Digitized by tine Internet Arciiive in 2008 witin funding from IVIicrosoft Corporation littp://www.arcliive.org/details/elementarytreatiOOtodhricli AN ELEMENTAEY TREATISE Ojr THE , THEORY OF EQUATIONS WITH A COLLECTION OF EXAMPLES. C^P^ BY I. TODHTJNTEE, M.A. F.R.S., HONOBAEY TEIiLOW OF ST JOHN's COLLEGE, CAMBRLDGE. Honlroti: MACMILLAN AND CO. 1885. \Thc Right of Translation is reserved.] STEREOTYPED EDFTION, 3^ f ^^ Cambrilrge : PRINTED BY C. J, CLAY, M.A. k SON^ AT THE UNIVERSITY PRESS. PREFACE. The present treatise contains all the propositions which are usually included in elementary treatises on the Theory of Equations, together with a collection of examples for exercise. As the Theory of Equations involves a large number of interesting and important results, which can be demonstrated with simplicity and clearness, the subject may advantage- ously engage the attention of a student at an early period of his mathematical course. The present treatise may be read by those who are familiar with Algebra, since no higher knowledge is assumed, except in Arts. 149, 175, 268, 308... 314, and Chapter xxxi., which may be postponed by those who are not acquainted with De Moivre's Theorem in Trigonometry. This work may be regarded as a sequel to that on Algebra by the present writer, and accordingly the student has occasionally been referred to the treatise on Algebra for preliminary information on some topics here discussed. In composing the present work, the author has obtained assistance from the treatises on Algebra by Bourdon, Lefe- bure de Fourcy, and Mayer and Choquet ; on special points he has consulted other writers, who are named in their ap- propriate places in the course of the work. The examples have been selected from the College and University examination papers, and the results have been given where it appeared necessary ; in most cases however, from the nature of the example, the student will be able immediately to test the correctness of his result. Vi PREFACE. In order to exhibit a comprehensive view of the sub- ject, the present treatise includes investigations which are not to be found in all the preceding elementary treatises, and also some investigations which are not to be found in any of them. Among these may be mentioned Cauchy's proof that every equation has a root, Horner's method, the theories of elimination and expansion, Cauchy's theorem on the number of imaginary roots, the researches of Professor Sylvester respecting Newton's Rule, and the theory of determinants. The account of determinants has been princi- pally taken from a treatise on that subject by Baltzer, which was published at Leipsic in 1857 ; this is an excellent work, distinguished for the completeness of its proofs of the funda- mental theorems, and for the numerous applications of those theorems which it affords. For the parts of the Theory of Equations which are beyond an elementary treatise, the advanced student may consult Serret's Cours d'Algebre Superieure : there, for example, will be found a demonstration of the theorem, that the general algebraical solution of an equation of a degree above the fourth is impossible. The article Equation, by Professor Cayley, in the ninth edition of the Encyclopce- dia Britannica should also be noticed. Valuable historical information, relating to the higher parts of the subject, will be found in papers on Approximation and Numerical So- lution, by Mr James Cockle, in the Ladys and Gentlemam's Diary for the years 1854 and 1855, and also in papers on Equations of the Fifth Degree by the same author in the same work, for the years 1848, 1851, 1856, 1857, 1858, and 1860. I. TODHUNTER Sx John's College, March, 1880. CONTENTS. PAoa Inteoduction 1 On the Existence of a Eoot 15 Properties of Equations 22 Transformation of Equations 31 -Descartes's Eule of Signs ...... 40 On Equal Roots 48 Limits of the Eoots of an Equation. Separation of the Eoots 56 Commensurable Eoots 73 Depression of Equations 79 Eeciprocal Equations 85 Binomial Equations 89 Cubic Equations 99 Biquadratic Equations 112 Sturm's Theorem 121 Fourier's Theorem 130 Lagrange's Method of Approximation .... 135 Newton's Method of Approximation with Fourier's Additions 142 Horner's Method 4^0 Symmetrical Functions of the Eoots .... 165 Applications of Symmetrical Funptions . . ' . . 174 Sums of the Powers of the Eoots 181 Elunination 193 Expansion of a Function in Series 206 Miscellaneous Theorems 214 Cauchy's Theorem 231 Newton's Eule and Sylvester's Theorem .... 236 Eemoval of Terms from an Equation .... 251 Introduction to Determinants . . . . . .256 Properties of Determinants 266 Applications of Determinants ..'.... 284 Trigonometrical FormulaD . . . . . . 296 Examples 308 Answers 326 THEORY OF EQUATES. _--_^, I. INTRODUCTION". UinVEESITT 1. The reader can easily obtain a general idea of tllS 6bject of the following treatise by a reference to the theory of quad- ratic equations which he is supposed to have already studied. The equation acc^ + bx + c = has two roots, namely, -b^Jb'-4:ac 2a ' and with respect to these roots, we know that their sum is , ft and their product is - ; that is, their sum is equal to the coeffi- h c cient of the second term of the equation cc' + - a;+ — = 0, with its ^ a a sign changed, and their product is equal to the last term of this equation. (See Algebra, Chap, xxii.) Now it may be said that the general object of the following pages is to establish results with respect to equations of a higher degree than the second, similar to those which have been established in Algebra respect- ing equations of the second degree. The results obtained will be useful in other branches of mathematics, and the methods of investigation will afford valuable exercise to the student, since they are not too difficult for a person who has gained a knowledge of Algebra, and at the same time have sufficient variety to oc- cupy his attention. 2. The words equation and root are already familiar to the student from their use in Algebra; but for distinctness we will give a definition of them. T. E. 1 2 INTKODUCTION. Any Algebraical expression which contains x may be called a function of cc, and may be denoted by f{x). Any quantity which substituted for x in /(«) makes f{x) vanish, is called a root of the equation /(a?) = 0. An expression of the form /U){ -^ aa;"4- Sx"-' + cx""-^ + ..,^+kx + l, A where 71 is a positive integer, and the coefficients a,!)^ c ...h, I, do not involve x, is called a rational integral function of x of the w*^ degree j and if we wish to find what value of x makes this function vanish we have to find a root of a rational integral equation of the ^* degree ; this is the kind of equation which we shall consider in the present treatise. In such an equation we may if we please divide by the coefficient of the highest power of Xf so as to leave that power with only unity for its coefficient; the equation then takes the form "We shall say that the equation is now in its simplest form ; as will be seen hereafter, some of the properties of equations can' be enunciated more concisely when the equation is in this form than when a;" has a coefficient which is not unity. If we do not wish to suppose the coefficient of o^ to be unity, we may conveniently denote it by p^ ; then the equation takes the form p^x" + pX~ ^ + p^"~^ + ...+ p,,_^x^ + p„_ !«; + K = 0. The term p^ is called the term independent of x. 3. It must then be remembered that by equation we mean rational integral equation ; an equation which is not of this form may often be reduced to it by algebraical transformations; for example, the equation ax^ + hx + c Jx =f may be reduced to a rational integral form by transposing c Jx and f and then squaring; it will thus become a rational integral equation of the fourth degree. Equations which involve logarithmic func- tions, or exponential functions, or trigonometrical functions, or irrational algebraical functions, will not be directly included in our investigations ; for example, such equations as tan a; - 6""= 0, INTRODUCTION. 3 or x\ogx — a = 0, will not be included. However, the theory which will be given of rational integral equations will indirectly throw some light on these excluded equations. And when we speak of any function /(x) we shall always mean a rational integral function of x, unless the contrary is specified. 4. A remark of some importance must be made with respect to the coefficients Pq, Pi, p^, -" Pni i^ *^® equation In the quadratic equation ax^ + 6ic + c = we are able to solve the equation without knowing what particular numbers are de- noted by a, b, c; all we require to know is that a, h, c are some numbers independent of x. If we have to solve the equation x^ -I2x+ 15=^0 we may either transpose the 15 and complete the square in the ordinary way, or we may take the general formulae given in Art. 1, and put in them a=l, 6 = — 12, c=15. If we wish to solve an equation without having the numerical values of the coefficients previously assigned, we are seeking what may be called the algebraical solution of the equation; and if we can effect the algebraical solution of the general equation of any degree, we may obtain a numerical solution of an equation of that degree, by substituting the numerical values of the coefficients in the general formula which gives the alge- braical solution. As we proceed we shall find that the algebraical solution of equations up to the fourth degree inclusive has been effected; but both in equations of the third degree and of the fourth degree, when we substitute the numerical values of the coefficients in a specific equation in the general formula, the result takes a form which is sometimes practically useless. And beyond equations of the fourth degree the general algebraical solution of equations has not been carried, and it appears cannot be carried. ^B But with respect to what may be called the arithmetical solu- tion of equations in which the coefficients are given numbers, Ire success has been obtained. Thus, for example, although _ 4 INTRODUCTION. we cannot solve algebraically the general equation of the fifth degree, we can by numerical calculation discover any root which an equation of the fifth degree with known numerical coefficients may have, or at least we can approximate as closely as we please to such a root. - 5. Let us denote by/^cc) the expression then the value of this expression when x = a may be denoted by f{a). We will shew how the numerical value of f(a) may be most easily calculatecl, supposing that the coefficients of f{x), and also a itself, are specified numbers. Take for example an expression of the third degree ^ then we wish to find the numerical value of Poa[+Pxa''+p,a+p^, First obtain " Pf/i ; add p^, this gives p^^a +p^; multiply by a, this gives p^a^- + p^a ; . add p^, this gives • p^a'' -^-p^a+p^; multiply by a, this gives p^a? -vp^c^^-p^; add p^ , this gives p^c^ + p^o^ + p^a + p^ . We may arrange the process in the following way; P<>. Vx P^ P^. Pf' ■ p^d^+p^a sp^a^ + p^a^ + p^a p,aA: pi p.d' + p^a +p^ p,d' + p^a^ -^p^a + p^ We may proceed in the same way whatever may be the degree of /{x). For example, required the numerical value of- 3x^ -2x"-^x+7 when x = 3. 3-2 0-5+7 + 9 +21 +63 +174 + 7 +^1 +58 +181 Thus the result is 181. INTRODUCTION. 5 6. If any rational integral function of x vanishes when X = a, the function is divisible by x — a,. Let f{x) denote the function; then we have given that f(^a) = 0, and we have to prove that fix) is divisible hj x-a. Divide f{x) hj x-a by common algebra until the remainder no longer contains x; let Q denote the quotient and J^ the re- mainder if there be one. Then f (x) = Q (x - a) + JR. In this identity put a for x; since Q is sl rational integral function of x it cannot become infinite when x = a ; therefore Q (x — a) vanishes when x = a. Ahof(x) vanishes when x = ahy supposition. Thus B vanishes when x = a ; but i? does not contain tc, so that if it vanishes when x = a it always vanishes. That is, ^ = and x — a divides /(a;). 