Infinite Series . 9 7 
equation, and ( a ) an approximate much more near to the root 
required (?r) of the given equation than to any other : write a for 
(,v)the unknown quantity fought in the fubfequent quantities, A, 
-iy, &c.; and let there refult the correfpondent quan- 
tities a, jS, y, 5, &c. ; then will n - a be a root (e) of the infi- 
nite equation + yg ? — h . ^ s &c. 
= 0, of which a root of the equations t Q e ~o y 
-J—ye 2 =z o, &c. will be an approximate. If two roots of the 
given equation are nearly = d, then it is neceffary to recur to 
an equation not inferior to a quadratic. 
9. The fucceflive approximate values found by thefe and like 
rules will ultimately be to each other in a greater than any geo- 
metrical ratio : for example, let be an approximate to a root 
of the quadratic -X 2 - (tf + j)* 4. as = o, then will the new ad- 
dition to the approximate to the root s found by the common 
method at the diftance n- 1 from the firft approximate be 
nearly, where 6 = 
a 
i o. Let an equation x n - P*”— 1 4 Qx n ~ 2 - R# K ~3 4* Sx n ~~4 — &e. 
*=o, of which the roots are a , b , r, d , & c, ; and an equation 
x* -px n ~~ l 4 qx n ~~ z — rx n ~i 4 sx n ~4 ~ &c. = o, where />, y, r, s, 
&c. differ from the co-efficients P, Q, R, S, &c. by very fmall 
quantities: affume the (0) equations 7r4p 4<r 4 t 4 &c.=/ — P 
#tt4 b(> 4 C(r 4 -dj 4- &c. = Pa — q +Q = /3, 4- b % 4- C z <r 4' 
4 &c. ~ P/3 — Qa + r — R = y, <2 V 4“ ^ 3 p 4- fV 4- ^ 3 t 4 &c. = 
Py — Q^/3 4 Kcc — j 4 S = <?, &c. ; and from them find the un- 
known quantities 7 r, g, <r, r, &c; then will tt, £ 4p, r4<r ? 
^4- r, &c. be nearly the 0 roots of the equation - px *““ I 4 qx"~ z 
Vol. LXXVI. O 
