THE UNIVERSITY OF ILLINOIS LIBRARY Sead Stee pal let | xe (2: Return this book on or before the Latest Date stamped below. University of Illinois Library BAY 22 1964 L161—H41 Sy | PREATISE ON ALGEBRAICAL EQUATIONS. ‘2 View “hY 4 A. ‘ Dll er i if YT Hip : A : we? Pe pu bee en YN i he Ba, UCB ae wae | Pirie | I y f ‘ ie * Get op ik i { ri . s : “9 + ' * 7 ; i i ‘ “4 - i Lstannin f - roe \ ee r ¥ z j if “ ’ * 2 ; - Ty . bony 4 ' ; * + « ’ \ \ f if . 4 " ‘ #. j _ ' j “ e 5 £ i. : - ‘ ' { - : j * hy Ay y ; + é. ‘ “ ‘ yore ha Ra : “e t? Mode seo “an a3 Ln 8 aA tre Nf " my ‘ - x1 Three ' i ee) ek a EAT Oy. GPE Ye Pie be 7 y; ne , t “— orn ! V 1 ’ y > POC TARA en ; Cte aR eg ma yt phe ee ag Te) a OT are te recut | Aly: hr ca as MATE e. LW Leth iat th a ee } 4 ¢ ? . ue Are E | aie * theatres hae bh a ne Art, We a, Sere et} > Ware 2 ARTICLE 1-3. 4-6, 7-9. 10-18. 19-21. 22 23-29. 30-31. 32-35, 36-40. MATHEMATICS DEPARTMENT CONTENTS. SECTION I. ON THE GENERAL PROPERTIES OF EQUATIONS. Form of the equation f(7)=0. Definition of a root ...........eseees If a be a root of f(7)=0, f(#) is exactly divisible by x—a. Ex- pressions. for the quotient, and remainder, of the division of f(a) by L—-A, a being any quantity.............cccscseceee Ws S0ete ch natesd cleyusnuab vas tes If a and b, substituted for x in f(a#), give results with different signs, one root at least of f(z)=0, lies between them. Such a finite value, may be always given to 2, as shall make the first term of f (#) exceed the sum of all the other terms.........:....ccccceessvesees Every equation has at least one real ‘root, unless it be of an even degree with its last term positive. Every equation has as many roots as it has dimensions, and no more. Impossible roots enter SUMPREMTIMLLDY. DRILS 0 ce, cin plan cues ve veddiss'vee ecw ehde de steosen Gee bilo douenatet ee chdes Relations between the coefficients and roots of an equation............... Examples. Complete solution of #+=1=0. Resolution into their factors, of a= 1, 2°*—2cos@a"+1, sin 8, cos0, and e?—2cos0+e-*. SECTION II. ON THE TRANSFORMATION OF EQUATIONS, Transformation of an equation into one whose roots are those of the proposed, with contrary signs, or each increased or diminished by the same given quantity. Form of the development of f(x#+h), f(a) being a polynomial of » dimensions. To take away any term of MERLE ens szcectintadsanvascceb esas er eesucesesstarvusts vs Poatand tees senvestareratree Transformation of an equation into one whose roots are those of the proposed, each multiplied by the same given quantity. Use of this SPERTIEE TIG/ OF ITACtIONAL COEMICICIUS. ..-.crcescesssedccsseosedsanesccevncnass Transformation of an equation into one whose roots are the reci- procals of its roots. Case of failure. Other transformations ......... ° SECTION III. ON THE LIMITS OF THE ROOTS OF EQUATIONS. Definition of limits of the roots.’ If two numbers, when substituted for x in an equation, give results with contrary signs, an odd number of roots lies between them; if they give results with the same signs, an even number lies between them, or none at all. Superior and in- ferior limits POET H Oem RHEE EERO EEE HE TORRE E HOSE EEE HEHEHE EEE HEE EEE ETE EEE EEE EEE SEH EEE SEH OES PAGE ] 33 3g 40 Vi ARTICLE 41-48. 49. 50-54, 55-58. CONTENTS. Various methods of finding superior and inferior limits to the positive and negative roots of equations, suited to different cases. Newton’s method, by finding by trial what value of 2 will make f(#v) and all its derivatives positive.............. Pee ed | obaledustcatanemepeecestel venus Seda Waring’s method of finding the number and situation of the real roots, by finding a limit less than the least difference of the real roots. If an equation have only one change of signs, it can have only ‘one Positive TOO... <.;..ss:sceecaseeet nteete velusestnee soceer-atene Es Se Properties of the roots of f’(#)=0. An odd number of them lies between each adjacent two of the roots of f(v)=0; and if they are known, the number and situation of the roots of f(2)=0 can be found.......... ceatewabessanpsceashpansoshshevetsscenaass gs teen ttn eaeegnssye Des Cartes’s rule of signs. No equation, complete or incomplete, can have more positive roots than it has changes of signs; and no complete equation can have more negative roots than it has con- tinuations of the same sign. Use of this in detecting imaginary roots in incomplete equations.......... ev esveceansssbesesnees siueutaen EemnaEEam wae SECTION IV. PAGE 48 56 57 62 ON THE DEPRESSION OF EQUATIONS, SOME OF WHOSE ROOTS HAVE PARTICULAR RELATIONS TO ONE ANOTHER} CULAR FORM. 59-63. (4-68. 69-71. 72-82. Equal roots. The equation f(a)=0 will have equal roots or not, according as f(#) and f’(#) have a common measure or not. De- composition of a polynomial having equal factors, into others that have only unequal factors ....2..:csecsssos.cescesseseunasss sae weseen pana Commensurable roots. Newion’s method of divisors, for discover- ing the integral roots of an equation. Mode of lessening the num- ber of divisors to be tried ........../..5... at rreaaee iiiesseeiee odtp otebeweeeees ae Reciprocal equations. They may be all reduced to the case of a reciprocal equation of an even order, with its last term positive; and the solution of this can be made to depend on that of an equation of half the number Of CimeNsiOns.......0......0sssseesseeeeceeceeees Binomial equations. Any imaginary root of a”—1=0, (when n is prime, ) will, by its different powers from 1 to n—1, produce all the imaginary roots. The solution of a’—1=0 (when n is a com- posite number) can be reduced to those of a7—1=0, a7—1=0, &c., where p, g, &c., are the prime factors of m. Gauss’s method of solving binomial equations. Any radical has as many different values as there are units in its index................ssssseseasss hes anee cemetees SECTION V. OR ARE OF A PARTI- 67 72 77 80 OF THE GENERAL SOLUTION OF EQUATIONS OF A DEGREE INFERIOR 83-86. TO THE FIFTH. Solution of a quadratic. Its roots are reducible to the form r (cos§+,/—] sin@). Solution of a trinomial equation in the form OF & ‘QUBATAEIGGeseteisasshcaysdvycouatonrestiesed eden scasdyeepsesthe Renn eit: 89 Lai ARTICLE CONTENTS. 87-92. Solution of a cubic by Cardan’s rule. It only extends to the case in which.two roots are impossible. Adaptation of the formule to LOMATICHMIC. COMPULALION ..........0..0rccesverscrscenccncerssoesssasversescsvsccece ses 93-99. Solution of a biquadratic wanting its second term, by Des Cartes’s method. Solution of a complete biquadratic. Huler’s method. In all cases, the reducing cubic will have all its roots real, unless two roots of the proposed equation are possible, and two impossi- 100-107. 108-113. 114-119. ON THE 120-121. 122-125. 126-139. 140-142. 143-147. SECTION VI. ON THE SEPARATION OF THE ROOTS OF EQUATIONS. Sturm’s method. When none of the auxiliary functions are wanting, if their first terms have all the same sign, all the roots are real; if they have not all the same sign, there will be as many pairs of imaginary roots as there are changes of signs................ Fourier’s method. The number of real roots of f (a#)=0 which lie between a and 4, cannot exceed the difference between the num- bers of changes of signs in the results of the substitutions of a and } for z, in the series formed by f(a) and all its derivatives. Des Cartes’s rule of signs included as a particular case. ...........ceeeeeeeee When two, or any number of roots, are indicated in any interval, it may be broken up into others, in which such of the roots as are real, are situated singly. The criterion of the reality of two indi- cated roots may be deduced from geometrical considerations. ......... SECTION VII. INCOMMENSURABLE ROOTS OF EQUATIONS, Properties of the polynomial f(x). Proofthat f(b)=f(a)+(b—a)f'(X), where X is some quantity between @ and 0, ..........secsecsecseceeeseuscees Newton’s method of approximation. Fouwrier’s limitations of Newiton’s method. Geometrical illustration......................cceeceee Continued fractions. To convert any rational fraction, or the irra- tional quantity 1V, into a continued fraction. To transform a continued fraction into a series of converging fractions. Limits of the error, in taking any converging fraction for the value of the continued fraction. Periodic continued fractions. ..................00206- Lagrange’s method of approximating to the roots of an equation, by continued fractions. This method may be advantageously com- MUMMIES CILP ITE $ CDCOLETO. « ccavscayfedorente cher sscheonecseesecenoccevtoucuet Solution of aa+4y=c, by continued Puctinns: Tnvestigations of the nature of the continued fraction which expresses V N. Solution of #?— Ny? =+ 1 eee vil PAGE 92 98 105 115 123 METHODS OF FINDING APPROXIMATE VALUES OF THE REAL 132 134 141 155 vill ON ARTICLE 148-150. 151-157. 158-162. 163-164. 165-169. 170-171. 172-175. CONTENTS. SECTION VIII. SYMMETRICAL FUNCTIONS OF THE ROQTS OF EQUATIONS. ; t . PAGE Definitions. Law of formation, and number annaee =(a™brct...). 169 Sums of similar powers of the roots expressible by the coefficients, and conversely. Particular method for, equations having a small number of terms. Method of approxirhating to the values, real or imagiriary; of the roots: .)....0<...0.0csesscossognesseaetaee neem seed eee | Every rational symmetrical function of the roots is expressible by the coefficients. General method of transforming equations by symmetrical, fUNctiONS.........2..0ss00sassesessievenes sansees ty eee se saan 180 Laplace’s proot that every equation of an even degree has at least one real quadratic factor. Method of resolving a polyno- mial into its quadratic. factOrs........0:05ssos-ssésosdepenses=s0al seeannene yes hg SECTION IX. ON ELIMINATION. General equation of the n™ degree between two unknown quan- tities. To find all the systems of values which will satisfy two such equations. Simplification of the general method .............+. 192 Mode of detecting false solutions, and of freeing the final equa- tion from them. Application of the process of elimination to transform , equatiONS .i.c0.'.as sede: uascgses vad potewous «5pse) spain Geen pediccenh es 196 Method of elimination by symmetrical functions; degree of the final equation. Method of rationalizing equations ......... Sasmnerhbse 202 SECTION X. ON THE GENERAL SOLUTION OF EQUATIONS OF ALL DEGREES, 176-178. Lagrange’s method, by forming a reducing equation whose root is y=2,+a%,+a°43+...4 a" xv, Application to equations of degree. Degree of the equation to be solved, in order to form the reducing equation........./s..é.scsss0s-sevess0s duce ene genera nna eabeng ore 209 Students reading this work for the first time, may confine their attention to the following Articles: 121; 23-34; 36—42;. 46—48; 51, 52, 55—57; 59,60; 64-66; 69, 70; 72—77; 82; 8388; 93, 94; 122; 126—138; 143, 144; 148—151; 165—167. Afterwards they may add: Art. 22. Exs. 1—15, 35, 44, 49, 54, 67, 89, 100—104, 123, 140, 154, 158—160, 163, 170. \ ERRATA. p- 7,1. 6, for yn read y’". p- 7, 1. 14, for a, read x", ? j the third and fourth degrees; and to binomial equations of any ies THEORY OF EQUATIONS. SECTION I. ON THE GENERAL PROPERTIES OF EQUATIONS. Definitions and Principles. ° 1. Every equation under a rational form involving the powers of only one unknown quantity #, may, by dividing its two members by the coefficient of the highest power of a, and transposing the terms, be reduced to the form a + pe + pea ® +... + Pn-1 2 + Da = 03 where x”, the highest power of «, is positive, and its co- efficient is unity; and p,, p.,.-. Pn, the coefficients of the . other powers of aw, are known quantities which may be positive, or negative, or zero. The equation is said to be of the number of dimen- sions, or of the degree, which is expressed by the index of the highest power of # which it involves; and to be com- plete, when it contains all the other inferior powers of #, and a constant or absolute term; otherwise, to be incomplete. . an! 2 Every quantity or expression, real or imaginary, which, when substituted for # in the expression a” + p,a"~'+4... + Pp,» makes the whole vanish, is called a root of the equa- tion a + pv) + pv? + 22. + Pa_ 2 +p, = 0. To solve an equation is to find all its roots. 2. To effect the general solution of equations, would be to find the expressions for all the roots of an equation of any assigned degree in terms of its coefficients, the co- efficients being general symbols. This has hitherto been done only for equations of a degree not exceeding the 4"; and even for cubic equations, it will be seen that the functions of the coefficients which express the roots, are insufficient to give the numerical values of the roots when they are all real; hence we are led to suppose that if we could obtain general formule for the roots of equations of the 5" and superior degrees, we should be unable to obtain from them the numerical values of the roots by the simple substitution of the numerical values of the coeffi- cients. It has therefore become necessary to invent me- thods for obtaining, either exactly or approximately, the roots of»numerical equations; and which, although only applicable to such equations, depend for their demonstra- tion upon certain general properties of equations, which it is the object of the following Articles to exhibit. The investigations constituting the Theory of Equations, which may in general be conducted by processes purely algebraical and elementary, besides affording a knowledge which will have its use in almost every branch of Analysis, will be found particularly serviceable as an exercise to the mathe- matical student. 3. If the signs of the terms of any equation be all positive, it cannot have a positive root; and if the signs of a complete equation be alternately positive and negative, it cannot have a negative root. 3 For, in the former case, every positive quantity, sub- stituted for w, will give a positive result, instead of making the whole vanish, and therefore cannot be a root; and in the latter case, every negative quantity, substituted for a, . will give a positive or negative result, according as the . degree of the equation is even or odd, instead of making . the whole vanish, and therefore cannot be a root. 4. If a quantity a be a root of the equation e+ pa") 4+ p.a-* +... + p, = 0, the first member is divisible by #—a@ without a remain-~ der, and conversely. Suppose the expression w+ p,x"~! + p,a"-? + 02. + Das which we shall hereafter denote by f(a), to be divided by @—a@; now since «—a is only of one dimension with re- spect to w, the division may be carried on till we obtain a remainder independent of v; let Q be the quotient, in which only positive powers of x will enter, and # the remainder ; f(t) =Q<(@-a) +R... (1). In this identical equation write a for x, then the first. member becomes zero, because a is a root of f(#) = 0; also the term Q.(wv — a) vanishes, since one of its factors vanishes and the other cannot become infinite; therefore R=0; and since R does not contain a, it is not altered by substituting a for #, and therefore zero is the value of R in equation (1), whatever be the value of w; that is, J (#) is exactly divisible by # — a. Conversely, if the expression x” + p,a”~' + ... + p, be divisible by #— a without a remainder, a is a root of the equation a + px"! + pia"? + 0. +p, = 0. For, f(#) = Q.(#—- a), where Q is a polynomial con- taining only positive powers of w; if therefore x =a, f(a) . = 0, or ais a root of the equation f(x) = 0. 4 5. Hence, since a is evidently a root of «” — a” = 0, x” — a” is divisible by # — a, whether 7 be odd or even ; and the quotient = #”~’ + aa"®~? + aa’ 4+... +a". Also, when 7 is even, w"—a” is divisible by # + a; since >] > in that case, — @ is a root of a2 — a" = 0. When 7 is odd, — a is a root of «” + a" = 0; thevefore in this case, #” + a” is divisible by 2 + a, but not by # —a. If n is even, #” +a” is divisible by neither # + @ nor # — a. 6. To find the quotient and remainder, when the ex- pression #” + p,v""'+...+ p, 1s divided by w —a, a being any quantity. Let the division be carried on till the remainder is independent of «, and let Q be the quotient and # the remainder ; | 7." + pe") + pax” +... + Pp = Q (w-—a) +h... (1), in which identical equation, since # does not contain w, and Q contains only positive powers of a, if we write a for a, we get a®+p,a°'+ pa"? +...+97,=R; that is, the remainder, after dividing f(#) by a — a, is equal to f(a), the value assumed by f(x) when in it a is written for «&. Next, substituting this value of R in equation (1) and transposing, we have we” — a” + p, (@"~' — a’) + pp (a"7* — a"~*) +... + Dr, (@ — a) = Q(w — a); but the quantities a” — a", #~!—a"~', ... are all divisible by # —a; therefore, effecting the division, we get (Art. 5.) wet aa? et a ee + p, (x"~* + aa’? + aa * +... + a") 4 oo« Daag Sees 5 or, arranging the result according to powers of a, Q=a2"4+ (a+ p,) a? + (@ + pat po) a + (a> + p,a? + poa + p3) vt? +... + (a> + pa" + PO" * + cow + Py_,)3 that is, in the quotient of f(x) divided by «#—a, the coefficient of the first term w’~} is unity; and the coeffi- cients of the other terms are formed, one from the other, by multiplying the coefficient of the preceding term by a, and adding the coefficient of that term in f(w), which involves the same power of 2 as the preceding term does. Existence of Roots and Factors. 7. Af two quantities a and b, when substituted for » in the expression /(), give results affected with different signs, one root at least of the equation f(«) =0 lies be- tween them. Suppose a N, and when «=b, P-—WN is negative or Pp (a4 a+ tet lop ae we shall of course have 2 > pa") + pra? +... + Da P + Dn 5 because in the latter inequality some of the terms may be negative, and no positive term is greater-than the cor- responding term in the former case. Now the inequality | : f : P is satisfied, if a = or >a” a" e-—l “—l a" >p ? or if #=or>p+1; therefore p+1, and every greater number, is a value of # which makes the first term of 7 a“ +p, ve" '+.,. +p, greater than the sum of all the other terms; or makes the value of f(x) positive. Again, let v=—y; then, according as m is even or odd, a+ pe +... + Pn_,v +p, becomes Y" —Pi Yo + vee — PnrY + Dns or — (y"— py”) +... + Pa-1Y — Pn): Now by what has already been proved, the value p+, and every greater value for y, makes the former expression positive, and the latter negative; but this value of y corresponds to — (p +1) for x, at the same time that the preceding functions of y correspond to f(#) ; therefore the value —-(p+1), or any greater negative value for a, makes #” +p,”7,.;+... +p, positive or negative, accord- ing as 7 is even or odd. This proposition shews what term in f(x) is the most important, when very large values are given to a; viz. the term which involves the highest power of w. 10. Every equation of an odd degree has at least one real finite root of a sign contrary to that of its last term ; and every equation of an even degree with its last term negative has at least two real finite rocts of different signs. First, let the equation be of an odd degree with its last term negative; then #=0 gives a negative result, and @2=p+1 gives a positive result (p being the greatest coefficient without regard to signs); therefore the equa- tion has at least one real positive root between 0 and p+i1.. If the last term be positive, then #=0 gives a positive result, and # = — (p+1) gives a negative result ; therefore the equation has at least one real negative root between 0 and — (p +1). Secondly, let the equation be of an even degree with its last term negative; then w = 0 gives a negative result, and each of the values, w= p+1,v©=—(p+1), gives a posi- tive result; therefore the equation has at least two real 8 roots, one positive between 0 and p +1, and the other ne- gative between 0 and — (p + 1). Oxns. The mere existence of the roots may be proved, without reference to Art. 9, as follows. In an equation of an odd degree, if the last term be negative, # =0 gives a negative result, and w= gives a positive result; therefore, there is at least one real root between 0 and @, or one positive root. Similarly, if the last term be positive, v=0, w= —o, gives results with different signs, and therefore include between them, at least one real negative root. If the equation be of an even degree with its last term negative, then w = 0 gives a negative result, and r= +0, a positive result; therefore the equation has at least two real roots of different signs. 11. If an equation have only one change of signs, it can only have one positive root. Since the equation has only one change of signs, it will have one or more positive terms, and all the rest will be negative; therefore (Art. 10) it will necessarily have a posi- tive root a; let p,#"~" be the last positive term, and let the equation be divided by x”~", and it will be ao” + Pye"? + 22. + Dy (Es +... + Pe ) = Ai) then when # = a, the two parts become equal, but if a> a, the first part increases and the second diminishes; and if # --+ Py = 8,5 Pr = as or, coefficient of 2nd term with its proper sign = sum of the roots with their signs changed ; coefficient of 3rd term with its proper sign = sum of the products of every two roots with their signs changed ; coefficient of the (r+1)"™ term with its proper sign = sum of the products of every 7 roots with their signs changed ; and the last term with its proper sign = the gern of all the roots with their signs changed. Or, if we choose, which is more convenient, to employ in the enunciation both the roots and the coefficients with their proper signs, we have — p, =sum of the roots, p, = sum of the products of every two, — p; = sum of the products of every three; and generally, (— 1)"p, = sum of the products of every r roots. 16 20. ‘These relations, which furnish » equations be- tween the roots and the coefficients, do not afford any immediate means of finding the roots; and if we wished to employ them to find one of the roots by the elimination of the m — 1 others, we should always arrive at an equation similar to the proposed. Let, for example, the equation be of the third degree, a” + p,v + p,& +p;=0, roots a, b, c; Py) (are p,=ab+ac+be, p; = — abe; to eliminate b and c between these equations, multiply the first by a’, and the second by a, and add them to the third, and we find a+ p,a + pa + ps = 0. 21. But although not leading to the determination of the roots, the above relations will enable us to discover many of their properties; and they are to be regarded as constituting one of the fundamental propositions of the Theory of Equations. At present we shall employ them to find the values of some of the more common symmetrical functions of the roots; that is, of functions in which each root is alike involved, so that their values are unaltered when any two of the roots are interchanged. (1.) To find the sum of the squares of the roots of iA) = 0. —p,=at+b+e4+... 48; p=0+P+e+... +? +2(ab+act+ber+...) = sum of squares + 2p, ; . sum of squares = p; — 2po. (2.) To find the sum of the reciprocals of the roots. (=. oie O¢.....2 + ac...) 4 Ob.) ee (—1)"p, = abe... 1; : 1 1 1 I 1 This=a(- 4 5 +...45)-140(< 454.45) 14 a ob l a / | Le 1 =(a+b+...4+/2) (-4+54...43)-% QOD l ; Pn-1 PiPn-i =(-p (-7=2)-n =PPe —n, RN aie, Dn 22. The following are examples of depressing an equation when one or more of its roots are known; or of forming it when all its roots are known; also of re- solving certain polynomials into their factors. (1.) To find the roots of #—1=0. One root is 7=1; dividing by # — 1, we get the quad- ratic # +a+1=0 containing the other two roots, and which gives for their values een 3 2 — (2.) To find the roots of #—- 9a#’+ #-9=0. The first member = «’ (av? — 9) + av — 9 = (a+ 1) (x - 9) = (wv +1) (#’-—# +1) (@ + 3) (#@- 3); ee / isd .. the five values of # are —1,-— 3, 3, say (3.) + 2at —-4ne* +1=0, may be solved as a quadratic; for it becomes successively 1 i (a? + =)*+ an (a? - =| =. “v av 3 1 1 ( ~ a)? + 4m(a® — ) + an? = a(n —1), & i ES Ws mae a? —— = 24/n? —1— 20, a — 20° (\/n? —1—n) =1; and by reason of the double values of the radical quantities involved, the eight roots are expressed by one formula wav (/n41—J/n) /n—14+VSn). Solution of Binomial Equations with the help of a Table of Sines. (6.) The preceding is an instance of what must happen whenever the general solution of any equation can be effect- ed, as stated in Art. 2. We shall next give an example of 1g an equation of the 7'> degree, where it is possible to get a formula expressing the » roots and no other quantities, viz. the binomial equation a” £1 = 0. This is the only extensive class of equations that has been solved by a general method; and the discussion of the nature and properties of the roots is of great interest and importance in the Theory of Equations. It is con- venient to consider the two cases #” —1=0 and a"+1=0 separately. (7.) All the roots of # —1=0 are impossible, except one when 7 is odd, and two when 7 is even. If we expel the factors w — 1 or a — 1 according as n is odd or even, the depressed equations are eee 4 hese are 1 "0, Me re waitiet: oar l= 0.5 of which, the former cannot have a positive root, and it cannot have a negative root since the proposed cannot have a negative root, 2 being odd; and the latter, since it contains only even powers of «#, can neither have a positive nor a negative root; therefore the depressed equa- tions have alls their roots impossible. Since the proposed equation is the same as #=1, the condition which it expresses is, that the arithmetical or algebraical values of w are such, that, being raised to the »™ power, they produce unity. On this account the roots of a? —1=0 are called the mn roots of unity. (8.) To solve the equation # —1=0. Since the equation can only have the real roots 1 and —1, we may assume # = cos@ + el sin@; for this value will coincide with the real roots when @ is zero or a multiple of a, and in all other cases will be imaginary. Then De Moivre’s formula gives xv” = cos n@ + incl sin n@; 20 therefore all values of @ determined by the condition cos nO +4/—1 sin nO =1, will give values of aw which are roots of the proposed ; therefore we must separately have sin n@ = 0, cos n@ = + 1, and consequently 7@ must be an even multiple of 7, =2A7 suppose, where ) is any integer whatever. Hence all values of w comprised in the formula 2X7 LMT PE oN, 2” = cos —— + —1 sin n ae OL th are roots of # —1=0, or are »” roots of unity. Moreover, this expression has ” different values and no more. ) For, taking ) from zero to 4 (m—1) or $m accord- ing as ” is odd or even, we find in the first case, one real value + 1 when A = 0, and 4 (m —1) pairs of imaginary values corresponding to values of X from 1 to $ (m—1), or m values on the whole; and in the second case, we find one real value + 1 when X = 0, one real value — 1 when A= in, and in — 1 pairs of imaginary values correspond- ing to values of X from 1 to 4m — 1, or m, values on the whole. And all these imaginary values are different from one another, because the series of angles involved in them, Qa Ao Og (n —1)7 (n — 2) cas . +. OL aes 3 5 eo UL nN Az n nr .. 2) are all different from one another, and all less than a. Also the formula (1) has no more than 7 values, For if we take X negative, the two values are not altered but only interchanged; and if we take \ = or>n, the effect is to add a multiple of 27 to one of the angles (2) which alters neither the cosine nor sine; and lastly, if we 21 consider the values of # corresponding to values of \, m and nm —m equally distant from 0 and n, we shall find them the same; for, taking \ =» —m, 2(n-—-m — , 2(n-m ee 7/7 x 2h — mda 4 _remn ~8nq . 2mqr 2m : a aaa isin = Cos HO sy) 1sin ; n n n the same as when \ = m; so that we can get no new values by taking ) greater than $n Therefore the formula can never assume any other values than the » different ones resulting from taking X from 0 to 4(m—1) or 4m, according as m is odd or even; since therefore the formula 2 ey nn equally expresses all the roots of #” —-1=0, and no other quantities, it is the complete solution of that equation. (9.) Hence we observe that for any value of d, except zero, or +m when m is even, the two corresponding roots are conjugate, and one is the reciprocal of the other; for 9 a 2 2nd — (cos as + ay a 1sin = (cos ~ AY ah 1 sin rat. n n n Also, since 2a — , 2rh74 Qa —~ . Qa\*h cos —— + Nl sin 7s ; : nv ” we observe the remarkable relation among the imaginary roots, that they are all powers of the first imaginary root Qrr — 2 x . : : a at corresponding Loe = 1, Viz. cos — + Sine sin ee so that n 22 if we denote this by a, the series of imaginary roots will be n—} n—2 2 25 As a’, a eee a. or at > 1 1 ] 1 To Tee Kg eect (ORNaas e rae (a a2 n therefore, since a? = — 1 when 7 is even, all the roots of #” —1=0 are contained in the series Leecteentts aes Comes aise (10.) We next come to the case of a +1=0, all whose roots are impossible, except one when” isodd. For | if nm be even, it is manifest that every real quantity, posi- tive or negative, when substituted for w gives a positive result, and therefore cannot be a root; and when # is odd, expelling the factor w + 1, the depressed equation is | abi ts =e as weet a + a — 1 ao +1=0, which cannot have a negative root (Art. 3), and it cannot have a positive root because the proposed cannot have a positive root, therefore all its roots are imaginary. (11.) To solve the equation 2 +1=0. As before, we may assume a = cos &£4/—1 sin 0, . 2" =cosnd ++/—1sin nd; hence all values of @ which satisfy the condition cosn0 +4/—1sinn@ = —1, will give values of « which are roots of the proposed ; hence we must separately have sinn@ =0, cosn@ = —1; there- fore 2@ must be an odd multiple of 7, = (2X + 1) + sup- pose, where ) is any integer whatever. Hence all values of w comprised in the formula YZ A+ Da 1 2X.4+1 vt are roots of 2" + 1 = 0, or are n™ roots of negative unity. Moreover, this formula will give for #, m different values and no more. For, taking ) from 0 to $(m — 1) or 4m — 1, according as # is odd or even, we find, in the former case, 4 (2 — 1) pairs of imaginary values corresponding to values of X from 0 to 4 (m — 1) — 1, and one real value — 1 for’ =4.(m — 1), or 7 values on the whole; and, in the latter case, we find tn pairs of imaginary values corresponding to values of X from 0 to $m-1, or m values on the whole. And all these imaginary roots are different, because the angles in- wolved in them 3n ba (nN - 2) (2 -—1)7 WT oo eee COE ee ear 3 ee ae or i> ie nied eae ©. el 8 n° v2 T ; v2 hr WL ( ) are all different, and less than x. And the above-mentioned # values are all which the formula can give for w. For if we take negative multiples of 2, the values of w are the same as if those multiples were positive; and if we take A= or>n, the effect is to add a multiple of 2m to one of the angles (3), which alters neither the cosine nor sine. IfrX=n-1, pe 03 ON DT 4 Sin eee = cos — & o/— =isin — = cos = /-isin=, the same as when A=0; and if X\=n—i-™m, In —-2m—-—1 In —-2m-—1 (2n m Ee rags AC m \or n # = COs 2m-+1 — 2m + 1 i SS —|] sin om tT UA) 9% 24 the same as when }\=m; so that values of A, equally distant from 0 and m—1, give the same values of a, and therefore we can get no new values by taking \ > $(m — 1). Therefore the formula can never assume any other values than the » different ones resulting from taking r from 0 to $(m-—1) or $n-—1, according as ~ is odd or even; since therefore the formula 2X + 1 ——~ ee osha (GGS Se aE creas sin eee n n equally expresses all the roots of w+ 1=0, and no other quantities, it is the complete solution of that equation. (12.) As in the former case, it may be shewn that if e e . Tv a denote the first imaginary root cos Bie “/ —1 sin ae all n the imaginary roots may be represented by b) ayes | ES ser Aaah oe. 5 a a a’ a” 2 a” 1 or, since a” =—1, and therefore a?”= 1, the lower line may be replaced by 2n— = n 2n—1 2n Bee inary ete or ate therefore, since a” = —1 when m is odd, all the roots of #”" + 1=0 are contained in the series -—3 29y—]) As a’, cece ae” b) a”” e It may be observed that the case of #” + 1 =0 ( odd), might have been reduced to that of y"-—1=0, by making v= —Y. We shall now give the resolution of # +1 into its factors. (13.) To resolve a —1 into its factors. Put 2? —1=0; “. @ =1=cos2A7 + RPL sin 2A7, A being any integer ; 2X1 aya e= a pair of values (except when 2d = 0 or any multiple of n, when there will be only one value) to which corresponds the quadratic factor 9 T xX” — 2x2 COS +1, where ) begins from 1. First, let 2 be even, then +1 and—tT are roots, and x —1a factors and, by taking X from 1 to $n—1, we obtain the other quadratic factors, 2 4: *. @—1= (wv —-1) (a —24 cos a1) (a — 2” cos — +1) na n n N—2 -.(@ — 2x cos take n +1). If we take ) greater than $m —1, or less than 1, the factors recur. Secondly, let 2 be odd, then + 1 is a root, andw—1a simple factor; and by taking \ from 1 to $(m—1), we obtain all the quadratic factors, 2 i a" —1= (#-1) (a’-2a cos — + 1) (a — 2a cos — + Wie. n n (n-—1)7 .-. (2 — 2a cos ae Thee 1). (14.) To resolve a+ 1 into its factors. Let #7 +1=0; @=—1= cos (2X41) 7 £+/-1 sin(2d +1)7, d being any integer ; os «2 eS ; kta helt nr .._t= 26 a pair of values (except. when 2A +1 is any multiple of , when there will be only one value) to which corresponds the quadratic factor 2X.