489. And hence laftly is learned to diftribute a given Equation into Periods, by looking upon the Co-efficient of the fecond Term as a Lateral, of the third Term as a Square, of the fourth as a Cube, of the fifth as a Biquadrate, &c. and the abfolute Number as a Power of the fame Denomination with the given Equation Ex. gr. the Equation. a-72a+1268α14593a+1000000=0, is thus diftributed into Periods, a*—72a3+1268a-14593a+1000000=0 And if any Co-efficient have not fo many Periods as the abfolute Number, it may be supplied with Cyphers to the left Hand for Integers and the right for Fractions, as in thefe, a2+01a2―072a-30753=0 aa—3.0aaa—0.8125—0 PROBLEM VII. 490. To take away the second Term nam-1 in any given Equation. Effection. 1. Divide the Co-efficient of the fecond Term n by the Number of the Dimenfions of the Equation m. n 2. Augment or diminish the Root a by the Quotient according as the m fecond Term nam-1 is Pofitive or Defective, and the Equation refulting will want its fecond Term. Q. E. E. Demonftration. Suppofe, Ex. gr. the fecond Term were to be taken away from the Cubic G Equation Equation a-6a2+17a-38=0, or a3-na2+17a-389: Make a=y-e. Whence If then in fumming up the above-written Values of a, the Terms -3eyz -n, or -ze-n=0. Whence -3e=+n, or +3e=n. Therefore 2; confequently a=y+2. And if the fecond Term of the given Equation had been na2, e would n or +2, and confequently for a in such Case might be fubftituted -2. Thus alfo in a Quadratic Equation e will be found = ± 5 , in a Biquadratic =±1, in an Equation of five Dimensions =± in an Equation of fix 4 2 n Dimensions e=, &c. Q. E. D. COROLLARY XXIII. 491. Hence it is plain that if the fecond Term p be taken away from any Quadratic Equation, it will be reduced to an Inadfected one, and may be refolved as fuch. Ex. gr. fuppofe a2+a-6=0, a given Quadratic Equation, confifting of a Pofitive and a Defective Root; make a=y, whence +a2= confequently y=2.5 a2 -|-a-6 (In. 146.) and a=y-2 the Pofitive Value of a. a-+3=0, therefore a=-3 the Defective Value of a. COROLLARY XXIV. Then a-2 492. Alfo by taking away the fecond Term, all Adfected Cubic Equations may be reduced to three Forms, viz. 493. To take away the third Term qam in a given Equation Ex. gr. Suppofe the Equation aa—3a3+3aa-5a-2=0, and make a=y-e, then -4e +6ee +eeee a*—3a3+3a3—5a—2=y* 4° 3° +90 g2 = y + zee =0 -3 +3 +5e -5 -2 Because the third Term in this Equation is +6ee9e+3, therefore make 6ee-+9+3=0, which divided by 6 becomes ee+e+o. Make e=u—1 And uu-=0, or uu=1, whence u and confequently e =-1, and a=y+1. And if for a in the given Equation be wrote y there will arife this Equation wanting the third Term Q. E. F. PROBLEM IX. 494. To take away the laft Term but one out of an Equation, where the fecond Term is wanting. This is performed by only fubftituting the laft Term divided by y for the Root fought. Ex. gr. Let it be required to take away the laft Term but one from the 8000 100 Equation a3+54—20=0. Make a=20, then a3+5a—20= + y 20=0; which laft Equation multiplied by yyy becomes 8000-10020 y y yo, and that again divided by 20 becomes 400+5y yyyy-o, or yyy —5yy—400—0; an Equation wherein y=- : 2, E. E. 20 a 495. TH CHAP. VI. Of the Limits of Equations. DEFINITION XXVI. HE Limits of an Adfected Equation are two Quantities, between which all its Roots are contained: or according to others, they are two Quantities, between which the greatest Pofitive Root is contained. PROBLEM X. 496. To find the Limits of an Equation. Effection from Sir Ifaac Newton. Multiply every Term of the Equation by the Number of its Dimenfions, and divide the Product by the Root of the Equation; then again multiply every one of the Terms that comes out, by a Number lefs by Unity than before, and divide the Product by the Root of the Equation; and so go on, always multiplying by Numbers lefs by Unity than before, and dividing the Product by the Root, till at length all the Terms are destroyed, |