Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990.Γ m $eUy promu 1 PRESENTED TO THE CORNELL UNIVERSITY\ 1870, BY The Hon. William Kelly Of Rhinebeck. MATHEMATICS LIBRARYA COMPENDIOUS TRACT ON THE THEORY AND SOLUTION OF CUBIC AND BIQUADRATIC EQUATIONS, AND OF EQUATIONS OF THE HIGHER ORDERS. INTENDED AS A SUPPLEMENT to his » TREATISE on the ELEMENTS of ALGEBRA:* .«y BV THE REV. BHÎRIDGE, B.D. F.R.S. FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE. SECOND EDITION, REVISED AND CORRECTED. LONDON: FOR T. CADELL, STRAND: AND SOLD BY DEIGHTONS, AND STEVENSON, CAMBRIDGE ; AND PARKER, OXFORD. 1833.PRINTED BY RICHARD WATTS, Crown Court, Temple Bar.ADVERTISEMENT. A short time after the present Edition of this useful little Tract had been sent to the press, the friends of thé worthy Author had to lament his loss. It had, however, been completely revised by him, with the assistance of a friend, who has taken the greatest care to have it correctly printed. Some particulars in the last chapter of the former edition not appearing, on revision, perfectly satisfac- tory, and the general difficulty of the subject being, in the Author’s opinion, much more than propor- tioned to its utility, he finally decided on omitting that chapter, which accordingly does not appear in this edition.CONTENTS. Chap. I. PAGE On the Construction and Properties of Equations, 1 Chap. II. On the Transformation of Equations .... 17 Chap. III. On the Method of finding Limits to the Roots of Equations.......................................39 SECT. I. On the method of finding quantities which shall exceed the greatest positive root, or the greatest negative root of any given equation..............................ib, II. On the method of finding a limiting equation to any given equation......................................49 Chap. IV. On the Solution of Equations.............................. 61 I. On the solution of equations by means of the divisors of the last term......................................ib. H. On the solution of equations whose roots bear some given relation to each other .......................67 III. On the solution of equations of the form xn±kn=0, 81 VI. On the general solution of Cubic Equations .... 89 V. On the general solution of Biquadratic Equations . .100 VI. On the solution of equations in general by the method of approximation ..................................107TABLE OF CONTENTS. Chap. V. On thet Symmetrical Functions of the Roots of Equations........................................... SEC i I. On the method of finding the sums of the powers of the roots of any given equation.................. II. On the method of finding the sums of certain combina- tions of the sums of the powers of the roots of any given equation .............................. III. On the method of transforming an equation into one whose roots shall be some function of the roots of the given equation ..................................... PAGE 121 ib. 134 141ON THE THEORY AND SOLUTION OF EQUATIONS. CHAP. I. ON THE CONSTRUCTION AND PROPERTIES OF EQUATIONS. 1. The several orders of Equations receive their respective denominations from the highest powers (or combinations of powers) of the unknown quantities contained in them. Equations which contain only one unknown quantity, when complete in all their terms, may be reduced to the follow- ing forms, by bringing all the terms to the same side of the equation, and arranging them according to the powers of x, beginning with the highest ; ax +6 =0, a simple equation. ax1 + h x + c =0, a quadratic. aæ3 + &a?2 + cæ + d=0, a cubic. aæ4+ 6æ3 + cæl+dx + e=0, a biquadratic. &c. &c. * axn + bxn 1+c*n 2+dxn 3+....................+/# + Sr=s°> b an2 CONSTRUCTION AND PROPERTIES OF EQUATIONS. an equation of n dimensions, where u, ò, c, &c. represent any known quantities whatsoever, positive or negative. Now, let each of these equations be divided by a, the coefficient of the highest power of x, and let - =p, - = a, a a d 6 ƒ -=r, ~ = s, I =v, £ = w, &c., then they become a a a a x+p =0 x2 + px + q =0 x3+px2 + qx + r =0 x*+px3+ qx2+ rx + s =0 &c. =0 æn+px'n_1 + (7æw'2 + ræ*_3 + &c...-buæ + w =0. And these are the forms to which the several orders of equations must be reduced, before we can conveniently pro- ceed to the investigation of their nature and properties. 2. Next, let 2—α=0, χ—β = 0, x— y =0, χ—δ=0, a?—e=0, &c. be n simple equations, in which the roots or values of x are respectively «, β, y, δ, ε, &c. and let them be multiplied together ; then ;)(#—β)..........................................=æ2—(« + Æ)æ-f «β = 0 t)(æ-—yô)(í»—y)..................................= æ3—(« + /Ô +ν)τ2 + («/ό + «γ + /δγ) «βγ=0 t)(a? — β)(χ—v)(x — δ)...........................= æ4 —(« + /β + y+ à)æ3 + (a/8 + «y -j-ad-j-^y + ^ + yd)#2 CONSTRUCTION AND PROPERTIES OF EQUATIONS. J « ^ ¿8 I J '■d £ G cr1 Oí <υ Λ -w G CD pG ^ β5 £*H- Ό g φ «S β Ή cd e« "g * ao <03 g G> SS •Sg J ** *5 •eo <03 60 -< s ^ G> G 00 Ö G •G *'s> Ιϋ « λ o — ü § ü H- SO ao I ■d G G O da + AO + ?*· + 02. o da «o AO ?*· 02. + d da to 02. « + d da AO 02. « £ T Λ β ζΛ c G O *G cö G è ■i 1 Pt η S è ' ss + g ·| co Η<ι *+0 Ο ^ G ^ ?" so «>. .ç^ re ■“ 43 1 « e co 1β u 0> > CD xn CD CO ° s co ^ S M .S^O ao o gs ^ S ^ ^ I ^ s b : 2 fi ^ ce *S «4H g 0 S» in 03 'g <03 2 s- G scs 'd SO o 8* & e « Λ %- "Ö <03 §» G co g <03 κβί ■i'S^S 'd s ^ 1 s ^ ^ oo #g ® S >> S c ^ Jr bo IS C > § ® g ï s ^ % +s O g3 *S .S . 03 ^ o ° CU >H g* 03 0 ,β 03 s >> Ö pO S TJ s? 4> >s >» § g- «*H .5 s 2 03 J=î tuo 03 .S s ■ri -2 § s H g M & O -p s <5 ^ o 03 C3 S t4 CONSTRUCTION AND PROPERTIES OF EQUATIONS. 3. In attending to the process by which the several sim- ple equations were multiplied together in the foregoing ar- ticle, it may be observed, That the quadratic equation x2—(«-}-£) is composed of the two simple equations *—«==0, x—β=0. That the cubic equation Λ?3 —(« + Æ + y) x2-f («β + αγ + βγ) X—αβγ — 0 may be considered as composed, either of the three simple equations x—« = 0, χ—β = 0, x—γ=0, or of one simple equation x—«=0 and the quadratic equation arising from the multiplication of x—β=0 into x-—y = 0. That the biquadratic equation x4—(α + yö + y + δ) x3 + (αβ+αγ+αδ + βγ + /3δ + y$)x2—(«/3γ + «βδ-f ay h + «/3γδ=0 may be considered as composed, either of the four simple equations x—a=0, x—β=θ9χ—γ=0, x—δ=0, or of the simple equation x—«=0 and the cubic equation arising from the multiplication of χ—β=0 into x—y=0 into x—δ=0, or of the two quadratic equations arising from the multiplication of x—«=0 into x— /3=0, and of x—y=0 into x—δ=0, respectively. And, in general, “ that the equation * of n dimensions “ may be considered as composed of all the different com- “ binations which can be made of the several equations m- “ ferior to it, in which the sum of the indices of the highest " powers of x in the equations belonging to êach combina- " tion amounts to n.” 4. On examining the different orders of equations as con- structed in Art. 2, it will be found that “ a quadratic equa- " tion contains two values, a cubic equation three values, a “ biquadratic equation four values, and an equation of n “ dimensions n values, of the unknown quantity ; and no " more? * The equation of n dimensions will, in general, be represented byCONSTRUCTION AND PROPERTIES OF EQUATIONS. 5 “ more." For instance, let «, β, y be substituted succes- sively for X in the expression (x—«) (χ—β) (a;—y), and it becomes, > when a is substituted for x, (*—«) («--β) β..............for (β—*) (β—β) (β-y); y..............fora·, (y «) (y— β) (y-y); and as in the first case «—«=0 ; in the second β—/3=0 ; and in the t/imZ y—y=0 ; it appears that in each case one of the factors becomes equal to 0 ; consequently, their pro- duct must in each case be equal to 0. But if a quantity a, which is not equal to either of the quantities «, β, y, be substituted for x in that expression, then it becomes (α—α) (α—β) (a —y); and as in this case neither of the factors α—-a, a— β9 a—y is equal to 0, their product cannot be equal to 0. Hence «, β, y are the only quantities which, substituted for x in the equation (a?—a) (x—β)(χ—y)=x3—(« + β + y) x2 + (<*β + «y + Æy)#—«/Öy=0,t will make that equation a true one, and consequently are the only values of x in that equation. In the same manner it may be shewn that «, β, γ, δ are the only values of x in the biquadratic equation which arises from the multiplica- tion of the simple equations «—«=0 ; x—β — O; x~y = 0 ; x—δ=0 ; and that the n quantities α, β} y, δ, ε, &c. are the only values of x in the equation (A) = 0. 5. Now let the roots «, β, y, δ, e, &c. be substituted suc- cessively for x in the equation (A)=z 0, according to the form in which it stands in Art. 2 ; then, since either of those quantities, t If a be substituted for x in the expression λ*3—(a+/3+y) #2“b(a£+<*7+|8y) æ~a\8y, it becomes a3—(a-h/J-by) cr+(a£+ay+Æy) α—αβy, which is evidently equal to 0 ; and the same will appear, if β or y be substituted for x.6 CONSTRUCTION AND PROPERTIES OF EQUATIONS. quantities, when substituted for x, will give a result equal to 0, all the following equations will be true ; viz. a"—panmml + qan 2—r«n 3+&C........±Va:Ft0=O βη—ρβη~*+ (¡βη~2—7*Æn~3 + &C......±νβψν> = 0 yn—pyn~í + qy1l-2—ryn-3+&c...........±vyq=i0=o δη—ρδη-1 + g δ””2—rhn~3 + &c...... ±ν^ψίυ=0 &c. &c.=0 Take, for example, the biquadratic equation a?4·—lla?3 + 41 æ2 —61a? + 30=0, which is the equation arising from the multiplication of the several simple equations x—1 = 0, x—2 = 0, x— 3 = 0, and x—5=0 ; the roots of this equa- tion are 1, 2, 3, 5 ; let these numbers be substituted suc- cessively for x in the given equation, and in each case the result will be found equal to 0. 6. Since «, β9 y, δ, e, &c. are the only quantities which, substituted for x in the equation (A) = 0, will give a result equal to 0, it is evident, that “ if for x in any given equa- “ tion, whose roots are all positive whole numbers, there " be substituted successively 1, 2, 3, 4, 5, 6, &c. ; all such " numbers as make the equation equal to 0, will be roots “ of that given equation.” For instance, if for x, in the equation æ3 —9a?2+ 23a? —15=0, there be substituted suc- cessively the numbers 1, 2, 3, 4, 5, it will be found that 1, 3, 5 respectively give a result equal to 0 ; but that of the numbers 2 and 4, the former gives æ3— 9a?2 + 23æ —15=3, and the latter a;3 — 9a?2 + 23sc — 15 = — 3 ; 1, 3, 5 are therefore the three roots of the given equation æ3—9æ2+ 23#—15 = 0. 7. Resuming the equation (A) xn—pxn~l+ qxn~*—rxn~*+ &c......±vx:F^=0; let this equation be divided by x—a, and let the quotient be Q, and the remainder R ; then, (# —«) Q + R=0 ; if « be substituted for x in this latter equation, then («-—«) Q=0,and consequently (since («—a) Q + iï=0) R must beCONSTRUCTION AND PROPERTIES OF EQUATIONS. 7 be equal to 0; i. e. “x—-«=0 divides the equation (A) = 0 without a remainder Γ In the same manner it may be shewn that X—>8 = 0, x—γ=0, x — δ=0, X—e = o, &c. divide the equation (A)=0 without a remainder. Now (A)=(x—a)(x—/3)(χ—y)(x — δ) &c.=0. Hence Iá} =(x-¡β)(χ-γ)(χ-S)&c. = 0,| -11)-^. = (a:-y)(ít-á)&c.=0,'l x u an equation whose roots Yr λ ' an equation whose &c· are β9 γ, δ, &c. J roots are γ, δ, &c. ) lèi = (x — «)( x — y\x—δ)&0. = 0,i .—íèl-= (x—α)(χ—δ)&0. = 0, Ì an equation whose roots VîC^--an equation whose &c· are «, γ, δ, &c. / roots are «, δ, &c.J =(x—«Xx——δ) &c.=0, Ì =1«Xx—/^)&c.=0,Ì an equation whose roots /; J an equation whose /; &c. are «, β, δ, &c. ) roots are «, β, &c.) from which it appears, that “ if one or more roots of an “ equation be known, and the given equation be divided “ by the equation which contains those known roots, the " quotient will be an equation which contains the rest of “ the roots of the given equation.” Example I. Take the cubic equation x3—9x2 +23x—15 = 0 (Art.6), one of whose roots is unity, Since 1 is a root of this equation, a?—1 = 0 will divide it without a remainder ; thus, x —l^t?3 —9x2+ 23x- x—y — 8a?2 + 23x — 8 a?2 + 8 x c—15Í x2—8a? + 15 = 0, whose roots V (Chap.V. EL·)* are 4 ± 1, or 3 and 5; the other roots of the given equa- tion, therefore,are 3&5. + 15a?-15 + 15 x— 15 * (El.) refers to the Author’s Treatise on the “ Elements of Algebra,” sixth or seventh edition.8 CONSTRUCTION AND PROPERTIES OF EQUATIONS. Example II. Two of the roots of the biquadratic equation a?4— 11 #3 + 41 x2—61 #+ 30=0 are 1 and 3 ; find the other roots of the equation. Since 1 and 3 are roots of this equation, each of the sim- ple equations a?