Infinite Series. 159 
Prom fimilar principles may be found tbe tangent of tne 
. r , ■ r 1 r i t + txrxb 
arc required from the equation — /*_ • — 
where the tangent t to rad. 1 of the arc L m = A is given. Let the 
fluent of the fluxion ( = A + B/ + Cr + IX ! + otc. ; 
and 
tA- i X b 
V ( 1 + * ) 
A' + B 7 + C 7 2 ]+ D 7 '’ + &c., the equation 
will become / + - S . “ — — x *’ + &c- — 
oa. 
:r A' 
= tt ; 
.•BzJrB'' 1 rBrUB' " ' ‘ (rUSdrrb'J 
from this equation inveftigate /= tt -f Ptt 2 + Qtt j 4- &c., and 
thence / + / the tangent required. 
In like manner may be found the fecant, cofine, ccc. or the 
arc required. 
14. The fame principles may be applied to cut an area de- 
feribed round any given point in a given curve equal to an 
area a. 
Let x be the abfeifs and y the ordinate of the given curve, and 
b the diftance of the beginning of the abfeifs from the given 
points, and let (A) be the area of the curve deferibed round the 
pointS, when the abfeifs is *, which differs very little from 
the given area (fi) > to find the value x J ?e oi the abfeifs, when 
the area = a. 
Let jy=X a funaion of x, and in X for x write x + e, and 
reduce the refulting quantity (X 7 ) into a feries X + Be + Ce + 
T)e 3 +'&c. proceeding according to the dimenfions of e ; then 
will the area J'yx— J'^x + e J'Bx + e~ J'Cx + e J'Dx + Szc. — 
A + he + ke* + le* + &c., and confequently A + be + he* + /e 3 +' 
&c. =t (bz*= 7 +e) x l = A + he + ke z + le l + &c. + 1 (b ztAf+Vj x ; 
"(X + Be + Ce z + D^ 3 + &c.) = A + i (J> — **) JtX+p+I^*^ 
B 
