364 On the Methods used for approximating 
been treated very ably and generally by Mr. Horner, in the 
Philosophical Transactions for 1819: but as the following in- 
vestigation appears to contain a more simple developement of 
the principle common to all the methods than has yet been 
given ; andis, moreover, drawn out in the form most familiar to 
persons of this country ; I request the favour of being allowed 
to insert it in your Journal. 
Let 2° -+- ha’! + i272 + ko" +12" Ke. &c. = N 
represent any equation with known coefficients extending to a 
given number of terms. 
Assume 7 as nearly as possible to some one of the roots of the 
equation with N transposed. 
And putr +. = xorthetrueroot. Then 
a —_ 7" 4+ nr} 7 € + y any He ] aad 2 
1 2 
+ n. n—1l.n—2 p32 3 Bo 
i 2 3 
ha = fr’ $A n—i ree AMT m2 2 
La he 
4+ Anal. a—2.N—3 yr 33 Ko. 
IF pat rab cd hanes 
A : - = 'N 
we? ar tne re 2. n—2 .2—3 yn-4, 2 
Ley 
4 7 M2 .n—3 - N—A prs 3 Be, 
ty, WSs cal ed 
Rat? = ket ke 3. pk R38 NTS ps? 
1 2 
4+ En—3.n—4.n2—5 yn-6 8 _ Ko 
&e. = Ke. + Ke. 
Here all the terms in the first column (which may be consi- 
dered as coefficients of <°,) and all the coefficients of ¢, of ¢’, 
of «°, &c., in the second, third, and following columns, are known 
quantities. Put then the coefficients of «°° = A, of: = B, of 
& = C, &e. Kc. And the equation will become 
A+B:4+C°+ D+ Es, &c. = N. 