7. The above demonstration is important and instrtfctive ; we may ho"wever prove the theorem in another way, which will moreover have the advantage of exhibiting the form of the quotient Q. Suppose f(x)=p^x''+pf:''-'+2y,x''-'+... +p„.,x'+p^_^x+p,^^, - then since /(a) = we have /(a;) =f{x) -f{a) = Po (^" - ^") +Pr (^""' - »""') +Ps^ (^""' - ^""') + • • • +Pn-, (^ - «)• Now the terms as" -a", a;""^— a"~\ ... are all divisible hj x — a (see Algebra, Art. 483). By performing the division we obtain for the quotieiit (Po if~^+ ^^""' + ^''^""^ + ••• + ^""'^ + »""') + pf (^"~' + aic"-' + a V-' + . . . + a"-') + ... We may rearrange the quotient thus : Po^""' + {Po^ +Px) ^~'+{Po^' +Pi(^ + P2) ^""^ + - +K«"~' +p,a"~' + --' +Pn-X> and we may denote it by q^x''-' + q,x''-' + q,x''-'+... +q„.,x + q^_^. G INTEODUCTION. The new coefficients are therefore connected with each other and with the old coefficients by the formulae 9'o=i'o> 9'i = «9'o+i^n 5'2 = «2'i+i?2, % = (^2+Pz^ ; that is, each Tiew coefficient is found hy midtiplying the preceding new coefficient hy a and tJien adding the corresponding old coeffir dent. It will be observed that these new coefficients are succes- sively determined by the process of Art. 5. 8. If -s. — 2i, divide f (x) which is any rational integral func- tion of X, then a is a_ root of the equation f (x) = 0. For let Q denote the quotient when /(a;) is divided by a; — a, then /(a?) = ^ (a? - a). In this identity put a for a?, then Q is not \ infinite, and therefore Q{x—ob) vanishes. Thus y (a;) vanishes y when x — a^ and therefore a is a root of the equation y* (a;) = 0. /|[ f— . . . ' f' 9. To find tJie remainder when any rational integral function of's.is divided hy ^—c, where c is any constant. Let /(a;) denote any rational integral function of ai, and divide y*(a;) by a; — c until the remainder is independent of a;; let ^ denote the quotient and R the remainder. Then f{x)-^Q{x-c)^jR. ' In this identity put c for a?, then Q is not infinite, and therefore Q{x-c) vanishes j thu9/(c) = R. That is, R is equal to /(c) when x=c, but R does not contain a?, so that R is equal to f{c) always. For example ; if 2>x* — 2x^ — 5x + 7 is divided by x—Z, the quotient is 3a;^+ 7aj^ + 21a; + 58, and the remainder is 181; see Arts. 5 and 7. For another example let us divide the same expression by a;- 4 : 3_ 2 0-5+7 + 12 + 40+160+ 6S?0 '+10 + 40 + 155+627 Thus the quotient is 3x^+ 10a:^+ 40a;+ 155, and the remainder is 627. This process is a particular case of Synthetic Division; see Algebra, Chapter LViii. INTKODUCTION. 7 10. Let f{x) be any rational integral function of a?, and suppose x-^y put for x ; then we propose to arrange f{x + y) according to powers of y, and to determine the coefficients of the different powers. Let fix) =p„a;" + p,cc""^ +p^x''~^ + ... +p„_^x -^p^ ; then f(x+y)=p,{x+yy+p^(x+yy-'+p^{x+yy-'+.., + p,_,{x+y) +p^. Expand (x + yY, {x+yy~\ ... by the Binomial Theorem, and arrange the whole result according to powers of y ; we thus obtain for /(x + y) the following series : P^x"+pX~'+P2x''~'+ ... +Pn-iX+P„ + y^np^x''-' + (n-l)p^x''-'+ (n-2)p^x"-'+... +p\ f^[n{n-l)p^x^'^ + {n-l){n-2)p^af'-'+...+.2p^_^\ + + ... |-|?^(?^-l)...(7^-r + l)JpX-'•+(7^-l)(7^-2)...(?^-r)^X"'■"'+••.| + .... + IH- The first line of this series is obviously y (a;). We shall denote the coefficient of y by f'{x)y the coefficient of ~-^ by /"(x), the coefficient of r^- by /'" (x), and so on ; this notation becomes inconvenient when the number of accents is large, and so in general the coefficient of j- '^U ^^ denoted hyf^x). Hence 8 INTEODUCTION. By inspection it will be seen that the functions /(a?), f{x), f"{x), f"'{x), ...f"{x) are connected by the following general law: in order to obtain /""^^(x) we multiply each term in. f\x) by the exponent of a; in that term and then diminish the exponent by unity. ^ 11. Let us suppose, for example, that y* (a;) is of the fourth . degree; let f{x) =p^x' + p,x^+p^+Ps^+p^. Then /' (x) = ip^x^ + Zp^x^ + 2p^x + p^ , f"(x) = 4:.3p^x'+3,2p,x + 2p,, r'{x) = ^.3.2p^x+3.2p,, r\x) = i.3.2.p^- If we suppose numerical values assigned to p^^ p^, p^, p^^ p^ and Xj we may calculate separately /(cc), /'(x), ... by the method of Art. 5 j we shall however hereafter, in explaining Horner's method of solving equations, shew how these calculations may be most conveniently and systematically conducted. For another example suppose that /(a;) =p {x + cy. Then /(«) =^L" + wcaj"-' +^?^^?^ cV-' + ... +«^^^ ; therefore f'{x) =p Lx-'+n{n-l)cx-%- '^^''~^'^^^~^^ c'x""+ ... + nc^'^ ; that ia /'(*) =P'^{x + c)"'^ : similarly f"{^ = pn {n — V) {x ■{- cY~^y f"'{x) =pn (n -l){n- 2) (x + c)"-^ and so on. Suppose that <^ (x) and \l/{x) are two rational integral functions of a;, and that f(x) = {x) + il/ (x) ; then it is easily seen that /\x) = <}i'{x) + xp'ix), and f\x) = <}>'\x) + xj^'^x), and so on. INTRODUCTION. 9 12. If we write the series iov f{x-\-y), beginning with the highest power of y, we shall have fix + y) =py+ {l\ + riv,x)y''-' + 1^,+ (7^-l)^^a; + ^"^ i^o^^'j f'^ ( , ^, (n-l)(n-2) , n{n-\){n-'l) ^ „_, + |;?3 + (^ - 2) p^x + ^ :^ -^ p^x- + -^ 1^^ -^ p,x'^ 2/" + ... ( / IN n(n-l)...(n-r+l) \ „_, + U)^ + {n-r + l)p^_^x+...+-^ '—r^ -Po^}y This may be seen from the form already given for /{x + y), or by j; expanding separately every term in /(cc + y), and arranging ac-y cording to descending powers of y. 13. The function /' (x) is called the Jlrst derived function of f{x), the function /"(a?) is called the second derived function of f{x), and so on. The reader, when he is acquainted with the ele- ments of the Differential Calculus, will see that each derived function is the differential coefficient with respect to x of the immediately preceding derived function, and that the expression for/{cc + y) m. powers of y is an example of Taylor's Theorem. Moreover, it must be observed that/''(a3) is deduced from /'(a?) in precisely the same way as /'(as) is deduced from /(a;). Thus f"{x) is the first derived function oif{x), and /"'(a?) is the second derived function of /'(aj), and so on. Hence by the preceding Article we have y' Similarly f"{x + y) =/'(x) + yf"{x) + j^/""{«) + •• .r-2 ,...-^/'(x).....j|^m And so on. 10 INTRODUCTION. >^ '14. In any rational integral function of jl arranged according to descending powers of x, any term which occurs may he made to contain the sum of all which follow it, as many times as we please, hy taking x large enough, and any term may he made to contain the sum of all which precede it, as many times as we please, hy taking x small enough. Let P(pG'' +PiX"~^+p^x'*~^+ ... +p^_^x'' +p^_^x + p^ be any ra- tional integral function of x; suppose for example that the r^ term jo^.jCc""'"^^ occurs; that m, suppose p^_^ not zero. Let q denote the numerical value of the greatest of the coefficients p^, p^+^i-.-p^. The sum of all the terms which follow the r^^ term cannot exceed 2'(a;" '■+«;" *■ ' + ...+X+1), that is, q -— . The ratio of the r* X — 1 term to this is P.^^z2}^^ that h,^r=^=^^^ . By taking x large enough, the numerator can be made as large as we please, and the denominator as near to q as we please ; thus the ratio can be made as great as we please. This proves the first part of the proposition. To prove the second part put a; = - , then we obtain the series 2/~" {Po +p,y+py + ••• +Pn-y~' +Pnyl' We have now to prove that by taking x small enough, that is by taking y large enough, any term p^y"" which occurs can be made to bear as great a ratio as we please to the sum of the teriMf Po + Piy+'-'+Pr-iy""'^ which precede it; this has been already proved in the first part. ^ 15. One of the first questions which can occur in the theory of equations is whether a root must exist for every equation ; and we shall now give some simple propositions which establish the existence of a root in certain cases. We shall require a theorem which is often assumed as obvious, but which may be proved in the manner shewn in the next Article. 16. Let /(a;) be any rational integral function of x, and /*(«), /(&), the values oif{x) corresponding to the values a and h of a- •. INTEODUCTION. 11 then as x changes from aioh the function f{x) will change from f{a) to/(6), and will pass through every intermediate value. Let any value c be ascribed to x, and let /(c) be the corre- sponding value of f{x) ; let c + h be another value which may be ascribed to x ; then by taking A small enough f{c + h) may be made to differ as little as we please fromy(c). For Ac + h) =/(c) +//'(c) + j^/"(c) + ... + j^ /-'(«) + 1/" {<=)■ Then, by Art. 14, by taking h small enough, the first term of the series hf'{G), ^ — n/'X^)* fo/^'W? ••• which does not vanish, can be made to contain the sum of all which follow it as often as we please, and by taking h small enough this term will itself be ren- dered as small as we please. Therefore /(c + h) -f{c) can be made as small as we please by taking h small enough. This shews that as X changes, f{x) changes gradually ^ so that ii f(x) takes any value for an assigned value of a?, it will take another value as near as we please to the former, by taking another value of x which is sufficiently near to the assigned value. Hence as x changes from a to h, the function f{x) must pass without any interruption from the value /(a) to the value/ (6); for to assert that there could be interruption would amount to asserting that f{x) could take a certain value, and could not take a second value as near as we please to the first value. / 17. We do not assert in the preceding Article that f{x) always increases from f{a) to /(&), or always decreases from f{a) to/ (6); it may be sometimes increasing and sometimes decreasing. What we assert is, that it passes without any sudden change of value, from the value f{a) to the value f{b). The proposition is one of great importance, and probably will appear nearly evident to the student on reflection. It is obvious that f{x) has some finite value for every finite value ascribed to x, also we have proved that an indefinitely small change in x can only make an indefinitely small change in/ (a;), so that there can be no break in the succession of values which /(a;) assumes. 12 INTRODUCTION. / 18. The student who is acquainted with Co-ordinate Geo- metry will find it useful and interesting to illustrate the present subject by conceiving curves drawn to represent the functions. Thus let /(a?) be denoted hjy, so that 2/=/(£c) maybe conceived to be the equation to a curve ; then by supposing this curve drawn for the part lying between x = a and x=^h, o. good idea is obtained of the necessary consecutiveness in the values assumed by f{x) between the values /(a) and/ (6). It must be observed that we do not restrict a, h,f(a), /{b), to be positive quantities; and by values intermediate between /(a) and /{h) we mean intermediate in the algebraical sense; that is, any quantity z is intermediate between f(a) and f(b) which makes z —/{a) and/ (6) — z of the same sign. 19. If two numbers substituted /or n in a rational integral expression f (x) give results with contrary signs, one root at least of the equation f (x) =0 lies between those values o/x. Let a and b denote the two numbers ; then /(a) and/ (5) have contrary signs. By Art. 16, as a? changes gradually from a to b, the express! on /(a?) passes without any interruption of value from f(a) to f{b); but since f(a) and /(b) are of contrary signs the value zero lies between them, so that /(x) must be equal to zero for some value of x between a and b ; that is, there is a root of the equation /(cc) = between a and b. We do not say that there is only one root. And we do not say that if /(a) and /(b) are of the same sign there will be no root of the equation /(cc) = between a and b. 20. An equation o/ an odd degree has at least one real root. Let the equation be denoted hj/(x) = 0, where /(x) =p^"-hp^x''-' + ... +p„_rX +p,, and n i"fe an odd number. < •. When X is large enough the first term of /(«), namely pjx"^, will be larger than the sum of all the rest by Art. 14, andf^there- fore the sign of /(x) will be the same as the sign of p^x". ' Thus, by taking x large enough, the sign of /(x) can be made the same INTRODUCTION. 13 as the sign of p^ when x is positive, and the contrary to that of p^ when X is negative. Since then /{x) changes its sign as x passes from a suitable negative value to a suitable positive value, there must be some intermediate value of x which makes /{x) vanish ; that is, there must be some real root of the equation /(x) = 0. We may determine whether this root is positive or negative. For when we put zero for x the sign of /{x) is the same as that of 2?