . F r • 
xL; E% £ 12 . . E 1 *- 1 $ F 7 , F 1 - 2 . . F 1 *- 1 , &c. denoting fuc- 
ceffive values of the quantities E, F, &c. refpe&ively ; and L, 
admitting of no divifor of the formula K x K/, where R- is.afuc- 
V ol« LXXVI, M # ceffive 