+1 a — Pico ee +1, n where A begins from zero. First, let m be even, then taking ) from 0 to 4m -—1, we have all the quadratic factors, 3 ey Le ~ 2" cos + 1) (a — 2% cos ie 1 n n (nx —1)7 . (a? — 2a CO; at eline 1). Secondly, let m be odd, then — 1 is a root, and #+1a simple factor; and by taking \ from 0 to $(m-1)-1, we find the $(m — 1) quadratic factors, T 3 * vw 4+1 = (v +1) (v’-2axc0s— +1) (a — 2x08 Bee Whee n n (2 — 2)a -. (a — 2a cos +1). We shall next give the resolution into its factors of another remarkable expression, which includes the preced- ing as particular cases; and deduce from it several import- ant results, (15.) To resolve #”—2cos@x” +1 into its quadratic factors. Solving the equation #” — 2 cos 0a” + 1 = 0, (1) we find ”—cos@ + / —1sin 0=cos(2A7+0)+ fod sin(2A m+ 6), being any integer ; 2r7 +0 ae alt 2 0 @ = 008 ———— ei sin eer Ne n a formula which gives the 2m values of w, and no other 27 quantities. First, taking \ from 0 to 7 —1, we obtain 2n different values; for if two of them were alike, for instance when X = p, A\=4q, as two angles cannot have their sines and cosines identical unless they differ by a multiple of 9 is 2a, it would be necessary that sue should be a multiple of 27, which is impossible, since p and q are both less than n. Again, supposeX =np +r, where p is any positive or negative integer, and 7 is a positive integer less than m, so that A falls beyond the limits 0 and m —1, and may represent any number whatever ; then Q — Q v= COS (2px +="™ +") co rf 1 sin (2pr+ 7+") n Q2ra7r +80 ee SP art gree ie Seer haf &bsin' oe = CO 5) the same as when X\=~r. Hence the formula can never acquire any other values than the 2m different ones which result from taking X from 0 to m—1; it is therefore the complete solution of equation (1). To the pair of values (2), corresponds the quadratic factor . Q7r+80 a“ — 2a cos ———_ + 1; n and by taking from 0 to m —1, we shall obtain the n quadratic factors required ; 0 a" —2cos 0a" + 1 = (a’ — 2% cos — + 1) n aqr’+ 0 | 2 +0 } (a* — 2a cos 8 + 1) (a —22 cos +1)... 2(n—-—1)7 +0 (n-1)r +O | V4) 0 (&* — 2a cos 1). 28 Oss. When % is even, since cos 2a +0 2r7 +8 ) flee is ————— = — COs (== +7] = —cos ee), n n it appears that the factors corresponding to ) and to x tin will only differ in the sign of the second term ; therefore, when we have obtained the first half of the factors by taking \ from 0 to $m—1, we ne find the next half corresponding to values of \ from $n ton -—1, by changing the signs of the second terms of the former. (16). Also, since a” — 2 cos 0a” + 1 remains unaltered when we change the sign of 0, its quadratic factors may be arranged in pairs under the general form ; 2r\7 +0 xv — 2H cos ———— + l, n where \ is to be taken from 0 to $n or 3(m — 1) according as m is even or odd; it being observed that each of the values \ = 0, X = 4m, gives only a single factor, and not a pair. (17.) To resolve sin 26, cos »@, into their factors, 7 being any integer. We know that these can be expressed by polynomials containing only powers of sin 0, and of which in certain cases cos @ is also a factor; our object is to determine the factors of those polynomials. First, suppose odd; then, both signs being taken, 0 ™__QcosOa"®+1= (a - 22 cos +1) n + @ ; Agr + @ +1 a — 22 COS +1}... n ( : Qo @° — 22 COS n~-1)r+0 ss (a? ~ 20 005 @= IFA" 4 1), 29 Oy T Now make 2 =1, and for — write 20, and for —, 2a: n n and extract the square root of both sides; -. sin n@ = 2"~* sin @ sin (2a + @) sin (4a + 8)... ... sin {(m —1) a+6}; and changing 7@ into = +70; that is, 6 into 0+ a, we get cos 20 = 2"~* sin (a+ 8) sin (a — 8) sin (3a + 8) sin (30-6) aac ... sin (7 —2) a — 0} sin (na + 6), : T or, since ma = 2 ’ cos 29 = 2"~* cos @ sin (a + 8) sin(3a+6)...sin }(n —2)a +6}. Now transform each pair of sines by the formula sin (8 + 0) x sin (G — 0) = sin® B — sin? 8, and we have the required resolutions of sin”@ and cos”@ into their factors (7 being odd), sin 29 = 2”~* sin 0 (sin® 2a — sin’ @) (sin? 4a — sin? @) ... . sin’ (n — 1) a— sin’ 0} cos 2 = 2”~* cos @ (sin® a — sin’ 9) (sin? 3a — sin? @)... .-. {sin’ (2 — 2) a — sin’ 6}. Similarly, when m is even, we find sin 20 = 2"~' cos @ sin @ (sin’ 2a — sin’ 9) (sin 4a — sin? 6) ... .-. sin’ (n —2) a — sin’ 9} cos 29 = 2”~' (sin? a — sin? @) (sin? 3a — sin’ @) ... .-- isin? (m — 1) a — sin’ 0}. (18.) Hence we can resolve sin @ and cos @ into their factors. If we change @ into = we have, n being odd, : ee ae ee 0 sin 9 = 2"~' sin — (sin’2q — sin’ -| (sin? 4a —sin*~]...; n n n 30 eRe sin 0 therefore, making .6 = 0, since in that case Q = My We gel sin — n n = 2""'sin? 2a sin? 4a... ; veo) . 0 sin? — ain: s e n n * sn @ =n sin—\ 1 —- ——— 1 eee ee n sin’ 2a sin? 4a T Now make nm =o, and observe that a = re and there- n fore that eed sin — sin — sin — n 06 Nn — a 3 = —: &e. sin 2a T qT sinta Qo Ge” 6 0? “iN: 4.9? 4.6? 4.9? Similarly, cos @ =(1 — = (1 _ =) (3 =) ees The same values of sin @ and cos @ may, of course, be obtained from the formule for sin m0 and cos 28 when 7 is even. : T . : ; : Putting @ = rH in the resolution of sin @, we get =£-3)0- Dla) (gy) o> BR cas OO Pace 2N .2Nn a a to 1.8:3.6.5.7 ... (2% — 1) (Oe which is Wallis’s formula. (19.) If we develope these values of sin @ and cos @ according to ascending powers of @, and compare the re- sults with the common formule 3 @? + eee cos @ = 1 co + eee 1.2.3 : 1.2 ; ? sin 9 = @- 31 we find 1 1 1 1 7 EET Ug ee Pa 1 1 1 Hi 7a =+—+—5.+ 5+..=—. Ate GF PSEA ore 8 (20.) By making wv =1 in the result of example 15, we may deduce two important formule: first, we have “Tt *) 11) 2(1 —cos@) = 2 (1 - cos 2 (: — cos 9 n n 2(1 = eo *™ +") este (1 - oon be +0) pea therefore, replacing 1 — cos @ by 2 sin’ a &c., extracting the square root of both sides, and changing @ into 26, we get ‘ . 0. wrt. 2r4+0 . (n-1)7+9 sin 0 = opie sin — sin sin cep dye Sa rabae kart 3 nN Vn 117 n and writing g + 6 for 0, w+20 . 37+20. 57 +20 —_—_—— slIn ——————- sIn ——————_ on Qn Qn . (Qn —1)r + 20 weal fo oes oe 2n cos @ = 2"7' sin The formule which result from these by putting 0 =0, are sometimes of use. (21.) Ifinthe expression « — 2racos @ + 7’, we write B x : 1 +— for w, and 1 — — for r, it becomes 2n 2n e\? id ONY =? (14+) -2(1- §,) cos p + (1- =) =2(1+2)) Qn An? Qn 4n? 2 2 - 2(1 -5) cos @ = aint ( + = cot? 2). an® 2 An 2 32 + If then @ = ee » the expression just found is the general form of the quadratic factor of ._ =) oe Smee a) 1 4+- — on — —|} cosd + ——]| 3; ( ‘ Qn ( An® ( Qn and this quantity therefore, taking X from 0 to 4m or 4 (n — 1), pg . 20 £90 = 4s1In° —.4 sin ae 2n 2n 2 x : 9 Be 927 + @ nv Qn An Qn Now make n=, observing that, in that case, g 2n ‘ rae (1 “5 =| =e" (e denoting the base of Napier’s system ) Ta of logarithms), 27 tan = = 9, and that, by putting z = 0, we have : . 27 0 | 2 sin —.2s1n oo) == 2 SIN 2n Qn 2 z wt C= Dinos 0 +e = meal af 1 PRIA: )( Z he | +) ( eee) or where both signs are to be taken. The particular cases of 0=0, 0=7, may be noticed. Several of the preceding results are useful in the higher branches of Mathematics, especially in the Integral Calculus. SECTION II. ON THE TRANSFORMATION OF EQUATIONS. 23. Iw discovering the properties of f(a) =0, and determining its roots, one method of great value is to trans- form it into another equation, d (y) = 0, whose roots have given relations with its roots. We thus, without knowing the roots of a proposed equation, make them undergo cer- tain changes, such as all to be increased or diminished by a given quantity, or all to be multiplied or divided by the same number; which render the determination of the roots easier, or the equation in its new form more con- venient for solution. The problem of transforming an equation is, in its most general state, to eliminate # between the equations J («) = 9; W (a, y) = 0, the latter being the equation which expresses the relation which the roots of the transformed are required to have with those of the proposed equation. At present, we shall confine ourselves to a few simple cases which are necessary in the actual solution of equa- tions, reserving the others to the chapter on Elimination. 24. To transform an equation into one whose roots are those of the proposed equation, with contrary signs. Let a, 6, ec, ... 1, be the roots of the equation xv + pv"! a pox” +... $0, = Oe n—-l Oot pa + He" +... + Dp = (w — a) (a — 6) (w-—c)... (vw — J). 34 In this identical equation change w into — y; then, whether m be odd or even, the result is y" — py + Pry” — «ee = Dp =(y+ a)(y+b)(y+e)...y+9, which shews that the first member, put equal to zero, is an equation in y, whose roots are — a, — b, —¢, ... — l. Hence, if the signs of the alternate terms, beginning with the second, of any complete equation be changed, the signs of all the roots are changed. Before this theorem can be applied to incomplete equa- tions, the deficient terms must be replaced by cyphers. Thus the equation «* — qa +7r= 0, or a + 0a + 0a —que +2 =0, is one whose roots differ from those of a + qe + r= 0 only in sign. 25. To transform an equation into one whose roots are those of the proposed equation, each diminished or in- creased by the same given quantity. Let a, b,c, ... 1, be the roots of the equation | x” + Dye + oa. + Pa @ + Pn = 05 10D" 6. +Py_s2+Pa=(@—4)(@-b) (=e)... (=I). n—tl In this identical equation, change # into y +h; w (yYth)y’ +p, (y thy +... + Pai (y +h) + Dn ={y-(a-A)t fy-(b-A)t... fy -@ AD}, which shews that the first member, put equal to zero, is an equation in y, whose roots area —h, b—h,...,1—h; or, expanding by the binomial theorem, and arranging it ac- cording to powers of y, we have the transformed equation yitnh) _, n(n—-1),,)\y*+...+h = 0. n h? a + p, ! Ha 2 + pih'-* +(n—1) ph + poh?-* + Dy a iia + Py_yh + Pn 35 _ To increase, on the contrary, all the roots, we must change w into y— A, and therefore we must take the odd powers of f in the above equation with a contrary sign. 26. If we arrange the transformed equation (y+h)*+ pi (y+h)""'4+ po(yth)-* +... + Pn—i(y +h) + D,= 0, according to ascending powers of y, we shall see the law of formation of its coefficients more distinctly; for we then have Ar + ph" + poh™? + 22. + Pugh? + Pp_ih + Dy + {nh + (mn —1) pyh”* + (n— 2) ph? + ... + 2D, oh + Daily + {n(n —1)h"-? + (n-1) (1-2) ph” > 4... + 2-2} [ to |e, where | denotes 1.2.3... 7. The first coefficient is the original polynomial with / in- stead of w, and will therefore be represented by ut (A); the second coefficient is derived from the first by multiplying every term in f(h) by the index of that power of 4 which it involves and diminishing the index by unity, and may be represented by /(h); the third is found from the second, by repeating the same process upon f’(h), or performing it twice upon f(h), and may therefore be represented by f”(h); and in like manner all the other coefficients, being formed by the same uniform law, may be represented by f(A), ... fe "“'(h); therefore the trans- 36 formed equation, arranged according to mee powers of y, 1s ) fa) +f Oy +f") +f" oh bia y f +7 (A) ae at 27. Hence it follows that if in J (a) we change x into «+h, the result, arranged according to A of h, is S(@t+hy=f(a) +f (ayh +f'@) = —— ay oe n—-1 PO =a hOT SF (2)s f’'(@), --- f"~" (a) being derived from f(«) according to the law explained above; they are called derived func- tions relative to the given function /(#), or derivatives of J (2). | Ozs. Those who are acquainted with the Differential Calculus, know that the above result can be immediately obtained from T'aylor’s theorem; and that f’(#), f" (2), &ec., are the first, second, &c., differential coefficients of J («), which all vanish after the ™, /(«) being here a ra- tional integral function of the n™ degree. 28. Hence, to increase or diminish the roots of a proposed equation, f(v)=0, by a given quantity h, we must write down /(«) and all the derived functions, f(a), S' (2), f' (2) .--f""(@), and substitute in them — A or +h for #, according as the roots are to be increased or dimi- nished; the resulting quantities are the coefficients of the transformed equation. Ex. 1. To transform «+ 5a*+ a ~ 162° — 207+ 16=0, into one whose roots shall be the same, except that each is increased by unity, 37 Here y=a+1, or v=y—-1; therefore the transformed equation is | ’" y y° AEDS D9 HONE + PCY EG y 7 4? (\— aii Soe but f (#) =a° + 50%+ a— 162° - 20@-16' of (—1)=-9. SJ (2) = 52° + 200° + 3a°- 324-20 8 f’(-1)=901 Je) = 200° + 6027 + 6x. — 32 Abs 1) a SF (2) = 60 x” + 120@ + 6 nf Oe 1) en SF’ (2) = 120a + 120 rie 1) tgs y y° iO FN SAE oe ap 23'0 at wig g ty or y°— 9y°+ y?-9 = 0, the roots of which were found (p. 17). Ex. 2. Increase, and diminish, by 3, the roots of a —2a°—#+2=0. The transformed equations are y> —11y’ + 38y — 40 =0, y° + Ty’ + 14y + 8 = 0. 29, One use of this transformation is to take away any term of an equation; by which means it is sometimes reduced to a form more convenient for solution, as in the former of the preceding Example$: Thus, to transform /(#) = 0 into one which shall want Pi > 112 the second ‘term, we must have nh +p, =0, orh = — and therefore v = y eel i. e., the roots must be increased , n by 2 » p; being the coefficient of the second term with its A oar § proper sign, and m the degree of the equation. We may arrive at this result immediately, by observing that if a, b, c,...¢ be the roots of 2+ p;a°"' +... = 0, and a+h, 38 b+h,...l+h those of the transformed equation y” + qy"~’ +...=0, then-g,=a+64+...d+nh=—p,+mnh; and if this = 0, then h = me , the quantity by which the roots are to be increased. To take away the third term, we must diminish the roots by a quantity h determined from the equation (Art. 25) i} ND SOD 1) op ee and, in general, to take away the (r+ 1)™ term, we must diminish the roots by a quantity h determined from the equation (Art. 26) r (7-1) *""(h) = 0, or. hk +—p hin + eh ee Wi ( ) ) Avi Tame. To take away the last term, we must have h” + p,h*~’ +... =0; i.e, we must solve the original equation. In effect, the transformed equation would have one root = 0, and therefore h = x. Oxzs. If all the roots of f(v) =0.be real, it will be seen further on that all those of #/”~"(w) = 0 are so; therefore the equation may be transformed in r different ways so as to want the (r+1)™ term; butif all the roots of f"~"(w) =0 be impossible, in. which case 7 must be even, f(x) = 0 can- not be transformed into another with real coefficients so as to want the (7+ 1)" term; and in the latter case, the proposed equation will have at least 7 imaginary roots. Ex. 1. To transform a’* — 62 + 42 — 7 =0 into one which shall want the second term. Pi Hiererpics — 0, tin 35 |". dame [ne =y+2; (y+2) -6 (y+ 2) +4(¥ +2) -7=0, or y — 8y — 15=0. Ex. 2.. Transform a’ + 522+ 82-—-1=0 into two others which shall each want the third term. 39 30. ‘To transform an equation into another, of which the roots are equal to those of the proposed, each multiplied by the same given quantity. In the identical equation ao + py) + pea? +. Py @ + Dp = (x -—a) (vw —b)... (v-J), change 2 into *, and then multiply both sides by m’, “y+ mpyy" + mpzy’ +... +:m"'p,_yy + m"p, = (y— ma) (y— mb)... (y — ml); therefore the roots of y+ mp,y" + m poy" +... + m™"p,_1y + mp, = 0, are ma, mb, mc,... ml; and it is formed from the equa- tion, supposed complete, whose roots are a, b, c,...1, by multiplying the coefficients, beginning with that of the second term, by m, m*, m*,...m”. The use of this transformation is to get rid of the co- efficient of the first term; or to make the fractional coeffi- cients of an equation disappear, without affecting the first term with any coefficient except unity. Thus, if ma” + p,2"~'+ p,2"~* + ... = 0 have roots Gs) O5..Cy Week, then my” + mp,y" | +m’? poy"~* or y” + py" + mp, y" "+ ... =0 has roots ma, mb, me, &c., and is of the usual form. +...=0 Also, if 2” + Pi yp- 71 Qe q3 roots a, b, c, &c.; and if m be the least common multiple of all the denominators 9,, q2., 93, &c., then (Pei ae ORS pasate .-. = O have m m? 9. y" + ‘eae sph + 2 =0 11 q2 40 has roots ma, mb, me,. &c., and all its coefficients are integers. Similarly, an equation may be transformed into another of which the roots are equal to those of the proposed, each divided by the same given quantity, by dividing the second, third, fourth terms, Re. (supposing the equation complete) by m, m*, m°, &c. respectively. 31. By taking m =the least common multiple of the denominators, we do not always get the transformed equa- tion with coefficients the least possible. All that is neces- sary is to determine m so that m, m’, m* ,.. are divisible re- spectively by q,, q25 435---3 and therefore that m, m*, m’, &c. contain the prime factors of q,, qo, q3, --. raised at least to as high powers as they occur in the respective denominators. A 3 5 Ex. 1. Ti le a eee 3 8 12 The transformed equation is y ae ne''t oe Tint 37.98” and the factors which the successive terms require in m are 3, 2°, 3.2, which is satisfied by m=12; and the trans- formed equation becomes | y — 16y” — 54y + 120 =0, where y = 12a. 5) 5 ff 13 Ex. 2. a —— ie" + — x’ — —_— fF - —_ = 6 12 150 900 The transformed equation is y' — 25y° + 375y" — 1260y — 11700 = 0, where y = 30a. 32. ‘To transform an equation into one whose roots are the reciprocals of the roots of the proposed equation. If in the identical equation aot p, er + ou. + Pai @ + Dy = (w—a) (a - 6)... (@-J), 41 1 ; we change a into fe we get 1 1 1 1 Bo Fs, (ta) (02) 3) A y y y y 0-9 9-}- Pp = (v-<) (v-5) - (¥- 3) ={y - y rahe y 7 ’ which shews that if we write : for #, and then multiply by y”, the resulting expression put = 0, has roots 33. This transformation fails if the transformed equa- tion be identical with the original one; that is, if the co- efficients be such that Pn» = PnPi> Pn-~2 = PnP2> &e., t= PnPn° Hence p, = +1; and according as we take the upper or lower sign, we have Pani = Py> Pn-2 = Pas &e. : OF Pa (Pi Paes = — Po» &e.; that is, the coefficients of corresponding terms taken from the beginning and end must be equal and of the same signs; or, they must be equal and of contrary signs; only it must be observed that if the equation be of an even number of dimensions 27, there will be a middle term p,#", and we shall have p, = p,p.,; which, for p., = — 1, gives . Pr = = Pro or Pr 95 P 42 so that when the equation is of an even degree and the corresponding coefficients have contrary signs, there must be no middle term. It is easy to see that when these conditions are satisfied, e e 1 « . the equation remains the same when — is substituted for « ; @ but when the corresponding coefficients have contrary signs it will be necessary, after the substitution, to change the signs of all the terms; and the above investigation shews that these are the only conditions under which an equation can have the property. Equations of this sort, that is, e e j e e 1 which remain the same when w is changed into -, are called Reciprocal Equations. 34. Every reciprocal equation will have its roots in ene a! : pairs a, ae b, 7; &e.; but when the degree is odd, there will, besides, be a root + 1 or — 1, according as the last term is negative or positive; and when the degree is even with the last term negative, there will be two roots +1 and — 1. e e 1 e For if a be a root of the proposed equation, — will be a a root of the transformed equation; but the transformed. & e e . 1 . equation coincides with the proposed, therefore — will be a a root of the proposed equation; and so on for every other root. Also, since 1 and — 1 are the same as their recipro- cals, each of these may enter any even number of times. Again, we may write the reciprocal equation of an odd degree, collecting the terms from the beginning and end, ao” 21+ p,a(a"~> £1) 4+... =05 and since every term is divisible by x + 1, it will have a 43 root +1 or —1 according as its last term is negative or positive. Also, the reciprocal equation of an even degree with its last term negative may be written we” —1+4 pe (2 *—1) 4+... =0, which is divisible by a — 1; therefore it has two roots + 1 and ar. In both cases, when the factor «+1 or a —1 is ex- pelled, the equation is reduced to a reciprocal equation of an even degree with its last term positive, which may there- fore be taken as the standard form of reciprocal equations, Ex. Reduce to a reciprocal equation of an even degree with its last term positive, each of the equations oe —— at — —a’ + are peed 1h 6 6 ; : BP pie ahi ae matte Oe Cee LF ES OD = 6 3 6 35. Various transformations may be effected by par- ticular artifices; we shall give one or two instances, of which the results will be useful to us in the sequel, and are of importance in discovering the existence of impossible roots in equations. (1.) To transform an equation into one whose roots are the squares of the roots of the proposed. Hf 2” + pa") + pv"? + pia? + 0. + py 18 + Dy =(#-—a) (#-bd)...(#@-J, then a” — p,a"-! + pp"? — pv +... Sp, 0 + Ds =(# +a) (a+b)... (~@ +7); (Art. 24) therefore, multiplying these equations together, we get Cpe ep, 4 +.) (pat + pa ey = (x — a’) (a — b’)... (a -P); 44 but the first member is: a + 2p." -* 4 (2p, + p,) a + wee hou ike (p?a** + 2p, pst" + 02) 5 therefore, replacing x by y, we have y” + (2p,— pi) y+ (Pe — 2PiPs + 2p,)y"—~ + «-- =(y-a) (y¥-®)...y-F); hence the transformed equation, whose roots are a”, b’, &c., is y" + (2p.- pi)y n=l + (p; — 2p:p3+2p,) yy" + + = 0. Hence the proposed equation (since its roots are the square roots of those of the transformed equation) cannot have more real roots than the latter has positive roots. Ex. v+e4+3a°+167+15=0. The transformed equation is wo + 2u° + 33a + 232? + 1664 — 225 =0, which (Art. 11) has only one positive root; and therefore the proposed has only: one real root. (2.) . To transform the equation 2° + qv +7 =0 into one whose roots are the squares of the differences of its roots. Let the roots of «°+qa+7r=0 be a, 6b, ¢; O=a+b+ec, q=ab+ac+be, —r= abe, and a’ + b? + c? = —2q, (Art. 21). Since one root of the transformed equation is 2abe (a-by=V+0 +0-e- 20+ ane ». € ° 2 2r ° if we assume y = — 2q — wv + —, then when w& assumes its @ three values,-y becomes equal to the three roots of the 45 transformed equation; therefore the required equation will result from eliminating w# between the proposed and a+(y+2q)u-2r=0; subtracting this from the proposed, we have ar ‘ y+q and if we substitute this value, and reduce, we obtain the transformed equation (y+q)#—-—3r=0, or v= 3 6 2 2 e q 108 -+ EELS = 0. y +0qy+9OQyr+ (= _ |. (tt ae Hence, if i + = is positive, the transformed has a real negative root (Art. 10), and therefore the proposed equation must have a pair of imaginary roots; since it is only when two roots are imaginary and conjugate to one another, that the square of their difference can be negative. 2 3 If Wi £e 7 = 0, then one value of y is zero; and there- fore the proposed has a pair of equal roots. rg If me oe is negative (and therefore g an essentially negative quantity) the transformed equation cannot have a negative root (Art. 3); and therefore the proposed has all its roots real. Ex. e TH +7=0, The equation of differences is y — 424° + 441y — 49 = 0, which cannot have a negative root (Art. 3); therefore the proposed. has all its roots real. SECTION III. ON THE LIMITS OF THE ROOTS OF EQUATIONS. 36. ‘Tue limits of any group of roots of an equation are two quantities between which the whole group lies ; thus +o and 0 are limits of the positive roots of every equation, and 0 and — © of the negative roots. But in practice we are required to assign much closer limits than these, usually the two consecutive whole numbers between which each root lies, so that the inferior limit is the integral part of the included root. This may be effected without knowing any of the roots of the equation, as will be seen in the following propositions. The roots spoken of in this section are the real roots. 37. Quantities between which the real roots of an equation taken in order lie, when substituted successively for the unknown quantity, give results alternately positive and negative. Let the real roots, arranged in order of magnitude, be a,b, c,...1, so that a is greater than b, 6 greater than c, &c.; the negative roots, if there be any, coming at the end of the series, and that being the least whose numerical value (neglecting the sign) is the greatest; then if f(x) = 0 be the equation, | J (#) = (@ - a) (v@ — b) («@- 0)... (@-D.p(@), where @(#) is a polynomial that remains positive whatever 47 real values be substituted in it for w, (Art. 15.) "Then. if we substitute for # a quantity a greater than @f “he result o (a) i is positive because every one of its factor§/is so; sit we positive. Again, a quantity between 6 and whole positive, because the two first factors a and the rest positive. Thus, quantities between roots taken in order lie,. when substituted for 2, iy r alternately positive and negative. —s 38. Again, suppose that a, 6, c,.../ are all the roots of f(#) = 0, which lie between two numbers a and #, of which a is the lesser, and that @(#) =0 is the equation containing the remaining roots; then substituting a and B successively for #, and dividing one result by the other, we get J (a) a (a - a) (a—b)... (a—- 2) @ (a) J(B) (8-4) (B-4)...(B-) P(B) Now all the factors in the numerator are negative, and all in the denominator positive; also @ (a), @(), must have the same sign, since @ (w) = 0 has no root between a and 6; therefore f(a), f(3) have different, or the same signs, according as the number of factors a—a, a-—b, &c. is odd or even. | Hence if two numbers, when substituted for x, give results with different signs, then one, three, or some odd number of roots lies between them; if they give results with the same sign, then two, four, or some even number of roots lies between them, or none at all. 39. If a number a can be found such that a and every greater number, when substituted for wv, gives a finite positive result, then a is greater than the greatest root, and is called a superior limit of the roots; for if there could be a root greater than a, then some number greater than a would give a negative result, which is contrary to the supposition. 48 Similarly, if a number 6 can be found such that ( and every smaller number, when substituted for #, gives a finite result with a permanent sign, that is, constantly positive or constantly negative, according as the degree of the equa- tion is even_or odd, (3 is less than the least root, and is called an inferior limit of the roots. Ozs. This supposes the greatest and least roots to occur singly; or, if repeated, to be repeated an odd num- ber of times; for if we had /(v) = (#—-a)’(#—-6)..., and a were between a and 6, then a and every greater number would give a positive result, without a being the superior limit. 40. Ifas many quantities can be found, which, sub- stituted for # in f(x), give results alternately positive and negative, as the equation has dimensions, it is plain that the odd number of roots which lies between each adjacent two of the quantities, cannot exceed one. But as it is seldom the case that so many can be found, the next point .to be determined is, whether all the real roots that exist have been discovered; this enquiry will obviously be nar- rowed if we find the limits beyond which the quantities, successively substituted for the purpose of separating the roots, need not extend, that is, the superior and inferior limits of the positive and negative roots; the principal methods of doing this are the following. Superior and Inferior Limits of the Roots. 41. All the roots of an equation lie between p +1 and —(p +1), p being the greatest coefficient without regard to sign. For it is proved (Art. 9) that » +1 and every greater number, when substituted for aw, gives a positive result, therefore p + 1 is greater than the greatest root; also that — (p + 1) and every greater negative number gives a result with a permanent sign, that is, constantly positive or con- 49 stantly negative, according as the degree of the equation is even or odd, therefore — (p + 1) is less than the least root. 42. 'The greatest negative coefficient increased by unity, is a superior limit of the positive roots of an equation. Let — p be the greatest negative coefficient; then any value of a2 which makes nu— a” —p(a"'+a""* +... + #2 4+ 441) positive, n n—1 w—2 2 a” — 1 or a" > p(w +a" +... +a +241) >p A ama will, a fortiori, make n—-2 a” + pa" ' + pv”? +... + p,_,2 + p, positive, or make f(x) positive; because in the latter case there will generally be fewer terms to be taken away from x”; and of these, not one is greater than the corresponding term in the former case. n 5 1g | Now the inequality 2” > p , 18 satisfied if “2-1 2 = or >a" Bear v—-1l1=or>p,or¢=or>p+l. ain 1 Since, therefore, p + 1 and every greater number, when substituted for v, will make f(«) positive, the numerical value of the greatest negative coefficient increased by unity, is a superior limit of the positive roots. Ozss. This result, as is easily seen, is included in that of the preceding article; for if all the coefficients were ne- gative, the substitution of the greatest of them increased by unity, and of every greater quantity, would give a posi- tive result; therefore, a fortiori, the result will be positive if some of the coefficients be positive; the limit however here determined will usually be less than that in the former article, and never greater. (i 50 43. In any equation, if p,2"~’ be the first term which is negative, and —p the greatest negative coefficient, 1 + Jp is a superior limit of the positive roots. Any value of w which makes -r+1 n n—r n—r—1l 4 a" erie vw > pla "+a te bot) Sp oe ee en 9 x will of course make w” + p,a"~* + pox" * + ... positive. n—r+1 ° ° e a 1 . . > . Now the inequality «" > p — more satisfied 1f 4 if Biot on oo ve" > p yor v”~'(v — 1)>p, or (w-1)'7'(@-1) = or >p, ea or (vw —1)" =or>p, or e = or > 1 + 4/gn Since, therefore, 1 + \/ p and every greater number gives a positive result, 1 + a/ p i is a superior limit. This method may be employed when the first term is followed by one or more positive terms. Ex: x - 112” mae? he es = 61 = 0. Here r = 3, and a limit of the positive roots 44. If each negative coefficient, taken positively, be divided by the sum of all the positive coefficients which precede it, the greatest of the fractions thus formed; in- creased by unity, is a superior limit of the positive roots. Let the equation be io” + pa" 4+ pou"? + (= pala ees soni (—pedare’ then, since (Art. 5) +... +p, = 0; Pm&™ = Py (@ — 1) (@"-) + a"-*F +... +41) + Dy, 51 if we transform every positive term by this formula, and leave the negative terms in their original form, we shall have 0 =(a - Wa + (a - 1)a"* + (a —1)a"~° dee + eal + 1 + pi (@—1)x"~* + p, (w-1)a"* +... + py (@-1) 4+ py + p,(v@—1)a"-" +... + po(w—1) + py aya” Ae Ae Se SET enon. , Now if such a value be assigned to » that every term is positive, that value will be the superior limit required ; in the terms where no negative coefficient enters, it is suf- ficient to have # >1; in the other terms, each of which involves a negative coefficient, we must have (1 +p, + pe) (~@-1)> ps, (1 + p, + pot... + Pri) (@-1)>p,, &e., iL Ps ~4+132 Py 1+ Pi + Pre ~+ 1; &e. or ®@ > > 1+ Pi + pot +. + Pr- If then w be taken equal to the greatest of these frac- tions increased by unity, this value, and every greater value, will make f(x) positive, and therefore will be a superior limit of the positive roots. This method gives a limit easily calculated, and generally not far from the truth. Ex. 1. 4a” — 8a + 234° + 105a? — 804% + 3=0. 8 80 8 80 The tractions are — and ——-+—_-—- , and — > —_; 4 44+ 23 + 105 4 132 therefore 3 is a superior limit. Ex.2. 42° -62°-—7a°4+ 8a'+72° — 23a°-—224-—5=0; here 3 is a superior limit. Ozs. The form of the equation will often suggest artifices, by means of which closer limits may be deter- 52 mined than by any.of the preceding methods; thus, writ- ing the equation of Ex. 1 under the form 16 4v* (@ — 2) + 230° + 105@ ( =; =) +3=0, \ we see that # = or >2 gives a positive result, therefore 2 is a superior limit. Similarly, by writing the example of Art. 43 under the form 61 x(a — 25) +11 (x - —) =0. we see that 3 is a superior limit. We have seen (Art. 12) that an equation of an even number of dimensions with its last term positive may have no real root; but we shall now shew that in any equation whatever, if the absolute term be small compared with the other terms, there will be at least one real root also very small. 45. In the equation pot" + pa" ' + &.+a—7r=0, where r is essentially Ne th and which may represent any equation whatever, if r<_ , where p is numerically 4(1 +p) the greatest coefficient, then there is a real positive root < 2r. By dividing by the coefficient of v, and changing the signs of all the terms, and of all the roots if necessary, every equation may be reduced to the form . —r+a+&c. + p,v""'+p0"=0 (1) where r is essentially positive ; let p be numerically the greatest coefficient, then any value of «<1 which makes pei —a* = 4 a> pia? 4 ehh ec. -+ at) a -—wv 53 will make the first member of (1) positive; and this condi- tion is fulfilled by a Bor > 5 or (l+p)e’-(+r)a+r=0, or 2(1+ p)e=(1+7)-/+rP?-47r(14+p); if then 47 (1 + p) <1, the radical will have a real value >r; and there will be for wv a real value less than ————— 2(1 + p) which makes the first member of (1) positive; and r= 0 makes it negative; therefore in any equation reduced to : 1 : Ae the above form, if + <-———_- there is a real small positive 4(1 + p) root < 2r. Ex. «+182 —21a*—12%+1=0 has a real root 1 between 0 and r+ 46. To find an inferior limit of the positive roots, we must transform the equation into one whose roots are the reciprocals of the roots of the former; and the reci- procal of the superior limit of the roots of the transformed equation, found by the preceding methods, will be the quantity required. Hence if p, denote the greatest coefficient of a contrary sign to the last term p,, an inferior limit of the positive roots is P “. For the transformed equation will be Pn aes Pr (Art. 32) : 1 ina a aces Nae O, of which i is the greatest negative coefficient; therefore Rn ee lis a superior limit of its roots; and consequently Pr Dp He aes oe sex : en an inferior limit of the positive roots of the proposed Pr + Dn equation. Ex. ve — 420% + 441a@ — 49 = 0. 49 hoa : Here p, = 49, p, = 441, ———, or — Is an In- 49 + 441 10 ferior limit of the positive roots. By putting a =-—, we may discover a limit closer to the roots; for the trans- formed equation is gy +2y— m0, or 9( +e a y —-9y +-y-—=0, or x (y-! ~|y-—)= y Mens: bene bey a 3A 7\Y ~ ae : which evidently has 9 for the superior limit of its positive 1 Cee Pe roots, and therefore the proposed has 7 for its inferior limit. 47. To find superior and inferior limits of the negative roots, we must transform the equation into one whose roots are those of the former with contrary signs (Art. 24); and if a, B, be limits, found as above, of the positive roots of this equation, then — a and — # will be limits of the nega- tive roots of the proposed equation. Ex. e-7e¥+7=0; putting # = — y, we get y’-7y—-7 =0, of which 1 + +/ 7 or 4 is a superior limit. A 1 1 Also, putting y =~, we get +e — og) or x? — — + 2*—— =0, of which = is a superior limit; 28 28 3 55 therefore the negative root of the proposed lies between — 4, and — 3. Newton's Method of finding Limits of the Roots. 