—1=0 and x—3 = 0 will divide it without a remainder ; or it may be divided at once by the quadratic equation æ2 —4x + 3=0, which arises from the multiplica- tion of x—1 = 0 by x—3=0$ thus, 4æ + 3^a?4—11 a?3-f 4 læ2 — 61 x+ 30 æ4— 4æ3-[- 3a?2 * — 7x* + 38x2 — 6lx — 7x* + 28x* — 2lx * + I0x2—40^ + 30 + 10 a?2—40 a? 4 30 x2—7 x +10 = 0, whose roots are -t-, or 2 and 5 ; the other roots of the equation, therefore, are 2 and 5 ; and the four roots of the equa- tion are 1, 2, 3, 5. * * * 8. Hitherto we have considered the roots α, β, y, δ, e, &c. of the equation (A)=0 as all positive; in which case the several simple equations which go to its formation are of the form x—«=0, χ—β=0, a?—y=0, x—S=0, &c., and the signs of the coefficients (p, q, r, s, t9 &c.) of that equation are alternately positive and negative.* If the roots be all negative, the simple equations, of which the equation (A)=0 is composed, will be of the form æ 4- «=0, χ + β = 0, æ 4- y=0, x -f- δ=ο, &c. ; in which case the coefficients of that equation will evidently be all positive, and the equation itself of the form æn+j9æn~1 + gæn”2 + ræ,l~3+ &c.....+væ+w?=0. 9. When * This form of the equation C¿0=0 ; viz. a?n—pxn~1+ÿd?n~;2—......................i væ^Ç-w—O will in general be adopted, as most convenient for the purpose of investi- gation, the coefficients p, q, r, Scc. being understood, as before, to repre- sent any known quantities, positive or negative.CONSTRUCTION AND PROPERTIES OF EQUATIONS. 9 9. When the equation (A) = Q contains both positive and negative roots, it will be found that " such equation will “ have as many positive roots as it has changes of the signs “ from + to — or from — to +, and as many negative roots “ as it has continuations of the same sign from + to + or “ from — to —, among its coefficients.” This is Des Cartes’ rule, and the truth of it will appear from the following in- vestigation. Let α, — β be the two roots of a quadratic equation ; then the two simple equations of which it is composed are X—«=0 and X + β=0, which being multiplied together give x2—a) >x—aj3=0, an equation in which the last term is ne- + Æ) gative, and the second term is negative or positive accord- ing as a is greater or less than β. The two general forms, therefore, under which quadratic equations, having both a positive and a negative root, may be exhibited, are x2—px—q=0, and x2+px—q=0 ; and in each of these cases there is but one change of the sign, (viz. from+to—), and one continuation of the same sign, (viz. from — to — in the former case, and from + to + in the latter). The rule is thus shewn to be true in quadratic equations. Into the two forms of quadratic equations containing a positive and negative root (viz. x2—px—q=0, and x2+px —q=0) let a positive root (y) be introduced; then we shall have, in the first case, -y) +py) where the second term is negative and the last term posi- tive, and the third term negative or positive according as q is greater or less than p y; and in the second case, (X +px q)(x y)—x —q + where the -y) T y) third term is negative and the last term positive, and the second term positive or negative according as p is greater or less than y. The general forms, therefore, under which c10 CONSTRUCTION AND PROPERTIES OF EQUATIONS. cubic equations, having one negative and two positive roots, may be exhibited, are X3 — p'x 2±q'x + r'=0, and xz±p'x2—gi'a? + r'=0 ; in which it may be observed, that (whether we take + g' or—q' in the first instance, or +p' or — p' in the second) there are two changes of the signs and one continuation of the same sign. If a negative root (-—y) be introduced into the equations x2—px—q=0 and x2+px—q~0, and the same process of reasoning made use of, it may easily be shewn that the general forms under which cubic equations, having one posi- tive and two negative roots, may be exhibited, are x*±p'x2—q'æ—r'=0, and X3 +p'x2± ç'x—r'=0 ; in which there is one change of the signs and two continuations of the same sign. We have thus established the truth of the Rule as applied to cubic Equations; and by introducing first a positive root (δ), and then a negative root (—$) into these different orders of cubic equations, the Rule may in like manner be proved for biquadratic equations; and so on for equations of the higher orders.* To exemplify the rule in two or three instances. In the biquadratic equation æ4+9æ3+9æ2—41x—42=0, whose roots are 2, —1,-3,—7, it may be observed that there is but one change of the signs from + to—, and three continu- ations of the same sign (viz. two from + to +, and one from — to —). Again, in the equation x5 + 7 a?4—15 a?3—115 a?2—46x+ 168 = 0, whose * This method of shewing the truth of Des Cartes’ Rule seems best adapted to an elementary treatise ; but as its application to the higher powers would, from the necessary subdivision into cases, become very long and embarrassing, the reader is referred for a general proof of the Rule to “ La Croiacy Complement d'Algebre, p. 67. ed. 3. or to “ Gamier, Elimens “ d'Algebre, chap. 34. ed. 3.”CONSTRUCTION AND PROPERTIES OF EQUATIONS. 11 whose roots are 1, 4,—2, — 3,— 7, there are two changes of signs, and three continuations of the same sign. 10. By referring to the Table in Art. 2, it is evident that when the equation (A)=0 contains both positive and nega- tive roots, the values of the coefficients p, q> r, s, &c., which are expressed in terms of the roots «, β, γ, δ, &c. will con- tain positive and negative quantities also. Thus, if the roots of that equation be α,— β, y9 δ, &c., we should have Ρ = α—/3 + γ + δ &C. q — —α/3-f «γ + αδ &C.—βγ—βδ &C. T = —«βγ — «/3δ &C. + αγδ &C.—/3γδ &C. &C. = &C. ; in which case, whenever the sum of the positive quantities is equal to the sum of the negative ones, the coefficient itself will be equal to 0. In the equation æ4—15 x2—10 æ + 24 = 0, for instance, whose roots are 1,4,—2, —3, since the sum of the positive roots is equal to the sum of the negative ones, the coefficient of the second term becomes equal to 0 ; this second term therefore does not appear in the equation. In applying the foregoing rule to equations thus deficient in one or more of their terms, the place of the deficient term may be supplied by ± 0 ; thus, by inserting ± 0 in the equation æ4—15a?2—10x + 24=0, it becomes a?4± 0 —15a?2—10 æ + 24 = 0 ; and the rule still continues true ; for whether we make use of the positive or negative sign in the term ± 0, we may observe that there are two changes of the signs and two continuations of the same sign. 11. If the last term (w) of the equation (A) = xn—pxn~l -f anda—then p=(a + 6) -f* (a—b)~2 a q=z(a + s/±b) (a—V±6) =a2T&, and consequently the equation itself is X2—2ax + a2Tb=0, an equation whose coefficients are integral* II. Let * That a quadratic equation whose coefficients are integral may have its two roots in a radical form, appears also from the ordinary solution of it (as in Art. 79. El.) ; for the two roots of the equation - y δ==:ç αβγ + αβδ-}- <ζγ$ + /0γδ=Ι* «βγδ = $ we have as many equations as unknown quanties for deter- mining the values of «, β, y, δ. Now,16 CONSTRUCTION AND PROPERTIES OF EQUATIONS. Now, by transposition, β + 'ν + δ==ρ—oc, .*. αβ -f ocy + αδ = «(ρ—oc)—jp« — oc2 βγ + βδ+ yδ=q—(#β+ ocy + ocà)=g — poc + oc2 βγ$=r — (αβγ + αβδ -¡-αγ$) = 'Γ—α(βγ + βδ + γδ) ==r—goc+poc2 —oc3. But βγ*= -, OC .·. ì-=r—qa -f p«2—«3, oc or oc4—poc3 -f goc2—r« + s=0 ; an equation which is identic 1 with the given equation. In the same manner it might be shewn that the equations for determining the values of β, y, or δ are/34—ρβ3 + g/32—r/3 + s=0,y4— py3 + gy2—ry + s=0, or $4— p$3-f-g$2—r$ + s=0 respectively; so that to find the values of oc, β, y, δ, by this method, requires the solution of an equation precisely similar to the given equation ; and the same is true for equations of all dimensions. Hence the direct proof of the proposition extends no further than to those orders of equations, of which the general method of solution is known. Our remaining ob- servations upon this subject must, therefore, be reserved for the end of the Fourth Chapter.17 CHAP. IL ON THE TRANSFORMATION OF EQUATIONS. By the transformation of equations is meant the changing of them into others, whose roots shall bear some given re- lation to the roots of the original equations. 17. If it be required to transform the equation (A)=xn— pxn~' + qxn~2—&c. . . . ±vχψιν=0, whose roots are «, β, y, δ, &c., into one whose roots shall be m times the roots of this equation, let y=mx, or ; then for x and its powers in the equation (^4)=0 substitute and its powers, and the resulting equation will be m ,n—2 jpyn 1. qym 1 n—1 "* /VY1 «—2 r y n VV + &C.... ± — T«)=0, m" m1 1 m" m'* or (multiplying by mn) yn—mpyn~1 + m2qyn~2—m3ryn~3 + &c.... ±m*~lvy =Fmnw=0, an equation whose roots are mot, m/3, my, me, &c. where m may be any quantity whatever, either integral or frac- tional. Hence this rule for transforming an equation into one whose roots shall be any multiple or part of the roots of the given equation ; “ Multiply each term of the given “ equation, beginning with the second, by the successive " powers of the whole number or fraction expressing the " value of such multiple or part ; and the resulting equa- " tion will be the equation required.” For example, let the 2d, 3d, and 4th terms of the equa- tion x3—2x2—2 = 0 (whose roots are 1,2, —l) be mul- tiplied by 3, 9, 27, the successive powers of 3, and the re- sulting equation will be x3—6x2— 9x + 54=0, an equation d whose18 TRANSFORMATION OF EQUATIONS. whose roots are 3, 6,-3, or 3 times the roots of the given equation. Again, the roots of the equation a?3—28 a? + 48 = 0 are 2, 4, —6; let the 2d, 3d, and 4th terms of the equation a?3± 0—28a? + 48 = 0 be multiplied by the successive powers of and the resulting equation is a?3—7 a?+ 6=0, an equation whose roots are 1, 2, —3, or half the roots of the given equation. 18. Let m= — 1, then the successive powers of m are — 1, + 1, —1,+1, &c. $ so that the equation whose terms are multiplied by the successive powers of m will have its signs changed alternately, beginning with the second. Now, when m = — 1, y = — a?, or the values of y are the same with those of a?, but with the signs changed ; i. e. if the values of x in the equation (A)=0 are «, —β> y, — &c., the values of y in the resulting equation will be — «, β, —y, δ, &c. ; hence this rule for transforming an equation into one whose roots shall be the same with the roots of the given equation, but with the signs changed ; “ Change the signs " of the terms of the given equation alternately, beginning “ with the second, and the resulting equation is the equa- “ tion required.” Take, for example, the equation a?4 + 9 a?3 + 9a?2 —41 a? —42=0, whose roots are 2, — 1, —3, —7 (Art. 9) ; change the signs of the terms of this equation alternately, beginning with 9a?3, and it becomes a?4 —9a?3 + 9a?2 + 41a? —42=0, an equation whose roots are 1, 3, 7, —2,' as will appear by multiplying a? —1, a?—3, a?—7, a? + 2 together. 19. By means of a transformation of this kind, we may remove the coefficient of the first term from an equation, or clear an equation of fractions, in such manner, that a given relation shall still subsist between the roots of the trans- formed equation and those of the given equation. I. Take,TRANSFORMATION OF EQUATIONS. 19 I. Take, for instance, the equation Aæ4—px3 + qx2—rx + s = 0 ; divide by A, and it becomes η*— {,3+ϊ,·-ί, + ;-° ; A A A A let this equation be transformed into one whose roots are multiples by A (Art. 17), and we have x*-px3-\-hqx2 — h2rx + A3$=0 ; an equation, the coefficient of whose first term is unity, whose second term is the same with the second term of the original equation, and whose other terms are derived from the original equation, by multiplying the terms of this lat- ter equation (beginning with the third) by the successive powers of A. The rule is evidently applicable to equa- tions of all dimensions ; and if the roots of the resulting equation be «, β, y, δ, &c., then (since they are multiples by A of the roots of the original equation) the roots of the original equation will be « β Y i &C ti ti til’ II. Take now the equation -~/r+HT “+ c* 5 and let it be transformed into one whose roots are multi- ples by hlcl, then the resulting equation (Art. 17) will be xn—klpxn~i -f h2kl2 qxn~2—A3¿3/2ra?”~3-f &c. = 0, an equation all whose coefficients are integral; and if the roots of this latter equation be «, β9 y, δ, &c. then the roots of the given equation will be J?___V_ JL &c hkV hkl9 hkV hkV ' For instance, if it be required to transform the equation T3—+ x+ 1=0 into one wrhose coefficients shall be integral, multiply its terms (beginning with the second) by the successive powers of 6, and it becomes X3 — 3r + 24.T + 216 = 0; let20 TRANSFORMATION OF EQUATIONS. let the roots of this latter equation be «, β, y, then the roots of the given equation *3— 2 #2 + tt+ 1 = 0 are -, -, -. 