Let the first term be a, the last x, and b the ommon difference; then a + b will be the feand, and x- b the laft but one, &c. Thus, a, a + b, a + 2b, a + 3b, a + 4b, &c. b, x- 26, x- -3b, x-4b, &c. It is plain, that the terms in the fame perpendiular rank are equally diftant from the extremes; id that the fum of any two in it is a +x, the im of the firft and laft. Hence of thefe four, s, a, r, n, any three being given, the fourth may be found. If n is not a small number, it will be mcft conveniently found by logarithms. Cor. 3. If the feries decreases, and the number of terms is unlimited, then, according to this no Thus.333, &c.: 증+ 10 II. Of GEOMETRICAL SERIES. DEF. When a number of quantities increase by fame multiplier, or decrease by the fame divi- r=10; therefore s they form a geometrical feries. This comon multiplier or divifor is called the common ratio. Prop. I. The product of the extremes in a geoetrical feries is equal to the product of any two ns, equally diftant from the extremes. Let a be the firft term, y the lait, r the common tio: then the feries is, a, ar, ar3, ar3, arf, &c. 3, 2, 4, 4, 4, &c. Each term in the upper rank is equally diftant om the beginning as that below it from the end; nd the product of any two fuch is equal to ay, he product of the first and last. Prop. II. The fum of a geometrical feries, wantthe firft term, is equal to the fum of all but the & term multiplied by the common ratio. = ༣ 3 ICO +, &c. y = ard 10 III. Of INFINITE SERIES. In many cafes, if the divifion and evolution of compound quantities be actually performed, the quotients and roots can only be exprefied by a fe ries of terms, to be continued ad infinitum. By comparing a few of the firft terms, the law of the progreffion of fuch a feries will frequently be dif covered. When this cannot be done, the work is much facilitated by feveral methods; the chief of which is that by the binomial theorem. THEOREM. Any binomial (as a + b) may be raiJed to any power (m) by the following rules. 1. From infpecting a table of the powers of a binomial obtained by multiplication, it appears, that the terms without their co-cfficients are a", am 1b, amb3, a” — 3b3, &c. 2. The co-efficients of thefe terms will be found by the following rule. The co-ctficient of any term multiplied into the exponent exponent of a in it, and divided by the exponent of b increased by 1, will give the coefficient of the next following term. The co-efficient of the firft term is always 1; and the coefficients of the terms in order are as follow: ,m X m I 2 -2 xm, &c. 3 or more conveniently expreffed thus: 1. Am, B m-I DX x" cx=2, D3, &c. the ca X 2 сх 3 which the dimenfion is equal to the fum of the dimenfions of the equations multiplied.-N. B. The term dimenfion, in this chapter, means either the order of an equation, or the number denoting that order, which was formerly defined to be the higheft exponent of the unknown quantity in ang term of the equation. If any number of fimple equations be multipl together, as x — a = 0, x − b = 0, x − c = n &c. the product will be an equation of a dim fion, containing as many units as there are linse equations. In like manner, if higher equations re multiplied together, as a cubic and a quadratic, one of the fifth order is produced, and so on. Converfly. An equation of any dimention is confidered as compounded either of fimple e. tions, or of others, fuch that the fum of their mentions is equal to the dimenfion of the gro one. By the refolution of equations thefe in rior equations are difcovered, and by invefugating the component fimple equations, the roots of y higher equation are found. Cor. 1. An equation admits of as many fol tions, or has as many roots, as there are fimple may be turned into an infi- quations which compofe it. IX I 1, let a = 1, b =r, and m =— 1; and the fame feries will arife as is obtained by divifion. This theorem may be applied to quantities which confift of more than two parts, by fuppofing them diftinguifhed into two, and then fubftituting for the powers of thefe compound parts their values. Thus, a + b + c = a + b + d3· PART II. Of the GENERAL PROPERTIES and RESOLUTION of EQUATIONS of ALL ORDERS. CHAP. I. Of the ORIGIN and COMPOSITION of EQUATIONS; and of the SIGNS and CO-EFFICIENTS of their TERMS. Cor. 2. And convertely, no equation can hav more roots than it has dimenfions. Cor. 3. Imaginary or impoffible roots mut ter an equation by pairs; for they arife from qua dratics, in which both the roots are fuch. Asi equation of an even dimention may have all ta roots, or any even number of them impote but an equation of an odd dimention must at ical have one poffible root. Cor. 4. The roots are either pofitive or n tive, according as the roots of the fimple c tions, from which they are produced, are pu tive or negative. Cor. 5. When one root of an equation is d vered, one of the fimple equations is found, which the given one is compounded. The s equation, therefore, being divided by this equation, will give an equation of a dimera lower by 1. Prop. II. To explain the general properties the figns and co-efficients of the terms of an ea tion. Let r - a = 0, giab = 0, gia cast d= o, &c. be fimple equations, of which CHE higher orders of equations, and their ge- roots are any positive quantities +a, + b + The lectione, are beft inveftigated by Con- . d, &c. and