[385 J 
AXVII. On the Summation of Series, whofe general Term is a 
determinate Function of z the Diflance from the firft Term of 
the Series. By Edward Waring, M. D. Lucafian Profeffor 
of the Mathematics at Cambridge, and Fellow of the Societies 
_ ¥ London and Bononia. 
Read May 20, 1784. 
see PR O| eB Loe M. 
| TA E fum 3 being given, to find a feries of which it is the 
UM 
1. Reduce the fum S into a converging feries, proceeding 
according to the dimenfions of any {mall quantities, and it is 
done. For example: let any algebraical fun@ion,S of an un- 
known or {mall quantity x be affumed, reduce it into a con- 
verging feries proceeding according to the dimenfions of x, and 
there refults a feries whofe fumis S.. 2. Let A, B, C, &c. be 
algebraical functions of «3 reduce the f Ax, f. Bx, f Cx, &c. 
into a’converging feries, proceeding according to the dimen- 
fions of «, and the problem is done. 
It is always neceflary to find the values of +, between which 
the abovementioned ferictes converge. Reduce the algebraical 
‘function S in the firft example, and the algebraical functions 
A, B, C, &c. in the fecond into their loweft terms; and in fuch 
‘a-manner, that the quantities contained in the numerator 
and denominator may have no denominator: make the deno- 
Vor. LXXIV. Eee minator 