^. Thus if p^ and p^ have the same sign^ so that — is positive. Pa there Avill certainly be a negative root of the eqjiation /(£c) = 0; 7) and if ^„ and p^ have contrary signs, so that — is negative, there Po will certainly be a positive root of the equation /(cc) = 0. Thus if an equation be of an odd degree, and be brought into its simplest form by dividing by the coefficient of the highest power of Xy iti will have a real root of the sign contrary to that of the last term._J 21. An equation of an even degree which is in its simplest/orm, and has its last term negative, has at least two real roots of contrary Let f{x) = be the equation; then when x is zero f(x) is negative by supposition. When x is large enough f(x) is posi tive, whether x is positive or negative. Thus there is some negative yalue of x which makes f{x) vanish, and also some posi- tive value of X which makes f(x) vanish. That is, the equation f{x)-Q has certainly one negative root and one positive root. J 22. ff the rational integral expression f (x) consists of a set of terms in which the coefficients are all of one sign, followed hy a set of terms in which the coefficients are all of the contrary sign, the equation f (a?) = has one positive root and only one positive root. By Arts. 20 and 21 the equation f(x) = must have one positiveTroot; we will shew that it has only one positive root. Let f{x) =p^x'* +p^x"~'' + p^x""'^ + ... +P„_tX -{-p^. Suppose the coefficients jp^, p^, ...^^ all positive, and the remaining 14 INTRODUCTION. coefficients negative; let ^,+, = -P^+j, ^,+3 = -i',+2, "-Pn = -P„' Then we may write /(aj) thus, /(a=)=a."-' |;,x+i^,»='"'+i"X-'' + -+;'.-% - ^---p-]- The expression Po^*" +pX~^ +^2^*^"^+ ••• +^, increases as aj in- creases, unless r = 0, and then it remains constant; the expression P P P . — ^ + — ^ + . . . + -~ diminishes as £C increases. Thus as x in- £c a; X creases from zero onwards, the two expressions cannot be equal more than once. That is, f{x) = has onl^ one positive root. The demonstration will be the same if we suppose the first set of coefficients negative and the second set positive. 23. To prevent any mistake it will be useful to draw attention to the precise results obtained in the last three Articles. In Art. 20 it is proved that the equation considered has at least one real root; it is not proved that it has one only. In Art. 21 it is proved that the equation considered has at least two real roots; it is not proved that it has only two. In Art. 22 it is proved that the equation considered has one positive root and onl^ one positive root : it is not proved that it has no negative root. 24. The propositions in Arts. 20, 21, and 22, as to the exist- ence of roots in certain cases, depend upon the fact that we are able to shew that f{x) undergoes a change of sign or changes of sign. We cannot infer conversely that if f{x) never changes its sign within a certain range of values for x there is no root of the equation f{x) = within that range of values for x. Take for example x^ - 6x+9 ; this expression never changes its sign, and yet it vanishes when x = 3: the expression is equal to (x — 3)^ But if the equation y(a3) = has no root within an assigned range of values for x we are sure thaty(a;) never changes its sign within that range of values for x. The following statements respecting the absence of roots will be seen to be obviously true : ON THE EXISTENCE OF A BOOT. 15 (1 ) If the coefficients in f{x) are all positive, the equation f^x) = has no positive root. (2) If all the coefficients of the even powers of a; m/(x) have one sign, and all the coefficients of the odd powers of x the contrary sign, the equation /(cc) = has no negative root. (3) lf/(x) involves only even powers of x and the coefficients are all of the same sign, the equation/ (cc) = has no real root. It is supposed in this case that there is a term independent of a;. (4) If f(x) involves only odd powers of x and the coefficients are all of the same sign, the equation /(x) = has no real root, except x = 0. It is supposed in this case, of course, that there is no term in- dependent of X. We say in the last two cases that the equation has no real root, and we do not say that the equation has no root, for we know that by virtue of some conventions an equation may in some cases have imaginary roots ; see Algebra^ Chapter xxv. And in fact we shall now proceed to shew that imaginary roots 7nust exist. II. ON THE EXISTENCE OF A ROOT. 25. We shall now prove that every rational integral equation has a root, either real or of the form a + b J- 1, where a and h are real ; such an expression SiS a + b J— ly where a and b are real, we shall call an imaginary expression. That is, when we use the term imaginary we shall always mean that the expression to which we apply this term is of the form a+bj- 1, where a and b are real. 26. The student is supposed to know that by virtue of certain conventions, imaginary expressions can be used in algebraical investigations, and theorems can be established respecting them. I 16 ON THE EXISTENCE OF A EOOT. Thus, for example, the positive value of the square root of a^ + h^ is called the modulus of each of the expressions a + hj-l and a — h J— 1 ; and with this definition we can shew that the modulus of the product of two imaginary expressions is the product of the moduli of those two expressions. For the product of a-^-h J- 1 and a' + 6' J- 1 is aa' — hh' + {ah' + a'h) J- 1, and the modulus of this is the positive value of the square root of {aa' - hhj + {ah' + a'h)', that is, of {a" + h') {a'^ + 6'^) ; that is, the modulus is the product of the moduli of the two given expressions. Also an imaginary expression a + hj—l is considered to vanish when a and b vanish ; that is, an imaginary expression vanishes when its modulus vanishes. Thus, by what has just been shewn, if the product of two imaginary expressions vanishes, the modulus of "one of the expressions must vanish; so that if the product of two or more imaginary expressions vanishes, one of the expressions them- selves must vanish ; and if one of the expressions vanishes the pro- duct vanishes. 27. The student who has not paid attention to the subject of imaginary expressions may consult the Algehra, Chap. xxv. The proof however that every equation has a root, real or imaginary, to which we shall now proceed, is somewhat difficult; the student therefore on reading this subject for the first time may assume this proposition, and reserve the remainder of the present Chapter for future consideration. 28. We shall first shew that a root, real or imaginary, exists for each of the following four equations : aj« = l, a;" = -l, cc'' = + y^, cc" = -^^ (1) £c" = 1. It is obvious that cc = 1 is a root of this equation. (2) £c" = — 1. If n is an odd number it is obvious that x = —l is a root of this equation. If n is an even number suppose it equal to 2w ; we have then to shew that there is a solution of x^'^= — 1 ; this amounts to shewing that tho-e is a solution of cc'"= '^J— 1, and is therefore included in the next two cases. ON THE EXISTENCE OF A ROOT. 17 (3) a;" = + J- 1. If 71 is an odd number it must be of one of the two forms im + 1 and 4m + 3 ; in the former case + J— 1 is a root, since (+^-1)'*"*^^ = + ^-1, and in the latter case-^— 1 is a root, since (- ^- 1)''"'"^^ = + J- 1. If w is an even number suppose it equal to mp, where m is an odd number, and p is some power of 2, say 2'. Put ^j = ocf, then the equation x"'^ = + J-l maj be written y"'- + J—^, and by what has been already shewn + J— I or - J- 1 is a suitable value of ?/, according as m is of the form 4r + 1 or 4r + 3. We have then to find a value of x which will satisfy af= + J-l or of = -J^, where p = 2^. The required value can be obtained by common Algebra. For take the square root of + J- 1 or of - J- 1 ; this will give an expression of the form a + (Sj-l, where a and y8 are real; take the square root of a + pj—ly which will give a similar expression; and so on: see ■ Algebra, Chapter xxv. Thus after q extractions of the square root we arrive at an expression a + bj-l^ such that (a + hj-iy (4) x''^- s/^. This case is treated like (3). If n be an odd number, - J^ or + J— 1 is a root, according as n is of the form 4«i + 1 or 4m + 3. If n be an even number suppose it equal to mp, Avhere m is an odd number and p = 2'', and proceed as before. 29-.^ Every rational integral equation has a root real or imaginary. Let /(cc)=PQa;" + PjCc"~^+^2^"~^+ ... +p^_^x^-^p^_^x+p^, where the coefficients p^, Pj, ...^?„_2, i\_ij P^ ""^^y ^ either real or imaginary ; we have to shew that the equation /(a?) = has a root either real or imaginary. If any imaginary expression be substi- tuted for X vcifix), we shall obtain a result of the form U+Vj-l, where U and V are real quantities, and we have to shew that an imaginary expression must exist which will make U=0 and F= 0. This we prove in the following manner. Since Z7^+ V^ is always a real positive quantity, if it cannot be zero there must be some Iue which is not greater than any other value, that is, there must T. E. 2 18 ON THE EXISTENCE OF A HOOT. be some value which, cannot be diminished ; but we shall now prove that if U^+V^ have any value different from zero we can diminish that value by a suitable change in the expression which is substi- tuted for x; so that it follows that C/'^+K" must be capable of the value zero, that is, U and V must vanish simultaneously. Suppose a particular value assigned to x, namely, a + hj— 1 ; let f{x) then become F+Qj—lj where P and Q are not both zero. Now put a + b J—l + h for x m/(x) ; the value which /(cc) then takes may be found by first expanding /(x + h) in powers of k, and then putting a + b J- 1 for x. Suppose then f{x + h)=X + hX'+flx"+ + uPo\3 where X, X', X", ... are functions of x; see Art. 10. Put a+bj^ for X, then X becomes F +Q J-\. Some of the coefficients X\ X'\ ... may vanish for this value of x, but they cannot all A" vanish, since the last coefficient, which is that of ^, is _Po |^. Suppose /i"* the lowest power of h for which the coefficient does not vanish, and denote the coefficient of 7^ by R+Sj-\, so that R and S are not both zero. Thus when a + b J— \ + h is substi- tuted for x the function /(ic) becomes P+QsFl+ (E + Sj^) ir + ... , where the terms not expressed can only involve powers of h higher than /A Denote this by P'+Q'J-\. Let h = €t, where e is a real positive quantity. By Art. 28 it is in our power to take t so that f may be -f- 1 or — 1 ; thus we can make P'+Q'J^i=P+Qj-l^{R + Sj^)^'^+..., 'so that . F = P^R^'+ ..., Q'=Q^Se"'+..., and F"+ Q"=r + Q'^ 2 {PR +QS)i"+ ..., ON THE EXISTENCE OF A ROOT. . 19 where the terms not expressed can only involve powers of e liigher than e'". Kow c may be taken so small that the sign of all the terms involving c in the value of F'^+Q'^ will be the same as the sign of ±2 (FE + QS) e"*, provided FH+QShe not zero; see Art. 14. We will first suppose that FE + QS is not zero. Then the sign o^F"'-¥Q"-F'-Q' is the same as thesign oi ^2{FR+QS)^\ when e is taken small enough ; and we can ensure that this sign shall be negative by supposing that ^'" is - 1 or +1, according as PRa-QS i^ positive or negative. We can therefore make F'^-^Q'^ less than F^+ Q\ Next suppose that FR + QS is zero. Then instead of taking f = ^\, take C = ^ J— 1. Proceeding as before we shall obtain F + Q'J^=:F + Qj^^{R + Sj^)e"'\/^+..., so that F'=F^Si"+..., Q'=Q^R€'^+..., and F"+Q" = F'+Q'^ 2{QR- FS) e"" + ... , where the terms Hot expressed can only involve powers of e higher than c'". Now (FR+QSy+{QR-FSy = {F'+Q'){R'+S'); and this cannot, be zero, because by supposition F^+Q^ is not zero, and R^+S^ likewise is different from zero. Thus since FR + QS is zero, QR-FS is not zero. Therefore the sign of 7^'+ Q'^-F^-Q^ will be the same as the sign of =>= 2 (QR-FS) e^ when e is taken small enough; and we can 'ensure that this sign shall be negative by supposing that f is —J -I or +^-1, according as QR—FS is positive or negative. We can therefore make F^^ + Q'^ less than F'-^Q\ We have thus shewn that when TJ^-^ V^ has any value different from zero we can diminish that value by a suitable change in the expression which is substituted for x; that is, U^+V^ is not susceptible of any positive value which cannot be diminished; 2—2 20 ON THE EXISTENCE OF A ROOT. lience, as we have already stated, it must be possible that U =0 and r= simultaneously. 