2 3 6 6 6 20. To transform an equation into one whose roots shall be greater or less than the roots of the first by the given quantity a, we must put y = *±a or *=y:Fa9 and for X and its powers in the given equation we must substi- tute y + a and its powers, and the resulting equation will be the equation required. For example, let it be required to transform the equa- tion *3—10*2 + 31*—30=0, whose roots are 2, 3, 5, into one whose roots are less by unity than the roots of the given equation. Make y=x—l or *=y +1, then *3= (y+i)3 =y3+ 3*/2+ 32/+ h 10*2= 10 (y + 1)2= - 10y2-20y-10 { + 31*= 31 (y + l) =....+ 3ly + 31 j y + 143/ -30 =........................-30' an equation whose roots are 1, 2, 4, or less by unity than the roots (2, 3, 5) of the given equation. To transform the equation *3—2*2—23* +60=0, whose roots are 3, 4,-5, into one whose roots shall be greater than the roots of the given equation by 2, put y = * + 2 or xz=y— 2, then x3= (y—2)3=y3-6y2+l-2y— 8^ — 2x2— — 2(y- — 23* =—23 (y- + 60 =....... an equation whose roots are 5, 6, —3, or greater by 2 than the roots of the given equation. -2)3=y3-6y2+12y- 8x -2)2= -2y*+ ay- 8i_ü3_8w*- -2) = ___—23y + 46| y by ..............+ 6 -H Oí co « 1 oí 1 g Y o oô « o Cm + -H ó o¿ I Ö Cri J II II II II II JU 1 8 H ©i 1 « % ¿¿ c-ι £ g g £ o 5ζ 1 + -H H- ~H Then this equation becomes yn + Pyn l + Qyn 2 + &c.. . , + Ty'2 + Vy+ JV—0, an equation which is of the same dimensions as the original equation (A)=0, and whose roots are «—α, β—α, y—a, δ—a, &c.22 TRANSFORMATION OF EQUATIONS. 22. In reviewing the coefficients P, Q, &c. . .. T, V, TV, taken in an inverted order, we may observe, “ that TV is the " given equation, having a instead of x ; that V is derived “ from W, by multiplying the coefficient of a in each term " by its index, and diminishing the index by unity ; that " T is derived from V, by multiplying the coefficient of a “ in each term by its index, diminishing the index by unity, “ and then dividing by 2. And if the next term were “ written down, we should find that it would be derived “ from T, by multiplying the coefficient of a in each term “ by the index, diminishing the index by unity, and then “ dividing by 3; and so on.” Take, for instance, the biquadratic equation #4—px3 + qx2—ræ-f-s=0, whose roots are «, β, γ, δ, and let it be required to transform it into one whose roots shall be less than the roots of the first by the given quantity a ; then if $=a4—pa3-\-qa2—ra-\-s, Β=£α3 — 3ρα2+ 2qa—r, Q=6a2 —3pa + (/, P=4a—p, the transformed equation will be p4 + P//3+ Qy* + Ry + $=0, and its roots «—α, β—a, y—a, δ—a. Here it may be observed, that S is the given equation, having a instead of x ; R is derived from Sy by multiplying each term of S by the index of a, and diminishing the index by unity ; Q derived from P, by multiplying each term by the index of a, diminishing the index by unity, and dividing by 2 ; and P derived from Q, by multiplying each term by the index of a, diminishing the index by unity, and divid- ing by 3. 23. If the value of a be such that any of the quantities P, Q, R, &c. are equal to 0, then the transformed equation will be deficient in its second, or third, ox fourth term, &c. ToTRANSFORMATION OF EQUATIONS. 23 To find such a value of a as shall make the equation de- ficient in its second term, we must put P=na—p=0, in which case na=p ora=^. Hence this rule; “Assume n " some other unknown quantity y, annex to it ± ^ (using the “ sign + when that of p in the equation (.A)=0 is negative, “ and the sign—when it is positive); then substitute y±- " and its powers for# and its powers in the equation (A)=0, “ and the resulting equation will be deficient in its second “ term of which the following are examples. I. Take the equation x'2—px + q = 0, in which n=2; substitute y + ^p for x, and we have x~=y2+py + lp\ —ρχ= -py-ïp2[=2/2-ip2+î=°> + q = + 3 + y + $-f &c. =Fn n 24. To * The upper signs correspond to the case where the sign of p in thé equa- tion (^)=0 is —, and the lower to that where it is +.TRANSFORMATION OF EQUATIONS. 25 24. To transform the equation (A)=0 into one that shall be deficient in its third term, we must put Q=TiiìÌIll\ar—(n—l)j9a+ç==0,ora2—?JL_ ==0; V 2 J K Jr 1 n n(n— 1) in which case, by solving this quadratic equation, we have a=P±K/Pl- η V n‘ n(n—1) Take, for example, the equation x3—5x*-|-3x—10 = 0 ; in which n = 3, p= 5, q=3, 2d ___. /25 _6_ /Γβ_ j4 ra* min—1) V 9 6 V g 3 ’ and\/- η . 3 V η,1 ρ__5 i 5Η- 4 hence a— _^=_=3 or i. 3 3 Now, let X *=?/ + 3, then x3=y3 + 9y2+ 27 y + 27 — 5 x'2= —5 y2— 30y — 45 + 3t =.......+ 3y+ 9 -10 =................-10 Let X =y then x3=y3 + if+ $y + L - 5x’= f + 3a? =......+ 3y + 1 -10 cr...............-10 In each case, therefore, it appears that the given equation is transformed into one deficient in the third term. |=2/’ + 4 y2-19=0 25. From what has been shewn in the two last articles, it may be observed, “ that an equation may be transformed “ into one deficient in its second term, by the solution of a “ simple equation ; into one deficient in its third term, by M the solution of a quadratic equation ; and by referring to “ the table of coefficients in Art. 21, it would be found, “ that it might be transformed into one deficient- in its wth “ term, E26 TRANSFORMATION OF EQUATIONS. “ term, by the solution of an equation of η—1 dimen- “ sions * ; but it cannot be transformed into an equation “ deficient in its last term, without the solution of an equa- “ tion of the same dimensions with it self \ inasmuch as “ __jpa«-ï _|_ qan~2_r a 4. &c.” This rule, therefore, does not afford us the means of trans- forming an equation into one of lower dimensions than its own ; for by Art. 11 it appears, that unless we can make the last term of the equation y”-fPyn“1+ Qyn~2+ &c........+ Ty2 + Vy+ W=0 vanish, we cannot divide the equation by the unknown quantity y, so as to reduce it to an equation of η—1 di- mensions. 26. To transform an equation into one whose roots are the squares of the roots of the given equation, let «, β, y, δ, &c. be the roots of the equation (A) xn—pxn~1 + qxn~2—rxn~~3 + sxn~i—&c. = 0 ; then (by Art. 18) the roots of xn + pxn~1 + qxn~2 + rxn~3 + sxn~* + &c. = 0 will be —a, — β, —y, —δ, &c. (where p, q, r, &c. as well as «, β, y, &c. are supposed to represent any quantities, positive or negative) ; if then these two equations be mul- tiplied together, there will arise an equation of 2 n dimen- sions of the form X2 n +p'#2n~2 -f- qfx2 n~1 + r'x2"~6 -f &C»=0, or, if æ2=p, of the form yn + p'y n_1 + q'y n_2 + r'y n~3 + &c.=0, the roots of which will be «2, β2, y2, $2, &c. as will appear from the following process. xn — # For in this case, Γ=ηαη-1-(w—1) pan-2 + (n —2)ça"-3— &ο.=0.:).(#—β) (X—y) (x — δ) &C. .) (a? 4 0) (a? 4 y) (æ + δ) &C.; TRANSFORMATION OF EQUATIONS. 27 O o H I) 83 ^ CO CO 4 4 η fo i I « « S- V. I 4 1 1 H « Oh Cw 4 4 1 I 83 83 A, S* I 4 Ό d8 I CV* I % 83 Oh ©* 4 83 "öi 83 Oh ©H Oh 83 8- 4 1 4 ci ci U ci ci ^83^ •S c8 «a <8 «8 II II J 4 1 1 4 1 T* 1 ci ci 1 8 1 δ 1 8 1 8 1 8 <% 0i % 83 Ci c* 83 'W J 4 •Kit 00 8* ©s ’^1 8» &. CO s 1 8 4 1 4 1 4- .§ a $ .2 c 2 8 ® W » il _T ® ^3 β. O* on jg u +* I ■5 ά M <* ce -c M o> CO 1 Ί3 ci S S •e*i 0 Sí JG <ü £ m JG bJO ST .S Cm 0 ;f *3 w #G • ^ H»T +2 ’ ^ C-M • H G O 1 H3 fi Co Oh ce G GM CD JG m 4 fti tlT s^s I h <3 o U a> JG o ■5 0 *> <*} G3 a 2)ÿs + (g!—2pr)y—r2=0 is the equation whose roots are a", β~, y‘. III. InTRANSFORMATION OF EQUATIONS. 29 III. In the biquadratic equation x4—px3 qx2 — ra? + $=0, whose roots are «, β, y9 δ, since all the coefficients of (A)= 0 may be considered as equal to 0, except p, q, r, s; we have p'—2q -p* q' = 2s — 2 pr + q* r' — 2qs — r1 ,v' = s2; . .·. y*-\-(2q—p2)y3 + (2s—2pr+q2)y2 + (2qs—r2)y + s2 = 0 is the equation whose roots are «% β2, γ2, δ2 ; and in this manner we may proceed to the higher orders of equations. By way of example, let it be required to find the equa- tion whose roots are the squares of the roots of the equa- tion X3 — 7 X 4- 6 = 0. Compare this with x3—px2 + qx—r=0, then —p=0 + + l)#2— (p+ 1)^+ j *—(p+ 1 )æ4 + +1)®* * — (p + l)a?2—jpæ —(p+ l)#2—(p + l)a? * ã+1 x + 1 * * From which it appears that the quotient is a recurring equation of an even number of dimensions ; and if the equation xb—px*-\-qx3—qx2’\-px—l =0 be divided by x—1 = 0, the quotient will be a?4——1) Λ?3 —jo-f-1)»*—(p— l)x+ 1=0, which is also a recurring equation of an even number of dimensions. III. InTRANSFORMATION OF EQUATIONS. 35 III. In like manner, by assuming a recurring equation of an odd number of dimensions in its most general form, and dividing it by x± 1, it will be found that the resulting equation is always a recurring equation of an even number of dimensions of one dimension lower. Hence it is evident that recurring equations of an odd number of dimensions may always be reduced to such as are of an even number of dimensions, by dividing them by æ± 1* Our remaining observations upon equations of this kind will therefore be confined to such equations as are of an even number of di- mensions ; and, to prevent a confusion of signs, we shall make use only of such equations as have all their terms positive, as the operations may be easily extended to equa- tions which have negative terms (conformably to the sym- metrical arrangement mentioned in No. I.) by substituting —p, — q, &c. for + p, + ®+1=0, let /)=0, and it becomes ¿ρ5+^λ?3+5,λ?2+1==0 ; divide by λ?+1=0, and the resulting equation is ¿r4—λ?3+(5'4,1)λ?2—#+1=0. Letp=0, 9=0, and it becomes #5+l=ss0 ; divide by #+1=0, and it becomes #4—#3+#3—#+l=0. In each of these cases the resulting equation is evidently a recurring equation.36 TRANSFORMATION OF EQUATIONS. Let this equation be divided by xn, and it becomes xn + pxn~x +oæn“2 +&C-+ ¿ + &C....4—2L + ~£~-f-L==0 : * xn~l xn~l xn which may evidently be written thus ; (*η+ΐ)+κ*η_,+jk)+?(y"5+^)+&c................+*=°· Hence it appears that the different orders of recurring equations of an even number of dimensions may be repre- sented in the following manner ; viz. x2+px 4-1 . ...........4-............................=0 x4+px3 + qx2+px + 1.....===(α;2+“2)+ί>(ίΡ + “-) + ?...= ° x*+px& + qx* + rx3 + qx2+px + l=fx34- —^ + p(x~+-^ + q(x + ~>\ + r=0 &c. = &c........................ =0. V. Now let , 1 x + -=z9 X thenfa^-^ =x2 + 2-f “ = ^2; x2+ -L=*2—2, v X/ X X2 (x + =x3 + 3Íx + Ιλ+ i_ = a?3+ 3 z + — = ;?3 ; or a?3+ -i=z3—3z; V xJ V xJ Xs X3 X3 9 &C. = &C. Let these values of x4* - , x2 + }~ a?34- -L, &c. in terms X x x of z, be substituted in the equations, at the end of No. IV., and we have x2+px +1..................=r 4-p...........................=0 xi+px* + qx*+px-{·1.......=(z2 — 2)+pz + q =z2+pz + (q—2)................=0 x* +p χδ + qx* + rxz + qx2 + p x +1 =(z*—3z) + (pz%—2p) + qz-t:r =2r34*jp^2 + (ÿ—3)z + (r—2p) . . . =0 &c. =&c..........................=0. VI. NowTRANSFORMATION OF EQUATIONS. 37 VI. Now the quadratic equation x*+px + l—0 may be solved by the ordinary method, and its roots are + and -ip-\/¡P*-1· For the solution of the biquadratic equation we have two conditions ; viz. x+ ì=r, or X2—zx + 1=0, X and x*+px'° + qx2 + px + l=z2+pz + (q — 2)=0 ; if, therefore, the two roots of the quadratic equation z2+pz + (q—2) = 0 be «, β, and we substitute them successively for 2 in the equation x2—zx + 1 = 0, we shall have two quadratic equations X2 — ax+ 1 = 0, T2—1 =0, the four roots of which will be the four roots of the recur- ring equation X* +pxz + qx2 +px + 1 = 0.* For * Take, for example,· the recurring equation , 35 , , 31 35 , , Λ a*—■- #34- ¿>2— ~ #4 1=0. o o o 35 31 Here p=— —, and q=-~ ; the equation jsr2 —2)=0 is there- ^ * fore z*—■— z + -r =0, whose two roots are —- and-; let these be sub- o 3 3 2 stituted successively for z in the equation x2—zx 4 1 =0, and there arise 10 5 two quadratic equations, x2— *- «2? 4 1 = 0, and x2— - #41=0, the roots 3 JL of the former of which are 3, £, and of the latter, 2, \ ; hence the four roots of the given recurring equation are 2, 3, 2, §.38 TRANSFORMATION OF EQUATIONS. For the same reason, if the three roots of the cubic equa- tion z*+pz* + (q—3) z + (r—2p)=0 be α, β, y, and we substitute them successively for z in the equation a?2—ZX+ 1 = 0, we shall have three quadratic equations, #2—«a? +1 = 0, X2—βχ + 1 = 0, #2 —yæ + 1=0, whose six roots will be the six roots of the recurring equa- tion x6+pxs + qx* + rx* + qx2+px + 1 = 0. From hence we infer, that a recurring equation of 2n dimensions may always be solved by means of an equation of n dimensions ; for if the n roots of the latter equation be «, β, y, δ, &c., then the several quadratic equations X2—«#+1 = 0, X2—βχ +1 = 0, #2—yT + 1 = 0, #9 —+ 1 = 0, &C. &C. will contain the 2n roots of the given equation. But in proceeding to the actual solution of these equations, we should anticipate the subject of the Fourth Chapter.89 CHAP· III. ON THE METHOD OF FINDING LIMITS TO THE ROOTS OF EQUATIONS. SECT. I. On the method of finding quantities which shall exceed the greatest positive root9 or the greatest negative root, of any given equation. 