48. The limits however, deduced by any of the pre- ceding methods, seldom approach very near to the roots; the tentative method, depending upon the following pro- position, will furnish us with limits which lie much nearer to them. Every number which, written for a, makes f(x) and all its derived functions positive, is a superior limit of the positive roots. For if we diminish the roots a, b, c, &c., of f(#) = 0 by h, that is, (Art. 25) substitute y + h for a, the result is Sy +h) = 0, or ) ion anid’ “iat pis, taeR J (2) ef Ch) +. +7 Ona! = 0. Now if we give such a value to f that all the coefh- cients of this equation are positive, then every value of y is negative; that is, all the quantities a—h, b-—h,c—h, &e., are negative, and therefore h is greater than the greatest of the quantities a, b, c, &c., or is a superior limit of the roots of the proposed equation. Similarly, h will be an inferior limit to all the roots, if the coefficients be alter- nately positive and negative. Ex. To find a superior limit of the roots of 5M + Te 1 = 0, The transformed equation, putting y + / for a, is 9 (1 —5h° + Th—1) + (8h? 10h +7) y+ Gh—-10)% + y°=05 in which, if 3 be put for h, all the coefficients are positive, therefore 3 is a superior limit of the positive roots. 56 Oss. ‘This method of finding a superior limit of the roots by determining by trial what value of # will make J («#) and all its derived functions positive, was proposed by Newton. Waring’s Method of separating the Roots. 49. If a series of quantities be substituted for 2 in J («), then between every two which give results with dif- ferent signs an odd number of roots of Fi (w) = 0 is situated; and between every two which give results with the same slgns an even number is situated, or none at all; but we cannot assure ourselves that in the former case the number does not exceed unity, or that in the latter it is zero, and that consequently the number and situation of all the real roots is ascertained, unless the difference between the quan- tities successively substituted be less than the least dif- erence between the roots of the proposed equation; since, if it were greater, it is evident that more than one root might be intercepted by two of the quantities giving results with different signs, and that two roots instead of none might be intercepted by two of the quantities giving results with the same sign; and in both cases, roots would pass undiscovered. We must therefore first find a limit less than the least difference of the roots; this may be done by transforming (as we have already shewn for a cubic, and shall hereafter shew generally) the equation into one whose roots are the squares of the differences of the roots of the proposed equation. Then if we find a limit & less than the least positive root of the transformed equation, / i will be less than the least difference of the roots of the proposed equation ; and if we substitute successively for # the num- bers s, s — J kk $s —2 J/ ky &c., (s being a superior limit of the roots of the proposed) till we come to a superior limit of the negative roots, we are sure that no two real roots, lying between the numbers substituted, have escaped us ; and that every change of signs in the results of the sub- stitutions indicates only one real root. Hence the number : 57 of real roots will be known (for it will exactly equal the number of changes), as well as the interval in which each of them is contained. Ozs. This method of determining the number and situation of the real roots of an equation was first proposed by Waring; it is however of no practical use for equations of a degree exceeding the fourth, on account of the great labour of forming the equation of differences for equations of a higher order. Ex. #°-7#%+7=0. The numbers 1 and 2 give each a positive result, but yet two roots lie between them. The equation whose roots are the squares of the differences is (p. 45) y® — 42y + 441y —49 = 0, an inferior limit of the nbs Jemned Li be. positive roots of which ie (Art. 46); therefore 538 less than the least difference of the roots of «*? — 7% + 7 = 0, and as ane substituting 2, ore the results are +,—,-+ 5; hence one i 5 5 4 value of «x lies between 2 and ef and one between = and a and, similarly, we find the negative root, which necessarily ; : 1 exists, to lie between-3 and-3 3; Use of the derived Functions in finding Limits of the Roots. 50. The first derived function f’(x) put equal to zero, gives an equation whose roots have remarkable relations with the roots of f(#) =0; these relations we now proceed to investigate, as well as to draw several important conclusions from them, beginning with the following proposition. 51. An odd number of the roots of the equation SJ (@)=n0" "+ (n-1) pia" + (2-2) pot? + 222+ Pa_1 =, lies between each adjacent two of the roots of f(#) = 0. 8 58 Let the real roots of f(a) = 0, arranged in descending order of magnitude, be a, b, c, ... J, in number n — 7; o. f(a) = (w — a) (@ —b) (w —)... (@ -))- P(@), where f(a) is a polynomial of x dimensions that cannot become negative for any real value of a. In this identical equation, for @ write y +h; “ flythy=(yth—a) (yt+h—b) Yt+h—oe)... .- Yth-).dtyt+), or f(b) +f). + f(t) = + ey! a ly weet {(h 6) (hc). cit) ee seri (h,~'a) (h — 0). +n ceh yt Ge a) eee (h— I) x {9 (0) + GOL + Ge) + ot v's therefore, equating coefficients of y, which in the first mem- ber is f’(h), and in the second member is the sum of the products of the constant term in each factor by the term in the other factor involving only the first power of y, we get 7 (h)= {(h—b)(h—c)...4+(h—a)(h—-c)...+(h—-a)(h—-B)...+...$ x O(h)* + (h-a) (h—-5)... A-D p (A) in which equation, if a, b, c, &c., be written for h, since the last term of the second member vanishes by these substitutions, and @(h) is always positive, the results will have the same signs as | (a — b) (a-c)..., (6-4) (6-C0)..., (ec —a) (c— 5)..., &e. which are alternately positive and negative, since they re- * It will be observed that the coefficient of (h) is the sum of a series of products, in forming which, each of the factors h — a, h—b, &c. is left out in turn, 59 spectively involve 0, 1, 2,... negative factors; a, 6, ¢, .. being arranged in order of magnitude. Hence an odd number of roots of f’(h) = 0, or of a) = na" * + (2 — 1) pa" * + 22. = 0, (since it is of no importance by what symbol we represent the unknown quantity,) lies between a and 6, an odd num- ber between b and ec, and so on; that is, an odd number of the roots of f’(~) = 0, lies between each adjacent two of the roots of f(x) = 0. 52. If the equation /() = 0 has n real roots, then nz”~' + (n—1)p,2""* +... =0, or f(z) = 0, has 2 — 1 real roots, for one of its roots lies between each adjacent two of the roots of f(«) = 0; and it is therefore in this case called the limiting equation of the proposed. 58, The equation f”(«) = 0, or Pee yr (nk 1) (n=) pare 2, Lo, being derived from f’(#) = 0 in the same manner as this latter is derived from f(#) = 0, will have an odd num- ber of roots lying between each, adjacent two of the roots of f’'(w) =0; and if all the roots of f(x) = 0 are real, all in f'(w) = 0, f'() =9, &c., are real, till we arrive at a simple equation. And, in general, the equations ST (2) = 9, f'(#) = 0, &c., have at least as many real roots wanting one, two, &c., as f(x) = 0. Hence if f(x) =0 has m —r possible, and + impossible roots, f”(#) = 0 will have at least 2 —7 —m possible, and therefore (being of 2—m dimensions), cannot have more than r impossible roots; which shews that though /(#) = 0 may have fewer real roots than several of its derived equa- tions, it has at least as many impossible roots as any one of them. 60 Ex. The equation. #’(«— 1)” =0 has all its roots real; therefore f"(x) = 0, n ,, N(nm—-1) n(n —1) or on. 2 SY ee ee 2n 1.2. 2n(2n—1) has 7 real roots, lying between 0 and 1. 54. If we know all the’real roots of f(a) =0, and substitute them, in order, in ST (@)s we may find how many real roots the proposed equation contains. For let a, B, y,---A be the roots of f’(#) = 0, ar- ranged in order of magnitude, and let o>, a; , Y> 2 cae be substituted for # in f(#), giving results +, f(a), f(B),---F), #3 then there can only be one root greater than a, and one less than X; for if there could be more, f’(#) =0 would have a root situated between them, that is, a root >a or <2, which is impossible, for a, 3, y, .-. are all the real roots of f’(#) =0 taken in order; also the other roots are situated singly between a and 3, Band y, &c. Hence if f(a) be positive, there is no root greater than a; for if there were two, these would include no root of f(a) =0, which is impossible; if negative, there is one root greater than a, and only one, for there cannot be three > a, for the same reason as before; if (8) have the same sign as F(a) there is no root between a and (3, otherwise there is one root, and so on: and if f(A) be positive for an equation of odd dimensions, or negative for one of even dimensions, there will be one root < d, otherwise none. It follows, therefore, that the number of real roots of J («) =0, will be exactly equal to the number of changes of sign in the results of the substitution of ©, a, B, y,---A; 61 — co for w; and can be exactly determined whenever we can obtain a solution of f’(v) =0., Ex. 1. To determine whether #°— qx +7=0 has all its roots possible. The limiting equation is 3a*— q=0; q q ete hy. WA: but f(«) = 2 (#- q) +7, and for both substitutions poh Fe 3 9 f(6)-- /2(-22) +r =2 (247 3 2 3 If then (=) < (*) » f(a) is negative; therefore there is one root >a; also f((3) is positive, therefore there is . | one root between a and (3, and another less than #. BN 3 If (-) > (") , f(a) is positive; therefore there is no root greater than a, nor one between a and #3, because J (£) is positive; but there is one root < #, that is, one ne- gative root which is the only real root. If \/q be written for a, the result is + 7; hence when. : = q all the roots are real, the greatest lies between ae g and - These results were obtained by a different method (p. 45). Ex. 25 #2 -— gr +7 =0. o. f (@) = na"~'— q=0, which has one real root aand nm —2 imaginary ones, or two real roots a and $3 and n — 3 imaginary ones, according jas m is even or odd. (p. 19). 62 In the former case, - 1 ] q\* 1a n— 1 qe fle <(2). fg78) #2 sora) ae n nr n which is negative or positive, according as (2) > or < n n—1 necessarily m — 2 imaginary roots, will have two real roots ra New ( ; therefore the proposed equation, which has n naj ; q \ or none, according as (2) > or < |= ‘ In the latter case, 1 1 mae, Bis r, F(B) =" q (y4 r, nr nm nr f(a) = which have different or the same signs, according as n - n—} > o< GE) n nm— 1 therefore the proposed equation (which has necessarily n —3 imaginary roots) will have three real roots, or one, according as . n n-1 7 (2) cerca aa n m-1 Des Cartes’s Rule of Signs. 55. No equation, complete or incomplete, can have more positive roots than it has changes of signs from + to — and from — to +; and no complete equation can have more negative roots than it has continuations of the same sign. Suppose the product of the factors corresponding to the ‘imaginary and negative roots of an equation to be already formed; then we shall obtain the first member of the equation by multiplying this product by the factors 63 v—a, « —b, &c. corresponding to the positive roots; and if we can shew that if any polynomial, whatever be the signs of its terms, be multiplied by w—a, the resulting polynomial will present at least one more change of signs than the original, the proposition will be established as far as regards the positive roots. Let the signs of any poly- nomial be +—-++4+----+-+4+ Me let it be multiplied by a factor « —a; then, writing down only the signs of the operation, we have 1 awe a ae Ree ete a Fe pag ee eee poeta i a eg s e tes and Pricotittenak: aste ah 68 for the signs of the result, the doubtful sign + being writ- ten where the addition of unlike signs in the partial pro- ducts is to be performed. Upon comparing this result with the original polyno- mial, we observe that (1) For every group of continuations there is a corre- sponding group of the same number of ambiguities. (2) The two signs, preceding and succeeding each group of ambiguities, are contrary. (3) There is a final sign superadded, contrary to that of the last term of the original polynomial. Hence, in the most unfavourable case, in which all the ambiguities become of the same sign, by (2) we may take the upper signs; therefore the signs of the result, excepting the last, are the same as in the original polynomial, or no change of signs is lost; and by (3) one more is introduced. Consequently, one change of signs at least is added, cor- responding to each of the factors w«-—a, « —b, &c., and none ever lost; and therefore no equation, complete or incom- plete, can have more positive roots than it has changes of sign. 64 To prove the second part of the proposition, change x into — y; then if the equation be complete, the continu- ations will be replaced by changes, and vice versd; and by the preceding proof the transformed equation cannot have more positive roots than it has changes; and therefore the proposed cannot have a greater number of negative roots than it has continuations of sign. Oxss. This is Des Cartes’s rule of signs, and it is applicable, as we see, to discover a limit to the number of positive roots of every equation; but not to discover a limit to the number of negative roots, unless the equation be complete, or unless we supply the deficient powers of xv, each of which we may consider as having + 0 for its coefficient. But for the negative roots, the best practical way, ls to write —y for a, and to find the limit to the num- ber of positive roots of the transformed equation ; and the theorem might be enuntiated thus; The equation f(#) = 0 cannot have more positive roots than f(x) has changes of sign, nor more negative roots than f(— «) has changes of sion. 56. When an equation is complete, since to each term reckoning from the second corresponds either a change or a continuation of signs, the sum of the numbers expressing the changes and continuations is exactly equal to the degree of the equation. Hence, when a complete equation has all its roots real, the number of changes is exactly equal to the number of positive roots, and the number of continuations to the number of negative roots. For if m, r, be respectively the number of positive and negative roots, and m’, 1’, the number of changes and continuations m +7 = m’' +7", each of these being equal to the degree of the equation; and as m cannot exceed m’, nor r exceed 7’, the only way in which this equation can exist is m =m’, r =71’. 57. In incomplete equations, the above theorem will often enable us to detect the presence of imaginary roots. 65 Ex.1. a@+qe+r=0. This equation has visibly (supposing q and y essentially positive) no positive root, and one negative root (Art. 10); if we complete it, it be- comes aw £0a°+ qe +r=0; and taking the lower sign there is only one continuation of signs, and consequently only one negative root, which is therefore the only real root of the equation. Ex. 2. «#—22°+1=0. A limit of the number of positive roots is 2; and writing — y for x, we get y? + 2y° — 1=0, a limit of the number of positive roots of which is 1, or the number of negative values of w cannot exceed 1: therefore the equation has at least two imaginary roots. Ex. 3. aw” + 5a°—3a° + 407 + 10a° — 44° -— 82° + 5=0 has at least four imaginary roots. 58. Every equation, which, otherwise complete, wants é consecutive terms, has at least ¢ impossible roots, if ¢ be even; and if ¢ be odd, it has at least ¢ + 1, or ¢— 1 im- possible roots, according as the deficient group is between two terms of the same, or of contrary, signs. Let the equation be Ot DO + et Path! 4 Qa + 1.4 p10 +p, = 0, where P and Q have the same sion; then writing that sign before all the intermediate evanescent terms, let s = number of changes and m — s = number of continuations presented by the equation, which are limits, respectively, of the num- ber of positive and negative roots. Now make the signs of all the intermediate evanescent terms alternately positive and negative, so that ¢+ 1 or ¢ fresh changes may be in- troduced according as ¢ is odd or even; thenn —s—¢—1 and m — s —# are limits of the number of negative roots. Hence there cannot be more than s positive roots, and n — s —~¢t—lorn—s-—t negative roots, or more than —¢—1- or 2 — ¢ possible roots; and therefore there are at least ¢ + 1 or ¢ impossible roots according as ¢ is odd or even. Simi- larly, if ¢ terms are wanting between two terms of different 9 66 signs, it may be shewn that there are at least ¢-—1 or ¢ impossible roots, according as ¢ 1s odd or even. If ¢=1,or if only one term be wanting between two terms of the same sign, then the equation has at least two impossible roots; but if a term be wanting between two terms of contrary signs, we cannot in this way conclude any thing respecting the nature of its roots. Generally, in an incomplete equation, if the deficient terms be replaced by cyphers, and first be taken with such signs as to make the total number of changes the least pos- sible, and = s; and secondly be taken with such signs as to make the total number of continuations the least possible, and = ¢; then the number of positive roots cannot exceed s, nor the number of negative roots, ¢; and the number of impossible roots is not less than n — s — @. Pros. Shew that the equation (@ — a) (w — 6) (a — c) — a? (w — a) — 6” (a — BD) —c?(w=c)=2abe, has for a limiting equation the quadratic to which it is re- duced by making any two of the quantities a’, b’, c’, vanish; and thence that all its roots are real. Write the equation under the form (w —c) {(@ —a) (w - 6) - &*} - Sa” (w —a) +b? (w —b) + 2a bch =0, and let a and (3 be the roots, taken in order of (wv — a) x (vw — b) — ¢? = 0, the depressed equation when a’ = D' = 0; then a is greater than both a and 5, and ( less, as will appear by solving the equation. Hence, substituting + ©, a, B, — ©, for w in the proposed equation, the results are t—farJa-a + b'/ ab}, + fa/a—B & V/b-B},-s therefore there are three real roots : one > a, another be- tween a and (3, and a third < p. SECTION IV. ON THE DEPRESSION OF EQUATIONS SOME OF WHOSE ROOTS HAVE PARTICULAR RELATIONS TO EACH OTHER, OR ARE OF A PARTICULAR FORM. Equal Roots. 59. Amone the cases in which an equation may be depressed by reason of particular relations existing among its roots, the most important is that where the polynomial ‘which forms its first member has equal factors, or where the equation has equal roots; because, both in the method of determining the number and situation of the real roots of an equation, and also in that of approximating to the values of its incommensurable roots, one condition either essential or advantageous is, that the roots should be all different from one another, i.e. that the equation should contain no equal roots. We must therefore shew how we may be assured that a proposed equation has no equal roots; and when it has equal roots, we must shew how they may be found, and, consequently, the complete solu- tion of the equation made to depend upon that. of one or several equations having only unequal roots. 60. If the polynomial f(#) and its derived function of the first order f’(7) have no common measure, the equation f(#) = 0 has no equal roots; but if they have a common measure, the equation has equal roots, every simple factor of the common measure occurring one more time in f(#) than it does in the common measure. 68 Let a, b, c,...l be all the roots real or imaginary of f(w) = 0; then, changing # into y +h, we have Sy +h)=yrh—a) yth—b)... Yth—-J; now if each member be expanded and arranged according to powers of y, the coefficient of y in the first member is /’(h) (Art. 26), and in the second member it is (h — b) (h-c)...(h-D+(h-a) (h-c)...2-J)) + (h -—a) (h—b)...(h —l) + &e., each of the factors h — a, h — 6b, &c., being left out in suc- cession ; therefore, equating these coefficients and replacing h by w, we get SJ (@) = (@ -— 6) (w-¢)...(v@ —1) + (w@ - a) (w-0)...(e@-J) + (a —a) (w —b)...(@ —1) + &e. Hence, if f(#) have only one factor = #-a, f’(a) is not divisible by #—a, because one of its terms does not involve v—a; and in the same manner it may be proved that any other of the unequal factors of f(#) is not a divisor of f’(w). Therefore, if f(w) be composed of un- equal factors, f(#) and f(a) have no common measure. Again, f’(#) = (@ -— a) (w—b) (w-¢)...(@- 1) x 1 1 1 1 +—— + hive sink tah e-a@ «2-b w#-C¢ v—l Now suppose the equation f/(v) =0 to have m roots equal to a, 7 roots equal to b, p roots equal to c, 2 f'(@) = (@= a)" (w= bY (@ 0)? ... (w=I) m T. p 1 ea Yd A R50) aT +=} Therefore f'(#) is divisible by (« —a)""(a —5b)'"! x (w —ce)?~'; and therefore, if f(w) has equal factors, f(«) and f"(«) have a common measure, formed by the product of all those factors, each raised to a power less by unity than that to which it is raised in f(a). 69 Ex. 1. f(#) = 0 -— 32° -9 + 27=0, S (#) = 32° -6" -9; the greatest common measure of f(#) and f’(x) will be found to be w—38; therefore the proposed equation hay, two roots equal to 3. Ex. 2. a — 2a° — 4a* + 122° — 34° —-184+18=0; it is the same as (a — 3)? (a —2”+42)=0. 61. Hence, if we know the value of one of the equal roots of an equation, we may find its multiplicity, that is, the number of times it is repeated, by substituting it in the derived functions taken in order; then the degree of the first of the derived functions which does not vanish _ by the substitution, expresses its multiplicity. For suppose the factor # — a to be repeated m times, o. f (@) = (a -— a)". (#), where d(x) has no factor = a — a. Change wv into a + h, then (Art. 27) 4 h vP h? h" .p(a +h) =/ (a) +f (2) = +f (a) ae + ... h” +f"(a)— +. +h ae _Now the first member is divisible by h”, but by no higher power of h; therefore the second member is so, and there- fore we must have f(a) = 0, f" (a) =0, &., f"—1(a) = 0; but f”(a) will be a finite quantity, because the coefficient of h” is so in the first member; that is, the first of the derived functions which does not vanish for x = a, is that whose order is m, the number of times the root a is re- peated. Ex. v + 2a —6a° — 4a? +137 -6=0; to find how often. the root unity is repeated. It will be found that f’”() is the first derived function which does not vanish, when 2 = 1; therefore the root 1 is repeated three times. 70 62. ‘To decompose a polynomial having equal factors, into other polynomials which have only unequal factors. Let L(@ig GA eee paghere X, denotes the product of the factors which enter “only once, X, the product of those which enter twice, &c., and X,, the product of those which enter m times; then if Ji\(2) denote the greatest common measure of f(a) and SJ (2); Ta) = X4iX gx, te Again, treating the polynomial f,(w) in the same man- ner as f(a) was treated if J: (w) denote the greatent com- mon measure of f(w) and fi'(2), SA@) = XGXG oe XS and proceeding in this manner, we shall at last come to’ In-(2) a Xn 3 beyond which, if the process be continued, we find f,,(w) =1, as X,, has only unequal factors. Hence, by division, we obtain Ee = XX, X;... X;, = pi(@) suppose, aa rad 2 Gir: Ce Dra f= p.(2), S#) i Ak Fiay . X= o3(2), festa) = Aha, Am = Dm _ (2), Ofce i(v) ey Fn He) Pi(2) _ x. Pl?) ple) Xx; ;(#) =X, ta Oder. a Hence m1 The solution of the original equation is thus reduced to that of the equations X,=0, 1, =0,... .X,,=0, each of which contains only unequal roots. 63. Hence the process of decomposing a polynomial J («) that has equal factors, may be thus represented ; S(@) — fil@) ---Fn-1(@) fn) Xi “Ag Ben hind In the first line, each term, beginning with / (a), is the greatest common measure of the preceding term and its de- rived function, and the last term f,() is unity; in the second line, each term is the quotient of the division of that term of the first line under which it stands by the following term; and in the third line, each term is the quotient of the division of that term of the second line under which it stands by the following term, and any term may equal unity. Then each of the functions X,, X3,...X, will, by its sub- scribed index, shew the multiplicity of the factors of which it is composed, in the original polynomial; and, by its de- gree, the number of factors that have that multiplicity ; and if any one of them .X, equals unity, then f(a”) admits no factor occurring 7 times. Ex. 1. f(#) = xv — 7a" —2a° + 118 a’ — 259 x — 834° + 612% — 108 x — 432, Ai(@) = v'— 70° + 132° + 3 — 18, J:(2) =wv— 3, J:(#) =1, di (a) = a* — 15° + 10x + 24, 2 (a) = #— 40° + « + 6, s(#) = 27-3, Ay= "+4, An =v — 4 -2, A3;=ue—3, os ff («) = (@ + 4) (a? — & - 2)? (a - 3)’. pelos s 2; > ile v — 1244 530° -— 920° — Ox! +212 2° -—153.a? — 1084 +108 =0; it is the same as (# — 1) (w — 2) (w + 1)? (w — 8° =0. Ex. 3. #—qe2+r=0 will have a pair of equal roots, n Rimsih Bis) = Gee Nt nm—I1 . e . e e n—] A. - The limiting equation 1s n®#” —q=0; 1 a n=] i ae (2) is the value of the equal roots, if the equa- n tion admits any; substituting it in the proposed, a (a"-'—q)+7r=0, we find n n—1 “(Ga)” v1) nm—1 for the relation among the coefficients, in order that the proposed may admit a pair of equal roots. Moreover, when nm is even, r must be positive, and the root which recurs 13 1 q\*>} ‘i. (2 ; when 7 is odd, r may be either positive or negative, n 1 n= I but in the former case the root which recurs is + (2) F n ak n—] : and in the latter case — (2) , as appears from (1). n Commensurable Roots. 64. Commensurable roots are those whose exact values can be expressed by finite numbers either whole or fractional, and therefore of course not involving in their expressions any irrational quantity. When the coefficients are whole 73 numbers, and that of the first term unity, the commensur- able roots are necessarily whole numbers, as will be proved ; in other cases they may be fractions; but in all cases they can be readily obtained, and the equation depressed. 65. If the coefficients of any equation be whole num- bers, the equation can have only whole numbers for its com- mensurable roots. . a . . . If possible, let 3% fraction in its lowest terms, be a commensurable root of the equation 2" + p 2" '+ pox"? + 00. + p= 0, ae ag (sy a n—2 nen (E + p; t +p, (5) +...+p,=0; therefore, multiplying by 6”~' and transposing, n a 73 = pya"-' + pa"-*b +... 1,000 that is, a fraction in its lowest terms is equal to a whole . . . . a ° number, which is impossible; therefore ; is not a root of the equation. If therefore the equation can be satisfied by real quantities, since they are not expressible in the form of a vulgar fraction, they must be either whole numbers or interminable decimals. Hence the commensurable roots can only be whole numbers; and the other real roots are incommensurable; that is, they cannot be expressed by finite rational numbers, either whole or fractional, and therefore can never be exactly known; but their values may be approximated to with any degree of accuracy, as will be shewn. Method of Divisors., 66. The commensurable roots of f(a) =0, which are necessarily whole numbers, may be always found by the following process, called the Method of Divisors, proposed by Newton. 10 74 Suppose a to be an. integral root; then, substituting a for v, and reversing the order of the terms, we have Dat Pa-1% + Poo +... + pia" '+a"=0; Ps ii Pn-\ r Pn-2 0 +... + po + a’ 1=0. a Py Hence o is an integer which we may denote by q,; substituting, and dividing again by a, we get GQ, + Pn-1 + Dnaot --» +p a"? + a"*=0. a Similarly, ane is an integer = gq, suppose; and proceeding in this manner, we shall at last arrive at aoa dem Pa Pa dinates a Hence, that a may be a root of the equation, the last term p, must be divisible by it, so must the sum of the quotient and next coefficient, g,+ p,_,; and continuing the uniform operation, the sum of each coefficient and the pre- ceding quotient must be divisible by a, the final result being always — 1. If therefore we take the quotients of the division of the last term by each of the divisors of the last term which are comprised within the limits of the roots, and add these quo- tients to the coefficient of the last term but one; divide these sums, some of which may be equal to zero, by the respective divisors, add the new quotients which are integers or zero (neglecting the others) to the next coefficient and divide by the respective divisors; and so on through all the coefficients (dropping every divisor as soon as it gives a frac- tional quotient), those divisors of the last term which give — 1 for a final result are the integral roots of the equation ; and we shall thus obtain all the integral roots, unless. the equation have equal roots, the test of which will be that 75 some of the roots already found, satisfy /’(w) =0; and the number of times that any one is repeated will be expressed by the degree of derivation of the first of the derived func- tions which that root does not reduce to zero, when written in it for w (Art. 61). It is best to ascertain by direct sub- stitution whether + 1 and — 1 are roots, and so to exclude them from the divisors to be tried. Hix. 1. e+ 30-87 +10=0. > 8 Here the roots lie between Fi + 1and — 11 (Arts. 44, 42), and the divisors of the last term are + $2, 5, 10{, a= oO ear —§ — 10 a = 5 — 9 —2 — 1 q,+(-8)= -3 -13 -10 -9 G2 = x x 2 x Gz2t3= 5 q3:= = Therefore — 5 is the only commensurable root; and it is not repeated since it does not satisfy the equation SJ (a) = 32° + 6x -8=0. Ex. 2. # —5a*° + a* + 16a — 204 +16=0. - 3 Here limits of the roots are 6 and — 4; and the com- mensurable roots are 4, 2, — 2. Ex. 3. wv 4+ 54° —2a° -—-674+20=0; w= —2, or —5. 67. The number of divisors to be tried may be less- ened by observing, that if the roots of f(x) = 0 were diminished by any whole number m, the last term of the transformed equation f(y +m) =0 would be f(m); if therefore a@ were an integral value of #, a—m would be an integral value of y, and would be therefore a divisor of | f(m). Hence any divisor, a, of the last term of f(x) is 76 F(m) _ a@—-m an integer, when for m any integer, such as + 1, + 10, &c., is substituted. to be rejected which does not satisfy the condition Kixeaie av — 5a*° —- 184 +72 = 0. Changing the signs of the alternate terms, we have 3 2 ’ 3 18 ao + 50 — 184 — 72 =0, or vw — 724 5@|x27— —]} =0; o therefore the roots lie between 19 and — 5. But (f(1) = 50, f(- 1) = 84, f(- 3) = 543 and the only admissible divisors of 72, which, when dimin- ished by 1, divide 50, are 6, 3, 2, — 4; also, all these divisors, when increased by 1, divide 84; but only 6, 3, — 4, when increased by 3, divide 54; 6,°8, —%, are the only divisors which need to be tried; and they will all be found to be roots. Ex. 2. ao —60° +1692 — (42)? =0. a=. 68. Ifa proposed equation have fractional cofficients, or if its first term be affected with a coefficient, since (Art. 30) it can be transformed into another equation with first term unity and every coefficient a whole number, this method will enable us to find the commensurable roots of every equation under a rational form. If the coefficients be whole numbers and the first term be p)x”, and we only wish to find the roots which are integers, no transforma- tion will be necessary ; only every divisor of the last term which is a root, will lead to a result — p, instead of — 1. Ex. 6a' — 25a° + 26a? + 4@ —8 = 0. It is the same as (a — 2)° (8a — 2) (Qa +1) =0. Solution of Reciprocal Equations. 69. These are equations which are not altered by s e 1 e changing # into —, and of which the roots are conse- a 1 1 quently of the form a, -, 6, R &c., together with + 1 or — 1 a several times repeated. The particular form of the equa- tion necessary to satisfy this condition, investigated at Art. 33, is such as to permit a great simplification in its solution; when the degree does not exceed the ninth, the solution can be completely effected. In Art. 34 it is proved that every reciprocal equation of an odd order will have 7 +1 or w« —1 for a factor, accord- ing as its last term is positive or negative ; and that every reciprocal equation of an even order with its last term negative (and consequently having no middle term) will have x2 —1 for a factor; and that if these factors be ex- pelled, the depressed equation, in both cases, will be a reciprocal equation of an even order with its last term positive; which may therefore be taken as the standard form of reciprocal equations. 70. The roots of a reciprocal equation of an even num- _ber of dimensions exceeding a quadratic, may be found by the solution of an equation of half the number of dimen- sions. Let the equation be re 4+ pa + ga + + kat) 4 la" + kat-' 4... +90 +pr+1=0; then, collecting the terms which are equidistant ftom the extremities in pairs, and dividing by 2”, we have 1 1 1 a + ae + P (a + =| + eee + k (« a -) + l = 0. v a i v i Let «7 + - =y, then because v 1 m t m—1 I etl ow .: + om ae + pol : making m =1, 2, 3...m—1, successively, and substituting in each equation from the preceding, we get arti 2 J tt 2 Q i takedeihe chit seed i ote YY —2)-y=y-3y ] a+ ay 9-39) — > 2) ae — eee eee ag eereeetreereee vee ees gee eeeseceeeeee a 4—=y"— ny” + &e. Hence, by substitution, the original equation will be transformed into an equation of m dimensions in y; any root of which, a, will give two roots of the original equa- : . 1 ; tion, by means of the relation 2 + —-= a; and a quadratic @ factor, v —av+1. HX4u: ve +a — Qa’ + 3a° — 8a -—8a' 4+ 30° -9e +741 =0. Expelling the root — 1, by means of Art. 6, we get for the depressed equation a+ (—141) a7 + (0-9) a + (9+ 8) aw + (— 12-8) at + (20 - 8)a#*° + (-12 + 3) a° + (9-9) 74+1=0, 1 Set : or o+—-9 (a+ 5) + 12 (w+) - 20 =0; a av op —4y +2-9 (y’ — 2) +12y—20=0, or y' — 18y° + 12y=0; 1. y=0, y=1; and the other roots are 3, — 4; 79 therefore the proposed equation is resolved into (v@ +1) (@ +1) (@& —@ +1) (a —- 341) (a +4a 41) = 0. Exe 2. 