let x + m = 0, x 4' n = c. be fimple equations, of which the roots are negative quantities -m, -n, &c. and let number of thefe be multiplied together, as following table : fidering their origin from the combination of inferior equations. In this general method, all the terms of any equation are brought to one fide, and the equation is expreffed by making them equal to o. Therefore, if a root of the equation be inferted inftead of (x) the unknown quantity, the pofitive terms will be equal to the negative, and the whole muft be equal to o. Dif. When any equation is put into this form, the term in which (x) the unknown quantity is of the higheft power is made the first; that in which the index of x is lefs by 1 is the fecond; and fo on, till the last into which the unknown quantity does not enter, and which is called the Abfolute Term. Prop. I. If any number of equations be multi- xq+m=0 plied together, an equation will be produced, of Xx-abc=0, acti +m &c. + ab -- abeXx-abem Let the equation proposed be x3- px2 + qx +abm o, a bi- ro, of which the roots must be diminished by e. By inferting for x and its powers, y te and its powers, the equation required is, +аст quadra -ant - bin cm From this table, it is plain, 1. That in a complete equation, the number of terms is always greater by unit than the dimention of the equation. 2. The co-efficient of the firft term is 1. The co-efficient of the fecond term is the fum of all the roots (a, b, c, m, &c.) with their figns changed. .The co-efficient of the third term is the fum of all the products, that can be made by multiplying inv two of the roots together.. O. Car. 1. The ufe of this transformation is to take away the fecond, or any other intermediate term; for as the co-efficients of all the terms of the tranf formed equation, except the firft, involve the powers of i, and known quantities only, by putting the co-efficient of any term equal to o, and refolving that equation, a value of e may be determined, which being fubftituted, will make that term to valih. Thus, let the co-efficient 3e-po, and e= The co-efficient of the fourth term is the fum, which being fubftituted for e, the new equati of all the products, which can be made by multilying together any three of the roots with their s changed; and fo of others. The laft term is the product of all the roots, with their figns changed. . From induction it appears, that in any equa ian, (the terms being regularly arranged as in the receding example,) there are as many politive gots as there are changes in the figns of the ms from to, and front to; and the maining roots are negative. The rule alío may e demonftrated. Nate. The impoffible roots in this rule are fupofed to be either pofitive or negative. Cr. If a term of an equation is wanting, the ptive and negative parts of its co-efficient must en be equal. If there is no abfolute term, fome the rootso, and the equation may be de effed, by dividing all the terms by the loweft wer of the unknown quantity in any of them. this cafe alfo, x-00, xoo, &c. may confidered as fo many of the component fimple mations, by which the given equation being divid, it will be depreffed to many degrees. CHAP. II. Of the TRANSFORMATION of EQUATIONS. on will want the fecond term. And univerfally, the away by fuppofing x=3 / Cor. 2. The fecond term may be taken away by the folution of a funple equation; the third by the folution of a quadratic and so on. Prop. 3. An equation may be transformed into another, of which the roots fhall be equal to the roots of the given equation, multiplied or divided by a given quantity. Prop. i. The affirmative roots of an equation x3- 19x3 comes 5, - 19x2 qay· 3 +5.. Let be the given difference; then ye, and ar a y a and its powers, and the equation be Cor. 2. If there are fractions in an equation, they may be taken away, by multiplying the equation by the denominators, and by this propofition the equation may then he transformed into another, without fractions, in which the co-efficient of the firft term is 1. In like manner may a furd co-efficient be taken away in certain cafes. Cor. 3. Hence alfo, if the co-efficient of the fecond term of a cubic equation is not divifible by 3, the fractions thence arifing in the transformed equation, wanting the fecond term, may be taken away by the preceding corollary. But the fecond term alfo may be taken away, fo that there fhall be no fuch fractions in the transformed equaIii tion, involves impoffible quantities; while at the fam time, all the roots of that equation are pofitie The reafon is, that in this method of folution, is neceffary to fuppofe, that x the root may be di rovided into two parts, of which the product is q But it is eafy to fhow, that in this, which is cal led the irreducible cafe, it cannot be done. co-efficient of the fecond term of the given equation. And if the equation ax3- px2 + qx ➡o be given, in which p is not divifible by 3, by fuppofing x = +, the transformed equation за reduced is 23. 3p2 + 997. × 2 − 2p3 + 9cpq 7a'r = o; wanting the fecond term, having for the co-efficient of the first term, and the co-efficients of the other terms being all integers, the co-efficients of the given equation being alfo fuppofed integers. General Corollary to Prop. 1. 2. 