29. Let the roots of the equation (A)=0 (being all possi- ble, and arranged according to the order of their respec- tive magnitudes) be «, β, — y, — δ, &c. ; a being the greatest positive root, β the next, and so on ; — y the least negative root ; —δ greater than —y ; and so on. Now let this equa- tion be transformed (by Art. 21) into one whose roots shall be a—ο, β—a, —y—α, —δ—a, &c. by making y—x—a or x=y+a9 and, substituting y + a and its powers for x and its powers in the given equation ; then if a be greater than the greatest positive root (a) of the equation (^4)=0, it is evident that all the roots of this transformed equation will be negative, and that all its coefficients will be positive (Art. 8), or the equation itself of the form yn + Pyn“"1 + Qyn“'2+ Ry^ + Stc______+Vy + W=* 0. 30. Hence, if such a value of a be found as shall make all the coefficients P, Q, R, S, &c. positive, it will be greater than the greatest positive root («) of the equation (A)=z0. Take,40 LIMITS TO THE ROOTS OF EQUATIONS* Take, for example, the equation x4—6a?3—4 X2 + 22 x + 9=0, and let it be transformed into one whose roots shall be less by a than the roots of the given equation ; then if y*+Py3 + Qy* + Ry + S=o be the resulting equation, we shall have (by Art. 22) S= a4 — 6a3 —4a2 + 22a +9 J?=4a3—18 a2—8 a +22 Q=6a2 — 18a —4 P—4 a —6. Substitute the numbers 1, 2, 3, 4, 5, 6, &c. successively for a in these expressions, and the first number which makes them all positive will be found to be 7 ; the num- ber 7, therefore, is greater than the greatest positive root of the equation a?4—6 a?3 —4a?2+ 22 a? + 9=0. Now, let the signs of the given equation be changed al- ternately, beginning with the second, and it becomes a?4 + 6 a?3—4 a?2—22 a?+ 9=0, an equation whose roots (by Art. 18) are the same with the roots of the given equation, but with contrary signs ; the greatest positive root, therefore, of this latter equation will correspond to the greatest negative root of the given equa- tion. Let this equation be transformed, as before, into the equation t/4 + P¿/3 + Qy2 + Ry + £=0, and we shall have S= a4+ 6a3—4a2—22a + 9 i?=4a3 + 18a2—8a —22 Q=6a2+ 18a —4 P=4a +6. Now, if 1, 2 be substituted successively for a in these ex- pressions, 2 will be found to make them all positive ; hence theLIMITS TO THE ROOTS OF EQUATIONS. 41 the number 2 is greater than the greatest positive root of the equation a?4+6a?3—4a?2—22a?+9 = 0, and consequently —2 is greater than the greatest negative root of the equation a?4 — 6 a?3—4 a?2 + 22 a? + 9 = 0. Thus the four roots of the equation a?4—6 a?3—4 a?2 + 22 a?+ 9 = 0 all lie between + 7 and —2; the number 7, moreover, is the least whole number which exceeds the greatest posi- tive root, and the number —2 the least whole number which exceeds the greatest negative root, of that equation. 81. Having found a quantity (6) which exceeds the greatest positive root (a) of the equation (^4)=0, whose roots are «, β9 —y, — δ, &c., let y=fr—a?, or a?=6—y, and * substitute b—y and its powers for a? and its powers in that equation, there will then arise another equation whose roots will be 6 — α, b—β, b + y, ò + δ, &c., which areali positive quantities. For example, the number 2 has been found to exceed the greatest positive root of the equation a?4 + 6 a?3 —4 a?2 — 22 a? + 9=0, which, from the nature of its signs, has evidently two posi- tive and two negative roots ; substitute 2 — y for a? in this equation, then a?4= 16 — 32 ?/ + 24 —8 ?/3 + y4>\ + 6a?3= 48—72y + 36y2 — 6ÿ3 I — 4a?2= —16+ 16y— 4 y2 \ = î/4 — 14 3/3 -f- 56i/2— 66y + 13 —22a? =—44 + 22 y j + 9=9 / an equation whose signs are alternately positive and nega- tive, and consequently all its roots positive. In this manner an equation having both positive and negative roots may always be transformed into one whose roots shall be all positive. G 32. The42 LIMITS TO THE ROOTS OF EQUATIONS. 32. The roots of the equation (A)= 0 being «, β, — y, —δ, ’ &c. we have (^4)==(ír—«)(# —β)(# + γ)(# + δ) &c. = 0. Now, let π, p9 ±σ, —τ, — χ, be a series of quantities, such, that π is greater than the greatest root « ; p greater than /3, but less than a ; ±2)=/?2—2 q48 LIMITS TO THE ROOTS OF EQUATIONS. III. Supposing the roots of the equation ( A)=0 to be the same as before, then«2β2 + «2γ2 + &c. -f β2y2 + &c. + y2h2+&c. (ton (^) quantities) = q2 — 2pr + 2s* ; if, therefore, « be greater than any of the other roots, n (^n~~ must be greater than ç2—·2pr+ 2s, and consequently «4 greater than gy-4pr + 4, of j n(n — 1) V n(n—l) But if δ be greater than any of the other roots, then δ4 will be greater than*2^ —4pr^+4s^ an(j conseqUen^y —\f ~ ~ a ne8at*ve quantity /ess than •For α2/32 + α272+&ο.+/32724-&ο.+γ2δ2+ &c. is the coefficient of the third term of the equation whose roots are the squares of the roots of the equation (^)=0. (Art. 26.) + With respect to all these rules, it may he observed, that “ if a be much greater than any of the other roots (or indeed if all the roots are “ not pretty nearly equal to each other) the quantities n Or1) a2, w«2, ·■*(¥) a4 will be much greater than q, p2—2>q, ç2—2jor+2s “ respectively ; and consequently Λ / 2 q /p2—2q 4 / 2?2—4jor + 4s V »(»~1)' V η ' V »(»-1) ' “ so much less than a, as to be no approximation to that root.”LIMITS TO THE ROOTS OF EQUATIONS. 49 SECT. II. On the method of finding a limiting equation to any given equation. 36. The roots of the equation (Â)=xn — pxn~l + qxn~'2 —rxn~3 + &c.... ±tx'2:fvx±w — 0, arranged according to the order of their respective mag- nitudes, being α, β, —y, —δ, &c., let this equation be transformed (by Artidi) into one whose roots shall be a—a, β—α, — y—af —δ—a, &c., and let the resulting equation be yn + Pyn~l + Qyn~2+ &c. . .. -f Sy3+ Ty2 + Vy + fV= 0, then V (the coefficient of the last term hut one of this equa- tion) is equal to nan~l—(n— l)jpan~2 +(w—2)qan~z—&c. .. . ±2ίαψν. But (by Art. 2) the coefficient of the last term but one of an equation of n dimensions is equal to the sum of the pro- ducts of any w—1 roots with their signs changed ; hence F= (a —a) (a—β) (a -f y) &c. -f (a—«) (a—β) (a Η-δ) &c. + (a—a)(a + y) (a-fà) &c. + (a—β') (a -f- y) (a + δ) &c. + &c. &c. &c. Now let the roots «, β, — y} —δ, &c. of the equation (A) = 0 be successively substituted for a in this value of F, then (since all but one of these combinations of roots become equal to 0, at each step of the operation) the results will stand thus ; When a is substituted fora, F= (α—β) (« + y) (« + δ) &c.= + /i β...........for a, V= (/3—a) (/3 + y) (/J-f&) &c.= —i —y...........fora, F=( —y—«)(—-y—β\—y + δ) &c.= + k — δ..........fora, F=( — δ — «)( — δ—β)( — δ + y) &C.=—l V= &c. &c. &c. = + &c. h where50 LIMITS TO THE ROOTS OF EQUATIONS. where the signs of the quantities h, i, k, l, &c. are deduced from those of the factors, as in Art. 32. 37. From hence it appears, that when the roots β, —γ, — δ, &c. of the equation = 0 are substituted suc- cessively for a in the equation F= 0, or for x in the equation nxn~l— (n—pxn~2 + (n— 2 )qxn~3—&c.... ±2t x:fv=z09 the results are alternately 4- and — ; the n roots of the equation (^) = 0 are therefore (by Art.32) limits between the (η— l) roots of the equation nxn~l—(n — \)pxn~2 + (n—2)qxn~3— &c. . . ±2tx:fv'=0, and vice versaf the (η—l) roots of this latter equation are limits between the n roots of the equation (A) = Q. Let these η— 1 roots be ?r, ±p, — σ, &c. and the arrangement is a β —7 —5 &c. 7Γ ± p — σ &c. This equation nxn~l— (n— l)pxn~2 + (n—2) qxn~3 —&c. . . . ± 2ίχψν = 0 is called the limiting equation to the equation (A)=: 0; and it is derived from (A) = 0 “ by multiplying each term of that “ equation by the index ofx, and then diminishing the index “ of that term by unity, in the same manner as V is derived u from W in Art. 22. 38. The limiting equation to the equation (F) = 0, found by this rule, is n(n — l) xn~2 — (n—l)(n—2)pxn~3 + (n—2')(n—3)qxn~i— &c.. ± 2¿=0> for it is derived from (V)=0 in the same manner as (V)=0 is derived from (A)=0. Divide this equation by 2, and it becomes w(w-i)r-»- («: 2 • Ute-T.gij,+ (n-lXn:r% &c... ±t=o, which (by Art. 21) is the coefficient (T)=0 of the trans- formed equation yn + pyn-1 + Qy-* + &C. .. S y* + Tif + Vy + W= 0, havingLIMITS TO THE ROOTS OF EQUATIONS. 51 having x instead of a; hence the equation (T,) = 0 is a limiting equation to the equation (F) = 0. In the same manner it might be shewn that the equation (S) = 0 is a limiting equation to the equation (T) = 0; and by pro- ceeding in this way we may obtain a series of equations, (F) = o, (I0=o, (S)=o, &c-(Q)= o, (P)=o, such that the roots of each succeeding equation are limits between the roots of that which precedes it. Take, for example, the equation x*+9x3+9x2—41X-42=0, whose roots are 2, — 1, —3, — 7 (Art. 9); the limiting equation to this equation is 4#3+27α?2 + 18jc —41=0, or (dividing by 4) x3+ 6.75a?2 + 4. 5#—10.25 = 0 ; this cubic equation will be found among the examples on approximation in Chap. IV., and its roots are .90018, —2.024, —5.626. The limiting equation to the equation 4^3+27t2+18t—41=0 is 6t2 + 27t+9=0 or (dividing by 6) τ2 + -|t4- -|=0, whose two roots are _4.137 4 4 The limiting equation to 6τ2+·27#+9 = 0 is 4# + 9=0, in which #=—-J= — 2.25. The series of limiting equa- tions to the equation #4 + 9æ3+ 9ar—41#—42 = 0, with their corresponding roots, may therefore be thus ar- ranged; Roots LIMITS TO THE ROOTS OF EQUATIONS. CO CM CO C5 I CO CM o m CM CM CO co o o 05 0 o o II II II. CM —' 05 Tf Tf I I I H ^ 5S i—I 00 ì> Tf «-H (M 1 + + W IN H « ** 05 N «O CM + 4 « 05 4 8< « o CO o o tí 0 o +* Λ f^ -t-i Il o ce ï* CM ¡É* . 0) • rH ς3 O-ι ¿6 S 1 3 J s I 8 0 S* ja u + § I Ί3 « ce « 3 O., CT1 05 T ° I -a c -H H P β rO Η- 14 Ô o 14 7 ^ η O ÖN Ö -o + c? HH • 1 8 8 « « * h3 ©N ^ © 8 3 β *o II nx 3 1 ^ 05 ' © 1 4 3 .2 X H as 3 sr 3 fi H- ct'o a) « O rj 3>®3 ~ .S « s „■£2 0) Ό ,δ^-Η ■1 o ¡j. ji ^ ■“ ^ o « - II ► .2 ^ o ·« N—' ¿ § e σ< .2 £ « s 50 ω 3 -a 5? -*-» s 41 β j* Y o II § e -H « o <ü ΙΛ g .& 2 .2 «2 r. £3 8 H § 8 §■ 3 ê « co qj 'O -O -η ~ ^ T -H >> ® A Æ if f - : 11 ü CJ OO I «' g i .2 CM *3 ce ¡g 3 O cr o; +1 ✓-N CM 1 o 1-3 -M β ncT s t+x ρ··Η N_/ Ή O (0 1 S 4- t-4 Y 1 o Ö 1 e »^r +j e cu Μ Ο 41 ·ρ> ö 1 § g +1 S .0.3 O 'U C*X í'8 g f. *2 S ■H Í3 3 Y * % J g Cvh .2 o I¡ Sx bo o u P-I *3 ►o Y 'e' These equations, of course, bear the same relation to each other as S, R, Q, P bear to each other in Art, 22.LIMITS TO THE ROOTS OF EQUATIONS. 53 equation (which is evidently of n dimensions) is also a limiting equation to the equation (^4)=0.* 40. In order to prove this, we must (in conformity with the process in Art8.36, 37,) substitute the roots of the equation (A) = 0 in the equation (A a) ± (Bbx)=0, and shew the nature of the results. When these roots are sub- stituted in the part (A a), the results in each case are 0 ; f the results therefore, which are to determine how far the equation (Aa)±(Bbx)=0 is a limiting equation to the equa- tion (A)=0, must entirely depend upon the value of the quantities which arise from the substitution of those roots in the other part ±(Bbx). I. LetQi) he positive, or the series a decreasing one ; and let the roots of the equation (A)=0 be all positive, or all negative. In this case, the equation in question is (^4α) + (.Βό#)=0. Now (by Art. 36) when the roots «, β, y, δ, &c. are sub- stituted in the equation (2?)=0, the results are + Λ, — i9 + k, * As the quantities a and b have been arbitrarily asssumed, it is evident thataiwS, ai.(n— l) b, ai(n—2) b, &c.. . αίδ, a, maybe any Arith- metic progression whatever. If the terms of this progression be all positive or all negative, no change will be made in the signs of the terms of the equation (A) — Ç) when multiplied by it. Under these circumstances, therefore, the equations (^)=0 and (Aa)±.