2u° — 5x? + 4a — 40° 4+ 5” —2=0. Expelling the factor x — 1, the depressed equation is Qa — 5a’ + 6a* -—5xH4+2=0, which may be resolved into (w — 1)? (2a° — # + 2) = 0. It may be observed that, by precisely the same process, the equation on—1l n—y merge. + loka + lat + hme) + hmia +... se sane® pm" x +m" =0, admits of the same reduction as the recurring equation which it becomes when m= 1; the formule to be used being m ae m n+l : (m n ey m n—1 e+—=y,e"t + | — =yia"+{—) b}—mia® 4 | : XL WH Hy ta AX | 71. The following are other instances in which equa- tions are solvable, on account of their roots being known to have particular relations to one another. Sn pes Tait Ex. 1. a! ~ = a + —— a — 850 +16 = 0, roots in geometrical progression. a a They are therefore of the forms ma? ? ats ais .. (Art. 19) a = 16 or a =2. ee i ; be Also ec xis — a+ a Far + air’: Y Yr” 357 Oe 1 ! ._—= (7+ + (" +5), which gives r = 2; 1 consequently the roots are a2 16, 80 Ex. 2. a" + p,v"~' + p, x"? + &c. = 0, roots in arith- metical progression. They are therefore of the forms a, a+ 6, a+2b, &c. -pi={2a+ (n-1)6} = = nay ZOD» ? pi —2p2=a°+ (a+b) +&e.= na’ + $14+24+3+...+(m—-1)}2ab + $1242? 4... 4+(n-1)73 B (22-1) (n-1)n 6 subtracting the square of the former from the latter equa- tion, we get 6°; and then a is known from the former, by substituting for b its value. =na+n(n—-1)ab + bts In general, the equation St (w) = 0 may be depressed, if we know a relation b = @ (a), between two of its roots, a and b. Write (x) instead of w in f(x), and let the resulting polynomial be F(x); then f(x) and F(a) are both reduced to zero hy making # =a; for a and b are roots of f(a) = 0 by supposition; and to write a for # in F(x), is the same thing as to write b for w in f(«); therefore f(#) and F(a) have a common measure a — a, which may be found; whence a, and b = @(a), become known, and the equation may be depressed two dimensions. Ex. o'+ 22° — Qa* — 22% —22=0; the sum of two roots is — 2, find the roots. Algebraical Solution of Binomial Equations. 72. These are equations of the form #” +a =0, con- taining only a single power of the unknown quantity, which may be reduced to reciprocal equations; for let a be the arithmetical value of \/a, and for a write aw, then the equation becomes a” +1 = 0, which is reciprocal. 81 Although we have already obtained the complete solu- tion of this equation (p. 18), so that with the aid of a Table of Sines, the numerical values of the roots’ may be’ easily found in the form a+ br/ ras approximately as can be desired; yet the solution by a purely algebraical process deserves attention, since in it additional properties of the roots are brought to light; and these roots, that is, the n'" roots of unity or of negative unity, are not unfrequently employed in several of the higher branches of Analysis. 73. In all cases of the equation # +1 = 0, having expelled the real factors if there be any, if we transform it . . ] e by the substitution y = # + —, so that a — ya + 1=0, since x y will be the sum of a pair of conjugate roots, y will always be real, as every value of & is impossible; and therefore the _ equation will be transformed into another of half the num- ber of dimensions having all its roots real. 74. If a be an imaginary root of 2 — 1 = 0, then a” will also be a root, m being any number positive or negative. For since a is a root, a” = 1; therefore (a")” = 1, or (a”)” —1=0; therefore a” is a root. Also, if a be an imaginary root of #” + 1 = 0, then a” is also a root, m being any odd number positive or negative. For a" = -1, .. (a’)" = (-— 1)”= — 1, since m is odd, Orta )-+.1 = 0; .2.a 18 root. In both cases, all the roots are manifestly unequal (Art. 60), for the derived function .”~' can have no factor in common with a” + 1. 75. The equations #—1=0, and #2 —1=0, can have no other common root, except unity, m and » being prime to one another. . For suppose, if possible, a to be another common root ; and let a and b be two numbers determined so as to satisfy 11 $2 the equation am — bm = 1, which can always be done (Art. 143), since m and # are prime to one another; then a” =1, a=10a"=1la”=1; .. by division we get nN as 4, er a=1, which is consequently the only common root. This would also appear if we sought the greatest com- mon measure of # —1 and #” —1, as we should find only % —1. e 76. The imaginary roots of «-1=0, m being a prime number, are the same as the several powers of a from 1 to # — 1, a being any one of the imaginary roots The quantities a, a’, a°,...a"~' are roots by what has been proved, and no two of them are equal; for, if possible, let a? =a", p and q being both less than », therefore a?~4=1; or, a is a root of #~’—1=0, and also of 2 —1=0, which is impossible, because p — ¢ and ” are prime to one another; therefore the roots of the equation are all contained in the series I, Qs a’, eee a’—!: and if it be continued, the roots recur in the same order, for ihe es bate Raye ah a’ =}, a 2=q".a’° =a’, kc. 7i- This property of producing all the other roots by its different powers, which, when 2 is prime, belongs to all the imaginary roots, is, in other cases, generally confined to the first imaginary root, a, determined by De Moivre’s formula, (as proved, p. 21,) or to its conjugate; or rather to any root a”, provided m be prime to », or to its con- jugate. If therefore 3 be any root, it is always true that any power of ( is a root; but not always true that all the roots can be produced by powers of /3. Thus, in the case a -— 1=0, or (a — 1) (@* +1) =0,; if we take pei mkhilied: 83 we can, by its powers from 0 to 5, only preduce the roots of &° —1= 0 twice over; but if we take 1++/-3 2. ear Se ads 2S = COS — + —i sim — p) 6 6; we can, by the powers of a, produce all the six reots. ® 78. ‘The solution of 2" — i =0, where ” is a compesite number, may be always reduced te those of #? —1 = 0, w!—1=0, &c., where p,q, r, &c. are all the prime factors of m. First, suppose 7 = pq; then #* —1 or #7! —1 is divi- sible, both by #? —i and #’— 13 therefore the roots of ?—1=0, Tie eri ah and the roots of #’ —1=0, 1, BB, ... Bs are roots of «"~i=0. Also, the products, formed by multiplying every quantity in the first row by every quan- tity in the second, are roots; for each one of these will be of the form a’ 8', also a” = 1, B"=15 .. (a"')" = 1, con- sequently a"/3' is a root. And no two of these products are alike; for, if possible, suppose a’B' = a® B73. a”? = B°-*; but a’* is'a root of 2* —1=0, and 3°‘ of «’— 1 = 0, therefore these equations have another reot im common besides unity, which is im- possible. ‘Therefore the pq products, formed by multi- plying every root of «? — i= 0 by every root of #’-1= 9, are the roots of the equation 2?! —1 = 0. In the same manner, if «= pqr, and a, A, +, be re- spectively roots of #? — 1 =0,#' —1=0, « —1 = 0, it may be shewn that a” B* y7 is the general form of the roots of &” —1=03 and it will give all the roots, if w, o, 7, as- sume all values from 0 to p — 1, g — 1, 7 —1, respectively. So that if the arithmetical values of the roots of the reduc- ing equations be known, these of the proposed equation will be known. 84 79. Secondly, suppose 2=p*; and let the roots »of wv? —1=0 be -1 ibs As a’, eee a’ 3 then these, as well as P/ > aye o aap 1, V/ as V/ a; eee J a ie are roots of the proposed; as is also the product of every one of the first row by every one of the second; therefore we have p* quantities of the general form a’ v/ a’, all satis- fying the proposed and all different from one another, which are therefore the roots of the equation ; but we obtain the arithmetical values of only p roots, supposing the reducing equation to be solved; for those of the second row can only be determined by solving the proposed equation. Similarly, if 2 = p*, the general expression for the roots will be a” A/a / a, a being a root of 2? -1=0. And if m = p’qr, and a, B, y, denote respectively roots of the equations #? —1 = 0, a’ — 1=0, a —1 =0, then the general form of the root of the proposed will be a™ .~/a?. P° .y7; and all the 2 roots will result by giving to 7 and p all values from 0 to p—1; and to o and 7 all values from 0 to g— 1, and x — 1, respectively. And, in the most general case, where m = p q” 77... the roots may in the same manner be found, by combining the roots of the equa- tions 7? —1=0, #f —1=0, &c. Gauss’s Method of solving Binomial Equations. — 80. The solution of #"—1=0, by what has been proved, can always be reduced to the case where is a prime number; and the case of » a prime number, by a method invented by Gauss, may be made to depend upon the solution of equations whose degrees do not exceed the greatest prime number which is a divisor of 2-1. The leading feature of Gauss’s method is to represent the imaginary roots by a series of powers of any one of them, 85 whose indices form a geometrical instead. of an arithmetical progression. Thus, if m be a number (and such, called primitive roots of m, can always be found) whose several powers from 1 to m —1, when divided by m, leave different remainders, and a be any imaginary root, then all the roots may manifestly be represented by 3 n= m m” m m " a ,a 5 a 9 eoe A 9 or, since m”~' n—2 &e., a” e e 2 = pn +1, where wu is an integer, by a, a”, a” , ‘ 81. The advantage of this mode of representing the roots is, (1) that they can be distributed into periods, each of which, when continued, will produce the roots of that period in the same order; and (2) that the product of any number of such periods will be equal to the sum of a cer— tain number of periods ; the importance of which properties will be seen in the use made of them. (1) Let m-—1=,7s, r being a prime factor of » — 1, and let m’=h; then the roots may be written in vertical columns, each consisting of + terms, as follows, h h? a? a a a eee a 2 s—] a qi qi es. git 2 2 272 2, s—1 a™ qi® h a” h ne a” h r—] r—1 r—] h2 r—lzs—1 eta a™ Se i and if any one of the periods formed by the horizontal rows be continued, the roots in that period will be produced in the same order; thus, if the first row were continued, the indices would be h'= m= m""!= nn +1, A’t!=m"t"= (un +1) m" =unh+h, &c.; and the corresponding roots, a, a”, &e. (2) Let any two of the above periods be represented by 2 s—1 Oa +a fe &e. ve at bh : 2 im a taht a &e. + as 86 and let us multiply them together, using each term of the lower line in succession as a multiplier, and starting at that term of the upper line which stands over it, and producing the upper line so as to supply the terms neglected at the beginning, the result is j j ae a+b ah+b ae at” +6 x es | a +a at” +B + &c. + 4 \ f 2 Sh QitOh 4 lahtoh att +byh (al? +B)k +&o+a ‘ Ay pe j B\R2 y 2 2\ 22 Lahm 2 htt Oe lakhs DP 4 al®+ OE Be 4 gah +O) Qt tn 4 glah+ one 4 glal? toy" 4 Be, 4 glial +o and therefore, collecting the vertical columus into periods, we get =(a’) Ea’) = D(a" "*) 4 Sa 4 Sa eta or the product of two periods is equal to the sum of s pe- riods; and consequently the product of any number of periods will be equal to the aggregate of a certain number of periods. Ex. 1 w—-1l=0; 6=3.2, .7r=3, s=23 also 3, 3°, 3°, 3', 3°, when divided by 7, leave different remainders, wiz. 3, 2, 6,4, 53; .. = 8, and the roots are py=at+a P2=a° +a" p,=a +a? and p, + ot p3=— le Also p,p. = a‘ + a? + a+ a= pr. + Ps P._pPy=atata+a’=p,t ps Pips= ata +a't+at=p,+ p, ~“s Pi po + Pops + Pips= — 2: and p,Pops= Py + Py + P2= 2+ Ps + Pyt Pr= bs 87 Therefore the cubte which has p,, p., p,, for its roots, Is p+ pP—-2p—-1=0. Ex. 2 #”—-1=0; 16=2.8, also the powers of 3 from 0 to 15, when divided by 17, leave remainders 139 1018 5 15 11 16 148 7 4 32 2 6, ~ peata t+a®+a’+a’+ae tai +a’ g=a@+a°ie +a%+a%+a’ +a"4 a; then p+q =-— 1, and pq=at+a®+a° ta oie aa i ALE padre eta +a ta’ t+a ta%ta?+ a ab+a’ +a?+a°% +a +e0° + 0a%+ a" eae PP TP Ee te Ga ss therefore p and ¢ are roots of s°+ ¥ — 4 =0. Next, the periods p, q, may be resolved, respectively into the periods ° > r=a game t=a +a saw se tae u=a ta +a’ +a? % ss-at+a’+a + QQ” ree ee + s&s = Ps and rs=al+a° +a° + 4 lI 14 2 16 a‘+a'tay+a | =p+q=-t; a tae t+a’+a therefore 7, s, are roots of x°—pz—1=0;3 and similarly t, u, are roots of s*—-qz—1=0. Lastly the periods r, s, ¢, w, may be resolved, respect- ively, into ’ ? : =a? +a! tie = i) 5 &= al +a t=a +a" 88 _ then 7,+ %=7, m= a+a"?+ae+a=t, eo *. 5 Ye are roots of s°-—rz+t=0; 4 : 1 _and 7,, the greatest root of this equation, =a + — = 2 cos ae a 82. Any radical has always as many values as there are units in its index, and these values are obtained by multiplying the arithmetical value of the root of the quan- tity under the sign, by each of the roots of + 1 or — 1. For, every root of the equation #” +a@=0, is an alge- braical value of af +a; but, whatever be + a, this equation admits 2 roots all different from one another; therefore the radical a/ a, considered algebraically, will have 7 different values. When a is real and positive, the equation xv” =a has always one real root a, and the m values of a/a will be obtained by multiplying a by each of the m values of J/13 in like manner, the values of a/ —a will result from multi- plying a by the values of at. Hence ./a x \/b will have r values, where 7 is the least common multiple of m and x. For, let a, , be the arithmetical values of the radicals, m+n then a/ a x x/b =a(1)™ ; m+n but if are be reduced to its lowest terms, the numerator will be an integer and the denominator will be r, the least. common multiple of m and 7; lla 1b So Oy which has 7 different values. SECTION V. ON THE GENERAL SOLUTION OF EQUATIONS OF A DEGREE INFERIOR TO THE FIFTH. 83. WE shall now direct our attention to those cases of finding the expressions for all the roots of an equation of an assigned degree in terms of its coefficients, the coeffi- cients being general symbols, in which a solution has been effected. These methods, which, as was before observed, succeed only for equations of a degree not exceeding the fourth, are the results of particular artifices; but they are all reducible to one principle, as will be hereafter shewn. The general expression for the roots of # + p,a”~! + &c. = 0, if it could be obtained, would consist, first, of a part affected with radicals, by means of the different values of which, it would be capable of representing ali the roots; and, secondly, since the sum of the roots is rational, of a rational part, h, which would be the same for every root ; hence, in taking the sum of the roots, the radical parts ’ ‘ a * must destroy one another, and we should have nh = — jipe 1 . ® . or h = — —p,, which is the value of the rational part of n every root. ‘The general solution of an equation wanting its second term will, consequently, be simpler than that of the corresponding complete equation, as it will have no part unaffected with radicals. Hence in the following instances we shall suppose the equation to be deprived of Its second term. 12 90 Solution of a Quadratic Equation. 84. Let the equation be reduced to the form w+pvtq=0;3 then this may be transformed into y’ = a, by taking away its second term. For, putting v=y—2p (Art. 29), we have 2 2 ye — py + + py os + q=0, eA a or yao ngs 2 Hence, if > the roots are real ; if S = q they are equal, and # + pa +q= (w+ 4p)’ is a perfect square ; 2 if c 9 Wii: p posed equation becomes v” —2rcosOu" +r? =0; or vw” —2cos@a" + 1=0, changing x” into wr, which has already been solved, (p. 26). Solution of a Cubie Equation by Cardan’s Rule. 87. Let the equation, by Art. 29, be reduced to the form v+igr+r=0; and put y=y+ 3%, that is, suppose w equal to the sum of two other unknown quantities ; . w= 8y2 (y+2) 4+ y+ 2%, and therefore the proposed equation becomes (Syz+q) (¥+2) +742 4+7r=0. Now since we have two unknown quantities, and have made only one supposition respecting them, namely, that y +.% =, we are at liberty to make another; let therefore 3Y%+q=090, 3 or x= — P) 5 PP HS Hence y’, 2°, are the roots of the equation q\° ert (2) = 0, i 3 since the second term with its sign changed is equal to 93 their sum, and the last term is equal to their product. Solving this equation, we get but w=y+ 2, oe TASS wih pee: r pe 3 r i 3 ' pee( tate) nd) sah is ae 2 4s 27 2 4 27 an expression which (since the cube root of any quantity has three values) contains implicitly the three roots. 88. This solution only extends to those cases in which the cubic has two impossible roots. Let m and be the arithmetical values of the two surds in the value of #, and 1, a, a’, the three cube roots of unity; then the three values of y are (Art. 82) m, am, a’m, and those of s are n, an,a°n. By combining these values two and two’to form y + x, we shall have nine values of w; the number being tripled by reason of our having employed 3 ae ys = — (2) , instead of yx = — 7 the relation arising im- 3 mediately in the process; and we observe that every com- ‘ 3 bination will satisfy y’s" = — (4) , but only three the given condition yz = — ; , which latter are the roots, viz. m+n, am+an, a’m+an; or substituting for a, a’®, their values, — sq + / 3) (p. 17), the three roots are m+n, and — 4 {m+n + (m—n) / 3}. 94 y* 3 Hence, as long as the expression J Ain a is possible, the values of m and m are possible, and the equation has one possible root, the numerical value of which, as also those of the two imaginary roots, may be obtained after somewhat laborious calculations from the above formule. But when the expression NA ‘i a is impossible, and m are impossible, and all the three roots appear under imaginary forms; whereas, the equation, being of an odd i 3 degree, has at least one real root; and indeed, sinee — + ms _is negative, it has (Art. 54) all its roots real; in this ¢ therefore the above formule, although algebraical expres- sions for the roots, cannot on account of the imaginary quantities which they involve, be applied to furnish the numerical values of the roots. Oxs. It is easily seen that the six superfluous values of w above mentioned would be the roots of the proposed equation, supposing g to be successively replaced by ag and a’g; since each of the relations yx = — Sat, Ye i a’q, leads to y*s*° = — a, g 3 a7 [. & 124/21 = 54° 991; cas ne & ~I Je) Nl 2 Or or —— ~w bd i J w = {55+54°9912° + {5554 991}° = 4°79..+0°208... = 4:'999..=5. 95 or «= ~ {5 4 (4-s92..)//=3} 2 ~] 1 Sees = - £5 + 34/— ; 2 Ex.2. 2 — 127 —65=0. 1 © = 5, Or — =)? z£3Y =i}. Pera a = 27 —5= 0. =2°0045..., or =— 1° 0472... 4 (1° 1362...) -1. 89. In the case of the roots being all real, which, for the reason just stated is called the Irreducible Case, that is, ® 3 3 : , q z when q is negative, and i = mu it may be observed that the assumptions in the process 3 yrsa—r, prea t G) : are inconsistent with one another; for the product of two real quantities can never exceed the square of half their sum. In this case we can shew that, in the expressions for the roots, the impossible quantities destroy one another, and the three roots are real. For let the values of m® and n’® be represented by a+ bi/-1; then, expanding by the bino- mial theorem, and taking P and Q to denote real functions of @ and 6, we have (a+bV/-1)'=P#QV/-1; ro 1 +90 = 2 P, m—n = 2Qr/-1; and the three values of # are 2P, and —-4.(2P+2Q / 3); which are all real. This mode of proceeding, however, is useless in finding the numerical values of the roots; for if we convert (4 +b/—1)% into a series, P and Q will be ex- 96 pressed by series which rarely converge, and from which we can never obtain the exact values of P and Q; and if we attempt to express the cube root of a +b Jan by an ex- pression of the same form, we shall have to solve a cubic of the same kind as the one in question. 90. Hence Cardan’s rule succeeds for the following forms, where q and 7 are essentially positive, xv +qa@ +r =0 in all cases, 3 ° a 0 when ag <—; af 4 2 3 4 ; “ Ue and it fails for # — qa +r=0 when Ls: ke 3 Vv —quv.+r all the roots of which are real. 91. There is one case in which Cardan’s rule succeeds for the equation w* — gx + 7 =0 when all the roots are real ; it is when two of them are equal, in which case also the roots of the reducing quadratic are equal; for then m =n, and the values of w are m+n,—-4(m+n),—4 (m+n). Ee 3 , Fy li this case, — = uh or - = (4) 9 Waa? Beil get ek A 27 2 3 7 2 g -7-- (4); nt is Ne and the roots are 2 3; 3 EM Atl A en / ns a 3° Be 3 Trigonometrical expressions for the Roots of a Cubic. 92. If in the expression Civ ie) Vi -50)+ a 97 hls : a i we put cot d = 2 (-) 2, it becomes ah, ; (— cotp@+ cosec)’*. Hence, reducing, the real root of 2° + qv +r =0 is PY x r/ - (tan' A cot” 2) ; 3 2 2 p 2 _-2 /1 cor 20 Similarly, the real root of #—qv+r=0, L< = tan’@, may be further transform- which, by putting tan ed into * (3\3 becomes (by putting cosec o =; (") » tan : = tan’ @), —2 /4 cosec 260. 3 3 y* Also, in the irreducible case, v?- qv +r =0, a Se ri 27 the expression r Ho. GF q\2 ae — QT Pr 2 ie le i Seg | en Aa Wy y Be ibad : G) (tal) “Yi - Za) 3 3\8 by making cos p=+ ; (") , becomes equally (2)" is cos fp £4/-1 sin p}, or (2) "feos (7 +) Bay a sin (7 = p)} ; and therefore the three values of x are q q ib —2 V4 cos ?, n/t cos (“>*). 3 3 3 a 13 98 Solution of a Biquadratic Equation by Des Cartes’s Method. 93. Let the proposed equation be reduced to the form w+qet+rat+s=0; and as the first member may be always regarded as the product of two real quadratic factors, we may assume it = (a +pu+f) (@-pxt+g) | =a! + (g+f-p’) a + (pg —pf)a+fs (effecting the multiplication), where the coefficients of the second terms, p and-— p, are equal and of opposite signs, because the second term of the proposed equation is wanting, that is, the sum of its roots is zero. Hence, equating co- efficients, ge+f—-p=q, pg—-pf=", fg=s, ogt+f=q+rp, g-f= ) tae =q+p?4— 2f=q+p tf Ss p- 4 py 2 4fg = q + 2qp* + p Re or p’ + 2qp' + (q° — 48) p? - 1 = 0, the equation for determining p, which rises to the sixth degree, because a polynomial of four dimensions, may have 4.3 é vetia (Art. 17) 79? OF x divisors of the second order. Also, because the values of p are the sums of every two roots of the proposed biquadratic, and because the sum of these roots is zero, and therefore the sum of any two is equal and of acontrary sign to the sum of the other two, there- fore the values of p will be in pairs differing only in sign ; this is the reason why the equation for determining p in- Come 99 volves only even powers of p, and may therefore be de- pressed to a cubic by putting p?=y. The reducing cubic is y+ 2qy' + (G? - 48) y— 1° = 0, which (Art. 10) has necessarily one real positive root; let this be e’, then the four values of x are contained in the quadratic equations ; ;( Lie ") ; a+ ee + —- em-j} = q\7 e : é 1 +") ; Pewee Oye: pb fe O. 2 q e Ex. a! — 3a? —-4247 —40=0. Here q=-—3, r= —42, s = — 40; and the reducing cubic is y° — 6y’ + 169y — (42) = 0, which has a root = 9, (Art. 67), .€=3; .. #4+3H7+10=0, 2 —-3r—-4=0: : 1 a the roots of these quadratics ~ si (s CS rf =3i) »-— 1, 4, are the roots of the proposed equation. 94. The reducing cubic will have all its roots real, unless two of the roots of the proposed biquadratic are possible and unequal, and two impossible. For the square of the sum of any two roots of the proposed is a root of the reducing cubic; if therefore the proposed have all its roots real, the reducing cubic will have all its roots real; or if the proposed have all its roots imaginary, and therefore of the forms at Br/ oats a B78, —-a+t B/—t, —a Basel, since their sum is zero, then the square of the sum of every two will be / real, and therefore the cubic will have all its roots real. 100 But if the proposed have two real unequal roots, and two _ imaginary roots, and therefore of the forms a+ B/—1, a- GN <1; —a+7y,-a-—y;, then the square of the sum of a real and an imaginary root will be imaginary, and therefore the cubic will have one and consequently two imaginary roots. As it is only in the latter case that a solution of the reducing cubic can be obtained, therefore Des Cartes’s method can only be applied to those cases in which two roots of the biquadratic are possible and un- equal, and two impossible. It will be observed that in the latter case, if the two real roots are equal to one another, 7. e. if yy = 0, the cubic will have all its roots real; but as two of them are equal, it can still be solved. 95. If the roots of the reducing cubic can be obtained, and are put under the forms (2a)*, (2), (2)*, then the four roots of the biquadratic are -(a+B+y), B+ty-a at+y-B, at+B-y¥. For, —3$q=a°+ 6’ ++’, and 1° = (8aBy)’; let p* = (2a)?, or p= 2a; therefore, taking the upper sign, 1 f= 5 (9+ pt 2) =~ (a + B+) +208 — 2By =a — (B+ +)’; therefore the first reducing quadratic is a +2an+a°— (B+)? = 0, which gives for w the values -(a+B+y), B+y-a; similarly, the other quadratic, taking p = — 2a, is a” —2av+a*—(B—--y)’?=0, which gives the other values at+y- fp, at+PB—-y+. 101 Hence the roots of the biquadratic are symmetrical functions of the roots of the reducing cubic; and whatever root of the reducing cubic is used in the process, the same values of x are obtained. Solution of a complete Biquadratic. 96. Let the equation be w+ per+qev’+re+s=0, and let it be supposed the same as p 2 (w+ 204m) —(kv+l)’ =0, where k, 1, m are unknown, and are to be determined so as to make the latter equation coincide with the proposed. Now 2 2 (a7 +204 m) =v + pax + (F + 2m) vu + pme +m J —(kx+l) = — kx’? - 2klx -—l’; therefore, by comparing this with the proposed, we have, to determine k, 1, m, the equations 2 . +2m—k?=9, pm—-2kl=7, m?-P=s. Substituting, in the second, the values of & and / ob- tained from the first and third, we get 8m? — 4qm? + (2pr —8s) mM— ps +4qs —7 =0, which will necessarily give one real value for m; then & and J are known; and we find the two reducing quadratics vu + (E+k)o+m+ino, ign (Ek) 4 m-1=0. 102 97. This method can be employed only when two roots of the biquadratic are possible and two impossible ; for suppose the roots to be a, B, y, 6; and suppose any two a, 3, to satisfy the first reducing quadratic, and conse- quently ry, 6, the second, . m+l=aB, m-l=yo; “. m=1(aB+-¥0), and the other values of m must be E(ay + Bo), 4(ad + By). | Hence if a, B, y, 0, be either all possible, or all im- possible, the values of m are real; but if two roots of the biquadratic be possible and two impossible, then two values of m will be impossible, and the reducing cubic may be solved by Cardan’s rule. In the latter case, however, if the two real roots are equal, the cubic will have all its roots real; but it may be solved, because two of them will be equal. Solution of a Biquadratic by Euler’s Method. 98. Let the equation be reduced to the form v+qa’+rxv+s=0, and assume *=y4+%4+4U; P=PrsPturet+2(yst+ yust zu), or a —(y +24’) =2(ys+yut+su); . ee 2a°(y? + 2° +u7) rr (y? +2" + u’)? = 4(y? 3" af yu + vu”) + 8yxsu (y+x+4+U), or, replacing y + s + u by #, and transposing, a* — 247 (y? + 8° + u*) — 8yxua + (y? +2" 4+ wv)? —4 (y's + yu’ + 2?u*) = 0. 103 In order that this may coincide with the proposed, we must have | q=-2(y +2? +), r=—8ysu, s=(y? +2? 4+’): —A(y’s? + yu? + 27 U*); | g— 4s: ory+?4+wv=- 2, ye” +yYurt+ ue = aan yy Py 7 SU = ——, OF Swe = y 8° Y 64. hence y*, 3°, wu, are roots of the cubic q’ — 4s Hee 16 64 bytes 9 Let #7, ¢”, ¢’”, denote the three values of ¢ in this equa- tion ; Rae tee t,t e's which six values, combined three and three, would give 8 values of y+ 2+ or «, instead of 4; the number being 2 ve doubled because we have used y’s*u? = 6x? instead of the : ie r ° given condition ysu = — r which only allows those values of y, x, u, to be combined which give, when multiplied together, a result with a contrary sign to ’. Hence if + be negative, there must be either two nega- tive quantities, or none, in every combination; and if r be positive, there must be either two positive quantities, or none, in every combination representing a root. Therefore, in the former case, that is, when r is negative, the roots are fe tot", Beets, 6. eae aes Hethiy fe: and in the latter case, when 7 is positive, the roots are t+t—t, t+t’-t, (+t -t, -t-f-0; and it will be observed, that the second set of roots results 104 from the first, by changing the sign of any one of the quan- SILIES S75 40,08 99. In this case also, the reducing cubic will have all its roots real, except when the proposed has two possible and two impossible roots. Since the last term of the reducing cubic is essentially negative, it will always have one real positive root 2°, and the remaining roots will be either both positive, both nega- tive, or impossible; that is, of the forms e?,¢%; —27,-— 7%; or p?(cos20 + / -1 sin 20). Hence, according as the reducing cubic has three posi- tive roots, two negative roots, or impossible roots, the biquadratic v+quv—re+s=0 will have its roots respectively of the forms t£(¢+2"), -t+¢ -7); taa/ui (tf —t"), -t&/-1 CHO t+2pcos0, — fen at 2p sin 0. In the case of the equation o+qe@+rat+s=0, we must change the sign of ¢ in the above expressions, and the results will be its roots. If 20 =z, or the two real roots of the biquadratic be- come equal, then, as before, the reducing cubic has three real roots, two of which are alike. SECTION VI. ON THE SEPARATION OF THE ROOTS OF EQUATIONS.. — 100. THE propositions in the preceding sections lead us to several important conclusions relating to the nature and the limits of the roots of every equation; and for equations of low degrees and of certain particular forms, the methods detailed in them (especially that of Art. 49) will actually determine the number and situation of all the real roots; that is, two quantities between which each of the real roots lies. They still, however, leave unsolved the main pro- blem, which is to discover the number and situation of the real roots of an equation of any degree. This we shall now endeavour to effect by the methods proposed by Sturm and Fourier, which are among the greatest improvements re- cently made in the Theory of Equations. Sturm’s method of separating the Roots. 101. By performing a process nearly the same as that of finding the greatest common measure of f(x), and its first derived function f’(«), a series of expressions may be obtained, in. which, by simply substituting a and 6 succes- sively for v, the number of roots of f(x) =0, which lie between a and b, may be exactly determined. The enun- ciation and proof are as follows. 14 | 106 Let f(x) =0 be an equation of m dimensions cleared of equal roots, f\(w) the first derived function of f(a); and let the process of finding the greatest common measure of J (wx) and f,(x) be performed with the condition that the remainder after each operation has its sign changed, and so modified is used for the divisor of the next operation*® ; and let f,(w), fa(w), ....f,(@) be the series of modified remainders; then the difference of the number of changes of sign, in the results of the substitutions of a and 6 for « in the series of quantities J (), A@), A), = fi(@)> (> expresses the number of real roots of f(v) = 0, which lie between a and 6. Calling the successive quotients q,, qo, &c., we shalt have the equations FO) = HA) -L(@) A (w) = mii (x) —fs(v) it, 1 (2) = tof (o) set 1 (wv) ie non In fale) wy (0), J.(#) being necessarily a number (Art. 60), since f(a) = 0 has no equal roots ; which shew, (1) That no value of # can make two consecutive functions, f,-:(7) and f,,(#), vanish; for then f,4,(x) and all the succeeding functions would vanish, which is impossible, since the last is a number. (2) That any value which makes a function, f,,(2), vanish, reduces the two adjacent ones to the same nu- merical value with different signs. * This changing of the signs of the remainders, which would be indifferent if the object was only to discover the greatest common measure of f (7) and f,(2), is essential to the method about to be explained. 