212

A GgNERAt RULE

TRACT 10‘.

4. Again, let the equation n = a” + n a" -1 x + &c, bo-multiplied by n — 1, and a" added to each side, so shall wehave

(«—1 ) n 4 - <f = tia" + (n — 1) + &e, for a divisor:

Also multiply the sides of the same equation by a and subtracta” + 1 from each, so shall we have

(if — a”) a = n a" x + n . - d'~ l x l + &c, for a dividend:

Divide now this dividend by the divisor, so shalln — a” n— 1 x r — 1 »—2 x 3 5

(n — 1)n + <j” 2 a T 2 3 a

Which will be nearty equal to x, for the same reason as be-fore ; and this expression is about as much too little as theformer expression was too great. Consequently, by adding

ct, we have® 4 - x or n” nearly = -—r—-, for a second

J (71 — 1) N.+ a" ’

theorem, and which is nearly as much in defect as the formerwas in excess.

5. Now because the two foregoing theorems differ fromthe truth by nearly equal small quantities, if we add toge-ther the two numerators and the two denominators of theforegoing two fractional expressions, namely

N + («— !)«’

a and

MN

the sums will be the numera-

na n ~ (w—l).N + a"

tor and denominator of a new fraction, w hich will be much

nearer than either of the former. The fraction so found is

n+ 1. n Hi— 1 .a” . . , y

“—r~ '■——- a ; which will be very nearly equal to n *

or a + x, the root sought; for, by division, it is found to he11— 1 nf 1 x-

■ 4 - &c, where the term' is

6 a

equal to a + x * -

wanting which contains the square of x, and the followingterms are very small. And this is the third theorem.

6. A fourth theorem might be found by taking the arith-metical mean between the first and second, which would be