(Bbæ)=iO will contain the same number of positive and negative roots (Art. 9). The quantities a and b might, however, beso assumed as to make the series «in δ, ai (η*-1) δ, &c. to consist partly of positive and partly of negative terms, in which case such a change might be made in the signs as to alter the fore- going relation of the roots of the equations (-¿f) = 0 and bæ) — 0 ; but this is a case which at present is not under consideration. j* For when we substitute the roots α, β, —yt —8, &c. in the equation (-¿0=0, the results areO; when the same roots, therefore, are substituted in the equation (Aa)=0, the results will be a X 0 = 0.54 LIMITS TO THE ROOTS OF EQUATIONS. + k, —l, &c. ; when the same roots, therefore, are sub- stituted in the equation (Bbx) = 0, or in the equation (Aa)+(Bbx)=0, the results will be + bah, —bfii, -\-byk, —bài, &c. ; and when —a, —β, — y, —δ, &c. are substi- tuted in (Bbx)=0, the results will be —bah, + όβί, —byk, + bài, &c. Hence it appears, that when the roots of the equation (A)=0 are all positive or all negative, and are substituted successively for a in the equation (Aa) + (Bbx)=0, the results will in each case be alternately positive and negative; the roots of the equation (A)=0 are therefore limits between the roots of the equation (Aa) + (Bbx)=0 (Art. 37), and, vice versa, the roots of the equation (Aa)+ (Bbx)=0 are limits between the roots of the equation (A)=0. Let the roots of the equation (Aa) + {Bbx)=0 be ±π, ±p, ±σ, ±τ, &c. then the arrangement, when the roots are all positive, will be + α +β +7 +δ &c. + 7Γ +jO q-σ +T &c. and when they are all negative, —a —/3 —7 —£ &c. —7Γ —p — σ —T &c.* II. Now let (&) be negative, or the series an increasing one ; and the roots of the equation (A)=0, all positive or all nega- tive. In this case, the limiting equation is (Aa)—(JB b æ)=0 ; and substituting as before, the results, when the roots are positive, will be —bah, ^-bfii, —byk, +όδΖ, &c., and when they are negative, the results will be +bah, —bfii, +byk, —bài, &c. Since the order of the signs of the results in this case is changed, the order of the roots will be changed, and * —a is here understood to he the least negative root, —j8 greater than — a, and so on, conformably to the order usually observed (Art. 29).LIMITS TO THE ROOTS OF EQUATIONS. 55 and the arrangement, in the case of the positive roots, will be + a + /3 +7 + δ & c. +7Γ + p +σ +T &c. and in that of the negative, —a —β —y —δ &c. —7Γ —p — σ — τ &c. III. iet (6)6e positive, or the series a decreasing one; and the roots of the equation (yi)=0 some positive and some ne- gative, as a, β, —y, -—δ, Substitute these roots as be- fore, and the results are + bah, — bfii, —byk9 + b%l, &c. ; of which it may be observed, that in passing from the sub- stitution of the positive root β to that of the negative one —y, there are two quantities (viz. —ò—byk) affected with the same sign; two roots or no root, therefore, of the equation (Aa) + (Bbx)=0 must (by Art. 33) lie between the roots β and — y of the equation (A) = 0. In this case it will be found that two roots lie between them. For the arrangement of the roots, «, β, —y, —δ, &c. and π, p, — σ, —τ, &c. must begin as if all the roots were positive; hence, by No. I. of this article, the arrangement begins thus ; a β &c. 7Γ p &C. Invert the order of the roots, and the arrangement will begin as if all the roots were negative, therefore (No. I.) the arrangement is —5 —y & c. —τ —σ &c. Hence, upon the whole, the arrangement is, a β —γ —S &c. 7Γ p —σ —τ &c. where a positive and a negative root (p and — ON THE SOLUTION OF EQUATIONS. 65 “ be then placed under each other, as in the table marked " (X), and select from them all the arithmetical progrès- (t sions (in a vertical direction) whose common difference is " unity; and the integral roots of the equation will be " found among the terms of these progressions which cor- “ respond to the supposition of æ=0. If the progression “ increases from the top to the bottom, the numbers must “ be tried with a positive sign ; if it decreases, they must “ be tried with a negative one as will appear from the following Examples. Example I. Take the equation xz — 5æ2-}-10# — 8=0. Results. Divisors. Aritli. Progr. Let x— I 2* 1, 2 1 x=0 8 1, 2, 4, 8 2 X = — 1 24 1, 2, 3, 4, 6, 8, 12, 24 3 Here the number to be tried is 2* which is found to suc- ceed ; the only integral root of the equation, therefore, is 2. Ex. II. Take the equation a?4—·8x3 + x2 + 82x—60 — 0 (in p.62.) Results. Divisors. Progressions. 16 1, 2, 4, 8, 16 1,2, 4,4 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 2, 3, 3, 5 132 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132 3, 4, 2, 6 The numbers tobe tried are 2, 3, —3, 5 ; of which only 5 and — 3 succeed; the only two integral roots of the equation, therefore, are 5 and —3. Ex. III. * As we only require the divisors of these results, they may be set down here without their signs. K66 ON THE SOLUTION OF EQUATIONS. Ex. III. Find the integral roots of the equation æ5 + æ4 — 14 a?3 — 6æ2-f 20 æ + 48 = 0. Results. Divisors. Progressions. Let æ=l 50 1, 2, 5, 10, 25, 50 1, 2,5 æ=0 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 2, 3, 4 æ= — 1 36 1, 2, 3, 4, 6, 9, 12, 18, 36 3, 4, 3 Here the numbers to be tried are 2, 3, — 4, all of which are found to succeed ; so that the equation has three inte- gral roots, viz. 2, 3, —4. The equation whose roots are 2, 3, —4 is (oc —2) (a? —3) (æ + 4)=æ3—æ2— 14a? «f 24 = 0; let the given equation be divided by it (Art. 7), and the quotient is x2 + 2 x + 2, whose roots are the five roots of the given equation are, therefore, 2, 3, —4, — - Ex. IV. Find whether the equation x4 + 3a?3-|-2æ2— 10æ+28 = 0 has any integral roots. Results. Divisors. Progressions. Let a?= 1 24 1, 2, 3, 4, 6, 8, 12, 24 3 Æ=0 28 1, 2, 4, 7, 14, 28 2 — 1 38 1, 2, 19, 38 I If this equation has any integral root, it must, by the rule, be —2, which upon trial is found not to succeed; the equation, therefore, has no integral root. All this accords with what has before been stated, viz, ts that there must be “ at least as many arithmetical progressions as there are “ integral roots but as there may be more, it will happen that the rule sometimes gives progressions when there are no integral roots; and this is a very ready way of deter- mining whether an equation has any such roots or not.ON THE SOLUTION OF EQUATIONS. 67 # SECT. II. On the solution of equations whose roots bear some given relation to each other. If two or more roots of an equation be equal to each other ; if its roots be in arithmetical or geometrical progres- sion; if it be a recurring equation; &c. &c.; it then ad- mits of a particular solution, the nature of which we now proceed to consider ; and first for equations which have equal roots. 48. Let the roots of the equation (A)=0 be a, β, y, δ, e, &c. and let this equation be transformed (by Art. 21) into one whose roots are «—α, β—a, y—a, δ—α, e—a, &c. by making y=æ—a or æ=y-f a,and let the resulting equation be yn + Py1*"1 + Qyn~2 + &c.... + Sy3+ Ty2+ Vy + W~0, where W= the product of all the roots with their signs changed =(a—a) (α—β) (a—y) (α—δ) (a—e) &c. If a = a, then IF==0, and the equation, being divided by y, becomes yn_1-f-Py”~24-Qyn~3 + &c.....+ $y2-f- Ty + V=0, where F= the product of the η— 1 remaining roots with their signs changed =(α—β) (a—y) (α—δ) (a—e) &c. If /3 = a, then F=0 ; and the equation, being divided by y, becomes yn~2 + Pyn~z + Qyn~4 + &c. . . . -p£y-f P=0, where T= the product of the n—2 remaining roots with their signs changed = (a—y) (α—δ) (a—e) &c. If y=a, then T— 0 ; and the equation, being divided by y, becomes yn~3+ Pyn~* + Qyn~b &c...............+S=09 where68 ON THE SOLUTION OF EQUATIONS. where S= the product of the n—3 remaining roots with their signs changed =(α—δ) (a—ε) &c., and so on. From hence it appears, that If «=β=α, or, if two roots of the equation (A) = 0 be equal to a, then if « be substituted in F, the result will be 0 ; and consequently a is a root of the equation F=0. If α=β=γ = α, or, if three roots of the equation (A)=0 be equal to «, then if « be substituted in jP, the result will be 0 ; consequently « is a root of the equation T=0. And in general, if m roots of the equation (A)= 0 be equal to «, then « will be a root of the equation which arises from making the coefficient of the (η —ra-f- 2)th term of the trans- formed equation {yn + Pyn~x + Qyn~2 + &c. = 0) equal to 0. 49. The equations F=0, 77=0, £= 0, &c. are the series of limiting equations, as they stand in Art. 88. The fore- going rule may indeed be immediately deduced from the nature of these limiting equations ; for the roots of the equation (A)=0 and of the equations F=0, T= 0, S=0, &c. may be arranged in the following order ; Roots of the equation (A)=0 ... « β y δ ε &c. F=0 ... tf iο σ 7 &C. 0 ... tf p' σ' &C. S = 0... tf" p" &C. If therefore the equation (^4)=0 has two equal roots («, β)9 then the root tf of the limiting equation F= 0, which lies between a and β, must be equal to either of them ; conse- quently « is a root of the equation F=0. If the equation (A) = 0 has three equal roots (<*, β, y\ then two roots (π, p) of the equation F=0 will be equal to a; consequently one root (π') of the equation T= 0 will be equal to «. And thus we may proceed through the whole series of equations. 50. SinceON THE SOLUTION OF EQUATIONS. 69 50. Since (by Art 48) the equation (J)=0 and the se- veral equations F=0, T~0, S= 0, &c. have each of them a root equal to a, these equations must all have a common measure (a? —a). To determine, therefore, the value of one of the equal roots in any given equation, we must find this common measure by the rule laid down in Art. 46 of the Elements, and then make it equal to 0 ; the simple equa- tion arising from this supposition will give the value re- quired. I. For instance, if the cubic equation a?3— pa?2 + ça? — r=0 has two equal roots, then this equation, and the equation 3a?3— 2 px + ç=0 have a common measure, which may be thus found, (EL pp.38, 39) 3 a?2—2px+ <7)3 a?3—3pa?2 + 3 ça?—3r^a? 3a?3 —2 px2 + qx —pa?2 + 2ça?—3r^ 3px2—2p2a?-fpç 3 3px2—6qx + 9 r (common measure) ^6ç—2p*^a?—^9r— pç) Let (6ç —2p2)a?—(9r—pç)=0, then x==^r~—^-7» which is the value of one of the equal roots ; and conversely, if ——PJL is substituted for a? in the equations 6ç —2p¿ ^ a?3— px2 + qx —r = 0, and 3a?2 — 2pa?-|-ç = 0, and in both cases gives the result 0, then will the equation a;3— px2 + ça?—r=0 have two equal roots. Example.70 ON THE SOLUTION OF EQUATIONS. Example. Find whether the equation #3 + a?2-— 16 a? + 20 = 0 has two equal roots ; and then solve the equation. Compare this equation with the equation x3—px2 + qx—r~ 0, 9r—pq ___ 6q— 2p~ then p= — 1 9 Ί f= 16 > — 20J -180—16_196__0 — yö —a 98 Substitute 2 for x in the equations a?3-f-a?2—16æ-f-20 = 0 and 3j?2 + 2 a?—16=0, and in each case 0 results ; the equation a?3+#2—16 x -f 20=0 therefore has two roots, each equal to 2 ; the third root is found by subtracting the sum of these two roots from p the sum of all the roots*, which gives that root == — 1 —4 =— 5 ; the three roots of the equation, therefore, are 2, 2, —5. II. Take now the biquadratic equation xi + qx2—rx+¿s=0 ; if this equation contains two equal roots, then the equations x4 + q x2 — r x + $=0 and 4a?3 + 2qx—-r=0 have a common measure, which may be thus found, 4a?3 + * For —p is the sum of all the roots with their signs changed (Art 2) ; consequently p is the sum of all the roots with their proper signs.ON THE SOLUTION OF EQUATIONS. 71 « (M 'β %- ce Cm CO 1 + SS c* ■M SS Cm s. fi fH 2 tS Jfi ^ H « JH «M O O O 00 00 rH r-i 4- I Si s* 00 00 + + H ce fi CT1 rf <ü O O S-t SS) s ;s ’S a a> u O •s Cm O O fi JH M ná fi 3 0) ce o o u O £ ■S g «s C0 fi ce Sh I C0 H « + « H O 00 H Tt< 00 + %* Ci + M « O (M I I H 00 H Ci + ”h O (M J *« S-i O .2 '■fi ná o* o ■n· ao oo 4- + c* « H H OS îO rr—7Γ» 2 + V3, ----y-, 2 3 + V5 2 + V3ON TH K SOLUTION OF EQUATIONS. 81 SECT. III. On the solution of equations of the form xn±kn=0. 55. Let x=ky, and substitute ky for x in the equation xn± kn = 0, then icnyn±kn=0, or (dividing by kn)yn± 1=0. If, therefore, the roots of the equation yn± 1=0 be a, β, y, î, &c. the corresponding values of x in the equation £«¿¿*=0 will be ah, yk, hkj &C. 56. Suppose now that a is some given number whose nth root is required to be extracted, then let x=nJa¡ from which we have xn=a or xn-~ a = 0. Compare the equation xn—a=0 with the equation xn— kn=0, and we have Jcn=a or k=nJa ; the roots of the equation yn —1 = 0, therefore, being «, β9 y, δ, &c. the iith root of a may be expressed by <*nJa, β nJa> ynJa> ànJa> &C. 57. To solve the equation i/n+ 1 = 0, I. Leti/ +1=0; theny= —1. II. . . . y2+ 1 = 0 ; then i/2 = — 1 ; and the two values of y are + \/— 1 and — n/—1. III. . . . y3+ 1=0; divide this equation by ÿ-h 1=0, and the quo- tient is y2—y + 1> whose two roots are ^----. the three roots of the equation y3+1 = 0, therefore, are —1, ~·^—?, -———?. IV. . . . y4+1 = 0 ; this is a recurring equation, which being compared with y*+py3 + qy2+py+ 1=0, we havep = 0, q== 0 ; the quadratic equation z2+pz + (q — 2)=0, therefore, becomes z2—2=0, whose roots are M ±\/2 :82 ON THE SOLUTION OF EQUATIONS. ± *¿2 ; hence the two quadratic divisors of the equation y4+l=0 are y2 + \/2y+l = 0 and y ¿—*/2y+ 1 = 0, whose roots are -ldV— 1 and 1±V-1 ” n/2 respectively ; hence the four roots of y4+ 1 = 0 are — 1 + n/ —1 —1—V—1 1 4* n/— 1 1—V—1 */2 V2 λ/2 λ/2 V. Let y5+1=0; divide this equation by y +1 = 0, and the quotient is y4—y3 + y?