30. It remains to be shewn that a and b in the expression a + h J^i which is the value of x that makes f{x) vanish, are finite. We have fix) =p^x'' [l + -^ + ^^^ ... + l'^\ , Substitute a + h J— 1 for x ; theny (:c) becomes Take any term of the series within the brackets, for example, that involving p^ ; we have P,{a+bJ-\y Poi^'+by pAo^'+br pA'^'+by = A +J3 J-1, say. Then it is evident that A and B diminish without limit as a and b increase without limit. Thus denoting the value of f(x) when x = a + b J- 1 hj 17+ Vj-\, we have U+Vj^ = p,{a+bJ-[Y{l+A'+B'J^], where A' and B' diminish without limit as a and b increase with- out limit. If we put a-b J- 1 for x we shall obtain a result which can be deduced from that just given by changing the sign of V-1 : thus u-vj:^=p^{a-bj:iy{i+A'-B^J~[}; therefore U' + V =pf(a'+by {(1 + Ay+ B"], and this increases without limit when a and b increase without limit; for the factor (a^+b^)" increases without limit, and the factor (l+Ay+B'^ tends to unity as its limit. Thus U'+V^ cannot vanish when a and b are indefinitely great, or when either of them is indefinitely great. I ON THE EXISTENCE OF A ROOT. 21 31. It will be observed in the demonstration of Article 29, that the coefl&cients of the proposed equation may be either real or imaginary. We shall however in the subsequent part of this book always suppose the coefficients to be real unless the contrary be stated. 32. The proof given in this Chapter of the existence of a root of an equation is called Cauchy's proof. The subject has recently been again discussed by mathematicians, and two memoirs will be found on it in the Tenth Volume of the Transactions of the Cambridge Philosophical Society, one by Mr De Morgan, and the other by Mr Airy; there is a supplement to the latter. It ap- pears from Mr De Morgan's memoir that the proof known as Cauchy's had been previously given in substance by Argand. We may briefly notice an objection which has sometimes been urged against Cauchy's proof. It has been said that it is conceivable^ until the contrary is shewn, that U^ + F^ Ttiay ap- proach indefinitely near to some limit greater than zero without ever reaching this limit. But this objection can be removed by the aid of Art. 30. Let z stand for W + V\ that is, for f(a + bj^)xf(a-hj^: then we know that z is finite for finite values of a and h, and infi- nite for infinite values of a and h. Hence the least value of z must occur when a and b have finite values; and if the least value of z were not zero the demonstration of Art. 29 would be contra- Jcted. m The student who is acquainted with the elements of Geometry of Three Dimensions will be assisted by supposing a, b, and z to be coordinates of a point in space, and imagining the surface deter- Lcd by the relation z=f{a + b J^ y^fia-b 7^). 22 PROPERTIES OF EQUATIONS. III. PROPERTIES OF EQUATIONS. /^ 33. Every equation has as many roots as the number which expresses its degree, and no more. Suppose the equation to be of the n^^ degree, and denote it by. fix) = 0, where/(a;) =pjid' + Pj^c""^ + jo^cc""^ + +i^„_i^ -^Pn- -^7 Chapter il. the equation/ (ic) = has a root either real or imaginary; let a^ denote that root. Therefore /(x) is divisible by x—a^, by Art. 6; so that f{x) = (x— a,)^^ (x), where ^ (x) is some integral algebraical function of x of the.(?^ - l)*** degree. Again by Chapter II. the equation ^^ (x) = has a root either real or imaginary; let a^ denote that root. Therefore 4>i{^) is divisible by x — a^, by Art. 6 ; so that (f)^ (x) = (x — a^ <^^ (x), where cfi^ (x) is some rational integral algebraical function of x of the (n— 2)'^ degree. Therefore f(oc) = (x-cCi){x — a^)^ (x) where ^^ (x) is a function of x one degree lower than /{x) j and the remaining roots of the equation f(x) = can be found if we can solve the equation ^ (x) = which is one degree lower than the equation /(x) = 0. Similarly if we know two roots a J and a^ of the equation /(^x) = we know that f(x) ^(x — a^ix — a^ cf)^ {x) where cf>^ {x) is a function of x two de- grees lower than/(ic); and the remaining roots of the equation f{x) = can be found if we can solve the equation ^^{x) = 0, which is two degrees lower than the equation /"(cc) = 0. And so on. 37. If f{^) be any rational integral algebraical function of x of the n}^ degree, we have shewn that f{x) must be capable of resolution into n factors of the first degree, so that IK /{^)=Po{^-^i) (^-aj (^-a„)y where a^, a^, a^ are either real or imaginary. It is to be observed that there is only one system of factors into which f{x) can be resolved; this has already appeared when the quantities I^a„,...a are all unequal, but it still remains to be shewn that when 24 PKOPERTIES OF EQUATIONS. some of the quantities a^,a^,...a^^, are equal, /(ic) cannot be formed in different ways in which the same factors occur with different exponents. If possible suppose that and also f{x) =p^ (x - a^y(x - a^y{x - a.^y Suppose r greater than p ; then dividing by (x - ajp we have Po (^ - ^i)'"'' (^ - ^2)X^ -%y =2'o(^- (^,Y{^ - ^i" !Nbw the left-hand member vanishes when aj = a^ , but the right- hand member does not ; the expressions then cannot be identical, j and therefore /(cc) cannot admit of more than one system of factors. 1\^ 38. If any rational integral function of a of the n* degree vanishes for nfiore than n different values ofx every coefficient in the function must be zero, so that the ftmction 7nust he zero for every value ofx. For if any coefficient in the function is not zero the function will not vanish for more than 71 different values of x, so that if the function does vanish for more than n different values of x every coefficient in the function must be zero. 39. The proof in the preceding Article makes the proposition depend upon the fact that an equation of the n^^ degree has n roots, and thus ultimately upon the investigations in Chapter 11. We may however establish the proposition by an inductive proof which does not require the investigations in Chapter 11. Suppose it true that when a function of x of the n*"* degree vanishes for more than n different values of x every coefficient in the function is zero ; and that we require to shew that when a function of x of the {n+ l)**" degree vanishes for more than n+1 different values of x every coefficient in the function is zero. Let fix) = q^x"*^ + q^x"" + q^x"'^ + 4- q^x + q„+^f and suppose that more than n+1 values of x make y (a?) vanish. Let a be one of these values so that /(a) =^ 0. Then f{x) =f{x) -f{a) = q^(x^^^-a'''-') + q^{x''-a") + q^{x-^-a-^) + +5'„(^-«). PPvOPERTIES OF EQUATIONS. 25 This may be written in the form f{x) = (x-a){x), where <}> (x) is a function of x of the r^^ degree. Since then there are more than n different values of x, exclusive of a, which make f{x) vanish, there are more than n different values of x which make <^ (x) vanish; therefore by supposition every coefficient in <^ {x) is zero. Now by Art. 7, ^ (^) - 2'o^" + fc^ + ^>""' + &^' + q,(^ + 3',)^"'" + ; thus q^^O because the coefficient of cc" is zero, then q^ = ^ because the coefficient of £c"~^ is also zero, then 5'^ = because the coefficient of a;"~^ is also zero, and so on. I Thus every coefficient infix) is zero. This establishes the proposition, since it is known to be true for expressions of the first and second degree. 40. 1^ f{x) be any function of x of the n^^ degree we have shewn thaty (cc) may be resolved into n factors of the first degree. Each of these factors will divide /(ic) so that/ (a?) will admit of n divisors of the first degree. Similarly as the product of any two of the factors of the first degree contained i-n.f{x) will be a factor of the second degree contained in f{x), it follows that f{x) will admit of — — —^ divisors of the second degree. Proceeding thus we see that/(:c) will admit of as many divisors of the r^^ degree as there are combinations of n things taken r at a time, that h,f{x) will admit of — ^^ '~ — divisors of the r*'* deofree. ^o' W Bat it must be remembered that the divisors of any degree, as for example the second, will not necessarily be all different^ be- cause the factors of the first degree m.f{x) are not necessarily all f^'^'^rent. The proposition however shews that there cannot he e than — LilS^ i different divisors of the r"' degr^ . , ee. 26 PEOPERTIES OF EQUATIONS. 41. In an equation with real coefficients imaginary roots occur in pairs. Let f{x) be a rational integral function of x in which the coeffi- cients are all real; then if a + pj -I is a root of the equation /(x) =0 so also is a- p J -1 a root. For when a+ /3 ^J — I is put for cc the function y (a;) takes the form P + QjS J —1, where F and Q involve even powers of ^. This is obvious, because if such an expression as ic*" be expanded, where x = a + (3 J —1, the even powers of P J - I will give rise to real terms, so that J — 1 will occur only in connexion with odd powers of 13. And as the coefficients in f (x) are supposed real J -I cannot occur except with some odd power of /8. If then a — jB J — \ be substituted for x in f{x) the result will be obtained by changing the sign of fB in the result obtained by substituting a + j3 J -I for x-y the result is therefore P -Qj3 J -I' Now suppose that a+ ^ J -1 is a root o?/(x) = 0; then P+QI3j^^=0, and, as a real quantity P cannot be equal to an imaginary quan- tity - QI3 J - I, this requires P=0, and^ = 0. And then a — j3 J — \ is also a root of f{x) = 0. 42. Thus if f{x) be a rational integral function of x with real coefficients, and have a factor x — a^ where a^ = a + ^ ^ — 1 , it has also a factor x — a^ where a^^ = a—P J — 1- The product of the two factors x — a — jS J — 1 and x — a + pj — l, is (a: — a) ^ 4- ^', or x^ — 2aic + a^ + jS^'j that is, the product is a real quadratic factor. 43. We have thus arrived at the result that any rational integral function of x with real coefficients may be regarded as the product of real factors, either simple or quadratic; and that there is only one such system of factors for any given function. Thus f(x) must be of the form (x — a){x — h){x - c) . . .{x - k) <^ (x), where PROPERTIES OF EQUATIONS. 27,- a, h, c,...k are all the real roots -of /(x) = 0, and c(> (x) is a function •*" *• consisting of the product of quadratic factors which cannot change its sign. '\. 44. In the manner of Art. 41 it may be shewn that if the ^ " coefficients of any rational integral function /'(a;) of x be themselves rational^ and the equation f(^x) = has a root of the form a + Jb ' where Jb is a surd, the equation has also a root a — Jb. Thus y (a?) , ' has a rational quadratic factor {x — df — b. > 45. To investigate the relations between the coefficients . of the ., ^ /•/unction f (x) and the roots of the equation f (x) = 0. ^" '^ Let /(x) = x" + p^x"''^ +p^x'''' + ... +/>„_, flJ +2^,/f and suppose that the roots of the equation /(x) = are a^,a^,. ..a^ ; then /(x) =:(x~ a,) (x -a^)...{x- aj. Since these two expressions for/(x) arc identically equal, relations t exist between the coefficients p^, Pzy-Pn^ ^^^ *^^® quantities- , ttj, »2,...a^; these relations we shall now exhibit. By ordinary multiplication we obtain ^^ {x — a^){x-a^)=x^-{a^ + a,^)x + a^a^, .^ ^K. (x - aj{x - a,J(x - a^ = x^ - {a^ + «„ + a^ x^ + (a^a^ + a^ttg + a^a^ + a^^g + «2«4 + %<^^ ^^ - («1 V3 -^- ^l«^A + «l«^3«^4 + ^2^4) ^^ + »l«^2«3^4 • ' Now in these results we see that the following laws hold : ^p I. The number of terms on the right-hand side is one more ^nan the number of the simple factors which are multiplied" together. II. The exponent of x in the first term is the same as the number of the simple factors, and in the other terms each exponent is less than that of the preceding term by unity. 28 PROPERTIES OF EQUATIONS. III. The coefficient of the first term is unity; the coefficient of the second term is the sum of the second terms of the simple factors; the coefficient of the third term is the sum of the products of every two of the second terms of the simple factors; the coeffi- cient of the fourth term is the sum of the products of the second terms of the simple factors taken three at a time, and so on; the last term is the product of all the second terms of the simple factors. We shall now prove that these laws always hold whatever be the number of simple factors. Suppose these laws to hold when n-1 factors are multiplied together; that is, suppose (x - a^) {x-a^)...