Infinite Series. 155 
x 2 ‘xf'xr + j' ' + &c., and denominator (r’ + ; and 
the co-efficient of a term t x <?“+i will be a fraction whofe 
m-Y 1 
numerator is 2/ x (« + i 
-* — I — — 771 2 W 
r- + 1 +m- j- 3 
r*+f 
Z» -f 
m — i 
771 1 ,z t 
. ™ x 2 X / X 
x 2V x r a + V 
2 3 4 5 
&c.) and denominator (/ + r) i " ,+2 ; » being a whole number. 
The continuation of thefe feriefes is too evident to need enun- 
ciation, as mu ft generally be the cafe when the number of 
factors contained in the fucceffive terms continues tne fame or 
increafes in arithmetical progreflion ; and the fa&ors them- 
felves increafe or diminifh in an arithmetical progreflion : the 
terms proceed alternately + and — by pairs. 
2. 2. Let the given fluxion be r , + 
2 te -Ye 1 xe 
[lie- Ye 2 yxe __ « 
2 te 
r z + t z {r’- + ? ) : 
r z + t L ~ (r z +?) 
{r z + t z — z z t l )i 
JPfiTy 
+ 
X £ + See. 
t 5 ] 3 x e ~ 
(Q) ; this feries becomes the fame as the preceding by fubfti- 
tuting in it e fort; in this feries e is confidered as variable, 
in the preceding t. 
The fluent of the former feries (P) 15 J - + ^ 
* ~ _ t r ~ + / _Z 2 l*J x - &c. : the fluent of the latter 
feries (Q) is the fame as of the former (P) except the ftrft 
term J'- *— : the co-efficient of the term e h in the fluents will 
be the fame as the co-efficient of the term e h ~ l in the feriefes 
(P) and (QJ divided by h. 
This feries, when the tangent of a given arc is known, finds 
the arc whofe tangent differs very little from the given tangent. 
ri. Let t be increafed or diminifhed by a quantity e in 
(P), where e bears a very fmall ratio to any root or value of t in 
Y" 2 the 