4 107 Now if in series (1) we make # = c, and then suppose e to assume all possible ascending values from — © to + ©, the resulting series of signs will have two states of permanence; one, as long as ¢ is nearer to — ©, and the other after c is nearer to + ©, than any quantity which makes any one of the expressions in series (1) vanish ; and between these states, whenever any of the expressions vanish, alterations in the order or number of changes of signs, or in both, will occur ; and we shall shew that when x passes through a quantity which makes one or more of the auxiliary functions vanish, it is only the order but not the number of changes which is affected; and that when # passes through a root of f(v) = 0, then a change of sign is lost. First, let #=c make only one of the auxiliary func- tions, f,,(v), vanish, without making f(«) vanish; then to discover the effect, upon the series of signs, of passing through c, we must compare the results of substituting ce — h and c +h for a, h being as small as ever we please; there- fore we may suppose / so small that neither f(#) nor any of the auxiliary functions can vanish for values between e-—handc+h, and that the sign of any series ascending by powers of 4 depends upon that of its first term. Hence the only part of series (1) in which the passage from ¢—h to ¢ +h can produce any effect upon the series of signs, is Sim-i(#)s Wi (7), Susit)s in which, if we write e—h for wx, expand the results (Art. 27), and reserve only that term of each on which its sign depends, we have | Sm-1(¢)> a hf mc); Ju+i(e)s which, since by (2) the extremes have different signs, give a change and continuation, whatever be the sign of the middle term; and these, by changing the sign of h, will be replaced by a continuation and change; i.e. the passage from c—h to c+h, through a root of f(x) = 0, causes 108 an alteration in the order but not in the number of changes. If the same value of w made an auxiliary function vanish in another part of the series, since by (1) adjacent terms can never vanish, the same considerations would shew that no change of sign could be lost or gained. Secondly, let 2 =c be a root of f(«) =0; the substi- tution of c—h for win f(x) and f(a), (taking h so small that the sign of the whole of each series depends upon that of its first term, and writing down only the first terms) gives -hf'(c), fic), or -hf'() fe), which have different signs; but if the sign of h be changed they have the same signs; therefore the two functions S(), fi(#), which for w=c-—h give a change, for v=c+h give a continuation; and therefore, in passing through a root of f(x) =0, a change of signs is lost. If at the same time that f(a#) becomes zero, any number of auxiliary functions vanished, since no two of them could be adjacent, it would follow, as before, that no change of sign could be lost in the parts of the series where they are situated. | Since then a change of signs is lost every time the substituted quantity passes through a root of f(#)=0; and since a change cannot be lost in any other way, nor one ever introduced; it follows, that the excess of the number of changes given by w=a, above that given by wv =b, (as f2(@)s fol)» ---Sa(®)> then if none of them be wanting, and their leading terms be all positive, (for the leading term of f(#) is necessarily so) the equation will have all its roots real; but if the leading terms are not all positive, the equation will have as many pairs of imaginary roots as there are changes of sign in them. But if some of the auxiliary functions are wanting, the number of real roots must be determined by substituting EE a eS Nee I ee ene 111 —« and + o for w in their leading terms, and taking the difference between the numbers of changes resulting from these substitutions. ‘This determines the nwmber of real roots. To determine their sitwations we must substitute 0, 1, 2, 3, &c., for «x in the series SI (2), SAi(2); S2(@),... fala); till we arrive at a number which gives the same number of changes as is given by + co; then, by noting the difference in the number of changes produced by the extreme sub- stitutions, we determine the number of + roots; and by noting those consecutive integers between which one or more changes are lost, we determine the integral limits between which the positive roots are situated, either singly or in groups; and in the latter case, we must substitute fractional quantities lying between the integral limits, smaller and smaller, till the complete separation of each group of roots is effected. In like manner for the negative roots, we must substitute 0, —1, —2, — 3, &c., till we arrive at a number which gives the same number of changes as is given by — co; then the total number of negative roots, and an interval in which each is situated, may be determined, exactly in the same manner as for the positive roots. And in order to diminish the labour of the process, it must be observed that when, in forming the auxiliary functions, we come to one (that of the second degree, for instance, when the conditions of Art. 84 are fulfilled) which is incapable of changing its sign for any real value of 2, we may take it for the last of the auxiliary functions. Ex. 1. f(#)=2°-Tax+7=0. SAi(®) = 34° — 7) 3a — 21a + 21 (# 8x°-— Tax — 1427 + 21, 112 or f,(v) = 24 -—3)6a—- 14 (3a + ‘ 60° — Ox Ow — 14 Q7 97 — — 2 ma & Q or fs(v) = +1. . Hence f(#) = a — 7x ¥ 7, Ai) = 3a" — rf Je(#) = 24 — 3, Js(#) rs Since the leading terms are all positive, and none of the auxiliary functions are wanting, the roots are all possible. Also, since 2 makes all the functions positive, the substi- tutions for the purpose of separating the roots may begin from thence; therefore, making # = 2, 1,0, —1,—2, &c., the signs are as follows: @) + ob ep tage + (1) + = = + (0) + = =~ e og B + + (- 2) f (— 3) (— 4) = + “ We may stop here, because the signs are the same as those given by — 0. Since the first line gives no changes, and the second line two, two roots lie between 2 and 1; also the last line has one more change than the preceding, there- fore one root lies between — 3 and — 4. + a « | + + + 113 To separate the two roots which lie between 1 and 2, let a v=-—, then 2 which has more change than the first line, and one less than the second, (whatever sign we give to the zero); there- fore one root lies between 1 and 1:5, and another between Psa and. 2. Ex. 2. f(x) =«"-qx+r=0, where q is essentially positive. Si(@) =nw—4q v n ao"~) — 7) v—-ge+r (= nN Sgr a 1 or f2(v)=a-a, rejecting the positive factor q (1 mt -| ‘ n nr and putting a = ——____ . eae (n-1)q But the remainder, after dividing n (ar _ 1) by v-a, n Paee'y Tee ee al, Art. 6 ic hea Fae => |e “Thi is (Art. 6) (a 1); J3() (<= 7) a (4 My Ay ac rejecting the positive factor 7 (=) ; Now supposing f;(#) positive, + ¢o gives no change, and — © gives two changes when is even, and three changes when » is odd. Hence if (4) > ( n 15 n—1 A -) , the proposed n— 114 equation has two, or three real roots, according as ” is even q n 7 n—l or odd. Similarly, if (4) << (- > + © gives one n m—1 change, and — o one, or two changes, according as 7 is even or odd; and therefore the equation has no real root, or one real root, according as m is even or odd. These results agree with those found at p. 61. Ex. 3. f(«) =2a'- 13a°+ 10x — 19 = 0, Site) = 407-138 +5; and we find SJo(a) = 18a" — 158 + 38. But the roots of 13a°— 154 + 38 = 0 are imaginary, be- cause (15)’< 4.13.38 (Art. 84); therefore it is sufficient to consider the above three functions, and since their leading terms give two changes for v7 =— co, and no change for ” = + 0, the equation has only two real roots. Ex. 4. f(#) = 2 — 4a°- 80 4 23 =0. Ji (2) = 42°- 12a°- 3, f,(x) = 12a + 9a — 89, SA3(&) = — 491 @ + 1371, f;(@) = — 7157932. Here there are only two real roots, of which, one lies between 2 and 3, and the other between 3 and 4. 107. It is manifest that in Stwrm’s method, the labour of forming the auxiliary functions increases very rapidly with the degree of the equation; since however they can always be formed, the method will enable us infallibly to determine, not merely a limit to the number, but the ab- solute number of real roots in any proposed equation, and the consecutive integers between which they lie either singly or in determined groups; as also the intervals in which no real root can be situated; but when two or more roots are indicated in any interval, if they lie very near to one an- 115 other, although the method leaves no doubt of the existence of the roots, it may be very difficult to subdivide the inter- val sufficiently to completely separate them. Fourier’s method of separating the Roots. 108. We shall now give another method of separating the roots proposed by Fowrier, which has the recommend- ation that the auxiliary functions employed in it are /(#) and its successive derived functions, which can be formed by inspection; so that the method can be applied nearly with equal ease to an equation of any degree; in parti- cular, the intervals in which no real root can be situated are, by Fowrier’s method, immediately assigned. The objection to this method is that, by its immediate appli- cation, we only find a limit which the number of real roots in a given interval cannot exceed, and not the absolute number; and that the subsidiary propositions by which this defect is supplied, are not of the same simple character as the original Theorem. The enunciation and proof are as follows. The number of real roots of f(#) = 0 which lie between two numbers a and b, cannot exceed the difference between the number of changes of sign in the results of the sub- stitutions of a and 6 for w, in the series formed by f(@) and its derived functions: viz. f(«), f’(«), f"(#), ... J" (2). If none of the equations SJ (2) = 05. fF (a) = 05) &e. have a root between a and 8, it is manifest that the substi- tution of a and 6, and of any intermediate quantity, in J (2), J (), &e., will always produce exactly the same series of sions; but if any of these equations have roots between a and b, then changes in the series of signs will occur in substituting gradually ascending quantities from 116 ato 6b; our object is to show that by such substitutions the number of changes of signs can never increase; and that one change will be lost every time the substituted quantity passes through a real root of /(#) =0; this we shall do, by examining, separately, each of the cases in which the series of signs can be affected ; namely, (1) when J (#) alone vanishes; (2) when some derived function, J” («#), alone vanishes; (3) and (4) when some group of derived functions, of which (2) either is not, or is, a part, alone vanishes; and lastly, when several, or all, of these cases of vanishing happen at the same time. First, suppose that w =c, (c being some quantity be- tween a and b) makes f(#) vanish, without making any of the derived functions vanish; then the result of sub- stituting ¢+h for # in f(#) and f(a), is h. f'(c) and f'(c) (supposing # so small that the signs of the whole of the two series which express f(c +h) and f’(c +h) depend upon those of their first terms, and writing down only the first terms) which have different or the same signs according as h is — or +3 therefore, in passing from c—h to c+h through a root of the equation, a change of signs is lost, but none gained*. Secondly, suppose that # =c makes one of the derived functions, f”(«), vanish, without making any other of the derived functions or f(x) vanish; then the result of sub- stituting c +h for # in S'(@)s f(a) f(a)» (these being the only terms which it is necessary to exa- mine) is SF" 0) h Bey orate OY: FAP (ec): * It is unnecessary to attend to the other terms of the series of derived functions, because h is supposed so small that not one of them vanishes by the substitution of any quantity between c—h and c+h; and therefore each has the same sign for c—h as for c+h. 117 If then the extreme terms have the same sign, there will be two changes when f is negative, and two continua- tions when f# is positive; if the extreme terms have con- trary signs, there will be one change, and one only, whether h be negative or positive; therefore, in passing from ¢ —h to e+ through a value which makes one of the derived functions vanish, either two changes or none will be lost, but none ever gained. Thirdly, suppose that «=c makes r consecutive de- rived functions vanish, without making any other derived function or S (#) vanish ; then the result of the substitu- tion of ¢c +h for « in the series SL (as fry vee fF (@)s f(s S"): (these being the only terms necessary to be examined) is ty Ont Or-o7y (¢); Sie (c), Ff"). If then the extreme terms have the same sign, there will be r or r+1 changes, (according as 7 is even or odd) when fh is negative, and no change when fh is positive; if the extreme terms have contrary signs, there will be ry or r +1 changes (according as r is odd or even) when h is negative, and one change when h is positive ; therefore, in passing from c — h to ¢ + h through a value which makes r consecutive derived functions vanish, 7 or + + 1 changes are lost (according as 7 is even or odd) but none ever gained, Fourthly, suppose the vanishing group to _consist of J (#) and the first r—1 derived functions (which corre- sponds to r roots =e in J (#) = 0); then the result of the substitution of ¢+h for w in f(x), f(a), ... f'~'(#), SJ’ (2), is hr h EP OnE LOr EL OLE: 118 in which there are 7 changes when / is negative, and none when f is positive; therefore, in passing through a root which occurs 7 times in the equation, 7 changes are lost, but none gained. Lastly, suppose the substitution of w =e to produce several, or all of the above cases at the same time; then because the conclusions respecting the effect of the passage through ¢ upon the series of signs in one part of the series of derived functions, are not at all influenced by what happens, in consequence of the same passage, at another distinct part of the series, by what has been proved, several changes will be lost, but none ever gained. Since then, in substituting gradually ascending values from a to b, changes of signs are generally lost for every passage through a quantity which makes one or more of the derived functions vanish, and invariably one for every passage through a root of f(v) =0; but none under any circumstances gained ; it follows that the number of roots of f(x) = 0, which lie between a and 6, cannot be greater than the excess of the number of changes given by w = a, above that given by w =5b. 109. Hence if the limits, @ and b, be —-© and+o, or any two numbers, the first of which gives only changes, and the second only continuations; and if in the series formed by f(x) and its derived functions, Se) J (x), J (2); ves J*(2); c be substituted for w and be then made to assume all values between these limits, the series of signs of the results will have the following properties; there will at first be 2 changes of sign, and at last no change, but n continuations; these changes disappear gradually as ¢ in- creases, and when once lost can never be recovered; one change disappears every time ¢ passes through a real une- qual root of f(#)=0; 7 changes disappear every time c passes through a root which occurs * times in ‘f (x) 0; 119 either two or none of the changes disappear every time one only of the derived functions vanishes, without f(x) vanishing at the same time; an even number p of changes disappears, every time an even group of p terms (not, in- cluding the first, f(a) ) vanishes; and an even number q +1 of changes disappears, every time an odd group of q terms (not including the first, f(#)) vanishes. Also if a value causes f(#) and the first 7-1 derived functions to vanish, and an even group of p terms in one part of the series, and an odd group of q terms in another part, to vanish at the same time; the number of changes lost in passing through that value, will be r+piq+1. 110. Hence if f(#) =0 have all its roots real, no value of w can make any of the derived functions vanish, and thereby exterminate changes of signs, without at the same time making /(#) vanish ; for if it could, since those changes can never be restored, and since a change must disappear for every passage through a real root, the total number of changes lost would surpass », which is absurd. Whenever, therefore, changes disappear between values of # which do not include a root of f(«v) = 0, there is, cor- responding to that occurrence, an equal number of imaginary roots of f(#) =0. Hence if « =c¢ produces a zero between two similar signs, or if it produces an even number p of consecutive zeros either between similar or contrary signs, there will be, respectively, two, or p, imaginary roots | corresponding; or if it produces an odd number q of consecutive zeros, there will be g +1 imaginary roots corresponding, according as they stand between similar or contrary signs; ec of course not being a root of J (x) = 0. Oxss. Since the derivatives which follow any one, J’ (#), may be supposed to arise originally from it, it is manifest that the same conclusions respecting the roots of J'(#) = 0 may be drawn from observing the part of the series of derivatives 120 SJ (2); SrtA le); ea fh (x); as were drawn respecting the root of /(#) = 0 from the whole series. 111. Hence we can shew that Des Cartes’ Rule of Signs is ineluded in Fourier’s Theorem as a particular case. When in the series formed by /(#) and its derived functions, we put w= — ©, there are m changes; and when we put v = 0, the signs of the series become the same as those of the coefficients Pas Pn-1 he Pris 1; let the number of changes in this series of coefficients = k, and therefore the number of continuations (supposing the equation complete) =n —k; also if we make r=4+ 0, the signs are all positive, and the number of changes = 0. Hence between # = — © and #=0, the number of changes lost is 7 —k; therefore in a complete equation there cannot be more than » —& negative roots, i.e. than the number of continuations in the series of coefficients; also between w= 0 and # = ©, the number of changes lost is #, whether the equation be complete or incomplete; hence in any equation there cannot be more positive roots than k, i.e. than the number of changes in the series of coefficients ; which is Des Cartes’s rule of signs. 112. Fouwrier’s theorem may also be presented under the following form. If an equation have m real roots between a and 5b, then the equation whose roots are those of the proposed, each diminished by a, has at least m more changes of signs than the equation whose roots are those of the proposed, each diminished by 6. The transformed equations would-be Sy +2) =9, fly +b) =0; 121 and if these were arranged according to ascending powers of y, the coefficients would be the values assumed by f(«), J (2), &c., when a and 6 are respectively written for w. Therefore, whatever number of changes of signs is lost in the series f(«), f(x), &c., in passing from a to b, the same is lost in passing from one transformed equation to the other; but the series for a has at least m more changes than that for b, therefore f(y + a) =0 has at least m more changes than f(y + 6) = 0. 113. To apply this method to find the intervals in which the roots of f/(#)=0 are to be sought, we must substitute successively for v, in the series formed by f (2) and its derivatives, the numbers ree 10,6— 1,.0;¢1, 10,72. +6 (1); (—a and + 3 being the least negative, and least positive number, which give, respectively, only changes and continua- tions) and observe the number of changes of sign in each result. Let h and & be the numbers of changes of sign when any two consecutive terms in series (1), a and J, are re- spectively written for vw; therefore h —k is the number of real roots that may lie between a and b; if this equals zero, J (2) = 0 has no real root between a and 6, and the interval is excluded; if h — k=1, or any odd number, there is at least one real root between a and b; if h —k =2, or any even number, there may be two, or some even number, or none; the latter case will happen when, as explained above (Art. 110), some number between @ and b makes two or some even number of changes vanish, without satisfying _ f(x) =9. Similarly, we must examine all the other partial - intervals; and when two or more roots are indicated as lying in any interval, their nature must be determined by a succeeding proposition. The two former of the following examples are extracted from Fourier’s work. 16 122 Ex. 1. f(a) = a°— 30 — 240° + 950° = 46a — 101 = 0 S (a) = 524 - 122° — 724° + 190e — 46 SJ (2) = 204° — 36a — 144 + 190 SJ” (@) = 60a — 724 — 144 SJ '\(a) = 120e — 72 *PSAG) = 120. Hence we have the following series of signs resulting from the substitutions of — 10, —1, 0, &c., for # in the series of quantities Di ho ih i (-10) — + “ + - + (-1) + - + - - + (0) = = + _ = + (1) - 1 + = 5 + (10) 5° 14. oye > penta ony a Hence all the roots lie between — 10 and + 10, because five changes have disappeared ; one root lies in each of the intervals — 10 to — 1, and — 1 to 0, because in each of them a single change is lost; no root lies between 0 and 1 because no change is lost between those limits; and three roots may be sought between 1 and 10 (because three changes have dis- appeared), one of which is certainly real; it is doubtful whether the other two are real or imaginary. Oxzs. When any value c of #, makes one of the de- rived functions, f”(v), vanish, we may substitute c +h in- stead of c, h being indefinitely small; then all the other functions will have the same sign as when w=c; and the sign of f"(c +h) will depend upon that of +hf"*'(c); 2. e. it will be the same or contrary to that of the following derivative, f”*'(c), according as h is positive or negative, or according as we substitute a quantity a little less or a little greater than the value which makes f"(a) vanish. 123 The use of this remark will be seen in the following ex- ample. Ex. 2. f(#) = 2*-4a°- 34+ 23 =0. SJ (a) = 4a - 1227 - 3 SJ’ (2) = 12a? — 240 J (@) = 24a — 24 ff" (a) = 24. TRL FG 2 KOs. oF oe" t v=0 T aa 0 _ + o=O0-~ h, + _ So _ + v=1 + = sa 0 + e=1 zh, + = - aa + = 10 =i a oF 43 136 Every value less than 0 gives results alternately + and —, therefore there is no real negative root; for # =0, we have a result zero placed between two similar signs, and therefore corresponding to it there is a pair of imaginary roots. There is no root between 0 and 1, but there may be two roots between 1 and 10. Ex. 3. f(a) = a° — 64° + 400° + 60”° —# -1=0. Here there is no root <—1; there is one, and there may be three, between — 1 and 0; there is one root between 0 and 1; and there may be two roots between 2 and 3. 114. The above process will determine the intervals in which the roots are to be sought, but not always their nature; when an even number of roots~is indicated, they may all turn out to be impossible. The series of magni- tudes, between — c and + o, to be substituted for w# in the derived functions, has been divided into intervals of two sorts, each contained by assigned limits, a and 6. The first sort of interval is one within which no root is compre- hended ; i. e. the limits of which, give the same number of 124 changes of signs in the series of derived functions. The second sort is one within which roots may lie; i. e. where the number of changes resulting from the substitution of 6, is less than the number resulting from the substitution of a, in the series of derived functions. This second sort of in- terval has two subdivisions, viz. cases where the indicated roots do really exist, and others where they are imaginary. When we have ascertained that a certain number of roots may lie between a and b, we may substitute c (a quantity between a and b) in the series of derived functions, and if any changes disappear, our interval is broken into two others; if no changes disappear, we may increase or dimi- nish c, and make a second substitution, and it may still happen that no change is lost; and so on continually; and we may be left after all in a state of uncertainty, whether the separation of the roots is impossible because they are imaginary, or only retarded because their difference is ex- tremely small. Hence when we know that two limits may include a certain number of roots, we must have a special rule for determining whether they are possible or impossible; this has been given by Fourier in the two following proposi- tions; in proving which, we assume that the development of f(«# +h) in Art. 27 may be put under the following forms, so as to exhibit the remainder. of the series, when we take only one, two, &c. terms ; J (@t+h) =f(a) +hf"(r) SJ (ath) =f(a) +hf' (a) + SRF"), and so on, where A, mu, &c., are quantities certainly situated between w and a+h, but of which the exact values are unknown, and for our purpose are unnecessary. 115. Having given that between two limits a and }, f'(#) = 0 has no root at all, f(v)=0 one root and no more, and that f(a) =0 may have either two roots or none, to discover whether these roots exist or not. 125 By what has already been proved, the series of signs re- sulting from the substitution of a in the series of quantities S (@); ST (2), SJ (@); awa Jy (@)s will present two more changes, than the series resulting from the substitution of 6; also, if we leave out the first term, there will be one more change for a, than for b; and if we leave out the first two terms, there will be exactly the same number of changes for a as for b. Therefore f(«) and f(x) will be both constantly positive, or constantly negative, for a and b, and for all intermediate values; and J (2) will have a sign different from that of f(«) and f’(«) when # = a, and the same as that of f(#) and f(x) when a =O. The two roots of f(x) = 0, indicated as lying between a and 6, will be real or imaginary, according as it is, or is not, possible to find a quantity c, between a and 6, such that f(c) shall have a sign contrary to that which is com- mon to f(a) and f(6). Let therefore, if possible, c=a+th=b-k be a quantity between a and b, such that J (2) J (®) or, expanding so that the terms of the second order may include the remainder of each series, and denoting by }, p, quantities intermediate to a and 8, £@ BLO _.. ENE (ahaa 2 F(a), . g. at, Feelin eh ya FO) 2°f) Or, since under the given conditions the last fraction = neg. ; + oH WMC) in each line must be positive, and also thy (0) Th (a) S () = MeL = pos., we must have 126 S (6) J (6) thy AE, I (@) TF GY FO) ae nie F(a) Fb) +h+k = pos. fo) f(a) on hak S— 71> = : + : a> gF () Ties If then this condition can be satisfied, a quantity c, be- tween a and b, may exist so as to make f (c) of a sign con- trary to f (a) and f (b); and if it can be found, the indicated roots are real and are separated: but if the condition is not satisfied, that is, if the difference of the limits be equal SM f © Fay FO’ taken without regard to sign, no such value of ¢ exists, and the indicated roots are imaginary. + h’= pos. —k = neg. ; to, or less than, the sum of the fractions It is manifest that if any three consecutive derivatives, S'(@), f"*'(#), f"*?(2), satisfy the prescribed conditions for a given interval, the same process will determine the nature of the pair of roots of f"(v) =0 indicated in that interval; and whatever number of impossible roots J'(e)=0 may have, f(v)=0 has at least as many (Art. 53). 116. When the above condition is satisfied, we must substitute a quantity ¢ between a and b in S («); if F (©) has a sign contrary to the common sign of f (a) and f (6), the separation is effected; if not, we infer that the limits are not sufficiently close to determine the nature of the indicated roots by a single process. In the latter case, ‘(c) necessarily differs in sign from one or the other of | S'(@, Jf'(6); choosing, then, that limit which makes J (#) have a contrary sign from f’(c), we must with it and ¢ repeat exactly the same process; and we are certain at last to discover either that no roots exist in the interval, or to separate them if they do. 127 Ex. wv —3.a' — 240° 4+ 95a” — 464 —101 = 0. a aaa alae dete CG eis i “- + + 2 1 0 (3) "= - ~ + + a Here, since there are two more changes for «& = 2, than for v=3; one more, omitting the first term; and the same number, omitting the two first terms; the equa- tion may have two roots between 2 and 3, and the con- ditions respecting the roots of f’(#)=0, f”(#) =0 are satisfied ; and since for the two limits, the fraction S (2) i 32 Fe. 43 the sum of which is greater than the difference, 1, of the limits; therefore the two indicated roots are imaginary. 117. In the next proposition, it will be necessary, for any proposed interval, to know the number of roots which each derivative, when formed into an equation, may have in that interval. The best practical way of doing this, is, in the two series of signs produced by the two limits, to write over each sign the number of changes presented by the series commencing with that sign and going to the end of the series; and then to take the: difference between each number in the upper line and the corresponding one in the lower. Applying the process to the foregoing example, we have 3 2 1 1 0 0 (2) - + - = + + 2 1 0 1 0 0 (A) ai + = a ‘" + 1 1 1 0 0 0 where the series of indices 2, 1, 0, 1, 0, 0, mark the number of roots which the equations f(v) =0, f/” (#) = 0, 128 J («) = 0, &e., may have, between the limits 2 and 3. Also, we observe that in this series (and indeed in every case, if we consider the way in which they are formed,) any index has immediately adjacent to it, either the same, or one differing from it by the addition of + 1. 118. When any number of roots of f(w) = 0 are in- dicated as lying between a and J, this interval may always be broken up into others, in which such of the roots as are real are situated singly. From observing the number of changes lost in the series formed by f(x) and all its derivatives, and also in the series formed by each of the derivatives and all those which fol- low it, in passing from a to b, let the number of roots which J () = 0 may have, or which the derivatives taken in order when formed into equations may have, between those limits, be determined; and let them be 0, 0, 0”, &c. Now sup- pose that in the series (where each function is accompanied by its index, 7. e. the number of roots which, when formed into an equation, it may have between a and 6) FO) f'(®) $0 @) ef) f(a) fO(@)s r) ny O” 2 1 € J'(#) is the first whose index is 1; then the preceding function has 2 for its index, for it cannot have 0, otherwise, since the first index is not zero, there would be some func- tion before f"(#) having 1 for its index. Now if ¢ be not zero, since f’(#) =0, f"*'(v) =0, cannot have a common root, two new limits a’, b’, may be found within the former, intercepting the root of f"(#) =0, but excluding every root of f’*'(#) =0. Hence the interval a, 6, will be bro- ken up into the three aa’, ab’, bb, the first and third of which give for /"(«) an index zero, and therefore an index 1 to some preceding function, and the second a’b’ will either make some preceding function have an index 1, or will allow f"(@) still to be the first function whose index is 129 unity for that interval, the indices of f’~'(#) and f"*'(a) being 2 and 0. Suppose the latter to be the case; then, by Art. 115, we may find whether /"—'(#) = 0 has two real roots or none between a and J’; if there are two real roots, then taking a quantity c’ between them, the interval a’b’ is divided into the two a’c and c’J’, each of which makes J ’(#), or some preceding function, have an index 1; but if the two roots of f”~'(«) = 0, indicated as lying between a and 0’, are imaginary, since every quantity intermediate to a and b’ will make f"~'(«) and f"*'(w) have the same sign, therefore in passing from a’ to 6’ through the root of J (2) = 0, since the adjacent functions have the same sign, two changes will be lost. Hence we may diminish the in- dices of all the preceding functions by 2, and _ proceed, relative to the interval a’b’, with that function preceding J '(«) which first has 1 for its index. Hence the proposed interval is replaced by partial intervals, in each of which the separation of the included roots is more nearly effected than in the original interval; and by proceeding with the partial intervals in the same manner as we did for a, b, we shall at last find only intervals in which the index of f() is either 0 or 1; and the separation of the roots of St (~) =0 which lie between a and J, will be completely effected. Ex. SJ (a) =a - a + 40° +4-4=0, Me Ee Ltd ded ae (— 10) + = + - + (- 1) + rs . 1 + (0) ~ + + ~ + 3 2 2 1 0 (1) + + + + + There is no root between — 10 and —1, and one root between — 1 and 0, also three are indicated between 0 and 1; but, forming the series of indices for that interval, we 17 130 see that f(a) = 0 is the first equation to which the cri- "(ax 6a? — 3a +4 terion can be applied; also Te becomes ———————— I’ (@) 12a — 3 ; 4 : : 4 glug which, for 2 = 0, becomes — Py and this neglecting the sign is greater than 1, the difference of the limits; therefore the roots are imaginary, and consequently there is only one root of the proposed equation between 0 and 1. Geometrical illustration of Fouwrier’s method of separating the Roots. 119. The criterion of the reality of two indicated roots In any interval may be readily deduced from geome- trical considerations. Let y = f(@) be the equation to a parabolic curve; then the portion of it between # =a, # =b, (supposing these limits to satisfy all the prescribed conditions,) must have the shape PCQ, (Fig. 2) O being the origin, ONM the axis of 2, PN, QM, the ordinates of its extremities having the same sign, C the single point where the tangent is parallel to the axis, and the curve through the extent PCQ being convex to the axis of xv, because for that in- terval f(#) and f(x) have the same sign. But if O’N’M’, a line parallel to ONM and cutting the curve in two points, be the axis of w,, the curve will have the ordinates of its extremities of the same sign, and will have its tangent parallel to the axis of # at one single point, and f(a) f(a) will have the same sign for all points between P and Q; hence, for any thing that yet appears, this construction will represent the function f(#) between # =a and a = b, just as well as the former; but it is manifest that when f(x) =0 has two roots between a and b, there will be two points of intersection with the axis of w, and the second is the true construction; and the former belongs to the case where there is no point of intersection, and where the abscissze of the points of intersection, that is, the roots of f(w) = 0, are 131 imaginary. If we knew the exact value, c, of OR, we might substitute it in Ja (w), and if the sign of the result was different from that of f(a) and (6), then f(c) would be represented by #’C, and we should be certain that there were two points of intersection; if the same, /(c) would be represented by RC, and there would be no point of inter- section. But if we can only find an approximate value of c, and the sign of /(c) is the same as that of f(a) and f(b), we are uncertain whether the points of intersection are ima- ginary, or so near to one another that our approximate foot of the least ordinate does not fall between them. Now in the case of real roots, that is, when O'N’M’ is the axis of w, and there are two points of intersection, if tangents Pt’, Qs’ be drawn at P, Q, it is manifest, that however near to one another the roots are, and however close the limits are to the roots, N’M’ must exceed N’?’ SH) f£@’ So f@ b — a, we know that the roots cannot be possible, and may pronounce them impossible. + M's’, orb —a must exceed if therefore we find either , or their sum, greater than But when we find the difference of the limits greater than the sum of the subtangents, we cannot conclude that the roots are possible; for this condition is satisfied not only by the axis N’ M’ but also by NM, as long as the tangents Pt, Qs, do not intersect between the curve and the axis. In the latter case, we must substitute a quantity d be- tween a and b for a, then if f(d) have a different sign from J (@) and f(b), the two indicated roots are real, and their separation is effected ; if not, KF ‘(d) will have the same sign either as f’(a) or f(b); let it be the former, then no root can lie between a and d; and we must now apply the cri- terion of the subtangents to the new and closer interval from d to b. ; SECTION VII. ON THE METHODS OF FINDING APPROXIMATE VALUES OF THE REAL INCOMMENSURABLE ROOTS OF EQUATIONS. 