—y + 1=0 ; compare this equation as before, and we have p= — 1, q= 1 ; hence the quadratic equation z2+jp*+(?— 2)=° becomes z2—z—1 = 0, whose two roots are ljy/5 . two qUa(jratic divisors of the equation y4—y3 + y2—y + 1=0 are therefore +1 = o, and +1 = 0, whose roots are l+*/õ±J— 10 + 2V5 ^ 1-V5 + λ/—10-2λ/5 - 4 an 4 respectively; the five roots of the equation i=o are, therefore, 1+75 + J— 10+2λ/5 1+^5—^-10 + ?V5 — 1, -------J > —' 4 ’ , 75 +λ/— 10-2V5 1-75-^-1°-3^5 --------4 * 4 VI.ON THE SOLUTION OF EQUATIONS. 83 VI. Letye4-1 = 0 ; divide this equation by y2+l = 0, and the quotient is y4—y2+l=0, whose roots are i l±*/-3 , . / 14:*/—3 . y =-^2—or y=± V —2—5 the roots of the equation y6 4- 1=0 are, therefore, + V—1, —*/—1, 4-y -------> ~“y g-----, + sJ I~g'-~3> for the roots of y2 -f- 1=0 are 4- V — 1., — V— 1 (No. IL) 68. To solve the equation yn —1=0, I. Let y—1=0; then y =1. IL ...y2—1 = 0; then y2=l, and the two values of y are + 1, -1. Ill___y3—1=0; divide this equation by y—1=0, and the quotient is y2 + y+l = 0, whose roots are ---±------; hence the three roots of 2 y3 — i = o are 1, -1 -f — 3 —1 -V —3 IV.... y4-—1=0 ; this equation may be transformed into (y2-i)Q/2+i)=o; its roots therefore (being the two Toots of y2 —1 = 0 and thè two roots of y24-1 = 0) are 4“ 1, —1, 4- V —1, —V—1. V. ...y8—1 = 0; the five roots of this equation, found by a process similar to that by which the roots of the equation y54-1=0 were found, are — I4V54- J— 10 — 2λ/5 — 14-V5— J—10—2*Tò 1, ..—.——— ---------,------------------—--f 4 4 —1—^5+ J—10+ 2 V5 — 1 —-v/5—J —10 + 2λ/5 4 ’ 4 VI.84 ON THE SOLUTION OF EQUATIONS. VI. Let ye—-1 = 0 ; this equation may be transformed into (y3-i)(y3+0=o; its six roots therefore (Arts. 57,58, Nos. III.) are . — lW— 3 — 1-V—3 1+V—3 1—V—3 + h —h-----~---, ----õ---» ---õ--* ---Õ VII...y8 -i=(y4—1)(y4 + 1)=° VIII. ..y10 l=(y5 l)(y5 +1)=0 IX...y«-l=(ÿe-l)(ye+l) = 0 f The roots of the equa- tions y8—1=0, y10—1 = 0, y12—1 = 0 are the several roots of the equations of / which they are thus com- posed ; and which have al- ready been found in Nos. IV. v! VI. of this and the ^preceding article. The equation y9—1=0 will be solved hereafter. With respect to the equations y7—1 = 0, yn—1=0, it maybe observed, that if each be divided by y—1 = 0, the quotient in the former case will be a recurring equation of ήχ di- mensions, in the latter of ten dimensions. From which it appears (p. 38) that the equation yT—1=0 may be solved by means of a cubic equation, but that to find the roots of the equation y11—1 = 0 would require the solution of an equation of five dimensions. 59. Now let p be any one of the impossible* roots of the equation yn—1 = 0. If p be substituted for y, then pn —1=0 or pn=l ; hence p2n=l, p3n=l, &c. ; if therefore, p, p\ p\ &c. ad infinitum, be successively substituted for y in the expression * The equation yn—1=0 can have no possible root except ±1 (n being even) or +1 (n odd). For the nth power of any possible quantity greater or less than 1 will itself be greater or less than 1 ; and consequently no such quantity, if substituted for y, can satisfy the equation yn—1=0.ON THE SOLUTION OF EQUATIONS. 85 expression .y"—1, the results (pn— 1, p2n— 1, p3n — 1, &c.) will in each case be 0 ; consequently p2, p3, &c. are roots of the equation yn —1 = 0 as well as p. From hence it ap- pears that the roots of the equation yn—1=0 may be ex- hibited under the following forms, 1, p, p 9 p , ... p ;p,p+,p^, occ. . .. P ; p2n,p2n+1, ρ2Λ+2, &c.... p3"”1 ; p3n, pin+1, p3n+2,&c. ad infinity where, if n be a prime number, all the powers of p w7ill be impossible, except pn, p2n, p3n, &c. which are each equal to unity t. For instance ; I. Let « be one of the impossible roots of the equation y3—1=0, then 1,«, <*2; or 1,«4, «5; orl,«7,«8; ori,a10,a11; &c. &c. ad infinit. are so many different forms under which the three roots of that equation may be represented. II. Let β be one of the impossible roots of the equation y5 —1=0, then 1,β,β29β3,β*ίθν l,/36,/37,/38/39 ; or l,/3n,/312,/313,/314,&.c.&c.adinf. are different forms under which the five roots of the equa- tion yb—1 = 0 may be represented; the several powers of « in the former case, and of β in the latter, being all im- possible quantities, These t If n be not a prime number, other powers of p, lying between 1 and pn, between pn and p2n, &c. &c. will be possible. Thus, if the equation 1+Λ/Ζ3 be y —1=0, and _, one of its impossible roots, (Art. 58. No. VI.) be taken for p, we shall have p3=—I (see Art. 57. No. III.), and conse- quently p9=— 1, p15=—1, &c. &c.86 ON THE SOLUTION OF EQUATIONS. These sets of quantities are indeed only the same roots under different forms. For «4=«3x« =lx« =« ).·. 1, «4, a5 are of the same a5=a x«2 = 1 x«2=«2) value as 1, «, a2; &c. &c. And /36=/35 X β =1 X β =β Λ β7=β5 χ/32 = 1 χ/32=/32 I .·. 1, β*, βΊ,β*, β9, areof the β* = β5 X >β3= 1 χβ3=β3 \ same value as 1,β,β2,β*> /39 = /35 Χ/34=1 Χ/34=/34 | β\ &C. &C. &C. &C. &C. / 60. Next, let 1, ρ, ρ2, &c.... ρ*-1 be the roots of the equa- tion yl—1=0, and 1, m“1 the roots of the equation j/m—1 = 0, Z and m being each of them prime numbers ; and let the l roots of the former equation be multiplied into each of the m roots of the latter, the re- sulting Im quantities will be 1, P> P2> pz> &c. . . . ... pl 1 «·> per, p2n ym—p =0/arem equations of m dimensions, for ym—pi =0\determining the n (or mm) values of &c. =0 V ?/ in the equation i/n—1=0. For instance, if 1, «, a2, be the three roots of the equa- tion y3—1 = 0, then ?/3— 1=0, y3—«=0, yz—a*=0, will be three cubic equations for determining the nine values of y in the equation y9—1 = 0. Now, by (Art. 56) the three roots• ON THE SOLUTION OF EQUATIONS. 89 roots of the equation y3—« = 0 (since «, fi, y in that Article correspond respectively to 1, a, a~ in this Article, and n=3,) are 3Ja, a3Ja, <*25Ja, and the three roots of the equation y3—«2 = 0, are 3Ja2, a3Ja2, «2^/«2; hence the nine roots of the equation y9—1=0 are L «> «% l¡<*> <*H<*29 «2i7«2· For the same reason, if 1, fi, fi2, fi3, fi* be the five roots of the equation y5—1=0, the roots of the equation y*5 —1 = 0 will be 1, fi, fi2, fi\ fi4 il A PôJfi> β2ϊ]β> β3^β> β*^β 5Jfi\ fi5Jfi\fi25Jfl\fi35Jfi\fl*5Jfi2 sJfi\ fi5Jfi\fi2òJfi\fi3òJfi\ fi*5Jfi3 5Jfi\ β5^β\ fi2òJfi\ fi3òJfi\ fi*5Jfi*. Τη the same manner, having found the m roots of the equation yOT—1 = 0 (m being any prime number whatever), we may write down the roots of the equation yn —1=0, where n is equal to the square of m. SECT. IV. On the general solution of Cubic Equations. 62. It has already been shewn in the second chapter, that the cubic equation x3—px2 + qx—r=0, whose roots are «, fi, y, may be transformed into another deficient in its second term, by substituting y + p for x in the given equation; in which case the roots of the transformed equation will be, a—\p, fi—\p, y—ip. Let «—ip=«'; β—τρ=β'·, y—ip=y'i then «=«'+ip; β=β' + $ρ; y=y' + ip ; if therefore the roots of the transformed equa- tion be known, the roots of the given equation will be N known90 ON THE SOLUTION OF EQUATIONS. known also. Hence the solution of a cubic equation com- plete in all its terms will be effected, if we can arrive at the solution of it in the form x3 + qx — r=0. 63. In the solution of the equation x3 + qx — r=0 by the rule generally known by the name of Cardans Ride, the unknown quantity (x) is supposed to be equal to the sum of two other unknown quantities (u -f- z) ; which being substituted for x in the given equation, we have æ3=(w-|- z)3=:u3 + 3u2z+ 3uz2 Λ-z3=w3+ 3uz (u + z) + z3\ qx~ +q{u + z) > = —r= — r) Now as another unknown quantity has been introduced into the equation, another condition may be annexed to its solution. Let this condition be, that 3uz + a=0, or z= — -2-, 3 u in which case the transformed equation becomes u3 + z3 — r =0 n3 or u3-----1---r =0 27 u3 or w6— ru3 — T1Tg3=0; which equation solved, gives w3 = Jr± ^ir' + ^Yq3, or u = HJr± + z3 — r—w3=Jr=F Jir'¿ + -2\q3, or z = + Hence ______________ oc=:u + +-£iqà + l¡%r—J%r2+ 1±rq3.* 64. Now * When the sign of */ig 4- in the value of u, it is — in the value of z, and vice versa ; so that in writing down the value of u-\-z, it is evidently unnecessary to annex ± to that quantity.ON THE SOLUTION OF EQUATIONS. 91 64. Nowlet jr + /7jr¿+2Vg3=a,& Jr—^Jr2 + and let 1, «, «2 be the three roots of ^3— 1 = 0 ; then (by Art 56) r2+ 2V93 may be expressed by J Gr «)ü/Uj α λ/®> and */jr—J^r2 + -^T(/3 may be expressed by ~ ^ «S/6, Hence ^ Jr + J¿r2 + -ÍT + «^/&." To shew that the whole of our reasoning on this subject has been correct, it may now be proper to prove the con- verse of this proposition, viz. “ that if the three roots of a “ cubic equation be “3Ja + 5Jb; *lJa + K2ljb-, <*23Ja+*llb, “ where a=^r + J^r2 + -±ïq3 and b = ^r—J “then that cubic equation will be x3 + qx—r=0;” which may be done in the following manner. i. To find the coefficient of the second term of the equa- tion, we have 5Ja+ II b « Ju + <* 23Jb a23Ja + a 3Jb Add these together, then (l + öc + α2) ^/a + (l-f α + «2)^/δ= the sum of the roots. But l + a + a2=0, since the equation y3—1=0 is defective in its intermediate terms ; the coefficient of the second term, therefore, is equal to 0. η. For the coefficient of the third term, we have ( >Ja+ 5JbX« 3> + «2W=« ’Ja* + (« + «yjab + a**]b* ( *Ja+ 3JÒ)(«23> + «^)=«^a2+(« + «2)^/aò + «^/62 (asJa + a?sJb)(a*lla + a sJb)= 3Jα2+(« + «2)^/αό+ ί^/ό2,sinceα3= Hence,Hence, by addition, (1 + «+«*) 1}α*+ 3 («+ «*) 3Jab+(l + « + «*) 3Jb2=the sum of the produrti of any two roots; which, since 1 +«+ a2=0 and α + α2=—1, is equal to —3 Ijab; But ^/aô=^î.+^ïr2+^T53)(ir-^Jr2 + -^g3)=î/-^= os ON THE SOLUTION OF EQUATIONS. i)9 CM > i« + Cm 2 c2 » 0) ¿3 s «4 s .fe g fe •+0 03 «S G O J ee .-G *- H & o o Λ CD ja te .5 *55 PÛ qT <2 o u o ja ir? + «* ^ a . the product vf the roots is a + (1+ a -+■ α2)^/α9ό + (l + a + aJ) 5Ja b2 -f δ =a + 6, since 1 -f- a + a2=0.100 ON THE SOLUTION OF EQUATIONS. SECT. V. On the general solution of Biquadratic Equations. 60. The same observation may be applied to biquadratic equations as was applied to cubic equations in Art. 62, that, since the equation x*—px3 + qx2—rx + s may be trans- formed (by Art. 23) into another which shall be deficient in Us second term, and whose roots shall bear a known re- lation to the roots of the given equation, the complete solution of a biquadratic equation will be effected, if we can arrive at the solution of it in the form æ4-f qxz—ræ + s=0. 70. In the solution of a biquadratic equation, by the method ascribed to Des Cartes, the equation X4 + q X2—ra? + s=0 is supposed to be the product of two quadratic equations (Art. 3.) x2-ïzx + t=Q and x2—zx + w=0, having the un- known coefficients z, t, wf and in which the coefficient of the second term of one is supposed to be z and of the other — z9 in order that when these two quadratic equations are multiplied together, the resulting biquadratic equation may be ^deficient in its second term. By actual multiplication, we have + zx + t) (x2—zx + w)=xi + (t + w—z2)x2 — (t—w')zx + tw= Let this equation be compared with the equation x4 -\-qz2—rx+ s*= 0, and there arise the following equations for determining the values of z, t, w, in terms of the coefficients q9 r, s of the given equation ; t -f íe — z2=(j, or t + w=q + z2 (t—w)z=r} or/—W3=- z /'ΙΓ = 6'. HenceON THE SOLUTION OF EQUATIONS. 101 Hence 2t = q + z2 + -i or t~ z q + z2 + - z 2w=zq + z'2—Γ, or w= 2 q+z2-- z From which we have tw~ (?+*2)2-Ι-2 = s, and ç2-f 2oz2+ 24— — =4s, z1 or z* + 2qz* + (jf—4s)*2 —r2=0. Let z2=y or z=±\/y9 then this equation becomes yz+%qy2+(q2—4»)y—r2=o, and if «, β, γ be the three roots of this equation, then the three values of z2 are «, β, y9 and the six values of z are ±rìa9 ±*Ιβ9 ±sly. Substitute these values of z2 and z in the equations exhibiting the values of t and w9 then q + a±JL ^ λ/ « 9+y±-f- pr . 2 2 vl a 2 ’ y + «T-f- ,r— or r î+yT-f- or ^ 2 °r 2 2 Hence the values of x 2 + zx + t=0, are i. |=o. n. a?2±Vy3x + V 2 y (*+β±7β' )=0. III. a2±Vyæ + V 2 > r^+y±^N )=o. V 2 > And102 ON THE SOLUTION OF EQUATIONS. And the values of x2—zx + w=x 0 are IV. X2=F fax+l |=0. \ < 2 / v. x2=fV£x+ 1 K 2 2 )=0. vi. x2:F'v/ya?