fuppofing e+nk+p−1xr=m+ns+pr, you have re+nk-m-ns, that is, to the leaft difference of the indices m+ns, e+nk,ƒ+nb, &c. provided that difference be a measure of the other differences; although there may be as many values of the first term of the series equal as there are units in p. Or, if that does not happen, r must be taken, as formerly, equal to the greatest common measure of the diffe

rences.

ton

$116. Suppofe that the orders of terms of the equation can be expreffed the first by x-Ayx P, the fecond by x-y× 2, the third by x- -Ayx L, &c. and fuppofe that Eymys is one of the first, Fyex one of the second, Gyfxb one of the third, and fo on then it is plain that, fubftituting for x the feries Aya+ Byn++ Cyn+27, &c. the lowest term that will remain in the first will be m+ns+pr dimenfions of y, the lowest term that will remain in the fecond will be of e+nk+qr, and the lowest term remaining in the third of f+b+lr dimenfions of y. For by the same reasoning as we used, in § 113, to demonftrate that, in the first order of terms x -Aynx P, the lowest dimenfions of y are m+ns+pr, we fhall find that, in the subfequent orders, the lowest dimenfions of y in the the terms x-xQ=Byn+r+Cy#+2r &c.]1×Q muft

must be e+nk- gn + qn + gre+nk+qr,

and fo of the other terms x

-Ayx L the

lowest dimensions must be ƒ + nb + lr. The indices therefore of the terms that do not va

m+ns+pr and e+nk+qr coincide: and if at the fame time r be a divifor of f+nb —m—ns, and be found in it a number of times greater than p-l, or if r be less than+nh

then r will be rightly affumed. In general, “ take

and either the least of thefe, or a number whofe denominator, exceeding pq by an integer, measures it and all the differences ƒ + nh m—ns, gives r;" fuppofing p, q, and integers. But if P, q, and I are fractions, you are to e+nk—m—ns

"taker fo that it be equal to

ƒ+nh~m~ns and fo that K and M may be

P-1+M

integers." Suppofe, for example, m+ns

P =

p = {; e+nk==;, q = 3; ƒ+nb=2, and

M=+K; whence it is easily seen that 5 and 11 are the leaft integers that can be affumed

I

6

for K and M. And that r= 1+x=/; and thereforem+ms+pr=332, e+nk+gr=13, and f+ab+lr=55. That is, the terms of the firft

12

series whofe dimenfions are m+ns+p+Kxr, m+ns+p+Mxr fall in with the first terms of the second and third series respectively *.

* See on this fubject, Colfon. Epift. in Animadv. D. Moivrei. Taylor Meth. Incr. Stirling Lin. iij Ord. s'Gravefande Append. Elem. Algebrae. Stewart on the Quadrature of Curves.

СНАР.

СНАР. XI.

Of the Rules for finding the number of impoffible Roots in an equation.

§ 117. T THE

HE number of impoffible roots in an equation may, for moft part,

be found by this

RULE.

"Write down a series of fractions whofe denominators are the numbers in this progreffion 1, 2, 3, 4, 5, &c. continued to the number which expreffes the dimenfion of the equation. Divide every fraction in the feries by that which precedes it, and place the quotients in order over the middle terms of the equation. And if the Square of any term multiplied into the fraction that stands over it gives a product greater than the rectangle of the two adjacent terms, write under the term the fign +, but if that product is not greater than the rectangle, write; and the figns under the extreme terms being, there will be as many imaginary roots as there are changes of the figns from to -, and from

T

to +.

Thus,

Thus, the given equation being x3 + px2+ 3p2x-q=0, I divide the fecond fraction of the feries,2,, by the first, and the third

by the fecond, and place the quotients and over the middle terms in this manner;

3

3

Then because the square of the second term multiplied into the fraction that ftands over it,

that is, × p*x+ is less than 3p*x* the rectan

3

gle under the firft and third terms, I place under the fecond term the fign -: but as

14

× 9p+x2 (= 3p+x) the fquare of the third term multiplied into its fraction is greater than nothing, and confequently much greater than - pqx the negative product of the adjoining tèrms, I write under the third term the fign I write likewife under ≈3 and the first and laft terms; and finding in the figns thus marked two changes, one from to and another from to +, I conclude the equation has two impoffible roots. In like manner the equation x3 6o has two impoffible roots;