to the Roots of Affected Equations. 355 
Assume s an approximate value of :; or as nearly as possible 
to the least root of this equation with N transposed. 
And put s + » = +s or the true root. Then 
noe. A 
Bs = Bs+Bn 
C2 = Cs?+2Cs1+Cr? 
D? = Ds°+3Ds°4+3Ds7°+D2' | are 
Est = Es*+4Es*-+ 6Es*x? + 4Es x? + Ent 
Fs? = Fs§+5 Fstn+10 F's* x? +10 Fs2 4? +5 Fsn'+ Fu’ 
&e. &e.  =&e. 2 2 2 2 
When as the horizontal ranks have the numerical coefficients 
of the binomial theorem to the indices 1, 2, 3, &c., the columns 
(as is well known) will have their numerical coefficients accord- 
ing to the orders of figurative numbers. 
Let the coefficients of the first column (or of 7°) = A, 
Those of » = B; those of x? = C, &c. ; then 
A, + Bn+ C+ Dr? + Ent &c. = N. 
Now assume ¢ an approximate value of 7, as s was assumed 
to « inthe former case; then make ¢+ # =, or the true 
root; and by repeating the same steps 
A,+B,2é+C,2+ D,2 + E,&, &c., = N, and so on. 
When the approximation has been carried to a sufficient 
degree of accuracy, or when the root has been strictly found, 
which will appear from the sum of all the terms involving « or » 
or 2, &c., becoming equal to nothing, (as indicated by A.A, 
.A, orA,, &c., becoming equal to N,) x will be equal to r + s 
+ t, &e. 
It is obvious that although the different terms are con- 
nected by the sign plus (+) each coefficient may be either 
positive or negative, or equal to nothing, and that the assumed 
quantities r, s, t, &c., may be positive or negative in the same 
manner. 
The methods best adapted for making the first approximation 
(r) are amply detailed in Maclaurin’s Algebra, Part II.chap. 9, 
in Wood’s Algebra, and in most elementary works. 
The other approximations may be made by assuming A+ Be 
2A2 