120. WueEwn all the commensurable roots of an equa- tion have been found, and all the incommensurable roots separated by the methods explained in the foregoing sections, the next step towards the solution of the equation is to find approximate values of the incommensurable roots; and to this we shall now direct our attention. It will however be necessary previously to prove certain properties of the polynomial f(x), which forms the first member of the equation. Since f'(v +h) — f(a) = fio) h+ f"@) = + sas os fof and as long as « is finite, none of the quantities /’(2), J (z), &c., being integral functions, can become infinite, therefore by taking / sufficiently small we may make the second member as small as ever we please; consequently if # increase continuously by insensible degrees between two limits @ and 6, f(x) will also vary continuously by insensible degrees between the same limits; and will go on increasing as long as f"(#) continues positive; and when f’(2) is nega- tive, it will go on diminishing. 121. Again, since S(@ i — f («) = f'(«) +f") > a ee ee 133 ; pal. meat and since by diminishing # we can make (a) = +... +h"! as near zero as ever we please, or always intermediate in value to + e and —e, where e is as near zero as ever we please; by taking A sufficiently small, we shall always have eee) >f' (4) -e < f(a) +e; if therefore w be always taken between a and 6, and if A, B, denote the least and greatest values which /’(#) can assume between those limits, d fortiori, the following in- equality may be satisfied, JS (a@+h)—f(#)>h(A-e) (a,- 4) (A -e) < (a,- 2) (B+ 8) Jes) SF) > CR ay Ss ha Os < mea rr hey Si 2 i FO wich 6 ~ ny (A ie < mc + TERE SOF therefore, adding, JT (6) —f (a) > (b - a) (A -e) < (6-@) (B+ e); or, since this is true however small e is, S (6) - f(a) > (6-4) A< (b-a) B. But as a changes by insensible degrees from a to b, f(a) will change by insensible degrees, and will assume all values between 4 and B, these being the least and greatest values which it can have in that interval. Therefore every quan- tity between A and B will be a value of f(x) corresponding to some value of w between a and b. Suppose therefore 134 , which we have shewn to lie between 4 and B, Sb) - f(@) b-a to be equal to the value assumed by f’(v) when w=); then J (6) =f(@) + (6-4) fA), where \ is some quantity lying between a and b. Newton's method of Approaimation. 122. When we know an approximate value of a root, we may easily obtain other values of it, more and more exact, by a method invented by Newton, which rapidly attains its object. We shall give this method, first in the form in which it was proposed by its author, and afterwards with the conditions which Fourier has shewn to be necessary for its complete success. Let f(«#) =0 be an equation having a root ¢ between a and b, the difference of these limits, 6 — a, being a small fraction whose square may be neglected in the process of approximation. Let c,, a quantity between a and b, be assumed as the first approximation to c, then c=c, +h, where h is very small; Ray ah ome iO or f(ce,)+f'(aq)h Gye + i ee Now since h is very small, h?, h®, &c., are very small compared with /; also none of the quantities f”(¢,), f°” (¢), &c., can become very great, since they result from substi- tuting a finite value in integral functions of #; therefore, provided /"(c,) be not very small (that is, provided f’(#) = 0 have no root nearly equal to c, or toc, and consequently J (#) = 0 no other root nearly equal to ¢ besides the one we are approximating to) all the terms in the series after the 135 two first may be neglected in comparison with them; and we have, to determine hf, the resulting approximate value of h, the equation SF (ey) + hfe) = 95 ll chm J (#) 5 ite (c) ay; alee and the second approximation is (2) Similarly, starting from c, instead of c,, the third ap- proximate value will be mat a att and so on; and if we can be certain that each new value is nearer to the truth than the preceding, there is no limit to the accuracy which may be obtained. Pere B20 —75.=>.0. Here, one root lies between 2 and 3, and the equation can have only one positive root; also, upon narrowing the limits, we find that # = 2 gives a negative, and # = 2°2 a positive result, therefore 2-1 differs from the root by a quantity less than 0-1, and we may assume c, = 2°1. Hence ge vo — 20 - ’) hd 0-061 C= M = Cal) yn — . 3a? —2 A 11°23” r=2°1 or c, = 2°1 — 0:0054 = 2:0946. Similarly, C, = 2°09455149. Ex. 2) oF. 7 @ —-7'=°0: There is only one positive root, lying between 3 and 3'1; and it equals 3°048917339. Ozs. To guard against over correction, that is, against applying such a correction to an approximate value, as shall 136 make the new value differ more from the root by excess than the original approximate value did by defect, or vice versd, we must be certain that each new value is nearer to the truth than the preceding; this gives rise to the follow- ing conditions, first noticed by Fourier. 123. For the complete success of Newton’s method of approximation, the following conditions are necessary. ~ 1.) The limits between which the required root is known to lie must be so close, that no other root of J (#) =0, and no root of f(w) =0, or f(x) = 0, lies be- tween them. (2.) The approximation must be begun and continued from that limit which makes f(a) and f’(#) have the same sign. Let ¢ be a root of f(«) = 0 which lies between a and 6, a Z Oty Als TO) 2 Now if f”() preserve an invariable sign between a and b, i.e. if f"(@) = 0 have no root in that interval, then /’(«) will increase or diminish continually from a to 6; therefore c, must be taken equal to that limit which gives f’(«) its greatest numerical value without regard to sign. First, let f’(w), f(a), have the same sign from a to b; then SF (2) increases continually in that interval; there- fore we must have c, = b, or we must begin from the greater limit. But f(b) has the same sign as f(¢+h)=f(c) +h f'(c)=hf'(c), or as f’"(c); therefore we must have ce, equal to that limit which makes f(#) and f(a) have the same sign. Secondly, let f(x), f’(a), have contrary signs from a to b; then f’(#) diminishes continually in that interval ; therefore we must have c,= a, or we must begin from the lesser limit. But f(a) has the same sign as (ce —h) =f(e) -hAf(c)=-hf'(c), or as —f"(c); therefore in this case, equally as in the former, we must have c, equal to that limit which makes f(w) and f”"(«) have the same slgn. These conditions being fulfilled, we have J (4) —l=-+., rps = J) h, (b hoee bF b or +3 C2 —C, therefore c, lies between ¢ and c,; hence the new limit ce, fulfils the requisite conditions, and we may with certainty from it continue the approximation. 18 138 124. ‘To estimate the rapidity of the approximation, we have error in Ist approximate value c,, = /h error in 2nd approximate value c,, = h —h,. But Se) +hf(e) +4 Vf" (Ww =; Tie) + Fey ee Gh) fC) +4 Reo orh ~—h,= eZ SF (a) Let the greatest value which f”(#) can assume between aand b (which will be either f(a) or f(b), if f"(#) = 0 have no root in the interval) be divided by the least value of 2f"(#) in that interval which will be either 2/(a) or 2 yk (6), and let the quotient be denoted by C’; then, neglect- ing signs, 7 h-h,) &c., q q i te 1 1 Rane bao eek Hence we see that, to reduce a vulgar fraction to a con- tinued fraction, we must proceed exactly in the same man- 143 ner as to find the greatest common measure of its numerator and denominator; taking care, however, first to divide the numerator by the denominator, so that when the numera- tor is less than the denominator, the first quotient, a, will be zero. And as the process of finding the greatest common measure of two numbers always leads to a remainder zero, and a quotient expressed exactly by an integer, we see that every commensurable quantity can be expressed by a continued fraction which terminates; and_ conversely, every terminating continued fraction is the expression of a commensurable quantity ; for by performing the calculations indicated, it can be reduced to an ordinary fraction. Ex. By performing the process of finding the greatest common measure of 743 and 611, we find the quotients. 1, 4, 1, 1, 1, 2, 3, 1, 3, and a remainder zero; 743 1 1 ] 1 1 1 herr) =— a os Peer ope a ey a 611 4+ 1+ 1+ 14+ 2+ 34 1+ 3 Hence, also, it results that an incommensurable quantity can be converted only into a continued fraction which does not terminate; of which we shall now give an instance. 129. To convert «WN (N not being a complete square) into a continued fraction. Let a* be the greatest square in N, so that N =a’ +b; . : , = ] then a is the greatest integer in /N; let —, be the ve remainder 3; — 1 o w=S/N=at+— Xv v= ; VN +a suppose t a = ————— £=a+4+— 3 NO a b a!” pp bf e e ° 1 e a being the greatest integer in a’, and —, the remainder. av 144 Suppose that, in continuing this process, we arrive at (r) J N +m 1 wv = ——————_ 1 jh + —"4 n y . 3 : l ‘ uw being the greatest integer in #”, and — the remainder ; y “ASN - (np -m) N-(np-my 7 me $2 2 -. ene if m= np — m, ye ta Ore (1) n n o ° 1S AN ee Y= St n 8 4 : ‘ 1 ‘ w’ being the preatest integer in y, and — the remainder. z a pe / N +m" 1 Similarly, s = ——,,-— =p" +-; &c. , n Uu m’, n”, being formed from m’, n’, »’, by precisely the same laws as m’, n’, were from m, ”, p, in equations (1); and »” being the nearest integer to x; and so on for the rest. Hence y, x, &c. and the quotients uw’, uw’, &c. will be found by an easy and uniform process, which must be continued till we arrive at a quotient = 2a; after which, the quotients will recur (as will be hereafter shewn) in the same order, beginning with a. Oss. Sincem’=np-—m, and nn’ = N-—(nu-m)’, or n’ = n+ 2um — np’, since nn = N — m’, J N +m? (—— being the quantity which precedes n / N + ) n 3 ~we see that m and n will always be integers, since they /N +0 /N +4. are so in the two first cases, aCe and A ; and it will appear that they are always positive. 145 Ex. 1. To express a/ 23 by a continued fraction. a/ 23 +0 “Te F gece fr 4, writing down only the integral part ; , 23-16 also m=-0,n=1, .°. m=4.1-0=4,n Si ae tie 7, © a/ 23 +4 3 eis gas ee om = 7.1 4 a 8, nl! 7 7 2343 gk V/ 28 + 3 ¢ mm!" = 1 ra LA 23 9 a 2 Jf 23 +8 , 28-16 ; a / 23 +4 k fio 10 ae 1 Tey iia a/ 23 + 4 ‘e Hence the quotients 1, 3, 1, 8 will recur; and conse- quently = 1, ] 1 1 1 1 eee ae —.> Ss —_— ——_—- — 1+ 3+ 1+ 8+ I+ 34+. 1 1 1 1 1 eg T= 2 141+ 14 44+ 1+. Converging Fractions. 130. Returning to the consideration of the expression 19 146 the fractions formed by taking 1, 2, 3, &c., of the quantities a, b, c, &e., are called converging fractions; thus a ¥e ae 1 abe+at+e & iy b bocie 1 be: 1a 4 b+- C are converging fractions. The converging fractions, taken in order, are alternate- a . ly less and greater than the true value of 2; thus ails too | e * small; a+ ‘ is too large, because a part of the denomi- deme nator is omitted; again, b + — is too large, and therefore ec Le: is too small, and so on. b+ Z c The quantities a, 6, c, &c., are called quotients; and any one with the quantity which must be added to it (sup- posing it were the last) to make the value of @ exact, is called a complete quotient. Thus, using the notation of Art. 127, I 1 a+—-,- b+-, c+-; &e, y bs U are complete quotients. 131. To transform any continued fraction into a series of converging fractions. J " Suppose us an oe , three successive converging frac- aaa | tions. Write down the quotients, and under them the converg« ing fractions, 144 a b Cc d mm m”’ m aab+labe+a+e ppp eed bode le) aang Now as far as we have gone we observe, that, having formed the first two converging fractions, and written them one row in advance of the quotients, the numerator of any fraction is formed by multiplying the numerator of the pre- ceding by the quotient that stands over it, and adding the numerator of the fraction preceding that; thus abe+at+c=(ab+l1)eiva; and the denominator, in the same manner, by multiplying the denominator of the preceding by the quotient over it, and adding the denominator preceding that ; b a thus be + 1 = bc + the denominator of wk Suppose the law to hold up to the quotient m, so that te being the fraction preceding 4 p=pm+p, g=qm+ qs then P = mee : q Gath 9 tr , Now fre differs from P only in taking in another quo- q q I tient, so that if m+ —-— be written for m, we have m (pm+p)m+p _ p'm'+ p (qm+q)m+q gm'+q " 1 if q (m+—) +4' which is the same form as the preceding; if therefore the law hold for the formation of any one converging fraction, it 148 holds for the formation of the next; but we have seen that it holds for the third, therefore the Jaw obtains generally. Cn Ex. 1. To find a series of fractions converging to oom Here the quotients are (Art. 128) 1, 4, 1, 1, 1, 2, 3, 4 1 1 5 1, 4; and the two first fractions are 7a and 1 + or 7 Hence, writing down the quotients and the two first frac- tions in the manner directed above, and forming the rest by the rule, we get A ES Bale RAAT Dh dice Bh A 3 1.5 6 LYO17 14671525107 743 ee 1455. O14 S71 eS oe 611° the last being the original fraction, and the preceding alter- nately greater and less than the true value. If the proposed quantity has no integral part, then the first quotient, as was before observed, will be zero, and the © F sheath) first converging fraction < Ex. 2. To find a series of fractions converging to J 23. The quotients are (Art. 129) 4, 1, 3; 1,:6paaeaeeeeeeee Hence we have hh BR Cd o 1 6 ean 4 5 19 24 211 235 916 1151 11'4 5 44 40° 1O1 Mesut Ex. 3. T'wo scales, whose zero points coincide, are placed side by side, and the space between consecutive divi- sions in one is to that in the other as 1 to 1°06577; to find those divisions which most nearly coincide. They are 15 and 16, 61 and 65, 76 and 81, &c. 149 132. The difference between any two consecutive con- verging fractions, is a fraction whose numerator is unity, and denominator the product of the denominators of the fractions. This is immediately verified with respect to the two first converging fractions, for ab+1 a 1 To prove, therefore, that it is generally true, it will be Sy Ne a sufficient to consider three consecutive fractions aria iery . 7 ah pp and to show that if the property holds for the two tied ! a it must hold also for P and ue : q q / 0 Our 0 eee P MPEP DP yp—-P9 q gq mqt+q qq (mq+q)q’ } : ; Sc taaaeee or pq? — gp’= +1; PONE ad or p'q~ q'p = 1. Oe eo: G 133. In a series of converging fractions, each fraction approaches nearer to the value of the quantity to which the approximation is made, than that which precedes it; and (the integral part, or zero, being the first converging frac- tion) all the converging fractions of an odd order are less, and all those of an even order greater, than the true value. 150 and to deduce the value of w, the quantity to which the ap- ' proximation is made, from that of P , it is sufficient to : : i replace the quotient m by the complete quotient m+ — =y 3 Me suppose, where y is always positive and greater than unity ; qyt+@ 0 pil ih isis Bt : P capi y ——-,—-@#= Sr q q(qy+q) @¢ ga (qy + 9°) Now @°< q, and y > 1, therefore on both accounts the 0 value of a — © is less than the value of ia —-« (not regard- q ing the signs); therefore the successive converging fractions : 0 approach nearer and nearer to «. Also since f — w and Lv — P have the same sign, the successive converging fractions q are alternately greater and less than the true value; but the ‘ : a. first converging fraction 7 is less than w; therefore all the converging fractions of an odd order are less than wv, and form an increasing series; and all the converging fractions of an even order are greater than a, and form a decreasing series. 134. All converging fractions are in their lowest terms. For if the numerator and denominator of the fraction Pp had a common measure, then from the equation pq — q'p = +1, it would follow that this common measure must di- vide unity. 135. The error, in taking any converging fraction for the value of the continued fraction, is less than unity divided 151 by the product of the denominators of that fraction and the following ‘one; and greater than unity divided by the pro- duct of that denominator and the sum of that denominator and the following one. : 1 For, since a — £ = = hopes pr q 4(qy¥+¢) and y is greater than m and less than m+ 1; therefore, leaving the sign out of consideration, Pp 1 1 -—~ <——____ > —_______ ; q g(qm+q) q(qm+qt+q) or, since g =qm+ q, v p 1 1 e-—-<— >——. q 9a 4g(q+q) 136. We can also obtain a superior limit of the error, depending only on the denominator of that converging frac- tion which we take for the approximate value; and an in- ferior limit, depending only on the denominator of the following one. For since q’ is always greater than q, Aft 1 1 — is less than =, and greater than —;_ there- 2q° qq +49) fore, a fortiori, These limits are to be preferred, on account of their ‘simplicity, to the former; and in most cases are sufficiently exact. Hence we may at any step measure the accuracy of our approximation. ‘Thus, in the examples of Art. 131, the , Loz : TA43 4 : fraction aa which converges towards aie differs from it 1 | ntity less than and greater than ——— ; by a quantity g 2(162)' 1 (125)? 152 1a PO16 d — .. and the fraction rae which converges towards / 23, differs ‘ : 1 from it by a quantity less than Gigi)?’ and greater than ( I 2 (240)? Ex. To find a series of fractions converging to the value of the ratio of the circumference of a circle to its diameter ; and to estimate the error with which each converging fraction is affected. The value of this ratio, exact to ten places of decimals, Is 3°1415926535; therefore, adding unity to the last deci- mal, the value of z will be comprised between the frac- tions 31 415 926 535 31 415 926 536 ~~ and ———______. 10 000 000 000 10 000 000 000 If now we perform the successive divisions for each fraction, we find the two series of quotients 3, Ts - bso leu 1202. 1 rn 3.7, (16s: ]y 202, ew Tene therefore, reserving only the quotients which are common to both, and which must belong to the continued fraction which expresses the value of 7, and forming the converging fractions, we get 8. 1175s 0s] eee eee lg 3 22° "3393355 *1T0S090 mae > b) 1? 7’ 106° 118° 83102° —s38166 These fractions are alternately greater and less than the true value of 7; thus os is too great; it is the ratio discovered by Archimedes, and differs from the true value by a ntity lying between : and : . ua , ss te __. ee zee Midi 7 x 106 7 (7 + 106) " 742 153 f/ 1 355 and —-. The fraction —— is the value dis pvered by 791 113 R Adrien Métius; it is also too great, but far nde, en that of Archimedes, since it only leaves an error co tised 1 1 : between ———— ‘and ————-. ~O 3740526 3753295 _.W 137. In order that the fraction P may differ from q the exact value of by a quantity less than a given quantity Barn. A 1 1 —, it is sufficient that we have phen 9) O8e Qi or >/a. a q a Hence we can always obtain, either exactly, or within any degree of approximation, the value of a quantity ex- pressed by a continued fraction; for if the continued frac- tion terminates, we then obtain its value exactly; and if it does not terminate, we can obtain a converging fraction whose denominator satisfies the condition g = or > /a, because the denominators of the converging fractions are integers, and go on increasing indefinitely. 138. In a series of converging fractions, each fraction differs less from the value of the quantity to which the approximation is made, than any other fraction in more simple terms. 7 Let ” be one of the converging fractions, and let = be s - : r another fraction whose denominator is less than g. If — be 8 one of the converging fractions, the proposition is manifest from what has been proved. But if i be not one of the s converging fractions, then it cannot lie between : and the 20 154 0 0 ‘ Dp 2. ear Pia oe preceding : ; for if it could, then + (% ~ 4 would be: q 7° 1 s . eee a - less than —, or + (p°s—q°r) < —, which is impossible, qq q because the first member of this inequality is an integer different from zero, and the second a proper fraction, since s < q. e r . p° P . « e Since then — cannot lie between — and '—, if if le to 8 q the right of : (supposing the three arranged in order of magnitude) it differs from # more than i does; and if it 2 0 0 lie to the left of a it differs from x more than 2 does, q and therefore a fortiori more than P does. q 139. Every periodic continued fraction is the ex- pression for one of the roots of a quadratic equation whose coefficients are commensurable. Let the continued fraction be 1 1 pee | —eee bd — a le St... UEUEY where y = 7 + so that a, b, c, ... 1 are quotients which do not recur, and 7, 8, ... v are those which recur indefinitely. , Let Bs ~, be converging fractions in the value of a, the last quotients comprised in them being, respectively, k andi; so that J and r are the quotients which stand over 155 them, when formed according to the method of Art. 131; fen id oe and let Q’ Q” be converging fractions in the value of y, the last quotients comprised in them being, respectively, uandv; then, as in Art. 133, _ Py +p re BY te dy+qg” Gy+@’ between which equations, eliminating y, we obtain an equa- tion of the second degree in #, which demonstrates the property announced. When we wish to find # under an irrational form, we must take the positive value of y in the equation Vy’ +(Q- P)y- P=0, and substitute it in the preceding value of w. Lagrange’s method of approximation by Gondeened Fractions. 140. ‘To approximate to the roots of an equation by the method of continued fractions. Let the equation f(v) =0 have only one root between ee 1 the integers a and a +1; then writing a+ - for 2, the y first transformed equation will be fa+ef@ ; st —, = 0 (1), y" ; Tees and since only one value of — lies between 0 and 1, y has only one value greater than 1; if therefore we substitute suc- cessively 2, 3, 4, &c. for y, stopping at the first which gives a positive result, the integer preceding that, is the integral part 156 | , ; the value of y. Let this be 6, and in (1) write b + 5 for y; then the second transformed equation will have only one root greater than unity, the integral part of which, as be- fore, will be the whole number next less than the one in the series 2, 3, 4, &c., which first gives a positive result when written for x; let this be ce, and in the second trans- : 4 1 , formed equation write c+ — for x, then the third trans- u formed equation will have only one root greater than unity, the integral part of which may be found as before, and so on. We thus obtain successively the terms of a continued fraction it 1 1 ] a a Db :0 td + icioe which expresses the required value of #; consequently we are able (Art. 137) to find this value to any required degree of exactness. If any of the numbers b, c, d, &c. is an exact root of the corresponding transformed equation, the process termi- nates, and we find the exact value of x. Also, if one of the transformed equations be identical with a preceding one, the continued fraction expressing the root is periodical ; for, after that, the same quotients will recur in the same order ; in this case a finite value, in the form of a surd, may be obtained for the root (Art. 139) by solving a quadratic whose coefficients are rational, both of whose roots will be roots of the proposed, (Art. 16) since the coefficients of the latter are supposed rational; consequently the first member of this quadratic will be a factor of the first member of the proposed equation, which may therefore be depressed two dimensions. Ex. To find the positive root of #-— 2#—5=0 under the form of a continued fraction. Comparing this with a — qw# +7 =0, we find that 157 Taya? wii 25 Sit “i tit —-—- — = — —-— 18 a positive quanti hi” iawn, P 4 i, therefore (p. 61) the equation has two impossible roots; and since its last term is negative, its third root is positive. Substituting 2 and 3, the results are — 1 and + 16, there- 1 fore the root lies between 2 and 3. Assume v=2+4+-, and the transformed equation is y® —10y°-— 6y -1=0, in which 10 and 11 being substituted give — 61, + 54. 1 e Assume y = 10 + —, and we obtain g 612° — 942°— 207 —1=0, whose root lies between 1 and 2. Proceeding in this manner we find ] 1 1 1 Te ee TOR Sie Ser ax the value of the root, in a continued fraction, which may be converted into a series of converging fractions. 141. When an equation has several roots between two consecutive integers, this method of approximating to them may be rendered easier by combining it with Stewrm’s theorem. Substituting 0, 1, 2, 3, &c., successively for w in the series of quantities (Art. 101) 7); Fi (@)s fi(a)s.2. f(a) OG) and noting between what substitutions, changes of sign are lost, and how many, we shall perceive between what integers the roots lie, and how many in each interval. For those roots which are situated singly between consecutive in- tegers, the process will be that described above; but for those which lie in groups between consecutive integers, we 158 must proceed as follows. Suppose several roots to lie be- e ] . J tween a and a +1; substitute a + — for « in series (1), and Y let the result be P(Y)> Pilly)» Poly) --- Pry) (2) then as many roots as f(a) = 0 has between a and @ + 1, so many positive roots greater than unity will @ (y) = 0 have ; and if we write 1, 2, 3, &c., for y in series (2), and observe between what substitutions changes are introduced, the con- secutive integers between which those values of y, either singly or in groups, are situated, will be determined. If there still be a group of values of y between consecutive in- ; es, , tegers b and b +1, put y = b+ — in series (2), and let the B result be Ni (x), Vi (z), Wo (2), vee W(x) (3) ; | then as many roots as @ (y) =0 has between 6 and b + 1, so many positive roots greater than unity will W(x) =0 have; and, as before, substituting 1, 2, 3, &c., for = in series (3), and observing where the changes are intro- duced, we may determine the situation of those values of z; and the process must be continued till we arrive at a transformed equation whose positive roots are situated singly between consecutive integers; the approximation to each of these roots, as well as to all those already par- tially found, may then be continued, as in Art. 140, to any degree of accuracy. Thus all the values of # between a and a +1 will be determined ; and the other groups of values of wv, if there be any, must be treated in the same manner. 142. It is manifest that Fourier’s method of separating the roots might be employed with similar advantages. For, being applied to the proposed equation /(v) = 0, it would enable us to. ascertain between what integers the roots lie, and how many in each interval. Next, being applied to the 159 I transformed equation f° (« + -| =0, or p(y) =0, (a and a y + 1 being one of those intervals,) it would point out be- tween what integers the values of y lie, and how many in each interval. Similarly, in the next transformed equation 1 wy (2 + -| = 0, or (x) =0, 6 and b +1 being an interval containing more values of y than one, it would shew the situation of the values of x; and so on for all the transform- ed equations which it might be necessary to obtain, to com- pletely separate the processes for approximating to each root of f(x) = 0. The following is an instance of the employment of Sturms theorem. Ex, S (#) = 60 — 1414 + 263 Si(2) = 64° — 47 SJe(@) =940 — 263 SJ3(a) = + 4) agai ae Dall (2) + ~ - + (3) + +, 2 “rie hence two values of w lie between 2 and 3; 1 ae Ms Zee — y p (y) = 290y°— 69y" + 36y + 6 di(y) = — 23y' + 24y + 6 pe(y) = — T5y + 94 op; (y) Shh 160 Hence two values of y lie between 1 and 2 ; and, putting L eee. y=1+-, it will be found that x has one value between g 3 and 4, and another between 5 and 6; and the roots must now be approximated to by separate processes. Solution of Indeterminate Equations of the first order by Continued Fractions. 143. Another useful application of continued fractions is to find the integral values of w and y, which satisfy the in- determinate equation of the first order, aw + by =c. We suppose a, b, c, to be integers positive or negative, and the two former prime to one another; for if they are not, ¢ must necessarily have the same divisor, since « and y represent integers. Let «=a, y=/3, be a solu- tion, then aa+bB=c; and therefore by subtraction, a (# -— a) = —b(y—); but since a and 6 are prime to one another, y — 6 must be a multiple of a = at suppose; therefore v —a = — Df, that is, w=a-—bt, y=P + at, where ¢ is any integer positive or negative. Now to find a ° ° . e a and 3, resolve ;, into a continued. fraction; and in the series of converging fractions, let P be that which imme- g diately precedes > then pb -qa= +1, according as g a Siete sh . cp.b-—cq.a=+Cc3 hence, comparing this with the proposed equation, if the second members have the same sign, B=cp, a= —cq; if different signs, (3 = — cp, a=cq. 161 Jy Ra 5a t+ Ty = 2. 1 ; : it eter and the converging fractions are > ~. 3.5—7.2=1, and 5.87 —7.58 = 29; *. w©=87-—T7t, y= — 58 + St. Ex. 2. llvw+13y=190. # =1140—13t, y= — 9504111. 144. When we wish to solve aw + by =c in positive integers, ¢ must be restricted in the general values « = a — bt. y=P+at. First, suppose a and b to be positive, and therefore c positive since # and y are to be positive; then we must have a-bt>0, B+ at > O06 t<- and > — e therefore only those integral values of ¢ which are comprised between the . e a . J ] ° limits — e j, are admissible. These limits are never contra- a dictory; for since q and ( are positive or negative integers which satisfy the relation aa + bB =c, we have aa + 63 >0, a ° ° . and .. 5 >- B ; but they may not include any integer, in which case the proposed equation has no solution in in- tegers; and in no case has it more than a certain number of such solutions. Secondly, let the equation be aw — by =c, a and b being positive; then #@=a+bt, y= + at; and in order that these p o28 a values may be positive, we must have ¢ > — ri and t > — ee hence we may give ¢ any value above the greatest of these limits, so that the proposed equation will admit of an infinite number of solutions in positive integers. In Ex. 1. (Art. 143) ¢ < 123 >114; .°. ¢ has only one value 12; and # = 3, y=2 are the only positive integral values. Similarly, in Ex. 2, ¢ has only one value 87. 21 162 The equation az + by +cz =d may be solved in posi- tive integers, by assigning values 1, 2, 3, &c. to the variable whose coefficient is greatest, and which consequently lies within the narrowest limits, and solving, as above, the re- sulting equations. Properties of the continued Fraction which expresses ./ N. 145. The last application we shall make of continued fractions shall be to determine the nature of the develope- ment of the square root of a number not a complete square, in that form; preparatory to which the following property must be demonstrated. Let oe ce ss os be the development / of a proper fraction —; then writing down the quotients and corresponding converging fractions, we have | bg) Cpe eae eae ee salt 2 Pp 2 pee a \ab 4h ae q ra q ray whence we obtain the following equations: A 1 Q ia m” @" as q's iy he fe. = Q Ad q {Lasts 7: q / 1 gama! 4.93; oth ta q m’ + q q ' q 1 q = mq + qs Tt dg » ees q q° ae 1 1 1 1 momen + m+ mt+.. O+ Sa " t that is, the development of #4 in a continued fraction (E, q “ange : : LP being the last of the series of fractions which converge to a) di gives the same quotients as the development of —, but in ; Q’ an inverted order; if therefore in any case q” = P, the series of quotients will be symmetrical, 7. e. the same taken from the beginning and end, or of the form a, }, c, ... rebel Sats a 146. If N be a whole number (not a complete square), then 4/N may be developed in an indefinite continued frac- tion whose quotients recur in periods, the last quotient in each period being double of the greatest number whose square is less than N, and the period, as to the other quotients, being the same taken from the beginning and end. In the continued fraction which expresses /N (formed as explained in Art. 129), let the series of complete quotients, partial quotients, and converging fractions, be wv N+e / N+m? / N+m / N+m’ / N+m,° / N+m 0 SSS LOC a as 1a aa 1 aa a 2. b n° [eA ar al ea) 7 0. 4 a, Ay. hes bs fho-- @5 My-- a Pp. Pp Pp Pr’. Pi 7? wk m9 eke are 0 q q q 91 vi ——_ N+m ., _ then any complete quotient —-—__— is formed from that n N —- 9. which precedes it by the law m= p°n°—m, n= 5p and we must first shew that all the quantities m, ”, m’, 164 n', &c., are positive integers. Suppose this to be the case up to m', n°, then all the partial quotients up to ,° are positive integers, and the converging fractions up to ? in- clusive can be formed in the usual way; and _ therefore, N+m . ; since’? 224i) ge ¢he complete quotient corresponding to n P, we have q aut *"\ ie P Washi: q (Ma) +g tL) JN = 3 which, by equating rational and irrational parts, gives PIN ON ae esa qp’)m = qq’ N — pp’ qm + qn =p (py — gp) n = p’— N¢@. But pq — qp°= +1 or — 1, according as PS une JN, q therefore » is a _ positive integer; also the equation Norte . q t (? ); d si 0 2G Td Pacers --—m); and since g> q, nm \q : — N n> P —m, and consequently n > / N-m; bur N+ q n n<«/N+m, which would be impossible if m were negative. Hence m and x will be always positive integers, since they are so in the two first cases. Ms We can now find the limits which m and m cannot sur- pass, however far the process be carried on; for the equa- tion N—m’=nm shews that m < / N, and therefore m cannot exceed a the nearest integer to /N; and since m+m= pn, 2a is the limit both of m° and »°. But since the continued fraction which expresses JN is unlimited, 165 and since there can only be a certain number of values of m and n, the same value of m must occur with the same value of m an infinite number of times, that is, the same complete quotient must recur; and whenever this happens, then the succeeding quotients will be the same as those before obtain- ed, and will recur in the same order; therefore the continued fraction which expresses »/N will (at least after a certain number of terms) be composed of a constant period of quo- tients, and we must now determine the point at which that period begins. Suppose the recurring period of quotients to be Fai: flo flo Ph gs cecece W 5 then since N-—m?