{x- a ,_ J = x"~' + g^oj""^ + q^x"~^ + ... + q^^^^x + q^^_^ , where q^ = the sum of the terms — a^, —a^, ... — a^_^ , q^ = the sum of the products of these terms taken two at a time, q.^ = the sum of the products of these terms taken three at a time. q,^_^ = the product of all these terms. Multiply both sides of this identity by another factor a; — a j thus {x-aj{x- eg ...{x- a) =02" + {q^ - a) x"-' + {q,_ - q^a) aj""' + fc-^/0^""'+ -^„-x«„- Now q^—a= — a, — a„— ... — a , —a ■II II 1 2 II — I n = the sum of all the terms - a^, —a^,...-a^; 9, - (Zi«„ = q, + a,{a^ + a^+... + «„_ J = the sum of the products taken two and two of all the terms — a,. —a„, ...— a : = the sum of the products taken three and three of all the terms — a^, -a^,...-a^; 9n-\^n - ^^® product of all the terms - a^ , - a^, . . . — a^. PROPERTIES OF EQUATIONS. 29 Hence if the laws hold when n-1 factors are multiplied together they hold w^hen n factors are multiplied together ; but they have been proved to hold when four factors are multiplied together, therefore they hold when five factors are multiplied together, and so on ; thus they hold universally. We have used the inductive method in establishing these laws; but they may also be obtained in another way : see Algebra j Art. 506. Since ii a^, a^j ... u^ are the roots of the equation the left-hand member is equivalent to the product of the factor^ x-a^^ x-a^, ...x-a^^, we have the following results. In anyW equation in its simplest form the coefficient of the second term is, equal to the sum of the roots with their signs changed ; the coeffi-\ cient of the third term is equal to the sum of the products of everj^ two of the roots with their signs changed ; the coefficient of th fourth term is equal to the sum of the products of every thre of the roots with their signs changed; the last term is th product of all the roots with their signs changed. Or we may enunciate the laws tjius : the coefficient of the second term with its sign changed is equal to the sum of the roots ; the coefficient of the third term is equal to the sum of the pro- ducts of every two of the roots ; the coefficient of the fourth term with its sign changed is equal to the sum of the products of every three of the roots ; and so on. Thus generally if p^ denote as usual the coefficient of cc""'' in the equation, (— l)'"p^ = the sum the products of every r of the roots. i 4G. It might appear perhaps that the relations given in the preceding Article would enable us to find the roots of any proposed equation ; for they supply equations involving the roots, and the number of these equations is the same as the number of the roots, so that it might be supposed practicable to eliminate all the roots but one and thus to determine that root. But on attempting this elimination "we merely reproduce the proposed equation itself. Take, for example, the equation of the third degree 30 PROPERTIES OF EQUATIONS. suppose the roots to be a^h, Cy then -a — h-c=p^f ah + hc + ca =p^ -ahc=p.^. In order to eliminate h and c and so to obtain an equation which contains only a, the simplest method is to multiply the first of the above three equations by aJ^, and the second by a, and add the results to the third. Thus — a^- a% — a^c + a^b + ahc + ca^- ahc =p^(i^ + p^a + p^ ; that is, a^+p^a^+p^a +^3=0; we have thus the proposed equation with a instead of x to represent the unknown quantity. And it is not difficult to see that we ought to expect a cubic equation in a, if we eliminate b and c from the relations we are considering. For the letters a, b, c represent the roots without any distinction of one root from the others; thus any equation which we deduce for deter- mining a ought to allow of three values for a, since a may stand for any one of the three roots of the proposed equation. Thus we may feel certain that we shall only reproduce the original form of the proposed equation by performing any algebraical operations on the relations which connect the known coefficients of the equation with its unknown roots, with the view of elimi- nating all the roots but one. 47. Although the relations given in Art. 45 will not de- termine the roots of any proposed equation, we shall find that they will enable us to deduce various important results with respect to equations. For example, if a^, a^, a^^ are the roots of the equation we have —p^ = a^+a^+a^+... + a^^, thus "^)«j^ - 2p^ = a/+ a/+ ^3^+ . . . + a/, TRANSFOEMATION OF EQUATIONS. 31 that is 'Px'- 2^2 is equal to the sum of the squares of the roots of the proposed equation. If then in any equ'ation f^— 2p^ is negative, the roots of the equation cannot be all real. 48. In the same manner as in the preceding Article we may deduce other relations involving the roots. Thus for ex- ample (-l)"~^/>^_^ = the sum of the products of the roots n -1 at a time, (- 1)"^„ = the product of all the roots ; therefore by division = the sum of the reciprocals of the roots. Also p. —'■=(«, + »„+ ...+a)(~ + - + ...+-) = 51+ -i + -• + ... + -" +-^+... ; therefore -^ + -^ + ... + -^ + -^ + ,,,=ULji^ -n. I Pn lY. TRANSFOEMATION OF EQUATIONS. 49. The general object of the present Chapter is to deduce from a given equation another equation the roots of which shall have an assigned relation to those of the given equation. It will be seen as we proceed that various transformations of this kind can be effected without knowing the roots of the given equation; and hereafter examples will occur shewing that such transformations may be of use in the solution of equations. ^r 50. To transform an equation into another the roots of which are those of the proposed equation with contrary signs. Let f(^x) = denote the proposed equation; assume y = — x, so that when x has any particular value, y has numerically the 32 TRANSFORMATION OF EQUATIONS. same value but with, the contrary sign ; thus x^ — y, and tlie required equation is /(- y) - 0. If f{x) =py+p^x'-' +p,x"-' +'... +p„_,x +;?,,, the equation /(-?/) = is Po (- yT + Pi (- yT~' +P2 (- 2/)""' + • • • -Pn-iV +Pn = ^» that is, py-py~'+py~^~'--^p„-,y^p^=^] thus the transformed equation may be obtained from the pro- posed equation by changing the sign of the coefficient of every other term beginning with the second. 51. The rule at the end of the preceding Article assumes that the proposed equation has all the terms which can occur in an equation of its degree, that is, it is assumed that no co- efficient is zero. But suppose we take an example in which this is not the case; thus let it be required to transform the equation x' + 3x'-4:X^-ix+7 = 0, into another in which the roots shall be numerically the same but with contrary signs. Put x = -y, and we get y^-3y'+4:y^ + Ay + 7 = 0. We may if we please write the original equation thus, x'+3x' + 0x*-Ax^ + 0x'-4:X + 7 = 0; then the transformed equation according to the rule in Art. 50, is 2/6 _ 3f + Oi/ + 4:y^+ Oy' + 4?/ -i- 7 = 0, that is, 2/'-%' + 42/' + 42/ + 7-0, as before. An equation is said to be complete when it has all the terms which can occur in an equation of its ' degree, that is, when no coefficient is zero. And we shall sometimes find it useful to render an equation complete by the artifice used above, that is, by introducing the missing terms with zero for the coefficient of each of them. TKANSFORMATION OF EQUATIONS. 33 52. To transform an equation into a/nother the roots of which a/re equal to those of the proposed equation multiplied hy a given quantity. Let fix) = denote the proposed equation ; and let it be required to transform it into another the roots of which are k times as large. Assume y = kx, so that when x has any par- ticular value, the value oi y is k times as large ; thus ^ = t , and the required equation is /( f ) = 0. 53. For example, transform the equation , Sx' 5x 2 . ^— 2^T-9 = ^ into another the roots of which are k times as large. Put a? = t and then multiply throughout by k^ ; thus we obtain 3 3ky' 5k'y 2k^ ^ This example will shew us an application which may be made of the present transformation. The coefficients of the proposed equation are not all integers; by properly assuming k we may make the coefficients of the transformed equation all integers. For instance, if ^ = 6, the transformed equation is 2/' -92/^ + 452/ -48 = 0. Generally, suppose the proposed equation to be x^'+p^x'*-' +p^x''-'+ ... +i?„_ia5+i?„ = 0, then if we put a5 = |, and multiply throughout by k", all that is necessary to ensure that the coefficients of the transformed equation shall be integers is, that for each term of the transformed equation ^^^'"y""'', every prime factor which occurs in the deno- minator of p^ shall occur to at least as high a power in ^^. T. E. 3 N^ 34 TRANSFOKMATION OF EQUATIONS. 54. To transform an equation into another the roots of which shall he less than those of the proposed equation hy a constaM difference. * Let f{x) = denote the proposed equation ; and let it be required to transform this equation into another the roots of which shall be less than the roots of the proposed equation by a constant difference k. Assume y=x — k^ so that when x has any particular value, the value of y is less by k] thus x = k + y^ and the required equation \&f{k+y) = 0. By Art. 10 the expanded form of the equation f{k + y) = is f(k)+yf\k)+^r(k) +^^r'{k) + ... -f 2/"*^=o. Thus if f{x)=py+p,x"-'+p^x''-''+...+p^_,x+p„ the equation /(A; + 2/) = when arranged according to descending powers of y is by Art. 12 + ... r / n\ 7 n(n-\) ...in-r + V) ,-)„_. ■^[Pr-^{n-r + \)p^_Jc+ ■^— '-y -Po^]y + ...+/(A;) = 0. A good practical mode of conducting the operation will be found in Chapter xviii. 55. If an equation is to be transformed into another the roots of which exceed those of the proposed equation by the constant quantity h, we use the method of the preceding Article. Let the proposed equation be denoted by/(aj) = 0, and suppose y = x + h', then x = y-hy and the required equation i3f(y-h) = 0. Thus we have only to put - A for ^ in the result of the preced- ing Article, and we obtain the required equation. But in fact this is included in the preceding Article; for that Article does not require ^ to be necessarily a positive quantity. 56. The principal use of the transformation in Art. 54 is to obtain from a proposed equation another which waaits am, as- TRANSFORMATION OF EQUATIONS. 36 signed term. Thus if we wish the transformed equation in 2/ to be without its second term, we take k such that p^ + npjc — 0, that is, k = — "^-^ . If we wish the transformed equation in y Wo to be without its third term, we must find k from the quadratic equation p, + {n- \)pjc + '^^^ > o^ = ^• And generally, if we wish the transformed equation in 3/ to be without its (r + 1)*'' term, we must find k from an equation of the r^ degree, namely r r(r-V) \r\n-r pu + "Lpk^-^ + "^y Y, p,k^-'^ . . . + ^= p=^: We shall see hereafter that the solution of an equation is some- times facilitated by first removing some assigned term. p 67. For example, transform the equation x^ - W + 4a; + 5 = into another without its second term. Here PQ = ^i p^ = —^ ') thus ^ = 2, and the required equation is (y + 2)»-6(2/ + 2)«+4(2/ + 2) + 5 = 0, that is, y'- 8^-3 = 0. I^P' Again, transform the equation x^—2x^— 4a;+9 = into another without its third term. Put y + k for x ; the transformed equa- tion is {y + ky-2{y + ky-4:(y + k) + 9 = 0, that is, y'+ y' {Sk -2) + y (3F- 4^ - 4) + ^»- 2^- 4yfc + 9 = 0. ^fe If the third term is to disappear k must be found from the equation 3A^ - 4yb - 4 = ; this gives k=2 or - f . With the value k = 2 the transformed equation is /+ 42^=4-1 = 0. With the value k = -^ the transformed equation is I 3—2 A .-A 36 TRANSFORMATION OF EQUATIONS. 58. To tromsform an equation into a/nother the roots of which a/re the reciprocals of the roots of the proposed equation. Let fix) = denote the proposed equation. Assume y = - , X so that when x has any particular value, the value of y is the reciprocal of that value ; thus x=~ and the required equation is 4)=»- Thus ^^f{x)=p^x''+p^x'"^+p^x''~^+...-\-p^_^^x+p^ the equa- tion / (-) = is y y iT 2/ ^^ that is, i?„2/"+;?„_y" '+i?„_y + ... +i?iS^ + i?o = 0. 