If in the first Example above +b+2, c==3, +d=+41 =ƒ= 5, then the Quadratic Equation (Step 5-) will be a+a6-0: the Cubic Equation (Step 9.) will be a3-3a-10424-0: the Biquadratic Equation (Step 13) will be a2a-25a-26a+120-0. And each of thefe Roots or Values of a fubftituted in the laft Equation will make the whole equal to nothing.

COROLLARY IX.

461. Hence whenever one Root of any Equation is found, if we divide by that Root, the Equation will ftill be reduced to one Dimenfion lower. Ex. gr. in the Equation a+2a5a-6-0, if we have one Root found, viz. a-1, or a+10, then dividing by a+1=0, the Quotient will be a2+a6o the Cubic Equation reduced to a Quadratic. And if again this latter Equation be divided by a 2 or a-2=0, the Refult will be a +3 Fo, or a3 the Quadratic Equation reduced to a Simple one..

462. In any Explicable Equation a±nam-pa-± qam~3 ±ram-4 &c. o from the given Coefficients n, p, q, r, s, &c. to find the Sum of the Squares, Cubes, Biquadrates, &c. of their Roots, under their proper Signs.

Effection.

For put A the Sum of all the Roots under its contrary Sign (In. 461.) and make B the Sum of the Squares of the faid Roots, G the Sum of all the Cubes, D= the Sum of all the Biquadrates, E the Sum of all the Fifth Powers, &c. then I fay that nn-2p-B, or nA—2p=B, nB—pA-\-39 =C, nC→pB+q4~4r=D; nDpC+qB—rA+5f=E, &c.

You have the Demonftration of this in the annexed Scheme..

SCHOLIUM

SCHOLIUM III.

463. But note, that because A the Sum of all the 1ft Powers is here taken under the contrary Sign (In. 461.) therefore the Sums of all the odd Powers, viz. C, E, G, I, &c. (or the 3d, 5th, 7th, 9th, &c. Powers) are by the foregoing Problem always exhibited under their contrary Signs.

Ex. gr. Suppose the Equation a-2a3—13a2+14a+24=0, whose Roots are—1+2—3+4=42, confequently B+1+4+9+16=+30, C=— 1+8—27+64=+44, D=+1+16+81+256=+354. E=1+32−243+ 1024 +812, F+1+64+729+4096=+4890, &c. But by the Coefficients A==-2, p=—13, q=+14, 7=+24, whence B―nA-2p= |—2×—2|—|—26=+4+26=+30 (In. 383, 392.) C=nB—pA+39=|— ×+30|—113×+2| +42=—60—26+42=-44: D=nG—pB+qA—4r —|——2—44)—|—13×30-|-14—2— 96=+88+390—28—96

=+354: E=nD—pC+qB¬rA+55—|—3×354--13—44 +14×30 −1+24×—2 +5×0=—708—572+420+48=—812: F=nE—pD+qC—rB +sA—61=|=2×—812-13×354|+|+14x=44|—24×30=1624+

4602-616-720 +4890: &c.

Again suppose the Equation a+2a-5a-6-0 whofe Roots are 1+2 and 3, whence A2, B14, C=-20, D=+98, &c. But by the Coefficients, A==+2, p=−5, q=~6, whence B=+14, C=+20, D= +98, &c.

Laftly fuppofe the Equation a-19a+30=0, or a3oa2-19a+30=0 whofe Roots are +2+3-5, whence A=0, B=30, C=-90, D=722, &c. But by the Coefficients A==o, B=38, C=90, D=722, &c.

COROLLARY X.

464. Hence we may learn a Method to find the Roots of Explicable Equa tions nearly, as follows. All 2d, 4th, 6th, 8th, &c. Powers are Pofitive Quantities, whether their Roots be Pofitive or Defective (In. 393.) confequently the Terms B, D, F, H, &c. found as above, are every one greater than the refpective Homologous Powers raised from the greateft Root of the given Equation, whether that Root be Pofitive or Defective: Or, which is the same thing Ba Dễ Ę* H 5. are every one greater than the said greatest

Root

Root (In. 22.) But D is nearer equal to it than B, F than D, Hễ than FK than H, &c. ad infinitum: Whence it is eafy to conceive, how the greatest Root of any given Equation may be approached to, nearer and nearer. Ex. gr. Suppofe the Equation a-2a3-13a+14a+94=0 were given to be refolved. Here B=30, D=354, F-4890, H=72354, &c. therefore B3=5. 4 &c. Da= 4.33 &c. F* =4.09 &c. H=4.04 &c. and confequetny if the Root be an Integer it cannot exceed ±4. I try therefore by fubftituting +4 for the Root which does not fucceed, but 4 fuccceeds And the Equation being divided by a+4=0 is reduced to this Cubic one a3+2a2—5a—6=0.

PROBLEM II.

465. In a given Explicable Equation, to find how many of its Roots are Positive, and how many Defective.

-

Effection.

When the Equation is prepared, as is directed above, begin at the Lefthand, and count how Changes there are in the Signs from --to-and fromto, and as many Changes as there are, fo many are the Pofitive Roots; and as many Succeffions as there are of the fame Sign without Change, fo many are the Defective Roots. Ex. gr. In the Equation a2a3-25a2-26a120=0 the Signs are +++, which fhews that there are two Pofitive Roots, because there are two Changes of the Signs, viz. from+to, and fromagain into +; alfo two Defective ones, because there are two Succeffions of the fame Sign; viz. ++ and➡: Again, in the Equation a—29a*+ 244a2-576=0, or a toas-29a4-0a3+244a2+0a-576-0, the Signs are ++++, which fhews that there are three Pofitive Roots and three Defective ones, because there are three Changes of the Signs, and three Succeffions of the fame Sign. Where note, that in this Cafe the Signs of the infignificant Terms, or those which are taken to fill up the Equation, as the Terms oa', oa3, oa above, muft always be of the fame Affection with the fignificant Term immediately foregoing.

COROLLARY XI.

466. If therefore the Roots of a given Equation be Rational, they may be discovered by seeking what two Factors, if the Equation be Quadratic; what three, if it be Cubic; what four, if it be Biquadratic, &c. do make a Product equal to the last Term, and a Sum equal to the Coefficient of the second Term with its contrary Sign: The Factors thus found are the Roots fought, and by fubftituting each in the Equation, will make the Whole equal to nothing.

D

Example