*57 
Infinite Series . 
r@xx 
(*”-h a 
+ r . x 
XX 
P; m 
'xx 
r+1 r -f 2 
(*‘+fl'y +l -r ’ ~ * ~ (*”+/) r ^ 
+ 
»y+ i ■ 2 
&c., where the quantity of -\-a n is increafed by a fmall quan- 
tity 13 ; then will ( ~ ~ * ~=J~-^- r ( -m + i 
( ■* 4 * # ~i~ ^ * 4 * ^ / nx a [ x -jr a J 

— r» x P)+ rS 
r . r+ I/3 2 
2 • 3 
2 . nxa 
in -j - 1 
„ ( ; &.y+i - (« + I -r +■ J«) X QJ 
f — - (m -f i - r + 2 n) x R) + 
n V , n , n\r 4-2 > ' ' 
. nx a v >(^+aJ~ 
r . r-f i . r-j-2 . 
2 . 3 . 4 . n xa 
( ■ ' ■ (a* + i - F+3«) x S) &c. The continuation of 
V (**+ 
the feries is evident; the letters P, Q, R, S, &c. denote the 
'Mtt* ’ &c ' 
From the length of the arc of an hyperbola or ellipfe given 
may be deduced by this feries the length of a correfpondent arc 
of an hyperbola or ellipfe, of which the equations expreffing 
the relation between its abfciffae and ordinates differ only by 
very fmall quantities from the equations expreffing the relation 
between the abfciffae and ordinates of the former. 
1 3. The fame principles may be applied to the refolution of 
algebraical, fluxional, incremential, &c. equations. 
Ex. Let LffzM, &c. (Tab. IV. fig. x.) be a given circle, 
whofe center is C and radius (r), and it be required to find an 
arc LM, fo that the area LSM defcribed round a given point S 
contained between the lines LS, SM, and the arc of the circle 
LM be equal to a given area (as). 
Find an arc L m = A nearly equal to the arc required LM, 
of which to radius 1 fubftitute s for the fine, and c for the co- 
fine, and write e for otM=LM — L m and b for SC; then will 
LM x f (A + <? x f j=±=SC x x fin. s arc : LM( b - x (A + e — 
A + <? 