59. To transform a/n equation into a/nother the roots of which a/re the squares of the roots of the proposed equation. Let/(£c) = denote the proposed equation. Assume y = x^j so that when x has any particular value the value of y is the xSquare of that value : thus x = Jy and the required equation \sf{Jy) = 0. Thus if /(ic)=i?o*'*+i?ii«"~^+i?a^"~^+ ... +^„_iaj + ;?^ the equa- tion f{Jy) = is « n-l n-a , py+p,y' +Ps^' +'-'+p„-y+Pn=^- By transposing and squaring we have / n n-2 n--4 Ns / «-l n-S Ns Kpy+p^y" +P4y' +•••/ =\Fi2/' +p^' +•••/• The equation will be in a rational form when both sides are developed, and by bringing all the terms to one side we obtain PoV+(^p,p.-p.') }/'-'+ {2p.p.+p,'- ^p.p>) t/"-'* - =0. TRANSFORMATION OF EQUATIONS. 37 60. These cases of transformation of equations might be increased, but we have given sufficient to explain this part of the subject. "We -will conclude with three examples which will illus- trate the use of some of the relations established in Art, 46. (1) If the roots of the equation x^+pa^ + g'a; + r = be a, 6, c, form the equation of which the roots are a h c V 6 + c' c + a^ a + h' Denote the required equation by Then we have, by Art. 45, ^ a b c -F = Y + + -,, o+G c+a a+o ah ho ca ^^{b + c) (c + a) ■*" (c + a) (a + b)'^ (a + h) (b + c)* (b + g){c + a) (a + b)' a+b + c = -p, ab + bc + ca = qy abc = -r. Thus we may now proceed to express the values of P, Q, and in terms of p, q, and r. For example i? = ^^K "" (b + c){c + a){a + b)' now by actual multiplication we find k(b + c) {c + a)(a+b)={a + b + c) {ah + bc+ ca) - a^ = -pq + r; T therefore R = . K T-pq Similarly we can express P and Q, 38 TRANSFORMATION OF EQUATIONS. But we may evade the trouble of this process by an algebraical artifice. We have b + c a + b + c-a —p — a' Thus if V = , when x takes the value a the value of y is ^ p + x' ^ , : and similarly when x takes the values b and c the values of y are respectively and c + a a + 6' Thus the required equation will be obtained by eliminating x X between the proposed equation and y = . Hence x-- y^— ; and by substituting this value in the pro- posed equation we obtain {\^yf^{\^yf 1+2/ ' . or r (1 + yj +pY (1 + y) -pqy (1 + yf-vY = 0, that is (r -pq) y^+{3r + p^- 2pq) y^ + (3r -pq) y + r = 0. Hence by this method we arrive indirectly at the values of P, Qy and E : we see that a b c _ Sr+p^— 2pq b + c c + a a + b r —pq ab be ca 3r-pq (b + c)(c + a) (c + a)(a + b) (a + b) (b + c) r-pq ^ abc _ r (h + c){c + a)(a + b)~ r -pq ' The last result has already been obtained by direct investiga^ tion. TRANSFOEMATION OF EQUATIONS. 39 (2) Required to transform the equation x^ + qx + r = into another the roots of which are the squares of the differences of the roots of the proposed equation. Let a, by c denote the roots of the proposed equation; then, by Art. 45, a + b + c = 0, ab + bc + ca=q, abG = — r; therefore a^ + 6^ + c^ = - 2q. The roots of the transformed equation are to be {a - bf, (b - cf, and {a-cy, now = -2g + (T: ^ c 2r thus iiy = -2q + of, when x takes the value c the value of y X is (a — bY; and similarly when x takes the values a and 6, the values of y are respectively {b — c)' and (c - a)^. Thus the transformed equation will be obtained by eliminating x between the proposed equation and y = — 2q + x^. Thus x^ + qx + r = 0, and x^+(2q + y)x-2r = 0; therefore {q + y)x-3r=0. 3r Hence x = ; substituting this value in the proposed equation and reducing, we have finally f + Qqy" + 9q'y + 277^ + 4^' = 0. Thus if 27 r^ + 4^^ is positive the transformed equation has a real negative root by Art. 20; and therefore the proposed equation must have two imaginary roots, since it is only such a pair of roots which can produce a negative root in the transformed equation. If 27r^ + 4^^ is zero the transformed equation has one root equal to zero, and therefore the proposed equation must have two equal roots. 40 DESCARTES'S RULE OF SIGNS. (3) Eequired to transform the equation oi? + 'pv? + g'as + r = into another the roots of which are the squares of the differences of the roots of the proposed equation. Put x = Qc! -^\ thus the proposed equation becomes that is, o;'^ + g-V + r' = 0, where ^ =g._^, y=^_^+^. Each root of the last equation exceeds the corresponding root of the proposed equation by -^ ; and thus the squares of the differences of the roots of the last equation are the same as the squares of the differences of the roots of the proposed equation. Therefore by the former example the required equation is y^ + e^y + ^(^Sj + 27r'^ + 4^'^ = ; that is, 8 o/o ^^ 2 /Q 2X2 (2/-9^g + 27r)V4(3g-/)^ . Hence if a, 6, c are the roots of o^ + poi? + gcc + r = 0, we see that (a-6y+(6-c)^ + (c-a)^ = -2(3g-/), (a-6)^(6-cr(c-<=-I,{(2/-9i7^+27rr + 4(3g-/)^}. Y. DESCARTES'S RULE OF SIGNS. 61. We have already in Arts. 21... 2 4 given instances of the connexion which exists between the signs of the coefficients in /"(aj) and the nature of the roots of the equation /(a?) = 0, and we now proceed to investigate a general theorem on the subject after some preliminary definitions. DESCARTES'S RULE OF SIGNS. 41 62. Wlieii each term of a set of terms has one of the signs + and — before it, then in considering the terms in order, a continioa- tion is said to occur when a sign is the same as the immediately preceding sign, and a change is said to occur when a sign is the contrary to the immediately preceding sign. Thus in the expres- sion x^ -{- 3a5^ - 4:X^^ 7x^ + Sx'^'^ 2x^-x^ + x+l, there are four con- tinuations and four changes j the first continuation occurs at —4:X^, the second at + 3x*, the third at + 2x^, the fourth at - a; ; the first change occurs at - Saj^, the second at + 7x^, the third at -x', the fourth at +1. It is obvious that in any complete equation the number of continuations together with the number of changes is equal to the number which expresses the degree of the equation j see Art. 51. And if in any complete equation we put —x for x, the continuations and changes in the original equation become respectively changes and continuations in the new equation. In an equation /(x) = which is not complete, the sum of the numbers of the changes of f(x) andy*(— a;) cannot be greater than the degree of the equation; because if terms are missing iny*(fl3), although it may happen that, the number of changes in f(x) or in f{-x) is thus diminished, it cannot be increased. We shall now enunciate and prove a theorem which is called Descartes's Rule of Signs. 63. In a/ny equation^ complete or incomplete, the number of positive roots cannot exceed the number of changes in the signs of the coefficients, a/nd in any complete equation the numher of negative roots cannot fxceed the number of continuations in the signs of the coefficients. We shall first shew that if any polynomial be multiplied by a factor x — a there will be at least one more change in the product than in the original polynomial. Suppose for example that the signs of the terms in the original polynomial are + + + - + 1-. We have to multiply the polynomial by a binomial in which the signs of the terms are + -. 42 DESCARTES'S RULE OF SIGNS. Then "writing down only the signs which occur in the process and in the result we have •+ + --- + - + -- + + - + + + - + -- + A double sign is placed where the sign of any term in the product is ambiguous. The following laws will be seen by inspection to hold. (1) Every group of continuations in the original polynomial has a group of the same number of ambiguities corresponding to it in the new polynomial. (2) In the new polynomial the signs before and after an ambiguity or a group of ambiguities are contrary. (3) In the new polynomial a change of sign is introduced at the end. Now in the new polynomial take the most unfavourable case and suppose all the ambiguities to be replaced by continuations ; by the second law we may then without influencing the number of continuations adopt the upper sign for the ambiguities ; and thus the signs of the original polynomial will be repeated in the new polynomial, except that by the third law there is an additional change of sign introduced at the end of the new polynomial. Thus in the most unfavourable case there is one more change of sign in the new polynomial than in the original polynomial. If then we suppose the product of all the factors corresponding to the negative and imaginary roots of an equation already formed, by multiplying by the factor corresponding to each positive root we introduce at least one change of sign.". Therefore no equation can have more positive- roots than it has changes of sign. To prove the second part of Descartes's rule of signs we suppose the equation complete, and put - y for x ; then the original conti- DESCARTES'S RULE OF SIGNS. 4lt nuations of sign become changes of sign. And the transformed equation cannot have more positive roots than it has changes ; and thus there cannot be more negative roots of the original equation than the number of continuations of sign in that original equation. 64. Whether the equation f(x) = be complete or not its roots are equal in magnitude but contrary in sign to the roots of y(_ x) =0, that is, the negative roots of/(fl3) = are the positive roots of /(— £c) = 0; and whether the equation be complete or not the number of the positive roots oif(—x)=0 cannot exceed the number of changes of sign iny*(— x). Thus the whole rule of signs may bo enunciated in the following manner : an equation f(x) = cannot have more positive roots than /(a:;) has changes of sign, and cannot have more negative roots thany(— x) has changes of sign. y^ 65. For example, take the equation x* + 3x^ + 5x — 7 = 0. Here there is one change of sign, and therefore there cannot be more than one positive root. And by writing — x for x we obtain the equation x* + Sx^ -5x-7 = ; here there is one change of sign, and therefore there cannot be . more than one positive root, so that the original equation cannot have more than one negative root. Thus the original equation cannot have more than two real roots. In this example we know by Art. 21 that there is one positive root, and that there is one negative root ; and we have just ascertained that there cannot be more than one of each. Again, consider the equation x^ + qx + r = 0, where q and r are both positive. Here there is no change of sign, and therefore no positive root ; this also appears from Art. 24. If we write — X for X, we obtain an equation with one change of sign, so that the original equation cannot have more than one negative root, and therefore the original equation must have two imaginary roots. Again, consider the equation x^ — qx + r = 0, where q and r are both positive. Here there are two changes of sign, and there- fore there cannot be more than two positive roots. If we write 44} DESCARTES'S RULE OF SIGNS. — dj for X, "we obtain an equation with one change of sign, so that the original equation cannot have more than one negative root. In this example we know by Art. 20 that there is one nega- tive root, and we have just ascertained that there cannot be more than one; whether the other two roots are real positive quan- tities or imaginary, we cannot infer from Descartes's rule of signs. But from Art. 60 it follows that the equation which has for its roots the squares of the differences of the roots of the proposed equation is y^ - Qqy^ + ^q^y + 27r^ - 4g'* = j and by Descartes's rule of signs, or by Art. 24, if 27r^-4g'" is negative, the last equation has no negative root, and therefore the original equation no imaginary roots; also if 27r^— 4$'^ is positive, the last equation has a negative root by Art. 20, and therefore the original equation must have two imaginary roots. 66. The student should observe that the results given in Art. 24, are all consistent with Descartes's rule of signs, and may all be deduced from it. Also the proposition in Art. 22 is included in Descartes's rule of signs; and we learn from this rule that such an equation as that considered in Art. 22 can have only one positive root, without repetition ; see Art. 35. 