+ | (=0. Each of these quadratic equations evidently contains four roots ; but Nos. iv, v, vi, are the same with Nos. i, n, m, respectively ; so that the equations to be finally considered are reduced to Nos. i, n, in ; and either of those equations will be found to contain the four roots of the given equa- tion x* + qx2—rx-M = 0. Example I. Solve the equation x4 — 25x2 + 60#—36=0. Compare this equation with x* + qx2—rx + s=0, then y = - 25) r = — 60 >.\ if + 2qyz + (ÿ2 —4s)y —r2=y3—b0y2 + 769 y —3600 = 0. s=-36j Since y(=22) is a square number, try the square numbers 1, 4, 9, 16, &c. successively, and 9 will be found to be one of the roots of this equation; divide the equation by y —9=0, and the quotient is y2—41 y + 400=0, whose two roots are 16 and 25; the three roots of the equation y3— 50y2 + 769y—3600=0 are, therefore, 9, 16, 25; hence the three values of 22 are 9, 16, 25 ; and the six values of z are ±3, ±4, ± 5 ; from which we have q + a±-L- -25+9T20 -------------=-18 or ‘2, q+β 2 2ON THE SOLUTION OF EQUATIONS. 103 ÿ + /î±-£- —25+ 16 T 15 VjO_______________ 12 or 3. 2 2 q+y±4- — 25 + 25T12 V y_______. 6 or + 6. 2 2 Make the proper substitutions in Nos. i, n, in ; and the two quadratic equations which are contained in No. i, are r2 + 3# —18 = 0, and x2—3 x +2=0; whose roots are 3, —6, and 1, 2, respectively. In No. η, are r2-f 4r — 12=0, and .r2—4a?+ 3 = 0; whose roots are 2, —6, and 1, 3, respectively. In No. in, are whose roots are 1, —6, and 2, 3, respectively ; so that the four roots of the given equation are 1,2,3, —6. Example II. Take the equation a?4—3æ2 + 6æ-{-8=0. Compare this equation with a?4 + , /— — V — σ), — 2π, — (>/—p + >/—0^, + (ü-2)(«-3)gaB-4_&Cj 2 2 2 5= n(7l‘~0(n l)(^—2)(r¿ — 3) an-4 , 2.3 2.3 ^ &c.=&c. 73. Now if a be very nearly equal to x9 i. e. “if y be a small fraction,” then the several powers of y(y29 y39 &c. yn~*9 yn~\ yn~l9 yn) will be so small as to make the quantity Ty2 + Sy3 + 8cc.... + Ryn~3 + Qy^ + Py"-1+ yn very inconsiderable with respect to W+Vy, and, conse- quently, to make the simple equation W+ Vy=0 a near approximation to the equation (X)=0 ; in which case an approximation to one of the values of y in the equation (X)=0 will be obtained by the solution of the equation W+ Vy=0 ; which gives W____________an—pan~l + qan-2—ran-3 + ,9a”-4-&c. nan~s —{n— l)pa*~2 + (?i — 2)qan~3—(n — 3)ran-4-{- &c. 74. Having y= V108 ON THE SOLUTION OF EQUATIONS. 74. Having thus obtained an approximate value of y, let it be called b, then a + b isa nearer approximation to the value of x in the given equation than a; let a + 6=c, and proceeding with c as we have done with a, we have ÌV___ cn —p c n~1 + qcn~~~ — rcn~3 + scn~4 — &c. _ V ncn~l—(n—l)pc”_2 + (^—2) qcn~3—(n—3)rcn~* + &C. let this value of y be called d, then c + d is a nearer ap- proximation to the value of x in the given equation than c or a + b ; and thus we may proceed till we have obtained the value of one root of the given equation to any degree of accuracy. 75. In the practical application of this rule we must endeavour to find two whole numbers between which some one root of the given equation lies ; and by substituting each of them for x in the given equation, and then ob- serving which of them gives a result most nearly equal to 0, we shall ascertain the whole number to which the value of x most nearly approaches ; we must then assume a equal to one of the whole numbers thus found, or to some decimal number which lies between them, according to the circumstances of the case. Now to find two whole numbers between which any root of a given equation lies, we may either have recourse to Art. 30, and find the least whole number ([k) which ex- ceeds the greatest root of the equation, in which case that root will lie between Jc and k— 1 ; or, we may find by trial two contiguous whole numbers, k and l, one of which, when substituted for x in the given equation, gives a posi- tive and the other a negative result ; and then (by Art. 33) one root of the equation will lie between k and L This, of course, goes upon the supposition that the equation contains at least one possible root. Ex. I.ON THE SOLUTION OF EQUATIONS. 109 Example I. Solve the equation x3 —-12# +8 = 0, by this method. The least whole number which gives x3—12a?+ 8 ) and 3a?2 —12 ) positive, is 4 ; the greatest root of this equation, therefore, lies between 3 and 4. Substitute 3 for x in the given equation, and the resiflt is — 1 ; substitute 4 for a?, and the result is + 24 ; this root, therefore, is much more nearly equal to 3 than to 4.* Let a=3, then(Art.73) y— — -----ί—= _L = .0666 8¿c.=b ; v 3a2—12 15 hence α + ό = 3.0666 &c., the first approximation. Let c=3.066, then c3 — 12c + 8 υ 3c2 —12 .0294914 16.201068 0018203 = d; and c-f·d=3.0666666—.0018203 = 3.0648463, the second approximation to the value of a?. This equation has already been solved (p. 97) by means of the cosines of circular arcs, and its greatest root, true to seven places of decimals, was found to be 3.0641776; it would therefore require two or three more substitutions before we should obtain as near an approximation to one of the values of a? by this method as by that of circular arcs; but the value now obtained is true to three places of decimals, and may therefore be considered as a near ap- proximation to the root. Ex. II. * For if any number be substituted for <2? in a given equation, and the result be 0, that number is a root of the equation; hence if two numbers be substituted for x in the equation, neither of which gives a result equal to 0, that number will be most nearly equal to a root of the equation which gives a result most nearly equal to 0.Example IL Solve the equation æ3-f 6.75æ2 + 4.5a?—10.25 = 0 (see Art. 38). By the mere inspection of this equation it will be readily seen, that, if 0 be substituted for x, the 110 ON THE SOLUTION OF EQUATIONS. O CJ O fG +* +* G e cö .§ rS te ^ gnO fG « ^ 05 •\ · fH co CM co lO I CM O 05 5 -4-> G hi li .2 H ^ • P«4 ÖD 0) -5 C4-. o O O hi CU fG H G ° «í do oo + s$ oo LO 00 + I % % 00 00 + + es ¿3 H ------- 6885162 102474 + 11.386#—10.25 --------- -------- + 11.386 ¿r-10.249 6.8865390324 10.24944948ON THE SOLUTION ÕF EQUATIONS. Ill Example III. Solve the equation a?3+23.3a?2 —39 a?— 93.3=0 (see Exam. II. Art. 44). The least whole number which gives a?3 + 23.3a;2— 39 a? — 93.3 3a?2 + 46.6a? —39 and 3a? +23.3 •positive, is 3 ; the greatest root of the equation, therefore, lies between 2 and 3. Let 2 be substituted for a? in the given equation, and the result is —70.1 ; and if 3 be sub- stituted for it, the result is +26.4; hence the root in ques- tion is more nearly equal to 3 than to 2. Let a — 2.8, then __ q3 + 23.3q2 — 39 a —93.3 2.124_ lg y 3α2+46.6α—39 ~ 115 Hence a?=a + y = 2.8— .018 = 2.782. Divide the given equation by a?—2.782 = 0, the quotient is a?2+ 26.082 a? + 33.56=0, and the remainder is so small as to make any further ap- proximation unnecessary. Now the roots of the equation a?2 + 26.082 a? + 33.56 = 0 are a?= —13.04±J\36.48 = —13.04+11.68 *=-1.36 or —24.72. Hence the roots of the given equation are 2.782; —1.36; —24.72; very nearly. Example IV. Find an approximate value of one root of the equation a?5—9x3 + 8λ?2—3a? + 4 = 0. In the solution of equations of high dimensions, it will always be worth while to substitute a few terms of the series 0, 1, 2, 3, &c. for x in the given equation, and see the nature of the results, before we apply the rule for finding the least whole number which exceeds the greatest root of the equation. In the present instance, Let112 ON THE SOLUTION OF EQUATIONS. Let t=0, and the result is + 4. x=l ..............+1. x=2...............—10. Hence one root of the equation lies between 1 and 2, and is evidently very nearly equal to 1. Therefore let o=l, then a5 — 9 a3 + 8 a2 — 3 a 4* 4_1 "5a4 —27a2-}- 16a —3 ~9 &c. = &. Henceα + ό=1.1111 &c. theapproximation. Let c= 1.11, then yz=z —-f 8 c2 —3c + 4__= — = —.008869 = 0 υ ôc4 —27c*+16c-3 -10.91634 the second approximation. And c + d=l. 111111 — .008869 = 1.102242 is a near ap- proximation to the value of one root of the given equation. Example V. Find whether it be possible to exhibit an approximate value of any root of the equation z4—3^3.h 12**— 13x+ 21 = 0. Let 0 be substituted for x in this equation, and the re- sult is +21; and if any positive or negative number what- ever be substituted for x, the result will be positive. From which we find, that there are no two numbers in the whole arithmetical scale, which, when substituted for x, give results alternately + and —. Hence we infer (Art. 33) that there are no two numbers in that scale, between which a root of the given equation lies ; its roots are con- sequently all impossible. This is indeed the case, for the equation arises from the multiplication of the two quadratic equations #2 — 2a?+7 = 0 and æa —æ+3 = 0; the roots of the former of which are 1±*/ —6, and of the iato-Ì*·7-11 2 76. We76. We now proceed to shew upon what the accuracy of the foregoing method of approximation de- pends. For this purpose, let «—«=«'; β—α=β' ; y—a=y'; δ—a=h'; &c. in which case the roots of the equation (X)=0 (p. 107) will be β\ y, δ', & c. ; and therefore, W=the product of all the roots................=αβ'y'S' &c.* F=the sum of the products of any η— 1 roots=«'/3'y' &c. + α'β'Ϋ &c. + ay'V &c. + jô'y'd' &c. ON THE SOLUTION OF EQUATIONS. 113 da 2© V ?» + d 2© + ó da V <0. Η" ö ¿a ¿o 2© + v8 c5 + da c5 V· da -f. V ô da + V o + da ó çq. da =*$ Si *© + ci =8 g' V ij . *© ¿8 /· o V β -Η 4-» ί- 03 Ρ-) -4- 60 Τ' ce ϋ ϋ ? J CQ. β © 0 + 5 ci g < £ ' lí ■V 02 « ο da *© V· + « 6 da da Ào 2© V V· <0. \s + + ô ô ¿a ÿ . ¿o *© v+ + **·$<§ 8?V «5. CQ. V« ve + - GO P* tb t5 <& S ce !* • Oi S- I £ o £ O o da ¿a í*· à© <0. V K ,v "e ^ & I « (M CO I I ss SS >» ^ a g ce ce <4H O o o ^ β Il 2 Ç*i «G ¿0 $0 <4H O O ^5 le > H O è β o rS <υ o s (D te « 0 1 <υ G le > <4-1 0 α> 3 1 Μ ** Ο bo & c S *§Ί βΛ g ω ^ S s ^ ce 02 ο V. I CD JG Q The signs are not taken into account in this and the following investigations, as not being material to the points in question.114 ON THE SOLUTION OF EQUATIONS. Now, unless «' be fractional and very much less than any of the other roots β', y 7 δ', &c. óf the equation (X)=0, the ratio fi'y'iï' &C. l a β'y' &C. + a β'}>' &C. -\-ay'ü' &C. + β'y'W &€. by no means approaches towards a ratio of equality. It is only, therefore, when a is a much nearer approximation to one root of the given equation than to any other, that the foregoing method of approximation can be applied with any degree of accuracy. 77. Hence it appears, that " when an equation contains " two or more roots which are very nearly equal to each otherwe must adopt some other method for finding the approximate values of y in the equation (X)=0. Now, if two roots, a and of the given equation be very nearly equal to each other and to a, then the two roots β' of the equation (X)=0 will be much less than any of the other roots y', δ', &c. ; and since, in this case, the ap- proximate value of V will be « y δ' &c. + β' y δ* &c.* and the approximate value of T will be y δ' &c.* we have W «'/3Vr&c. ,Λ, τ~ F_«yy&c. + /3yy&c.__. T y^'&C. So that the approximate values of a and β' (being the two roots * For V= a^y Sec.^-α'β'δ' &c.+o'7'8' Sec.+jSy'S' Sec. ; and since o', jS'are supposed to be fractional, and much less than y, 8', &c. the terms a'^y Sec.-\ra'β δ' Sec. (which contain the product of «' and β*) must be much less than the terms ay' 8' See. -bfi'y'ô' Sec. (which contain only o'or β'). Also, T=a'¡3' &c.+a y Sec.+α'δ' Sec. +j8'y' &C.+β' 8' &c. +7'8' Sec. ; /. the terms which contain one or both of the very small fractional quan- tities a' or β' must be very small when compared with the term 7'8' See., which contains neither of those quantities.ON THE SOLUTION OF EQUATIONS. 115 roots of the equation y2+(«' -f β') y + αβ'=0) will be deter- mined from the solution of the equation y>+ry+tr=Ty*+Vy + }V=0. --α'β'Υ, 78. If three roots, «', β', y, of the equation (X) = 0 be much less than any of the rest, then the approximate value of F will be a β'S' &c. + «'y'S' &c. + ,3'y'S'&c. ;* of T, will be a'y &c. + /3'S' &c. + y y + &c. ;* and of S, will be S' &c. ;* which gives JF_«'/3yS'&c.. S S'&c. ν_«'βΎ &C. + a'y'y &c. + /3'y'S' + S. S'&c. 7 7 T a'S'&C.+ /3',;S' &C. + y S' &C. , s---------r&r-2—=a +7 ; so that the approximate values of α', β\ y' (being the three roots of the cubic equation y3 + (0/+ β'+ y') ÿ 2 + + a/y/ + ^V) 3/ + σ! β* y —0^ will be determined from the solution of the cubic equation *■+£•+5,+^ or Sy3+Ty2 + Vy + fV=Q. In the same manner it may be shewn, that, if the equa- tion (X)=0 contain m roots very much smaller than its other n—m roots, the approximate values of y in that equa- tion may be determined from the solution of the equation of * For in this case, a & y &c. (containing the product of a, β', y) must be much less than the other terms in V (which contain only the product of a and β' &c.) ; and a β' &c., a'7' &c., £'7', &c. must be much less than other terms «'θ' &c., β'δ' &c., 7'5'&c. in T ; also, £= a &C.