=nn, and N-m*=nn,, «. n°=n7,°3 also since m = p2n°® — m°, m = wn, — m,°, *, m — m,° = 7° (u° — w). But the equation 0 ” 0 qm + gn =D, gives maT —" neato nim since Le being an approximate value of / N, can only differ q t Aor from a by a small fraction —; therefore, since g’ eve Y? GF As 203 as, Gs eee 2B; as 2s As &e. which law would be yet more regular, if the first quotient: were either 2a, or zero; 7. e. if the irrational quantity de- veloped were / N +a instead of JN. 167 Solution of the indeterminate equation of the second Order, x* -Ny’ = +1. 147. Every converging fraction, an which corresponds q to the quotient 2a in any period, is such that p° — Nq’ = +1. For since » = 2a, the equation m +m’ =n, in which nei- ther m nor m’ can exceed a, will necessarily give m = m'=a, and n =1; therefore the equation (pq° — qp°) n = p> - Nq’ becomes p* — Nq* = + 1 according as Poor< JN. q Hence the equation «2° — Ny* = + 1 may be always solved in whole numbers (at least with the upper sign) what- ever be the number MN (provided it be not a perfect square), in an infinite number of ways. If the number of terms in the period a, (3, ... 8, a, 2a, be even, all the fractions in the different periods corresponding to 2a will be > / N, and we shall obtain solutions only of a? — Ny’ = +41; but if the period consist of an odd number of terms, then the first fraction which corresponds to 2a will be < / N, the second fraction corresponding to 2a > ,/ N, and so on; so that all fractions corresponding to 2a@ which stand in odd places will satisfy vw” —Ny’ = — 1, and those in even places the equation xv ree Ny’ =+ 1. Le Mae i — 254 = 1. For 1/23 we have (p. 148) the quotients and converging fractions, p= 24,y¥=5; or = 11, y = 240, Ke. 168 Ex. 2. a? —18y?= +1; With the upper sign w= 18, y=5; with the lower a = 649, y = 180. The preceding investigation of the properties of the continued fraction which expresses ws N, is taken from Legendre’s Essai sur la Théorie des Nombres. SECTION VIII. ON THE SYMMETRICAL FUNCTIONS OF THE ROOTS OF AN EQUATION. 148, A symmernricat function of the roots of an equa- tion, as was before observed, is an expression in which each root is alike involved; and which is consequently made up of all the roots in such a manner that if any two be inter- changed, its value is not altered. Thus —pp=at+b+ct+... +1, po=ab+ac+bc+ &e, and, in general, all the coefficients are symmetrical functions of the roots; for in these expressions, if b were written in every place where a occurs instead of a, and a in every place where b occurs instead of b, or if any other two of the roots were interchanged, the values of the expressions would not be altered. 149. We shall first consider the elementary cases where, in each term, only one, two, three, &c., of the roots are involved ; viz. i ac bh” = ce” 4 d”™ oe &c., a™b? + ac? + bc? + &e., ab? ef at a ce? bf rt b” a? ef + &C., The first is formed by taking the sum of the roots each raised to the same power m, and consists of 7 terms. The-second is formed by taking all the permutations of the roots taken two together, and affecting the first letter 22 170 in each product with the index m, and the second with the index p; and it consists of m (m — 1) terms. The third is formed by taking all the permutations of the roots taken three together, and affecting the first letter in each product with the index m, the second with the index p, and the third with the index qg; and it consists of 2 (7-1) (w —2) terms. Similarly, the symmetrical function each term of which contained 7 roots, would be formed by taking all the per- mutations of the roots taken r together, and in each product affecting the first letter with the index m, the second with the index p, the third with the index gq, and so on; and it would consist of ~ (m— 1) (n — 2)... (n —7 + 1) terms. (The above supposes all the indices m, p, q, &c., to be unequal; we shall afterwards return to the case where some of them are equal). Since, therefore, in the above cases, any term being given, all the others may be deduced from it, by forming all the permutations of the letters which compose it, and affecting the letters in each with the indices taken always in the same order; we may denote them by the symbol > prefixed to any one of the terms, thus a (a"), Barb"), = (a"bret); and the first, that is, the sum of the m™ powers of the roots may be indifferently expressed by > (a”) or S,,; we shall generally employ the latter, as the sums of similar powers of the roots are the simplest sort of symmetrical functions, and are the quantities by which all others are expressed. 7 150. The value of every rational symmetrical func- tion of the roots of an equation can be expressed by the coefficients, without knowing the actual values of the roots, as we shall shew. But it will be necessary to consider only the case of integral functions; because when the terms of a symmetrical function are fractional, we can, by reducing 171 them to a common denominator, express the function by a single fraction whose numerator and denominator are inte- gral symmetrical functions. ab ac be Thus — + — + — —3abe, which is a fractional sym- 2c? «2h? 2a? 3 J metrical function of the three quantities, a, b, c, becomes ob +ac + bc — 6a°bc’ by reduction y 2a’ b* c? Newton's Theorem for the Sums of the Powers of the Roots, 151. To express the sum of the m™ powers of the roots of an equation in terms of the coefficients, and the sums of the inferior powers. We have (Art. 60) ee ee Ae J (2) = therefore, effecting the divisions, which can all be exactly performed, since a, 6, ... J are roots of I («) = 0, we get (Art. 6) aw ; ta) = 14 (a+ p) v7? + (@ + pia t+ pp.) +... C— ea + pa Pe, Pa) Oo ot &e. £O) | @ hese wv”! + (b+ p,) yr? + (6? + p,b + p.) a”~° gery pb"! + 7,6"? + ... + Pala” + &e. = os JS (2) = yn} + (1 +p;) a”? + (0? + pil + po) v~ ae anne a—l Sippy Ot pl” +e + Pm) Oo + &e. 172 Hence, adding these quotients together, we have S (#) =n" 4 (S, + np ,) ot ty (ise pS, we N Po) "+... TF (S', + Pi S'in—1 + PoSime2+ --. + Pm Sit ND) a art, But f'(a) = na"~) + (mn —1) pa" + (n — 2) pow" * +... woe + (2 — mM) pa” 1 + &e. 5 hence, equating the coefficients of corresponding terms in these identical expressions, we get S,+np,=(n—-1)p,, or S,+ p,=9; So+ pS + rpo= (nm —2) p., or S,+ p,S,+ 2p.=0; &e., Sin + DP, Sn-1+ PoSm—o+ e¢+ Pm-, Sy + NPm= (N — M) Pn OF Sin + Py Smn—-1+ PoSm—o+ se2 + Pm—1S) + MDm = 0, the formula which gives the sum of the m"™ powers (m being less than 7) of the roots, in terms of the coefficients and the sums of the inferior powers; and by means of which the sums of all similar powers whose index is less than the degree of the equation, can be successively expressed by integral functions of the coefficients. But if m be equal to or greater than n, multiplying the equation by #”~”, we have ) / in — 1 f / Tak, f a — oo ° e+ pe + pot + pe ie therefore, replacing a successively by all the roots a, 6, ¢,.. ..1, and taking the sum of the results, we have Sat Pim- i+ Po Smn-ot «-- + Pa Smop = O- ly Hence, making m=n, n+1, n+ 2, &c., successively, we find, observing that So= a + 6 co ee nN, S,+ pySn_) + PpoSn—2+ Arte Np, = 0, Snap t+ Py Sat PoSn-1 + . 2 pas, =O} &c. .—— 173 Hence the sums of all similar powers whatever of the roots, can be expressed by integral functions of the co- efficients. Ozs. The sums of the powers of the roots of any equa- tion S, S,, S83, &c., form what is called a Recurring Series; that is, a series in which an equation of the first degree with constant coefficients holds,good between a certain defi- nite number of consecutive terms, in whatever part of the series they be taken; for, any one of them, S,,, depends, when the equation is complete, upon the m preceding, by the equation (1), in which the constants of relation are the co- efficients of the equation. 152. To find the sums of the negative similar powers ‘ 1 of the roots, we must write — for w, and apply the above @ formule to the transformed equation in y. 153. We may observe that by the preceding method the value of @ (a) + P(b) +... + (i), (where @ (x) de- notes any rational algebraic function) may be readily found. For, Be 24) 90)" ote) S (#) C-a @&— mah therefore, performing the divisions, and reserving only the remainders, (Art. 6) Aw’'+ Bar?+&e. dla) f(b) p (2) Seem epee OHNE ORE a0 ROC ; S («) e—-a, «av—b v—l food ena + &c. =a"! } (a) +p(b)+..p(J)i + &e. . O(a) +o (6) +...+ (1) = A = coefficient of the highest wee of w, in the poranindes of the division of f'(x).p (#) by f (2). 154. In practical applications to equations of a low de- gree, or consisting of a small number of terms, we may, 174 instead of calculating the sums of the powers successively from one another, express them immediately in terms of the coefficients of the equation, by the following method. . 1 e ° e e For «v write in the identical equation y wv + pa"! + pa"? + pyar" 3 +... + p= (w — a) (w — BD) x io, 10) an ee a “T+ Py + Pry t+ Psy? + o-. + Pry" = (1 — ay) C1 — by) x (1 -—cy)... (1 — ly). Hence, taking the Napierian logarithms of both sides, PLY + Pr ya ns | y>+ Ds | y' + &e. ene — Pi Pe — Py Ps Teh Coa ou + py" po | = ip! => 9S, - SYS. — 3y°Ss — Gy'Ss — &e. 5 therefore, equating coefficients, S,=- pi S.= ~2p, + p,” S3 = — 3p3 + 3p; Po — py Zz T — 4p, + 4p, ps + 2p.” — 4p," po + py’, &e. 5 and, in general, S,, = coefficient of y” in the expansion, by ascending powers of y, of — m log) yf (=) t y Pixies: e+rvxv+s=0. Let a + rv +s =(#- a) (w—b) (a@-c) (a&-d); .1t+y (7 +sy) = (1 - ay) (1 — by) (1 — cy) (1 — dy); o (r+sy)—sy (7 + sy)’ + &e. = —yS, - SYS, - SYS; — 79'S. — 59S — 5y° Ss = Sees 175 hence, equating coefficients, we have s,=0, 8,=0; S,= — 3r, §,= —49; S;=0, S,= 3r’. Ex. 2. The sum of the m™ powers of the roots of we” —1=0is m, when m is a multiple of 2; and zero in all other cases. Let #*° -—1=(#-a) (w~-b) ... (@ -l); » T-y = (1 — ay) (1 = by) ... (1 - Ly); 1 é Te Vy + hye tet rat +&e. =yS,+ 4y'S, + 2 Yin t Be. wine RY a iS, = coe = 0, S,= 2n — eee = J, = 2 teens.) | To express the sum of the mh powers of the roots of a quadratic in terms of its coefficients. Let v—pe+q=(a#-a) (a —b); 1-y(p—qy) = (1. — ay) C1 - by); therefore, taking the Napierian logarithms of both sides, and writing down only the terms which, when developed, will involve y”, we have m—1 m—2 oe m y ell m-1 7] 4 as m—~2 ae ties (PT 99)" + (p— ay)” + &e. 1 = —y"S, + &e. aye, m+ Cc therefore, equating coefficients of y”, : m{in- 3)" 4), Sm =p" — mp" "gq +—_— p g’ - _m(m ar ~1) (m —r—2) —(m —Q2r + Ni moet Gee Rey 1.2.3: 38% (ral) Ex. 4. ao” —px"-' +g =0. S., = coefficient of y” in expansion of — m log §1-y(p—qy"~')} m(m=2N+1) mn oyg2_ M(m—IN+2)(M—3N41) aon 4 Ke, oO" —m m—n + | a 1.2 T.2ra 176 155. Similarly, we may express the coefficients im- mediately in terms of the sums of the powers. For since (Art. 154) log. (1+ piy t+ Poy? + Psy? + --- Pry") = — YS, — 3Y'Se — ty S; — &e. 1+piy +79? + psy + &e. see ye eee =] — y 8, en 4S, y” = 18; y — &c. + $(S))° + 25,8, poe? — =(5})° hence, equating coefficients, p= —s; 1 Pe = — a2 + Le (S,)? 1 1 2 1S Se ee Ps Sek rg te ieet 1) _ Ex. a + poa'+ pow? + py? + pr@ + py = 0. Here §,=0, and proceeding as above, and in the de- velopment of the second member reserving only powers of y as far as the sixth, we have : —hy*Sohy®Sg—... 1+ poy? + psy + Piy't Psy t+ poyr=e ey =1-— (FS.+ Fy Ss+ ty Sit ty Ss + ty*Sh) y’ y" + 7g Sat FySs+ 2y°Si)'— —, Ss)" 1.2 Lew 1 : Lge Ps= —3854 5 2528s KY rs. — (4-93) Dom = ah ak sedi (S') "7 1°o08 G Se)" ae 177 Use of the Sums of the Powers of the Roots in approai- mating to the Values, real or imaginary, of the Roots. 156. The employment of the sums of similar powers of the roots, was first pointed out by Newton as a method of approximating to the greatest root, in the following pro- position. If in the series S,, S,, S;, &c., formed by the sums of the powers of the roots of an equation, each term be divided by that which precedes it, the successive quotients continually converge to the greatest root, provided it be real. Suppose a, b, p (cos 0+ Ay aie sin 8), &c., to be the roots, arranged in order of numerical magnitude, each pair of imaginary roots being estimated in that respect by its modulus (Art. 85), then Sinai amtt4 bt? 4 20"*1 cos (m +1) 0 + Ke. — iS a" +b" + 2p" cos m@ + &e. b\ +} p m 1+ (-) + (4) 2 cos(m+1)0 + &c. a a b m™ ™ 1 + (=). + (2) 2cos mO + &e. a a supposing the greatest root to be real; b m L m+) which (since the fractions (=) 3 (-) , &c., may, by the a increase of m, be made as small as ever we please) approaches Sme1 S m the greatest root, provided it be real, becoming closer and closer as m increases. But if there be a pair of imaginary roots whose modulus exceeds the greatest of the real roots, then to a as its limit; and therefore is an approximation to a m+ b m+ 2cos(m+1)6+4+ (*) + (-) + &e. p Sint1 =p i ° im s m b m Sn 2cos mO + () + a + &c. p p 178 and therefore, by the increase of m, approximates to cos (m+ 1)0 Om aI which may evidently have any value. 157. Again, according as the two greatest roots are real or imaginary, we have 2p” cos mO c” ee SE ee te -- pee See * a” + 5” a™ 4.6" or = p” f cos m@ + (<) + (") + te}. P cE Hence, whether a and b be real or imaginary, provided they be the two greatest roots, we shall have, by the con- tinual increase of m, : S,, = (a" + b") (1 + S,;, = a" + b” nearly, Soe att + OMtt, Sngg= at? + bet”, each equation being nearer to the truth than the preceding ; ae Ds ‘ Sin+2 “aie Nae] a (ab)” (a ae b)?, (Un) 21,59 aes inte = (ab)"**(a ng b)*, (Um +1) and the quotient of the latter divided by the former = ab. This shews that if from every three terms of the series S,, S2, S3, &c., another series & (2,,) be formed by subtract- ing the square of the mean from the product of the extremes, then the quotients obtained by dividing each term of the new series by that term which precedes it, continually con- verge to the product of the two greatest roots. When the two greatest roots are real, since the first is already known, the second becomes known by the process just described. When they are imaginary, as their product is known, it remains to determine their sum, which may be done as follows. We have Sin S43 — Sint Sy as *. ab" (a on b) (a =a b)’, (v,) 179 and dividing this by w,, we get a result = a+ 6; which shews that if from every four terms of the series §,, S593, Sy, &e. another series = (v,,) be formed by subtracting the product of the means from that of the extremes: then the quotients obtained by dividing each term of this series by the corre- sponding term of the series ¥ (w#,) continually converge to the sum of the two greatest roots. Hence the product and sum of the two imaginary roots being known, each of them can be found. This is the only method yet known for approximating, with tolerable facility, to the real and imaginary parts of impossible roots, Ex. 1. «#-—102°-6r7-1=0 = (S;,,) = 10, 112, 1183, 12512, 132330, 1399555, 14802042, &c. 14202042 = —___—__ = 10°576. 1399555 EX. 2 @-—#4+40e+0—-4=0. =(S,,) =1,—7, — 14, 29, 96, — 34, — 503, — 347, 2083, 3838, — 6159, E (Um) = — 63, — 399, — 2185, — 10202, — 49444, — 241211, — 1168158, 2 (0n) = — 69, — 266, — 2308, — 11323, — 50414, — 245363, — 1207713; the first series being a divergent one, shews that a and b are Imaginary ; h fe , 1168156 nd gives ab = ———__ = eee 8 241211 ’ 1207713 d third gives a + b = ———— = 1.03; the second an g 4 eo 1 mt 1 as 1 a= (1.03 + 4.34/~1), b= = (1.08 SR Gai The remaining roots c and d may be found from the equations + e+d+1.03=1, cd = — ——.. 180 Theorems for expressing Symmetrical Functions of the Roots by the Coefficients. _ 158. Every rational symmetrical function of the roots of an equation can be expressed by the coefficients of that equation. First, to find the value of the double function = (@”b’). If we multiply together the two equations Sa" +b" + 07 + es #15 S,=a +b? +e? +... +P, we have SS, = a"*? +. 077? + ce") ae + a™b? + ac? + ba? + &e. Now the first line is equal to S,,,,; and the second consists of all the permutations of the roots taken two together, the first letter in each being affected with the index m, and the second with the index p, and is therefore equal to the double function S (a”b?); | SS) = Say pct > (GRO as or 3 (a"b") = SySp— Snap (1): Next, to find the value of the triple function = Be c’), Multiplying together the equations = (a"b?) = ab? + ac? + ba? + &e., S,= a+ ‘bi ce! + &e., the result will consist of three different partial products ; (1) the sum of products of the form a”*‘b? = & (a”*"b"), (2) the sum of products of the form a”b?*4= 3 (a”b?*4), (3) the sum of products of the form a”b’c! = > (a™b?c’) ; = (a"b*) . S, = 2 (a"*1b%) + = (a"b’*") + Sa bee): Hence, replacing 2 (a"b?’), = (a"t"b?), S(a"b?**), by their values obtained from formula (1), we have = (aM bP cl) = SnS_Sq— Sma pSq— SnqSp— Sp+q Smt 2Snepeg: 18] In the same manner might the quadruple function = (a"b?c'd"), or the sum of any succeeding combinations, be expressed by the sums of the powers; and as these latter are expressible by integral functions of the coefficients, it follows that all the above symmetrical functions can be expressed by integral functions of the coefficients. And as every sym- metrical polynomial in a, b, c, &c. must be composed of the assemblage, by addition or subtraction, of several symmetri- cal functions of the form > (a"b’c!...), it follows that the value of every rational symmetrical function whatever of the roots of an equation (without the roots being known) can be expressed by the coefficients of the equation. 159. The above expressions for the elementary sym- metrical functions will require to be modified, when any of the indices become equal. Thus, if m=p in the for- mula S (a"B?) = SyS'p = Snips since a”b’ = b"a?, the terms in = (a"b") will become equal two and two, and > (a”b?) will be reduced to 2 (a"b") ; ove = (a™b”) e 4 (9? - Som): Similarly, if m=p=qin S (a"b’c"), the six combina- tions formed by interchanging a, 6, c, in a’b"c" are reduced to one, and = (a”b’c") is reduced to 6E (a"b"c") ; e. S(a"b"e") = 1S, — ES om Sig +E S3m 3 and in general, if ¢'of the exponents become equal, the gene- ral formula must be divided by 1.2.3...¢. If =(a"b’c’...) have 7 roots in each term, it will consist, as we have seen, of n(n —1)...(~—7 +1) terms; and if ¢ of the indices n(n—1)...(n—7+1) bu 2r3s set become equal, it will consist of terms. Ex. 1. Let the roots of a? + pa’*+ qu+r=0 be a,b,c, to find the value of = (a6). 182 > (a*b) = 8, S,— S; = (— p) (p’— 2q) — (- 8r + 8pq — p*) (Art. 154) = S8r—pq. Ex. 2. Let the roots of «?—1=0 be a, b, ©, &e., to find the value of = (a”6"). 5 (al?) = SpSp— Snir Hence the value is m?— _, when m and p are both mul- tiples of 2; and —, when m + p only is a multiple of n; and zero, in all other cases. (Ex. 2, Art. 154). Transformation of Equations by Symmetrical Functions. The theory of symmetrical functions will enable us to transform an equation, whose roots are unknown, into another whose roots are all the combinations, formed after an assigned law, of the roots of the proposed, taken two, three, &c. at a time. We shall exemplify the method in the following transformation, as being the most convenient practical one of solving a problem of considerable interest. 160. To transform an equation into one whose roots are the squares of the differences of its roots. Let a, b, c,... 1 be the m roots of the proposed, then the roots of the transformed equation will be (a — b)*, (a—c)*, (b-c)’, &e., in number 4 (m— 1), since they include all the combina- tions of the » quantities a, 6, c,...1 taken two together ; hence the degree of the transformed equation will be n(n — 1) =m suppose. Let the transformed equation be y+ ny” | + gy”? + gay” 3 + 22. + Im= 9, 183 and let s,, s.,...8; denote the sums of the first, second, &c., qh powers of its roots; then (Art. 155) all the coefficients may be expressed by these sums, thus N= — 81> Q=— 5 (8+ 95)> = — $+ (G28, + M1824 83), Ke. 5 therefore it only remains to calculate s,, s., &c. Now if SS, Sz, &c., denote, as usual, the sums of the powers of the roots of the proposed equations, and k be any positive in- teger, we have (a —a)* + (wa —b)¥ +... + (we — Dt = next —k Sa k(k-1 A abeamidal 1.2 Weis —seaee tT (- eye Therefore, changing w successively into a, b, c.../, and taking the sum of the resulting equations, we have (a—b)* + (a—c)* +... + (a—-1)* + (b-—a)* + (b-c)* 4... + (b—1)* +... k-1 = nS; — HD 1),-) + Se aac ae Sei ae (- 1)‘nS,,.. Now if & be an odd number, each member of this equa- tion is separately zero; but if & be an even number and = 27, then the value of the first member is 2s;; and in the second member, the terms are equal, taken from the beginning and end ; 2i (2i —1) re 8; = nS.; ae 245, So5-1 + yg Sa Sai-2 poate G16 {24 Qe 1)... G+ 1) Te 2 ks Bae ( — 1) Si. Hence to form the equation whose roots are the squares of the differences of the roots of v*+ p,v""'+ p.v"-* +... + p, = 0, or, as it is called, the equation of differences, we must first calculate §,, S,, S;, &c., in terms of p,, p., &c. ; and next s,, S:, 83, &c., by the formula just investigated ; and lastly 1, 92, q3, &c., the coefficients of the required equation, by the method of (Art. 155). 184 161. We have seen one use of the equation of differ- ences (Art. 40), viz. to determine a limit less than the least difference of the roots of a proposed equation; another is to determine, within certain limits, the number of impossible roots which the proposed equation contains. If the transformed equation be complete and have no continuations of sign, it cannot have a negative root ; and therefore the primitive equation has no impossible roots, because a pair must give rise to a real negative root in the equation of differences; but if the transformed equation have continuations of sign, then it has either impossible or nega- tive roots; and as these can only arise from impossible roots in the proposed equation, it follows that this latter- has im- possible roots. Also if the proposed equation have p pos- sible roots, and if the difference of no two of its imaginary roots be a real quantity, the transformed equation will have ae tf, : ae positive roots, and the rest will be either negative or imaginary; hence if the last term of the transformed equa- tion be positive, is even; and therefore p, which Pp (p- 1) 4 must be of the same parity as m, will be of the form 4m or 4m +1, according as m is even or odd. Similarly, if the last term of the transformed equation be negative,- it may be shewn that the number of real roots in the proposed equation will be of the form 4m +2 or 4m + 3, according as 7 is even or odd. | Ex. 1. w&—2av—5=0. The equation of differences is 7° —12y" + 36y + 643 =0, which has not all its roots po- sitive ; therefore the proposed has impossible roots. Ex. 2. To transform w+ ra+s=0 into one whose roots shall be the squares of the differences of its roots. Here, by Ex. (Art. 154), and by the formule of Arts. 160 and 155, we have 185 S, = 0, S, = 0, S; = — 37, S, = — 4s, S; = 0, S, = 3r*, S,=7rs, S, = 48°, S, = — 3r°, Si) = — 10r’s, Si,=-l1rs’, Sy, = — 48° + 3rt; §,=0, &=-168s, s,=—78r, 8,= 5768", 8, = — 407’s, S, = — 7936s + 2190r'*; 1=9, = 8s, qs = 267°, qs = — 1128”, qs = 2167's, Qe = 2568° — 274". Hence the transformed equation is y° + 8sy* + 267? y® — 1128? y® + 2167? sy + 25683 — 277‘ = 0. : re BN ea LEN Hence if the last term be positive, or (=) > (=) , the number of real roots in the proposed will be of the form of 4m; but there cannot be more than two, therefore there are : Bef h\G none. If the last term be negative, or (=) < (=) the num- ber of real roots of the proposed will be of the form 4m + 2, and therefore there will be two. These results agree with those found at p. 62. Ozss. The absolute term of the equation of differences, which put equal to zero expresses the condition for the pro- posed having equal roots, and in most cases is the only one wanted, is a symmetrical function of the roots of the first derived equation; and from that consideration may be ad- vantageously calculated. For since f’ (a) = (a — 5) (a—-c) ... (a—-1), daa f (a) f(b) = SF ().- But if a, B, y,.-. A, be the roots of f"(x) = 0, then since f (a) = (— 1)" (a—a) (b- a)... (/—a), a F(a). f(B)...- fW=(— I) (@- a)(a— B) ... (@- 2) x (b-a) (6-8) ... =A) 2 (epee = f@. fe) f'O; 24 186 ren I (a). ST (PR) * £09). Ex. pee we Pack at kh. J (2) Soe + qs ab ma, ee q, = 3° (a’ +qatr) (B' + qB +7) ; = 8° {(a 8) + qaB (a° +B’) + qaB + gx (a+ B) +7 (a° + B) +7} 3 2 2 -#(()-8- pte AS jp oes g 8 = 49° + 277°, as at p. 45. 162. Besides’ the’ method of Art. 160, we may also transform an equation by means of symmetrical functions, as follows. | Suppose | that each root of the transformed equation is to be a rational function, @ (a, 6, c, &c.), of any number of the roots of the proposed equation; then having formed all the combinations p (a, b, ¢, &e.), p (a, c, d, &c.), &e., the transformed equation, resolved into its factors, will be Sy — p (a 6, Cy Be.) )t fy- (a, 6, d, &e.)} .. and as this product is not altered by interchanging a, b, c, &c., among themselves, (for the only effect of that is to place its imate in a differént order) we are certain that, after multiplication, the’ coefficients of the different powers of y will be symmetrical functions! of a, 6, c, &c., and’ ‘may therefore be expressed by the) coefficients: of the. cB poaa equation. Ex. 1. To transform 2* + pa? + ame +72 +8 =0, roots a, b, c, d, into one whose roots shall be ab+cd, ac+ bd, ad+ be; the transformed equation is 187 Sy —(ab+ cd) . Sy —(ac+4 bd) ‘ Sy — (ad + be)} =0, or y’ — gy’ + (pr -48)y— (7 — 498 + p’s) = 0, (1). Ex. 2. Hence, also, we can transform the proposed equation into one whose roots shall be (a+b-c-d)’, (a+c-—b-d)’, (a+d-b-c)’. For let x = (a+ b—c—d)’ , =(a+b+c+d)-4(ab+ac+ad+be + bd+cd) +4(ab+ cd) =p’ —4q 4+ 4y; Bp + 4g. BG bes a, =e) and substituting in (1), the transformed equation in x is 2° — (3p — 8q) 8° + (3p — 16p’q + 16q° + 16pr — 64s) — (p* - 4pq + 87)? =0. Ozs. Either of these transformed equations may be em- ployed i in the solution of the proposed biquadratic. Thus, in the first case, let a be a value of y, then ab+cd=a; abcd=s; therefore‘ab and ¢d are ‘known; also ab (c + d) + cd(a+b)=—r1r, (a+,6)+(¢+d)=—p; therefore a +b, and ¢ + dare known ;, hence all the roots are obtained from one root of the reducing. cubic. In the second case, if we know the three values of zx, by means of these, and the equa- tion a+b+ce4d=~— p, we can find the. roots of the biquad- ratic merely by addition and subtraction. Quadratic Factors of Equations. _ 163. Every equation of an even degree has at least one real quadratic factor. Let the proposed equation be w+ pv" 1+ pv"? + ...4+ p,=0, having roots a, 6, c, &e. ; and let m = 2p, Being an odd number. Let-it be trans- 188 formed (Art. 162) into an equation whose roots are the combinations of every two of its roots, of the form y=a+b + mab, m being any number ; also let the transformed equa- tion be ¢,,(y) = 0; then its coefficients will be symmetrical functions of a, 6, c, &c., and therefore rational and known functions of p,, p., &c; and its degree will be en (Ge) which is odd; therefore @,,(y) =0 will have at least one real root, whatever be the value of m. Hence, making m =1, 2, 3,... {u(2u—1) +1}, successively, each of the equations d,(y) = 0, d2(y) = 0, &c., will have at least one real root; that is, we shall have »(2u — 1) +1 real values for combinations of two roots of the proposed equation, of the form a+b+mab; but there are only u(24—1) such com- binations which are differently composed of the roots a, 6, c, &c.; therefore two of these combinations, for which we have obtained real values, must involve the same pair of the quantities a, b,c, &c.; let this pair of roots be a, 6, and a, a, the real roots of the corresponding equations Pn(y) = 0, Dm (y) = 0, so that a+b+mab=a, a+b+m'ab=a’; therefore a+6 and ab are real, and the proposed equa- tion has at least one real quadratic factor, and two roots, either real, or of the form a + BY -1. Hence every equa- tion whose degree is only once divisible by 2, has at least one real quadratic factor. We shall now prove that if it be true that every equation has at least one real quadratic factor when its degree is r times divisible by 2, or when » = 2’u where p is odd, the same is true when the degree of the equation is r + 1 times divisible by 2. For let m = 2"+'»; then the degree of the transformed equation will be 2"4(2"*' -— 1), which is only r times divisible by 2; therefore, by supposition, the trans- formed equation, ¢,,(y) = 0, will have two roots, either real, or imaginary. If they are real, then exactly in the same way as for the preceding case of the index being only once 189 divisible by 2, it may be shewn that the proposed equation has at least one real quadratic factor. If they are ima- ginary, we shall have y=a+3»/—1, each of which ex- presses the value of some one of the combinations a+ b+mab, a+c+mac, &c. Suppose therefore that we have a +b +mab=a+t BV -1 ; then, as shewn above, we can give m such a value m’, that @,,(y) = 0 shall have a root corre- sponding to the combination of the same letters, so that a+b + mab =a’ + B'4/—1; from which equations we can obtain values of ab and a + b under the forms atb=y+ PY Ay a ab=y + SRA : . @—(y+ S4/-1) a+ yy + 54/1 is a factor of f(x); but if any real expression have a factor of the form M + N/-1, it must also have one of the form M — N\/-1; we (y - Sof 1) & +r - 5 4/-1 is a factor of, F'(#)'; if therefore these two expressions have no simple factor in common, their product will be a biquadratic factor of f(a), (a? — yu ty’)? + (da - 0)’, which can always be resolved into two real quadratic fac- tors (Art. 93). If they have a factor in common, since they may be written a —yatry —/-1 (da—0), a —yauty +/—-1 (o7-0), it can only be of the form #—e; and the factors themselves become (w —«K+A/—1) (a —e), (w@-K«-dV/=1) (a - €); and therefore the proposed equation admits the real quad- ratic factor (w —_ Kk)? + pp 190 Hence an equation whose degree = 2"t', will have a real quadratic factor, provided an equation whose degree = 2'u, has one; but we have proved. this to be the case when r = 1, therefore it is universally true that every equa- tion of an even degree has at least one real quadratic factor. If now this factor be expelled, the depressed equation will have its coefficients real and its degree even, and will. there- fore, as, before, have one real quadratic factor, Hence the first member of every equation of an even degree may be re-, solved into real quadratic factors. 164. Hence if we divide the first member of any equa- tion w+ pa") + pov" +... +p, =0 by a + av +6, admitting no terms into the quotient that have w in the denominator, we shall at last obtain a re- mainder of the form 4a + B, A and B being rational func- tions of a and b; and in order that «?+a«+6 may be a quadratic factor 3 the proposed equation, it is necessary and sufficient that this remainder should equal zero for all values of #, which requires that we separately have 4 = 0, B=0. The different pairs of values real or imaginary of a and b which satisfy these equations, will give all the quad- ratic factors of the proposed ; and as the number of these factors is 4m (a —1) (Art. 17), the final equation for deter- mining one of the quantities a, b, obtained by eliminating the other between the two preceding equations, will*be of the degree 4 (nm — 1), which exceeds n, if m > 3; therefore the determination of the quadratic factors of an equation will generally present greater difficulties than the solution of the equation. As the proposed equation has necessarily $7 or 4 (m — 1) real quadratic factors, according as m is even or odd, there will always exist the same number of pairs of real values of a and 6, satisfying the equations d=0, B= 0; and if any of these pairs of real values be commensurable, they may be 19] easily found; and the commensurable quadratic factors be- ing known, the equation may be depressed. Ex. 1. To resolve #*— 62° + nx —3=0 into its fac- tors. Dividing by #°+ ax + b, we find a remainder ‘(n + 2ab + 6a +a) & = (vb = b’— 6b + 3); therefore, to determine a and b, we have n+2ab+6a—-—a@=0, a®b — 6? —-66+3=0. Solving the former with respect to b, and substituting in the latter, we find (a? -— 4)? =n” —64, or a= eke a a/ n? — 64; from whence 8, and the other quadratic factor v’—-axv+a—b—-6, Z may be determined. Ex. 2. The resolution of a+ pa?+ qi? + r«e +s into its two quadratic factors w+ mew+n, x + m'e +n, may be effected by the following formule : m=4(p+/2); m=4(p—V/z), r—qm+pm—-m , r—gqm +pm”?—m® n = eehroi eM = 5) : p—2m p— 2m’ where # is a root of the equation (which has necessarily a real root) cs ee — 89) x° + (8p" — 16p?q + oes se Otros) ey ar Soma 2g titolo SECTION IX. ON ELIMINATION. 165. AN equation between two unknown quantities a and y, supposed to contain no term which is fractional or irrational, is said to be of the degree which is expressed by the sum of the indices of # and y in that term where the sum is the greatest. The general equation of the 2" degree between # and y ought to contain all the terms in which the sum of the indices does not exceed 7; therefore, when com- plete and arranged according to descending powers of a, it will be a,x" + (b, + by) a"! + (c, + Cy + Coy’) 2°? +... +(l,+hy+hy +... + 1,y") =0. When an equation is incomplete, that is, when it does not contain all the terms which belong to its degree, we must suppose, in the general equation, the coefficients of the deficient terms to be equal to zero. Although we are always at liberty to divide an equation by any one of its coefficients, we cannot in the above general equation suppose a, = 1, aye then it would not comprehend _ those “equations which want the term involving 2”. . After having divided the equation by any onesof its coefficients, there will remain as many indeterminate constants as there are terms, wanting one; the number of these constants will therefore be 24+ 34+44...4+ (+1) =4n(n +3), 193 which expresses how many conditions an equation of the n™# degree may be made to satisfy, by a suitable determination of its coefficients. 