67. It is shewn in the proof of Descartes's rule of signs, that on multiplying a polynomial by the factor which corresponds to a real positive root, one change of sign at least is introduced ; it may be observed, that the number of the changes of sign introduced must be an odd number. For suppose in the first place that the last sign in the original polynomial is + ; then since the first sign is 4- , the whole number of changes of sign in the original polynomial must be an even number or zero ; and the sign of the last term of the new polynomial is -, so that the number of changes of sign in the new polynomial is an odd number. Therefore an odd number of changes of sign must have been introduced. Next suppose that the last sign in the original polynomial is — , so that the last sign in the new polynomial is + ; then there must be an odd number of changes of sign in the original polynomial, and an even number of changes of sign in DESCARTES'S RULE OF SIGNS. 45 the new polynomial. Therefore an odd number of changes of sign must have been introduced. 68. When all the roots of an equation f (x) = a/re real, the number of 'positive roots is equal to the number of changes of sign in f (x), a7id the number of negative roots is equal to the number of changes of sign ini{—x.). Let n denote the degree of the equation, m the number of positive roots, and m' the number of negative roots, fx the number of changes of sign in /"(a?), and fi the number of changes of sign in/(— a;). Since all the roots of the equation are real m + m' -^n. Also m cannot be greater than fx, and m' cannot be greater than /x', by Art. 63. Therefore fjL + fx^ = n, for the sum of /a and /a' cannot exceed n. Thus m, + m' = ix + fx'. And m cannot be greater than fx; nor can m be less than fx, for then m' would be greater than fx\ which is impossible. Thus m = fx, and m' = fx. In this proposition we assume that f(x) has a term in- dependent of X, so that the equation f(x)=0 is not satisfied by a; = 0. A root zero cannot properly be considered either positive or negative. If we wish to introduce the consideration of zero roots we may proceed thus: suppose the equation to have m positive roots, m negative roots, and the root zero repeated r times. Then we have m + m' + r = nf so that m + m,' = n — r. And we can shew that fx + [X can be neither less nor greater than n — r; so that fx + fx' = n — r. Then as before m = fx and m' = fx'. 69. Suppose fx the number of changes of sign in f{x), and fx' the number of changes of sign in /(- x). Then the equation f(x)=0 cannot have more than /x positive roots, and cannot have more than fx negative roots, and therefore cannot have more than fx + fx' real roots. Hence if n is greater than jx + fx the equation f(x) = must have at least n- fx-fx imaginary roots. In the next two Articles we shall shew more definitely what inferences we can draw as to the number of imaginary roots of an equation when that equation is not complete. 46 DESCARTES'S RULE OF SIGNS. 70. If any group consisting of an even number of terms is deficient in a/ny equation there are at least as many imaginary roots of tJie equation. Suppose the 2r terms which might occur in /(a?) between a;'" and a;"'"^"^ to be deficient; then the equation /(a?) = will have at least 2r imaginary roots. Let A and B denote the co- efficients of cc"* and cc'""^''"^ respectively in f{x), and suppose the deficient terms introduced with coefficients q^, q^^ ^si-'-i ^^^ ^^" note the new function by F {x). Then in the expression Ax"" + q^x""-' + q^x""-' +...+ q.x'"-''- + Bx'"-'"-' the number of changes of sign together with the number of continuations of sign is 2r + 1; in other words the number of changes of sign in this expression, together with the number of changes of sign which it would present if the sign of x were changed, is 2r + I. But now let the hypothetical terms be re- moved; then if A and £ are of contrary signs there will be one change of sign for f(x), and no change of sign for f(- x) ; and if A and JB are of the same sign there will be one change of sign fory(— £c) and no change of sign ior f{x). Therefore in both cases the loss of 2r terms ensures the loss of 2r from the sum of the number of changes of sign in F{x) and in i^(— £c). And this result holds for every deficient group consisting of an even number of terms. Thus there are at least as many imaginary roots of the equation f{x) = as the sum of the num- bers of terms in such deficient groups. 71. If any group consisting of^a^ odd number of terms is deficient in any equation^ the equation has at^ast one more than that number of imaginary roots if the deficient group is between two terms of the same sign, and the equation has at least one less than that number of imaginary roots if the deficient group is between two terms of contrary signs. Suppose the 2r-h 1 terms which might occur in f{x) between cc"" and aj"'"^''"^ to be deficient. Let A and B denote the coefficients of aj"* and a;'""^''"^ in f{x) respectively ; then if A and B are of the same sign the equation f(x)=0 has at least 2r + 2 imaginary DESCARTES'S RULE OF SIGNS. 47 roots; if A and B are of contrary signs the equation /(x)=0 has at least 2r imaginary roots. Suppose the deficient terms introduced with coefficients q^j q^y ^3,...; and denote the new function by F(x). Then in the ex- pression Ax"^ + q^x"^-' + q^x^-^ + . . . + ^^r+i^""""' + ■5^"'""""' the number of changes of sign together with the number of continuations of sign is 2r + 2 ; or in other words the number of changes of sign in this expression, together with the number of changes of sign which it would present if the sign of x were changed, is 2r + 2. But when the hypothetical terms are removed there will be no change of sign either for f{x) or /(- x) if A and B have the same sign, and there will be one change of sign for f{x) and one change of sign for /(- x) if A and B have contrary signs. Therefore the loss of 2?* + 1 terms from F {x) ensures the loss of 2r + 2, or of 2r, from the sum of the number of changes of sign in F (x) and in F (- a;), according as the deficient group is between two terms of the same sign, or of contrary signs. And this result holds for every deficient group consisting of an odd number of terms; therefore there will be at least as many imaginary roots of the equation f{x) = as the sum furnished by considering the deficient groups. 72. Thus as an example of Art. 71 we see that if a single term is deficient any where in f{x) between two terms of the same sign, there must be at least two imaginary roots; if a single term is deficient between two terms of contrary signs we cannot deduce from this fact any inference as to the number of imaginary roots. It will be observed that when in consequence of the deficiency of terms the sum of the number of changes of sign in f{x) and f{—x) falls short of the number which expresses the degree of the equation f(x) = 0, the difierence is always an even number. This appears from the examination of the two possible cases in Arts. 70 and 71. That is, with the notation of Art. 69, the number n- fi — fi is always an even number. This might have been anticipated from Art. il. 48 ON EQUAL ROOTS. VI. ON EQUAL ROOTS. 73. It is sometimes convenient or necessary to know whether a proposed equation has equal roots, as we shall see in the course of the work. We shall therefore now explain how we can de- termine whether an equation has equal roots, and how we can remove factors which correspond to the equal roots when they exist, and thus reduce the equation to one which has only un- equal roots. We have first to prove a property concerning the first derived function of a given function. 74. Let f (x) he any rational integral fu/nction of x and f ' (x) the first derived function; then will •^ ^ ' x — a x-b x — G x — fc where a, b, c,...k, are the roots real or imaginary of the equation f(x) = 0. For let Pq be the coefficient of the highest power of a; in /(a;), then we have identically by Art. 33, f(x) =p^{x-a){x-b) {x-c)...{x- h). (1) Put 2/+ « for x; thus f{y + z)=p^{y + z-a){y + z-h){y+z-c)...{y+z-k)] expand each side in a series proceeding according to ascending powers of ^; then the left-hand side becomes by Art. 10, /(y)+/(y)^+/"(2/)i^ + -' Thus the coefficient of z is fiy), and therefore f'{y) must be equal to the coefficient of z on the right-hand side, that is, to that is, to /(y) , f(y) , /(y) , ___ , /(y) y-a y-b y-c '" y-k' ON EQUAL ROOTS. 49 And as it is immaterial what symbol we use for a variable which, may have any value, we may change y into x; thus we have X— a X /'W=^-^z^-t^ — & (2) The result here obtained is true if among the quantities a, h, c,...k, there should occur one or more equal to a, or equal to 6,... and so on. Suppose that on the whole a occurs exactly r times, h exactly s times, c exactly t times,... ; then (1) may be written /{x)=p,{x-ay{x-by{x-cY..., and (2) may be written f (x) = "i^ + ^-^ + ^ + ... *^ ^ ' x — a x-h x — c 75. The equation f (x) = has or has not equal roots a>ccording as f (x) and f ' (x) have or have not a common measure which in- volves X. Suppose a, 6, c,...^ the roots real or imaginary of the equation fix) = 0, so that y* (cc) =p^(x-a)(x-b) (x—c) ...{x-k)'j then f{x) =^Pq{x -b)(x-c)...{x- k) +Pq{x - a){x- c) ...{x~k) -^ ... If a, h, Cf...k are all unequal, none of the factors aj — a, a; -5, x-Cj...x-k will divide /' (x), for (x - a) for example divides every term in /'(a;), except the first; and no product of any number of them will divide f\x). Thus if f{x) has no equal factors /(x) and/'(ic) have no common measure. Hence i{ f{x) and /'(x) have a common measure the factors oif{x) cannot be all unequal. tNext suppose that the equation /(x) = has equal roots ; pose that a occurs r times, that b occurs 5 times, that c occurs t times, and so on. Then •'(x)=^.(x-»r(.-6)-(.-c)'...{^+^+^^+...}. In this case the factor (x - ay~^ (x - b)''^ (x - cY~^ . . . occurs in every term oif'{x). Thus if/ (a?) has equal factors, /(oj) and /'(a;) have a common measure. Hence if f{x) and /'(^) ^^"^® ^^ com- In measure /(cc) has no equal factors. T. E. 4 50 ON EQUAL ROOTS. 76. For example, consider the equation f{x) = a;* - lla;« + 44a;' - 76a; + 48 = 0. \ Here /'(a;) = 4a;' -33a;'+ 88a;-76. It will be found that f {x) and f'{x) have the common measure x-2'j this shews that (x — 2)^ is a factor oi/(x). It will be found that /(x) = (x-2y{x'-7x + 12) = (a;-2^f (a;- 3)(a;- 4); thus the roots of the equation /(a;) = are 2, 2, 3, 4. Again, consider the equation f{x) = 2a;' - 12a;^ + 19a;'- 6a; + 9 = 0. Here /(a;) and /'(a;) will be found to have the common measure a; - 3 j and /(a;) = (a; - 3)" (2a;' 4- 1). Thus the roots of the equation /(.) = 0are3,3,+ y(-g, -^(-i). • 77. In the enunciation of Art. 75, the words *' which involves x " occur at the end. We mean to indicate by these words that we do not regard the factor p^, although that may in a certain sense be considered as a common measure of /(a;) and /'(a;). As we are here for the first time making an important use of common measures of expressions it will be convenient to in- troduce a remark on the subject. It is usual to consider the theory of common measures and of the greatest common measure in works on Algebra; but the theory is not necessary at an early stage of mathematical study, and becomes more intelligible after the result has been obtained which we have given in Art. 33. Let f{x) and ^ (a;) denote two rational integral functions of x) then /(a;) and «^ (a;) may be resolved into factors, so that f{x) =;?„ {x - a,) {x - aj (a; - ^3) . . ., cf> (x) = q,{x- h,) {x - y (a; - 63)...; and each of the functions can be thus resolved in only one way. Hence the function of x of the highest degree which will divide ON EQUAL ROOTS. 