3. If the roots of any of these transformed equations be found by any method, the roots of the original equation, from which they were derived, will eafily be found from the fimple equations exexpreffing their relation. Thus, if 8 is found to be a root of the transformed equation ≈3 +232 2 + 2 5 - 696 = 0. Since x = the correfpond ing root of the given equation 53-6x+7x 8+2 2. It is to be obfer5 300 muft be The fimple literal divisors of —2a3b are a, b. : 2b, any of which may be inferted for x. Supp ing xa, the equation becomes -za b a3 — 3a3 + 303-206} which is obvioufy: ba2 + 3a2b = The reafon of the rule appears from the p perty of the abfolute term being the product all the roots. To avoid the inconvenience of trying many vifors, this method is fhortened by the follow Rule 2. Subftitute in place of the unknown qu tity fucceffively three or more terms of the greffion, 1, 0, I, &c. and find all the divif of the fums that refult; then take out all the rithmetical progreffions that can be found amo thefe divifors whofe common difference is 1, a the values of x will be among those terms of t progreflions, which are the divifors of the refult rifing from the fubftitution of x = o. When t feries increafes, the roots will be pofitive; when it decrçafes, the roots will be negative. 10x+6= 4, 3, 2; and therefore 3 is a root, and it is 3, fince the feries decreafes. In this example there is only one progreffion, tity. If the refult has the fame fign as the given abfolute term, then from the property of the abfolute term either none, or an even number only, of the positive roots have had their figns changed by the transformation; but if the refult has an oppofite fign to that of the given abfolute term, the figns of an odd number of the positive roots must have been changed. In the first cafe, then, the quantity fubftituted must have been either greater than each of an even number of the pofitive roots of the given equation, or lefs than any of them; in the fecond cafe, it must have been greater than each of an odd number of the pofitive roots. An odd number of the pofitive roots, therefore, muft lie between them when they give refults with oppofite figns. The fame obfervation is to be extended to the fubftitution of negative quantities and the negative roots. It is evident from the rules for transforming equations, that by inferting for x, +1, the refult is the abfolute term of an equation, of which the roots are lefs than the roots of the given equation by 1. When xo, the refult is the abfolute term of the given equation. When for is inferted 1, the refult is the abfolute term of an equation whofe roots exceed the roots of the given equation by 1. Hence, if the terms of the feries 1, 0, 1, -2, &c. be inferted fucceffively for X, the refults will be the abfolute terms of fo many equations, of which the roots form an increafing arithmetical feries with the difference 1. But as the commenfurate roots of thefe equations must be among the divifors of their abfolute terms, they muft alfo be among the arithmetical progreffions found by this rule. The roots of the given equation therefore are to be fought for among the terms of thefe progreffions which are divifors of the refult, upon the fuppofition of *= 0, becaufe that refult is its abfolute term. If from the fubftitution of three terms of the progreffion, 1, 0, -1, &c. there arife a number of arithmetical feriefes, by fubftituting more terms of that progreffion, forne of the feriefes will break off, and, of courfe, fewer trials will be neceffary. II. Example of QUESTIONS producing the EQUA TIONS. By the former rules, the roots of equations, when they are commenfurate, may be obtained. Thefe, however, more rarely occur; and when they are incommenfurate, we can find only an approximate value of them, but to any degree of exactnefs required. Of various rules for this purpofe, the following by Sir Ifaac Newton is the the moft fimple. Lemma. If any two numbers, being inferted for the unknown quantity in an equation, give refults with oppofite figns, an odd number of roots muft be between thefe numbers. This appears from the property of the abfolute term, and from this obvious maxim, that if a pamber of quantities be multiplied together, and if the figns of an odd number of them be changed, the fign of the product is changed. For, when a pofitive quantity is inferted for x, the refult is the abfolute term of an equation, whofe roots are lefs than the roots of the given equation, by that quan. From this lemma, by means of trials, it will not be difficult to find the nearest integer in a root of a given numeral equation. Let the equation be x3 -2x- 5 = 0. 1. In this cafe a root is between 2 and 3; for thefe numbers being inferted for x, the one gives a pofitive and the other a negative, refult. Either the number above the root, or that below it, may be affumed as the firft value; only it will be more convenient to take that which appears to be neareft to the root. 2. Suppofe = 2 +f, and fubftitute this value of in the equation. 5 = 5 f If ƒ is lefs than unit, its powers ƒ and ƒ3 may be neglected in this first approximation, and 1of 1, or o.I nearly, therefore x = 2.1 nearly. 3. As fo.I nearly, let f.1+g, and infert this value of ƒ in the preceding equation. 0.001 + 0.038 +0.2g2 + g3· 0.06 +1.25 + 6.8* +108 I -=- I ƒ3 + 6ƒ2 + 10ƒ· I = 0.061 + 11.238 + 6.3g+go, and neglecting g2 and g3 as very small, 0.61 +11.2380, or g= 0.061 11.23 |