C 757 ] 
CIII. The Invention of a General Method for 
determining the Sum of every 2 d> 3 4/$, 
or $th. See. Ter?n of a Series , taken in 
order ; the Sum of the whole Series being 
known . By Thomas Simpfon, I\ R. S. 
Read Nov. 16, /k S the doctrine of Series’ is of very 
TV great ufe in the higher branches 
of the mathematics, and their application to nature, 
every attempt tending to extend that dodtrine may 
juftly merit fome degree of regard. The fubjedt of 
the paper, which I have now the honour to lay before 
the Society, will be found an improvement of fome 
confequence in that part of fcience. And how far 
the bufinefs of finding fluents may, in fome cafes, 
be facilitated thereby, will appear from the examples 
fubjoined, in illuftration of the general method here 
delivered.. 
The feries propounded, whofe fum (S) is fup- 
pofed to be given (either in algebraic terms, or by 
the meafures of angles and ratio’s, &c.) I fhall here 
reprefent by a bx -f- <rx z -|- + e &c. and 
jfhall firfl; give the folution of that cafe, where every 
third term is required to be taken, or where the feries 
to be fummed is a 4- dx l T g * 6 -j- kx 6 y &c. By 
means whereof, the general method of proceeding, 
and the refolution of every other cafe, will appear 
evident. 
Here, then, every third term being required to be 
taken, let the feries [a 4- dx l 4- &c.) } whofe 
value 