-H8' &c. + y' &c. + δ' &c. of which a &c., β' & c., y & c., will be very small with respect to S' Sec.116 ON THE SOLUTION OF EQUATIONS. of m dimensions which arises from cutting off m +1 terms from the equation W+ Vy + Ty2 + Sy3 + &c.... + Ry+ Qy'-* + Py*'1 + y ■=0. 79. In the practical application of these rules, we must therefore find the m roots of this equation of m dimensions. Let these roots be 6, c, d, &c. ; then, since x=a + y9 the m roots of the given equation, which are very nearly equal to each other, will be a + b9 a + c, a + d9 &c. Example. The equation #4 —17.lx2 — 6x+90=0 has two roots very nearly equal to each other ; solve the equation. The least whole number which gives —17.1 X2—6#+ 90 4 Æ3 — 34.2# —6 and 6x2—17.1 positive, is 4 ; one root of the given equation, therefore, lies between 3 and 4 ; substitute 4 for x9 and the result is +48.4 ; substitute 3, and the result is —.9 ; hence this root is very much nearer to 3 than to 4. To find whether it has also another root very nearly equal to 3, substitute 3 in the equation 4 æ3 — 34.2x— 6=0 ; the result is only —.6; one root of this latter equation, therefore, is also very nearly equal to 3 ; from which it appears (by Art. 48) that the given equation has two roots very nearly equal to 3. To find the value of the coefficients, in the equation y2+ from the solution of which we are to determine the ap- proximate values of y, we have a=3, 1F=«4—17.1 a2 — 6a+ 90 = —.9 F=4a3 —34.2 a —6=—.6 T=z6a2— 17.1 = 36.9. HenceON THE SOLUTION OF EQUATIONS. 117 Hence W__ .9 T 36.9 r____.e , T 36.9 JL = —.02439 41 JL =-.01626 123 and ÿ2 + ^y + ^=y2—.01626y—.02439 = 0, whose roots are .00813 + .156384, or .164514 and —.148254; so that b==. 164514, and c= — .148254; .·. « + 5=3.164514, anda+c=3-.148254=2.851746. The approximate values of the two roots of the given equa- tion which are very nearly equal to each other are, therefore, 3.164514 and 2.851746. The other two roots are impossible.* 80. If an equation contains two roots very nearly equal to each other, it may, however, be solved by means of the equation ^= —Art. 73), provided that the quantity a be so assumed, that the corresponding roots («', β') of the equation (X)=0 shall have the same signs. Now, if abe assumed less than both the roots α, β, then a and β' will both be positive ; if greater, a and β' will both be negative. In this case, by Art. 77, we have W__ a'Py'ìr &C.____________ <χ'β V ay'ìÌ &C. + β'γ,δ/ &C. <* + β Hence * For the quadratic equation which contains the two roots just now found is —6 #+9 = 0 very nearly ; divide the given equation by this quadratic equation, and the quotient is #2+6#+9.9 = 0 very nearly ; and the two roots of this latter equation are — 3+ sf —.9.118 ON THE SOLUTION OF EQUATIONS. Hence the approximate value of «' : its real value : : -_— : ±a' :: — -— ±(«'+¿0 *'+P And the approximate value of β' : its real value : : “ -: + β' : : —~—z¡ ·±&+Ρ) «+β β’ gt From which it appears (since —-—, and-----are necessa- α'+β *+β rily fractional) that an approximation to the value of either of the quantities «' or β' may be obtained by this method, but that it will not be a near approximation to either of them unless one of the quantities be much less than the other ; in which case it will be an approximation to that lesser* quantity. But the case is quite different if one of these quantities be positivera) and the other negative (*—£'); i.e. if the " quantity a be so assumed as to lie between the two roots “ which are very nearly equal to each other for then W_ a'β' m V ¿-β'1 so that if «' and β' be also very nearly equal to each other, W the value of — may be so great as to be no approximation whatever to the value of For instance, in the equation which was solved in Art. 79, the quantity a was assumed equal to 3, a number lying between the two roots which were very nearly equal to each other ; in this case the value of W was found to be — .9, and * For instance, if a be much less than the ratio of , : 1 approaches a ·ή-p to a ratio of equality, and we obtain a near approximation to the value / / f a of a : if β be much less than a, the ratio of -, , , : 1 approaches to a u+β ratio of equality, and we obtain a near approximation to the value of j8'.ON THE SOLUTION OF EQUATIONS. 7/17 119 given equation· But we know, from its actual solution, that the equation contains no such root. 81. In a similar manner, it might be shewn, that if an equation contains m roots which are very nearly equal to each other, an approximate value of one of those roots may be obtained from the solution of the simple equation W+Vy=0 ; but to avoid the error which might arise from the quantity (a) being assumed of such a value as to fall between those m roots, it will always be best to have re- course to the solution of the equation of m dimensions cut off from the equation (X)=0, according to the rule laid down in Art. 78. Indeed, if a great degree of accuracy be required, there is no other mode of solving such equation. On reviewing the several methods given in this chapter for the solution of Equations, it may be observed, i. That in solving a cubic equation by Cardan's Rule, although the rule at first exhibited the roots of the equa- tion under nine different forms, they were immediately re- duced to three, by rejecting such of those forms as were not consistent with the hypothesis upon which the solution depended. lí. That in the solution of a biquadratic equation by Des Cartes' rule, we obtained as many quadratic equations as would have afforded twenty-four roots ; but these were instantly reduced to three pairs of quadratic equations, each pair containing the four roots of the given equation, arranged in different orders. hi. That120 ON THE SOLUTION OF EQUATIONS. hi. That the rule for solving cubic equations by means of circular arcs, and the rule for finding the roots of the equation yn —1 = 0, gave an infinite variety of forms under which the roots of the given equations might be respectively represented ; but at the same time it was shewn that this infinite series of roots might be divided into different sets, each set containing only the three or the n roots of the given equations. In reference, therefore, to the conclusion* we came to at the end of the first chapter, it may be further remarked, that, so far as we do possess the means of solving equa- tions, the truth of the proposition that “ every equation has as many roots as it has dimensions, and no morehas been completely established. * This conclusion was, that f< the direct proof of the proposition that “ every equation has as many roots as it has dimensions extends no further “ than to those orders of equations of which the general method of solu- “ tion is known.”121 CHAP. V. ON THE SYMMETRICAL FUNCTIONS OF THE ROOTS OF EQUATIONS. 82. By the symmetrical functions of the roots of equations are meant those quantities, in which the roots of any given equation are similarly involved. For instance, let «, β, y, 5, &c. be the roots of the equation (A)=xn — pxn~1 + qxn"2—ræ"~3+ &c.......±tx' :fvx±tv=0, then, oC + β2 + y2 + + &c. ; «m + βΜ + ym + S’" + &c. ; 1 + I + l- + \ + &c·; a β y ο +^y} + ^- + &c. ; δ y β a a"β~ 4- «2y ' + δ" β y 2 “l·β2ΰ2 + y ~δ* + &c. ; à1 β + a2 y + «~δ + <*·β~ +· «y2 + «δ2 + β2 y + /32δ-|-βγ~ -f βδ~ + y* 5 + y à2 +■ &c. &c. &c. &c. &c. are symmetrical functions of the roots of that equation ; for each root, or the same power of each root, or the same combination of the powers of each root, occurs in each term of these quantities respectively. SECT. I. On the method of finding the sums of the powers of the roots of any given equation. 83. Previously to the investigation of the general theorem for finding the sums of the powers of the roots of an equation, it will be necessary to shew the manner in which the equation (^)=0 may be divided by any one of the factors a?—·«, x—β, a?—y, a?—δ, &c. R ForFor this purpose, we shall take for an example the biquadratic equation 122 ON THE SYMMETRICAL FUNCTIONS Γ5 « « c3+ {8—p)x'2+(82—p8 + q)x + (83—p8i + q8—r') = 0; c3 + (y—p)i’i+(y'i—py+q)x+(y3—py‘ + qy—r)=0; c3 + (i—p) x2 + (52—p S +q) x+ (à3—pà2 + qS— r)= 0 ; respectively.84. Assuming now the equation (d)=0 in its most general form as it stands in Art. 82, and per- OF THE ROOTS OF EQUATIONS. 123 H- « H“ <5. -H 4H 14- +1 -î £ 53 O P-. 53 (Λ G O <υ P-. o tao 5 '8 tuo · A. J A + ^ ^m+1 ± w ^=0, an equation of n + m dimensions, which has n roots «, β9 y9 δ, &c. and m roots equal to 0. Let «, β, y9 &c. be substituted successively for x in this equation, then «n+m —p a ”+m~1 + q a n+m-2—Ta ”+m~3 + & C.... T V « m+1 ± W a m == 0 finJtm—pfin+*n-1 + qfin+m-2—rfin+m-* + &c....Tvfim+1±wfim=--0 y— p y + q y - r y n+m-3 + &C.... T V y ± W y m=0 &c. &c. = 0 p (Tn^_m_i H“ g 2 3 “f* &C. ... “F V i WP'5 + r(T3—s ít2=:954 + 66 — 9= 1011 + 3 r /'substituting, at each step, the x 03 e « SE o o o o ¿3 . c O ce L- + II « g fi « ¡3 » * - ^ S 10 ·, + c « W α o •■S -H &.§ »3 È rS & « 2 ·*^ ο,) «e .‘í °1 « ” ® 0 e *=> 2 §o g £ g § í oí Cm X ^ o ce 2 8 (U m3 É o Ph ¿ J & ^ t, fi c>% ce T3 3 c+H υ O ‘43 _ ti fi fi Cm O fi tu α> X* Η a Pm 2 «! « » Ον Μ CU 0) Pu * ce ® co O O ,£33 fe H a cr &c. ... .ir2»4-w*-1’~?~1)aw-26-w(m~1) (”*~2) a—»W-&+*c. 2 2 · 3 (a-s/—b)m=am—rnam~W-b- 7^Z1la’—b+ m(m~1) (”*~2) a”-36V-0+&c. v 2 2.3 .·. their swm=2am—ra(m— 1) aOT“26 + &c. Hence the sum of themth powers of the roots of this equation (] **· hi. The quantity βι + β* * + y* + β* yl "Jr βι y* &c. which contains the sum of the products* arising from the multiplication of each of the terms of *■»· V. If m=Z, then this latter quantity becomes 2(α*βι7ι + αιβ*γ* + αιβιγ*+&&) a quantity, in which one of the terms of is combined with two of the terms of ; this function of lie multiplied together, and «' + ^ί+γί=σ,) r & then, «*+í+/3*+í +γ*+*.................. + α*βι + αιβ*+«*yi + «iy*-|-/e*y' + /3*y* ) ^<Γ*ίΓ" And, adopting the notation of Art. 97, N°. in, we have e-*+j + [F,] Le. ot*^2“boc2^*-bot*y*-j-osiy* + ^*Y^“t·β2 y ~g 99. ΙΠ=£, then a2* + /32* + Yí*+2(«A/3‘+«*vi+/S*y*) = (i = > or, [FJ—Vi+m°i + 2 °k + 2(Γ4+2ί, * 2 î.e. akfiïyl + alfikyl + alfilyh is equal to ffA (σζ)2 — 2 <τ^σζ—a [F2]+2* (Art. 102.) 2.3s v — 6y2~ 12 t 6s 104. The foregoing Theorems afford a ready method of finding the sums of the reciprocal powers of the roots of any given equation. For, i Λ By Art. 90, we have '-*?p‘+6rp»_6s 1 p’-l2,rp+6ç,) s.nce, and , are +9?ί +3Λ( each =0. Hence (3—30tf2| p'1+24qrp— 24qs + 12s j +12q3 + 2 w being the term (since /a=w—1, Thus, [Art. 97, N". n.) ^+i+i + ¿ + &c.= ti«K «■* y3¿ y- S¿ w ±+'+λ'+ι+&&-£φ α3 /83 y3 δ3 ΜΓ _L+_L+ 1 + 1 I grc 4 W4 &c. = &c. Example I. Find the sum of the reciprocal cubes of the roots («,β,γ) of the equation x*—px2 + qx—r=0. Here 4+i+A-®l· n3 * β y :(^)_!z^(Art.99.) 2r v 7 _73 + 3r3—j3 ge 139 ) 6s __r2 — 2qsm^r2__2q s2 s2 s.'OF THE ROOTS OF EQUATIONS. 141 SECT. III. On the method of transforming an equation into one whose roots shall be some function of the roots of the given equation. 105. The transformation of equations into others whose roots shall be some function of the roots of the given equa- tion is immediately connected with the subject under con- sideration ; as will appear from the following examples. Example I. Let «, β, γ, ò, &c. be the roots of the equation (A) xn—pxn~l + qxn~'2—r#n'3 + &c.. ..± w?=0 ; it is required to transform it into one whose roots shall be the mth powers (am, β™, ym, $m, &c.) of that equation. Let the transformed equation be 2/n — Pyn~x + Qyn~2— Rytl~3 + &m _(y2 + 2 «Æy = [PJ e-2^ + 2r + 2r (Art. 98.) = (p2—2ç)p—(p3—3gp-*-3r) + 2r =pç—r* Hence the transformed equation is y3—2py*+(p*+q)y-(í>q—r)=o. Example V. To transform the equation x3— px2 + qx—r=0 into one whose roots shall be («—β)2, («—y)3, (/β —y)2. Lst the transformed equation be y3—Py2 + Qy — P=0; and let 'Σί be the sum of the roots, 23 the sum of the squares of the roots, and 23 the sum of the cubes of the roots, of this equation. Then, (by Art. 86) 2j=P 22=P21-2Q 23=P2í-Q21+3P. And, by transposition, P=2I o=^5iH^_2 V 2 _b„Q21-P2, + 23 3 The values of P, Q, P will therefore be known, if the values of 2U 22, 23 be known. Now144 ON THE SYMMETRICAL FUNCTIONS Now 2,—+ + ==2(«î+^, + yî)_2(«/3 + «y + ;3v) = 2 ff.z — 2 q 2. i=(«— /3)4 + («— y)4 + (,e—y)4 = 2(a4 + $4 + y4) — 4:(α3β + αβ3 + α3γ + αγ3 + β3γ + βγ3 +6(>î/8î+ay+/îV) = 2o\ +£3·-£Ι (Art··98·09·) = 3<γ4—4(T3ff1+3(