166. To eliminate between two equations of any de- gree involving two unknown quantities, is to obtain an equation containing only one of the unknown quantities, and which gives all the values of this unknown quantity, which, together with the corresponding values of the other unknown, can satisfy the proposed equations. This equa- tion, involving only one unknown quantity, is called the final equation, and its roots are called suitable values. In what follows, we shall suppose the polynomials which form the first_ members of the equations to be freed_from any_common divisor which they may_admit; for if they had a common divisor containing both variables, it might be reduced to zero, and therefore the proposed equations might be satisfied, by an. infinite number of systems of values of « and y; or_if they had a common divisor con- taining only one of the variables, there would be_a limited _ number of values of | ePRA ES and an unlimited number of values of the other, by which the proposed equations might be satisfied; so that in both cases there could be no final equation. ee { Method of Elimination by the greatest Common Measure. \ 167. To determine all the systems of values which will satisfy two equations between two unknown quantities, each being of any degree. Let F (#, y) =0, f(a, y) =0, be two equations, re- spectively of the m' and n™ degrees, admitting only a limited number of pairs of values of # and y, and their first members consequently having no common divisor, involving either both or only one of the variables. Then in order that any value y= 3, may be a suitable value, it is neces- sary that there should exist one or more values of 2 which, substituted in the polynomials F (a, 3), f(a, 8), will re- 25 194 duce them to zero; these polynomials must therefore have a common measure a function of #, which equated to zero, will give one or more values of «x, that, jointly with y = A, satisfy the proposed equations. If therefore we perform the operation for finding the greatest common measure of F(a, y), f («, y), (which we suppose arranged according to descending powers of x) introducing or suppressing factors, functions of y, so that no quotient shall have any term with y in its denominator, we shall at last arrive at a remainder independent of 2, which put equal to zero will give the final equation Wy (y) =0. For if 8 be a root of this equation, and (w, y) be the last divisor, since y = 6 makes the re- mainder vanish, @ (#, 3) is a common measure of the poly- nomials F' (x, 8), f(#, 8); therefore (#, B) = 0 will give values of # which, jointly with y = {, satisfy the proposed equations. Oxss. In those cases where the process for the common measure requires neither the introduction nor suppression of factors, we are certain that the last remainder put equal to zero, or vy (y) = 0, will furnish all the suitable values of y, and no more; but we cannot affirm this in other cases, unless we are certain that the last remainder is unaffected by the factors that have been rejected or introduced; and it frequently happens that in the final equation, values of y are found which are foreign to the problem, and others are defi- cient which belong to it. On this account the method of elimination by the greatest common measure is imperfect, but it is still the most convenient practical one for numerical equations. 168. In particular cases we are able to find all the systems of values which satisfy two proposed equations, by easier methods than the one just described. Thus, whenever we are able to solve one of the equations with respect to one of the unknown quantities, # for in- stance, we have only to substitute the resulting expressions for « in the other equation, and we shall obtain equations 195 containing y only; and if we substitute the values of y given by these equations in the corresponding expressions for a, we shall obtain all the pairs of values required. Also, if the two equations are of the same degree with respect to the variable which we wish to eliminate, we may, by intro- ducing factors, if necessary, and subtracting, depress one of them to an inferior degree. And if the first members of the equations are, or can be, resolved into their factors, then the solution of them is reduced to the simpler case of finding all values of x and y which reduce at the same time a factor of each to zero. 169. In all cases of elimination between the equations F (x,y) =0, f(«,y) = 0, besides expelling any common factor which the polynomials admit, the application of the general method may be simpli- fied by previously ascertaining whether either has factors containing only one of the variables. This may be done by arranging each, first according to powers of y, and finding the greatest common measure of the coefficients of the several powers of y in it; and secondly by arranging each according to powers of x, and finding the greatest common measure of the coefficients of the several powers of w Let X, Y, be the factors thus discovered of F(a, y), and M its remaining factor; and let X’, Y’, N, be similar quantities for (wv, y); then the proposed system may be replaced by XYM=0, X’Y’N =0, which will be satisfied by simultaneously putting any factor of each equal to zero, provided we do not take X and X’ together, or Y and Y”’ together; for X and X” cannot be reduced to zero by the same value of w, unless they have a common factor; and that they cannot have, since by the supposition F(a, y), f(#,y), have no common factor. Hence, with the exception of M=0, N=0, each of the systems into which the system F(a, y) = 0, f(a, y) = 0, is 196 resolved, has at least one of its equations involving only one unknown quantity; and therefore its solution is attended with no other difficulties than what belong to equations of that description. But the system M=0, N=0, whose first members contain both variables, but have no factors depending on x only, or y only, will require the process of elimination by the greatest common measure to be applied to them, in order to reduce their solution to that of equations containing only one unknown quantity, as we shall now more minutely explain. 170. To examine the consequences of introducing or suppressing factors in the process of elimination by the greatest common measure, and to investigate ve means of obtaining an exact final equation. Let M@=0, N=0, be two equations between # and y, of the m™, and n™ degree, respectively ; the polynomials M and N being arranged according to descending powers of a, and not admitting a common divisor, and neither of them having a factor composed of w only, or of y only ; and let m be greater than. Divide M by N, and let Q be the quo- tient (containing no term with y in its denominator) and R the remainder, so that M=QN+R; then all values of # and y which satisfy M=0, N =0, also satisfy N=0, R=0; but if the division cannot be per- formed without putting powers of y in the denominator of the quotient, 2. e. if Q be of the form = where X contains y, then we cannot affirm that all values which satisfy the proposed system, also satisfy N=0, R=0; for the equation H M=—N+K shews that values which makes = 0, N=0, may make 0 K =0, so that se -N may assume the form 5” the real value 197 of which, and therefore of R, may be finite or infinite, in- stead of zero; and, conversely, values which make N = 0, R = 0, may still not make the second member equal to zero, and therefore not make M equal to zero. To avoid frac- tional quotients, we must use the same means as in finding the greatest common measure; that is, we must multiply M by the coefficient of the first term of N, or by certain factors of that coefficient ; then no common factor will have been introduced into both polynomials; and if P, a func- tion of y, represent this multiplier, Q the quotient, and R the remainder, we shall have PM=QN+R, which shews that the solutions of N=0, R-=0, are the same as those of PM=0, N=0. But these latter equa- tions resolve themselves into the two systems M =0, N=0; P=0, N=0. Therefore, besides furnishing the solutions of the proposed equations, the system N=0, R=0, will furnish those of P=0, N=0. Hence we must solve the two latter equations, one of which P=0 contains only y, and substitute all the resulting pairs of values of # and y in M@=0; then those pairs of values which do not satisfy it must be rejected, and we shall thus obtain those solutions of N= 0, R =0, which belong to the proposed system M=0, N=0. The remaining solutions of the proposed system are con- tained in the equations N = 0, R = 0, # being a polynomial of smaller dimensions than N. Now if # have factors con- taining only one of the variables, (which may be discovered by seeking the greatest common measure of its coefficients, when arranged according to the powers of each variable in succession,) so that R= XYR’, then the system MN = 0, R =0, may be resolved into the three systems, N=0, X=0; N=0, Y=0; N=0, R =0; the two former of which present no difficulty, because one equation in each contains only one variable; and the third N=0, R' =0, is exactly of the same nature as the one we = Oes started with; for N, R’, have no common factor, otherwise M and N would have the same common factor, which is contrary to the supposition, and neither N nor R’ admits a factor containing only one of the variables. This system then, by exactly the same process, may be replaced by an- other similar system R’ = 0, R” =0, the latter being of a lower degree in # than the former; and the system R’ = 0, R” = 0, by another, of which the second equation will be of a degree in w, inferior to that of R’=0. In continuing these uniform operations, we shall at last arrive at a re- mainder not containing vw; suppose this to be R”, then the solution of the proposed system is reduced to that of R’ = 0, R” = 0, and is thus made to depend upon the solution of an equation containing only one unknown quantity. 171. In ascending from R’=0, R” =0, to the pre- ceding system N =0, R’=0, it may happen that some so- lutions will have to be added, and others suppressed; and, similarly, in ascending from N = 0, R’ =0, to M=0,N=0; and so on, if there was a greater number of successive divi- sions. ‘This method then, as we perceive, will not always lead to a single equation in y, but to several, some of which may give unsuitable values for that variable. When we have recognized all those which really enter into the solu- tions common to the two proposed equations, we may, if necessary, combine them into one final equation. It may be observed that, since M and MN are prepared so as to admit no common measure, we can never find zero, but we may find a number, for the last remainder R”’; in that case, the final equation, R” = 0, is absurd; and the proposed equations (unless solutions have been suppressed in the process) are incompatible with one another; i. e. in- capable of being satisfied by finite values of # andy. It is easy to form equations of this sort; such for instance are P=0, PQ+k=0; P and Q being integral functions of x and y, and & a number; for the condition expressed by the former, reduces the latter to & = 0, which is absurd, since k is a number. 1y9 Also from the final equation R” = 0, we can never deduce a value 3, of y, which will reduce the preceding divisor R’ to zero independently of the value of x; for in that case, R’ would have a factor, y — 3, which is impossible, because in the process each remainder, before being employed as a divi- sor, is cleared of factors containing x only, or y only; but y = [3 may destroy some of the terms in R’, and so cause R’ = 0 to furnish a greater or smaller number of correspond- ing values, of «, or none at all if y = 8 reduce R’ to a num- ber. Of the above peculiarities, and of the application of the general method, the following are instances : seni yx? —(y® — 8y-—1)e%+y=0, v—y+3=0. The first division gives the remainder #7 +y; and the division of x — y’ + 3 by # + y gives the remainder 3. The proposed equations are therefore incompatible. Ex. 2. (y— 1) a°+ (y?+y) & + (8y? + y— 2) & + 2y =0, (y—1)a°+ (y'+y)a + 3y?-1=0. The final equations are y—-1=0, y¥-1)e+2y=0; the former gives y= +1; but the value y = 1, furnishes no corresponding finite value of #, since it reduces the latter to 2=0. Ex. 3. w—3ya°+ (3y°-y+1l)a-y'+y’?—2y =0, v—2Qyv+y—y=0. ot -2ye+y'-y) a —3yx°+(3y—-y+la-y ty — 2y (ay a —2yu?+(y’?—-y)a —ye + 2y + 1Ia-y+y-2y —ya +2ya-yr+y @ —2y) a —2ya +y° —y(v uv —Qyx v= 200 therefore the final equations are v-2y=0, y—-y=0, which give y=0 ) y=1 e=0 v=2s and as no factor has been introduced or suppressed, these two solutions are those of the proposed system. Ex. 4. (y —2) #-24+ 5y-2=0, yu —5a+4y =0. Multiplying the dividend by y, yu — 5at ay) (y — 2) ya” —2ya + 5y’?—2y (y —2 (y — 2) ya” — (Sy — 10) w + 4y° — By (3y — 10) @ + y* + 6y. Next multiplying the dividend by (3y — 10)’, (Sy —10)v+y> + 6y) (3y — 10)’ya?— 5(3y—10)?2+4(3y—-10)y ((ey- 10) ya+&c. (3y—10)’ya* + (3y — 10) (y'+ 6y) yx — (8y—10)(y?+ 6y’+ 15y — 50) xv + 4(3y —10)*y — (3y— 10) (y°+ 6y?+ 15y — 50) a— (y+ 6y) (y2+6y’+ 15y—50) y+ 12y* + 87y — 200? + 100y. Therefore the final equations are (suppressing the factor y, since the solution y=0, #=0, does not satisfy the proposed system, and is due to the factor introduced in the operation) (3y —10)v7+y’+ 6y =0, y' + 12y° + 87y’— 200y + 100 = 0, which contain no false values; for the only false value which the final equation in y could contain, would be 2, which is 201 impossible, since all the coefficients of that equation are integers. One pair of values is y=1, w=1; the other solutions can be obtained only approximately. Ex. 5: wv (4y? + 3) — 8ay=0, 4y (3 -— 22°) —4y?+ 3=0. Here we can solve with respect to one of the variables, and we find for the final equations Ex. 6. y= (vr), v= (y). The final equation resulting from the elimination of either of the variables between simultaneous equations of this form, admits of a remarkable reduction. For, suppose a to be a root of «— d(x) = 0, and put e =y=a in the proposed system of equations; then they are evidently satisfied ; there- fore «=a satisfies the final equation v= @ Sp (w)}, or J («v) =0; therefore every factor of «—@(w) is a factor of f(a); . f(a) = {¢ (#) — wh f(a), which leads to the reduced final equation f;(w) = 0. Thus, let the system of equations be 16v(i-x) — 16y(1-y)*. G+ ay) 1G +) then | w(1+)‘-16a2(1 —- v)?= a4 (a - 24 + 5) (@’+ 6a — 3) is a factor of the final equation. 26 202 172. It was observed (Art. 23) that the problem of transforming an equation, in its widest sense, required the general methods of elimination. This is especially the case where each root of the new equation is to be composed of several roots of the primitive equation. Of this use of the methods of elimination we shall now give some instances. To transform an equation into one whose roots shall be the differences of every two roots of the proposed equation. Let f(#) =0 be an equation of m dimensions, having roots a, b, c,...1; to obtain another equation whose roots are the differences between all the roots of the proposed and a, we must make y=a2-—a orv=a+y, and the substitution of this value for w in f(#) =0, will give f(a + y) = 0, the required equation; or, developing (Art. 27), 2 S(a) +f’ (@)-y +f" (a =. PM i) Since, by the supposition, a is a root of the proposed, J (a)=0; therefore the preceding equation has y for a factor, or admits a root zero, corresponding to the difference a — a; suppressing this factor, we have "P y —- Sa +f" @) +--+ 9" =% 2); an equation having for its roots the difference between a and the m —1 other roots of the proposed equation. If in this equation we replace a by b,c, &c., successively, we shall form equations whose roots are the differences between 6 and the m — 1 other roots, between c and the n — 1 other roots, and so on. Hence it follows that the differences of every two of the roots of the proposed equation are the values of y furnished by the equation f(a) +f") A+ + y""= 0, when we substitute successively in it, for w, all the roots of the equation f(#) = 0; which amounts to solving the system 203 formed by the above equation, and f(x) =0. If therefore we eliminate & between the equations CaS (2) ta) ee iy Os the resulting equation in y will be the one required. The proposed equation being of the n"" degree, the transformed equation will be of the (2 —1)™ degree; for the number of its roots is equal to the number of permutations which can be found with the quantities a, b, c,... 2, taken two and two together; also the transformed equation will con- tain only even powers of y, for if it have a root a — bit will also have the root b— a; so that its roots are equal two and two, and of contrary signs. Hence if m(m —1) =2m, and y= %, the transformed equation will be of the form Pets Gok yeh) eb a = 05 and the values of x are the squares of the differences of every two roots of the proposed equation. Ex. o+gqr+r=0. ° ” aA y 2 : In this case f'(2#) +f (x). er + y° = 0 gives 8a°+q+3any+y'= 0. 8a7+ 8Hy + y'+9q) 3a°4+ 38qa 4+ 3r(a—y 3a? + qut... 2(y+get+yr+qyt sr 2 +qgrt+y+qyt sr] 6 (y° +9)’ a" + 6(y? +9) yr + 2(y'+9)° (sa (y?+qQ) +... 6(y°+q) a? +3(y?+q) yet... A(y’+ q)°-3(y+qy+3r) (yi +4y— 37) 204 therefore, equating the last remainder to zero (since the factor y’+ q put equal to zero, reduces the last divisor to 8r, which is different from zero), we have the equation of differences | 4 y+ 6gy'+ ogy’ + 49° + 277" = 0; and putting y’= x, the equation of the squares of the dif- ferences is (as at p. 45) + 6q2°4+ 99's +4¢° + 277’ = 0. By similar reasoning it may be shewn, that to trans- form f(x) = 0, into one whose roots shall be the sum, ‘product, or ratio of every two of its roots, we must eliminate x between f(x) = 0, and h SJ (a) +f (#) Ee BN te ait h'-* = 0, where h=y— 2a, ee x, and wy — a, respectively ; taking x in the two former cases the square root of the result. 173. To eliminate one of the unknown quantities be- tween two equations containing two unknown quantities, by means of symmetrical functions. Let xv" + p,a"-'+ p.a"-*+ ... +p,=0 (1) oT Pg eet gga ot 2 es (2) be two equations respectively of m and m dimensions in «& and y; so that p,, p., ... p, are functions of y involving respectively no power of y above the first, second, &c., nm; and qs Yo» ++» Ym functions of y involving no power of y above the first, second, &c., m. If we can solve the first with respect to w, and deduce m values, a, b, c, &c., functions of y, then upon substituting them in the second, we shall have, for determining y, m equations not contain- ID 2, Viz. 205 teres: ge eas +4 agar” ne wc 4 Gn = 0 Bb” + gy") + quo"? 4... +. = 0 e+ 9,07 * + gc" 2 +... +, = y) 43 if Ay gy But in general the solution of (1) 1 s object must be to obtain a final equation eee, aD differently all the suitable values of y, and this we shall do by multiplying together the above m equations; for the result will be satisfied by every value of y derived from any one of them, and by no other quantity ; and to every one of these values of y there will correspond a value of & such that the pair will jointly satisfy (1) and (2). For suppose a value of y deduced from the first of equations (3) to be 8, and let the equation #-—a=0 give, by making y =, =a; then it is manifest that x =a, y = (3, will jointly satisfy the proposed equations. But in the result of this multiplication, the factors only change places when we interchange in any manner the quantities a, b, c, &c.; therefore the product will only involve rational and integral symmetrical functions of these quantities, which may be expressed by means of the coefficients of equation (1); and we shall so obtain the final equation in y. The cal- culations required by this method are in general tedious; but it has the recommendation of giving the final equation with all the roots it ought to contain, and no others. 174. When _we eliminate one of the unknown quan- tities between two equations containing two unknown quan- tities, the degree of the final equation cannot exceed the product of the degrees of the two equations between which the elimination is performed, To prove this, we must examine to what degree y may rise in the symmetrical functions composing the product of equations (3). Each term of this product will be itself the product of terms, one taken out of each of the equa- tions (3), and will therefore be of the form h ha i ~ 4 h pk / Yn -n@ - Ym —K? X YIm—1l ia Be au Yn h vas Tn—-k os Yn- t+ xa b C wee 206 But the product of the m equations, being symmetrical, must-contain all the terms of the same form which we can make with the above quantities; consequently it will con- tain all the terms represented by Ym—h Ymn—-k Umi + = (a" b¥c! at, ) (4), and we must now ascertain the dimensions of this expres- sion. Now the degree of y iN Yn_ys Ym-y &C- cannot exceed m—h, m—k, &c., respectively ; therefore in qn_;, Ym_—z Ym-1 .-. it will at most be equal to mn -h-—k-—J1-&c. Also if we refer to the formulz which give the values of the double, triple, &c. functions in terms of the sums of the powers of the roots, we see that in > (a"b*c’ ... ) the term of highest dimension in y will be found in §,8S,S,..., but the equations which give S$), S;, &c., in terms of p,, po, &c., (since these quantities do not involve powers of y exceeding the first, second, third, &c. respectively) shew that the degree of y in any sum S), cannot exceed h 3 there- fore the degree of y in = (a"b‘c'...) cannot surpass h+k+1+ &c.; consequently, in the expression (4), the degree of y will at the most be equal to mn. 'The same thing may be similarly proved of all the symmetrical functions whose sum makes up the product of the 2 equa- tions. Therefore, lastly, the degree of the final equation cannot exceed the product of the degrees of the two equa- tions from which it results by the elimination of one of the unknown quantities. Although the degree of the final equation cannot exceed mn, in particular cases it may be less than mn. If we extend the process to any number of equations, we shall have the general theorem discovered by Bezouwt, viz. that if between equations equal in number to the unknown quan- _tities, we eliminate all except_one, the degree of _the final equation will be at_most equal to the product of the degrees. of the several equations. 207 Ex. To eliminate # between the equations yu? —5x+4y = 0, (y —2)a°-24+5y-2=0. Let a and b denote the values of wx given by the first equation; then substituting them in the second equation, we have : (y — 2) a@ -—2a+ 5y-2=0, (y —2) b> -2b4 5y-2=0; the product of these equations, which will be the required final equation in y, is (y — 2)° 2 (a°b*) — 2 (y — 2) E (a°b) + (y — 2) (5y - 2) S, — 2(5y — 2) 8, +42 (ab) + (5y- 2) =0. 5 But p, = 3 Pp. = 4, ps =0; 5 25 — 8y” 125 — 67’ ~S=-, S,=———, S35 = - am Y y Y3 20 Z(ab)=4, > (a*b) = Aye = (a°b*) = 16. Hence, substituting and reducing, we find for the final equation (as at p. 200) y' + 12y° + 87y" — 200y + 100 = 0. 175. As all the preceding methods suppose the equa- tions to which they are applied to be rational, it is of im- portance to be able to reduce an equation involving radicals, to a rational form. ‘The extermination of radicals, con- sidered generally, is only a case of elimination, as will appear from the following example. 208 To reduce a —«/x —1++/# + 1=0 toa rational form. Let yor—l, 8 =v4+1; . @-Yy+rx=0;5 this gives y =a +2, and therefore y® = # — 1 gives 2+2sn+a2°-#7+1=0, and it remains to eliminate x between this and x* — #7 —1=0. Using the process of the greatest common measure, we find for the exact final equation, vw —32°+8a'+a°4+ 72? —-7H+2=0; a result that may also be obtained directly from the proposed equation, by successive involutions. —" SECTION X. ON THE GENERAL SOLUTION OF EQUATIONS. 176. A REMARKABLE application of the theory of sym- metrical functions is that made by Lagrange to the general solution of equations; by that means he solves the general equations of the first four degrees, by a uniform process, and one which includes all others that have been proposed for that purpose, the common relation of which to one another is thus made apparent. It consists in employing an auxiliary equation, called a reducing equation, whose root is of the form Rta Ma aa tS Dn denoting by 2, #2, ... #, the 2 roots of the proposed equa- tion, and by a one of the m™ roots of unity; and the prin- ciple on which it is based is as follows. Let y be the un- known quantity in the reducing equation, and let y => a, P, + AeWVo + eee + AnDLns @ 5 G2, --. a, Aenoting certain constant quantities; then if m —1 values of y, and suitable values of the constants a, a, --- a, can be found, so that we may have m — 1 simple equations; these, together with the equation —p, =e+%t+... + Pay will enable us to determine the 7 roots. 27 210 Now, supposing the constants in the value of y to pre- serve an invariable order a), a, &c., since the number of ways in which the z roots may be combined with them to form the expression a,#, + a,v, + &c. is the same as the number of permutations of 2 things taken all together ; therefore the expression for y will have m (m — 1) ... 3.2.1 values, and the eqpation for determining y will rise to the same number of dimensions, or will be of a degree higher than that of the proposed equation; hence the method will be of no use, un- less such values can be assumed for the constants a,, ao, ... @y as shall make the solution of the equation in y depend upon that of an equation at most of m—1 dimensions. Now this may be done (at least when m does not exceed 4) by taking the 2™ roots of unity a°, a, a”, a®, ... a®' for ayy az) .2. Gps so that y=at,+ ate +... ta” 'a,+ aH, + ... +a" 1d, For, in the first place, with this assumption, the re- ducing equation will contain only powers of y which are multiples of 7; for, since a” = 1, n—?r n—T nm—ri1 n—-r—-l a Y=a ata? ie B+ eos ied ai AX, e+ cee +a 1 —--? 0 n= or a’ which is the same result as if we had interchanged w, and &, 419 Ly and a,45, &c., so that if y be a root of the reducing equation, a”~"y is also a root; therefore the reducing equa- tion, since it remains unaltered when a” "y is written for y, contains only powers of y which are multiples of 2; if therefore we make y® = zs, we shall have a reducing equa- tion in x of only 1.2.3... (m—1) dimensions, whose roots will be the different values of x which result from the per- mutations of the m—1 roots #, &3, .-. @, among them- selves. We shall now have, expanding and reducing, Bap 2 n—I in which w,, 2%), U2, --- U,_, are determinate functions of the roots, which will be invariable for the simultaneous changes - of 2, into #,,,, % into w,,., &., since x = (a’y)"; and Oe 21) when their values are known in terms of the coefficients of the proposed equation, we shall immediately know the values of the roots. For let x,, 2), %, ... %,-1 be the different values of x, when 1, a, 2, y, ... A, the roots of y”—1=0, are substituted for a; then since y = a/ %, we have Oy + Oy +... +, = S38, nest | % @eeeveereevr eos eeeavees eee 880 ee ® + AM +... + AL a, = by Aue therefore, adding, and taking account of the properties of the sums of the powers of 1, a, B, yy, &c. (Art. 154), we get may = /% + 9/ twee tA Bae Again, multiplying the above system of equations re- spectively by 1, a” ', B"~’,... \”~’, we get Ny = a/ % + Queay st Bie Ae ON A/ ay and so on for the rest. Hence, since — p, = A/ Bs and (— pp)’ =% =u ++ --. + U,_), the problem is re- duced to finding the values of u,, w., ... u nm—\° 177. When 7 is a composite number, the above general method admits of simplifications. For let have a divisor m so that n = mp, and let a be a root of y*—1=0; then since a” = 1, a”*' =a, a”** =a’, &., a” =1, a” *! =a, &c. we have y n—-1 @, + at, +a°X3 + dant Hate ie A tak wbaikait: ame ok. where X, = @, + @mgy t+ Vomir + +++ + ©, m4) and consists of p roots ; ll .s=y"=uU,t+ Wat Wat... + hot Ata where w,, %, &c., are known functions of X,, X,, &c.; and when they are found in terms of the coefficients of the proposed equation, we shall be able to determine imme- 212 diately the values of X,, X,, &c., as before. ‘To deduce the values of the primitive roots 7, X, &... Pig We must regard separately those which compose each of the quantities X,, X,, &c., as the roots of an equation of p dimensions. Thus let the roots whose sum is .X,, be those of the equation a? — Xie 4+ La? — Ma? “oe ee where L, M, &c. are unknown; then the first member of this equation is a divisor of the first member of the pro- posed, since all its roots belong to the latter. Hence, effect- ing the division and equating to zero the coefficients of x~’, w?-*, &c. in the remainder, we shall have p equations in X,, L, M, &c., of which the p—1 first will give the values of L, M, &c., in terms of X, by linear equations. It will then remain to solve the equation so formed of p dimensions. Similarly, substituting the value of .X, in place of that of ,, we shall have an equation giving the next group of roots @, #,,,, &c.; and so on. Exel, a —pe+qe—1r=0. Let the roots be a, 6, e, and let y=a+ab+arc; . s=yp=e4+ +04 6abce+3(7b4+be+ea)a + 3(a’c + b’a + c’b) a’, | =U + Wa + Ua”. But w,, u,, are roots of the quadratic ur — (u, + Un) U + UU, = 0, and w+ U, = 32 (a’b) = 3pq —9r (Art. 159), U, U2 = 9 jabeS; + = (a°b*) + 3a°b?c*} = 9q°+ 9 (p* — 6pq) r + 817’. Hence w,, w,. are known, and .*. #%, = p> — (wu, + uz), is known. 213 Hence, denoting by z,, x, the values of x when a and a’ are respectively written for a, we have 3 G+b+c=p a+ab+ce=r/z, atab+ac=r1/x; from which we obtain the values of a, b, and c, viz. a=4(pt+V/x,+/%) b=1(pta /z, + ar/x) L(ptar/x, + a? /3;). Fix) 2: v— pa +qa*—ra+s=0. c Since 4 = 2.2, let a be a root of y’ — 1 = 0, sothat a= 1; then y= a, + av, +%,+aa%= X,+aX,, if X, = a + 23, Ag = Ha + %3 ._S= YP =U&y+ au; where uw = X} + X3, u,=2X,X,, and wm + um = 2% =p’. Hence wu, = 2 (a, + #3) (#2 + #), by interchanging the roots among themselves, will admit the two other values 2 (a, +4) (w3+ 4), and 2 (x, + a,) (a, + #3), and will therefore be a root of an equation of the form ui — Mu, + Nu,—- P=0; _the coefficients being symmetrical functions of x,, a, ¥35 X45 and consequently assignable in terms of p, q, 7, s. It is easily seen that if we make w, = 2q — 2u, we shall have an equation in w whose roots are Bg + Vo X45 By Lg + UgMyy VL, + yy 5 and the transformed equation is (Art. 162) u— qu? +(pr—4s)u— (p’-49) 8-7 =0. 214 Let «’ be a root of this equation, then w,=2q—-2u ; hence, making a= Hy B) =U, — Uy = p? — 2, = p’ — 4g + au 5 eX, +X, ips Xin, Knee =4 (p+ Vx), X,=4(p- VJ). Hence 2, #3, may be regarded as roots of a quadratic av — X,a + L=0; dividing the proposed by this, and put- ting the first term of the remainder equal to zero, we find X,-pXi+qme—t a 2X, —p therefore x,, v3, are known; and 2,, x,, will result from the same formule by interchanging XY, and X,, or by changing the sign of the radical J 3: a” —1 xs ina 0, » being a prime number. If 7 be one of the roots, and a be a primitive root of the prime number 7, (that is, a number whose several. powers from 1 to m —1, when divided by , leave different remain- ders) it is Reon (Art. 80) that all the roots of Ee equa- tion may be represented by n—2 2g 3 i Wek Ok iad wis ie est ig PA n—2Z Let y=7 + art + a’r™ + 0. + a"? 7" a being a root of the equation y”"'—1=0. Therefore, observing that a”~' = 1 and r” = 1, gay =u, + at, + au, +... +a" Ut,» (1), Uy, U,, &c. being rational and integral functions of r which do not change by the substitution of 7%, 7“, r®, &c., in the place of r; for these quantities, regarded as functions of 2,5 Vy, V3, &c., do not alter by the simultaneous changes of v, into #,, x, into a, &c., nor by the simultaneous changes of a, into #3, 2, into a,, &c., to which correspond the changes of r into 7%, into r@; &e. 215 Now every rational and integral function of 7, in which ¢" = 1, may be reduced to the form Meepy + Or + Dr +... +N. 6 the coefficients 4, B, C, ... N being given quantities inde- pendent of 7; or, since in this case the powers 7, 7°, 7°,....7"7' may be represented, although in a different order, by r, r2, yo", ... rt" "we may reduce every rational function of r to the form A+ Br+ Crt 4+ Dr? +...4 Nr", Therefore, if this function is such that it remains un- altered when 7 is changed into r%, it follows that the new form A+ Bre+Cr® 4+ Dr®4...4 Nr coincides with the preceding ; ~. B=C,C=D, D=kh, &i; N= B; and therefore the function is reduced to the form At B(rtrt +r? +... +r%”), or A — B, since the sum of the roots = — 1; hence each of the quan- tities w,, U,, U., &c., will be of the form 4 — B, and its value will be found by the actual development of x = y”~’; so that we have the case where the values of u,, u, u,, &e.; are known immediately, without depending upon the solu- tion of any equation. Hence if we denote by 1, a GB, y, &c., the — 1 roots of the equation #~' —1=0, and by 2, &,, 2, &c., the value of x answering to the substitution of these roots in the place of a in equation (1), we shall have, as in the former cases, 7 — 1 >) an expression for one of the roots of the equation #" — 1 = 0; and the other roots are r’, 7°, &c. 216 Thus the solution of «* — 1 = 0 is reduced to that of the inferior equation y"~' — 1 = 0, of which 1, a, B, y, &c. are the roots; also since 2 — 1 is a composite number, the de- termination of a, 3, y, &c. will not require the solution of — an equation of a higher degree than the greatest prime num- ber in 2 —1; that is, the solution of # —1=0 (m prime), may be made to depend upon the solution of equations whose degrees do not exceed the greatest prime number which is a divisor of — 1. Ex. 4. ”—-1=0. The least primitive root of 5 is 2; for the powers of 2 from 1 to 4, when divided by 5, leave remainders 2, 4, 3, 1; .y=rtart+artar also a* = 1, =1, andr4+r4r744+ 7 = -1; .s=y'=-—1+4a4+ 140’ — 16a". But the four roots of y* — 1 = 0, are | 1, —1, n/ ai sie R= 1, %, = 25, % = —15 +200 = 1, Ris = 20A/ aes w= h§— 14/540 -15 420/140 — 15 -209/-1}. 178. For the proof that, in the general equation of the n‘” degree, the formation of the reducing equation will re- quire the solution of an equation of 1.2.3...(—2) dimensions, PAs Sonn 7) 4 R —______—_————— dimensions, (m —1)m (1.2.3... p)” when 7 is a composite number, and = mp, where m is prime; and that, consequently, the method fails when m exceeds 4; the reader is referred to Lagrange’s T'raité de la resolution des équations numériques, note xili., from which the matter of this Section is taken. when 7 is prime ; and of THE END. efile” ® i ; im LJ a : y : wl { ’ f r UI] 082576