51 both f{x) and <^ (a;) is the product of all the common factors of the first degree in x, and this we may call the greatest common measure oi f(x) and (x). Here we have taken no notice of p^ and q^; but we may if we please find their greatest arithmetical common measure if they are numbers, or if they are both functions of another quantity, as 2/, we may find the greatest common measure of these functions of y. 78. Suppose / {x) =p^(x- ay(x - hy{x - c)*. . . ; then we have found in Art. 75 that y (a;) and f'{x) have the common measure {x - ay~^(x — by~^(x - c)'~\ ... Thus the common measure involves all the equal factors which occur in /(a;), but the exponent in each case is less than the corresponding exponent in /(x) by unity. If we divide /(cc) by the common measure oif{x) and/' [x), the quotient involves all the factors which occur in f{x)^ each factor occurring singly. Thus the equation obtained by putting this quotient equal to zero contains without repetition all the roots which the equation f(x) = has. 79. "We see that if the factor {x — aY occurs m/{x) the factor (x - ay~^ occurs in /'(a;); so that the equation /'(a;)=0 has r-1 roots each equal to a. ^owf"(x) is the first derived function of/\x); thus if r - 1 be greater than unity /' (x) and /'' (x) will have a common measure, and the equation f"{x) = will have r - 2 roots equal to a. Thus in this way we can shew that if (a; - ay is a factor oi f{x) then the derived f unctions /'(a;),/'' (a;),.. /''"^ (a?), all vanish when x = a. This may also be proved in the following way. Let f{x) = {x- ay <^ (x), where (x) is a rational integral func- tion of X which is supposed not to contain the factor x — aj put x = a + zj thus 4—2 52 ON EQUAL EOOTS. As the left-hand member of this identity is divisible by ^ the right-hand member must be so too. Therefore we must have /(a) = 0, /'(a) = 0,...../'-(a) = 0. And as the left-hand member is not divisible by a power of z higher than a** the right-hand member cannot be, and therefore /""{a) is not zero. Thus the number of terms in the series f{x), f\x)j /"(x), . . .which vanish when x = a, is the same as the exponent of a3-a in/(£c). For example, suppose f(x)=x'+2x* + 3x^ + 7x' + Sx+3; here it will be found that /'" (x) is the first of the series f{x),f' {x), . . . which does not vanish when x--\] thus the factor (a; + 1)^ occurs in/(a;). It wHl be found that/(aj) = (a; -f- Xfix^ -x + ^). For another example we will investigate the conditions which must hold in order that the equation X* + qx^ + ra? + s = may have three equal roots. Here f{x) = a;* + qx^ + rx-^8, f'ix) = ix^ + 2qx + r, /"{x)=12x' + 2q. Hence from/" (a;) = we obtain ^=-1 «• Substitute this value in /(a;) = and /'(a;) = 0: thus --^^+rx + s = (2), ^(-^+2^)+r=0 (3). From (3) we obtain -I ^ «• ON EQUAL EOOTS. 53 and substituting this in (2) we have I' *-i?-^=o (5)- And from (1) and (4) r'='-% (6). Hence (5) becomes 5 = -^ (7). Thus (6) and (7) express the required conditions. Conversely if (6) and (7) be satisfied, it will be found that 3r /{x), f'{x) and fix) all vanish when » = — — . 80. "We will briefly indicate another way in which the test for equal roots may be investigated. If the equation f{x) = has more than one root equal to a, then it follows that if f{cc) be divided by cc — a the quotient will vanish when x = a. Hence by taking the form of the quotient given in Art. 7, we must have npy-' + {n- l)p,a-'+... + 2ap^_^ + p^_^ = 0; that is, /' (x) vanishes when x = a. -^^81. It appears then that when we wish to determine the equal roots of an equation f(x) = 0, we may begin by finding the greatest common measure oi f{x) and /'{x); then we equate this greatest common measure to zero, and we have an equation to solve which has for its roots those roots of the equation /(a;) = which are repeated. As this greatest common measure may be itself a complex expression, involving repeated factors, it is useful to have a systematic process by which the roots may be obtained with as little trouble as possible. This we shall now give. H^ 82. Suppose f{x) = to be an equation which has equal roots ; and let /(«)=Js:,x/Jr/x,^..x„"•, where the product of all the factors which occur singly in/(£c) is denoted by Xj, the product of all the factors which occur just twice is denoted by A'^", the product of all the factors which occur just three times is denoted by X^, and so on. Any one or more of the quantities X^, X^, Xg,... will be unity, if thero is 54 ON EQUAL ROOTS. no factor in f{x) wMch is repeated just the corresponding number of times. ITow form the first derived function /' (oc) of f{x)j and then obtain the greatest common measure of /(x) and f'{x). We will denote this greatest common measure by /^{x)^ so that Next obtain the greatest common measure of /j(cc) and its first derived function //(a;), and denote it by /^{x), so that Proceed in this way and form in succession /.(x)= x^...x„-'. Now form a new series of functions by dividing each term of the series f{x),f^(x),f^{x),...f^{x) down to/^_^(a;) by the immediately succeeding term. Thus we get /■(*) = XjX^ ...X^, = ^{x) say, ^= X,...X„, = ^{x)=x'-5x'-hi, (l>,{x)=x^+2x'-x-2, ^(x) = x-l, X^ = x-2, X3 = a;' + 3x4-2, X=x-l, Therefore f{x) ^{x-2) (x' + 3x + 2)' {x-iy = {x-2){x+lY{x+2y{x-iy. Thus the roots of the equation /(x) = are 2, - 1, - 1, - 2, - 2, 84. When the coefficients of an equation are all commen- surable quantities the expressions X^, X^,.., of Art. 82 have likewise all their coefficients commensurable. Hence if one and only one of the roots of an equation, with commensurable quanti- ties for coefficients, is repeated r times, that root must be a com- mensurable quantity; for it will be determined by an equation X. = which involves no incommensurable quantities. ^K Hence we can deduce the following results : If an equation of the third degree with commensurable quan- tities for coefficients have no commensurable roots it has no equal roots. For if an equation of the third degree have equal roots, there must be either one root occurring three times, or one root occurring twice and another root occurring once; and in either 56 LIMITS OF THE KOOTS OF AN EQUATION. ca.se, as we have just seen, if the coefficients are commensurable quantities so also are the roots. If an equation of the fourth degree with commensurable quan- tities for coefficients have no commensurable roots it cannot have either one root occurring four times, or one root occurring three times and another root occurring once. If then such an equation have equal roots it must have two incommensurable roots each repeated twice. Thus if f{x) = be the equation f{x) must be a perfect square. If an equation of the fifth degree with commensurable quan- tities for coefficients have no commensurable roots it has no equal roots. For it will be found on examining eveiy case which can exist that if there be equal roots there must be one or more com- mensurable roots. Suppose, for example, that the equation has two roots each occurring twice and another root occurring once ; then if the coefficients are commensurable quantities the unrepeated root must be a commensurable quantity. YII. LIMITS OF THE ROOTS OF AN EQUATION. SEPARATION OF THE ROOTS. 85. In the present Chapter we shall first investigate some theorems which will shew between what limits all the real roots of any proposed equation must lie; and we shall then consider to some extent the possibility of discovering limits between which the real roots separately lie. The advantage of such a Chapter arises from the fact that the algebraical solution of the general equation of any degree above the fourth has not been obtained; and as we shall see hereafter, the numerical solution of equations is a systematic process based on the supposition that we have some knowledge of the approximate values of particular roots. It is to be observed that unless anything to the contrary is specially stated, the whole of the present Chapter relates to the real roots of equations. LIMITS OF THE EOOTS OF AN EQUATION. 57 86. When we say that a certain quantity is a superior limit of the positive roots of an equation, we mean that no positive root can be greater than that quantity. 87. The numerically greatest negative coefficient increased hy unity is a superior limit of the positive roots of an equation which is in its simplest form. Let f{x) = be the equation ; suppose it of the n^^ degree. Let p be the numerically greatest negative coefficient which occurs \iif{x). Then if such a value be found for x that /(a;) is positive for that value of x and for all greater values, that value is a superior limit of the positive roots of the equation f{x) = ; now if any positive value of x make x''-p{x"-' + x"-'+x''-^+ ...+X+1) positive, it will a fortiori make f{x) positive. That is, f{x) is a;" - 1 . positive for a positive value of a; if cc" -^ — — r is positive, and cc" - 1 therefore a fortiori if x^'-l-p :j- Ls positive, that is if ^a;''-l)( 1 ^-~\ is positive; and the last expression is positive if cc - 1 is greater than p. Thus f{x) is positive if x is equal to p + 1 or greater than p + \; that is, jo + 1 is a superior limit of the positive roots of the equation f{x) = 0. W 88. In the equation f{x) = put - y for x, and if n is an J odd number change the sign of every term so that the coefficient of 7/" may be + 1. Let q be the numerically greatest negative coefficient of the equation in this form ; then q +\ is a limit of the positive values of y, and therefore -{q+1) is a limit of the nesrative values of x. I Hence all the roots of the equation f{x) = must lie between 1 and -{q+ 1). 58 LIMITS OF THE ROOTS OF AN EQUATION". Hence a fortiori if m be the numerical value of the greatest coefficient in an equation without regard to sign, all the roots of the equation lie between m + 1 and — (m + 1). 89. In an equation of the n*^ degree in its simplest form if p he the numerical value of the greatest negative coefficient, and x°"' the highest power of x which has a negative coefficient, 1 + ^p is a superior limit of the positive roots. Let f{x) = be the proposed equation ; since all the terms which precede a;""*" have positive coefficients f{x) will certainly be positive for a positive value of x if a:"- jo («;""'■+ £c'*~'"~^+ ... -\-x^ + x + l) be positive^ that is, if x^'—p = — be positive. Hence, sup- X — i posing X greater than unity, f{x) will be positive a fortiori if x"" -p =- is positive, that is if a?" {x — 1) - joa;""'"^^ is positive, that is if aj*""^ (a; — 1) — ^ is positive, that is a fortiori if (aj— l)"" is equal to or greater than p. Hence if x = \ + Up or any greater value, f{x) is positive, that is 1 + J^lp is a superior limit of the positive roots of the equation f{x) = 0. 90. If each negative coefficient he taken positively and divided hy the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased hy unity, is a superior limit of the positive roots. Let the equation be f{x) = 0, where f{x) denotes p^x''+p^x'"^ + p^x''~^-p^x''~^+p^x'"*-¥ ... -p^"'-}- ...+;?„. Now we have a;"'= (a; - 1) (a;'"-'+ a;"'-'+ ... + a; + 1) + 1; let all the terms of the equation with positive coefficients be transformed by means of this formula, and let the others remain unchanged. Thus f{x) becomes LIMITS OF THE ROOTS OF AN EQUATION". 59 + p^{x- l)x"~^ + p^{x- l)a;"~'+ ... +p^{x-l) +pj + p,{x-l)x''-'+... +p,{x- 1) +p,^ + ... Consider now the successive vertical columns of this expression. Where there is no negative coefficient the value of the column is positive if x is greater than unity. To ensure a positive value of the columns in which a negative coefficient occurs we must have y ''■ iPo +Px-^Po){x-^) grea^r than p^ , (i^o+i^i+i^2+ ••• +Pr-i) (^-1) greater.than^^, Therefore x must be greater than ^ + 1,... and greater Po + P,+P. than ~- + 1, ... Therefore if x be taken equal Po-^Pi+P^+'-'+Pr-, to the greatest of the expressions thus obtained, that value of x, or any greater value, will makey(£c) positive; that is, the greatest of the expressions is a superior limit of the positive roots of the ^^^uation fix) = 0. 91. We will now illustrate the rules by two examples. First, take the equation aj' + 8a;* - 14a;' - 53a;' + 56a; - 18 = 0. By Art. 87 we have 53 + 1, that is 54, as a superior limit of the positive roots. By Art. 89, since 7i = 5 and r = 2, we have 1+^53 as a limit, so that 9 is a limit. By Art. 90 we have to take the greatest of the following ex- pressions ; - — - + 1, ^j — ^ + 1, - — - — -~ + 1, that is, we must take ^ ' 1 + 8 '1 + 8 ' 1 + 8 + 56 ' 53 -Q- + 1 ; so that 7 is a limit. GO LIMITS OF THE ROOTS OF AN EQUATION. Again, take the equation x' - 5x* - Ux^ + 2x'' + x- 70 =^ 0. Here Arts. 87 and 89 give 70 + 1 as a limit; and Art. 90 gives 70 -— + 1, so that 19 is a limit. 4 Thus, in both these examples, Art. 90 supplies us with the smallest superior limit. It is easy to see that Art. 89 always gives a smaller limit than Art. 87, except when r= 1, and then the two limits coincide. Art. 89 is advantageous in general when several positive coefficients occur before the first negative coeffi- cient, so that r is large. Art. 90 always gives a smaller limit than Art. 87, except when the greatest negative coefficient is preceded by only one positive coefficient, namely that of the first term, and then the two limits coincide. Art. 90 is advantageous in general when large positive coefficients occur before the first large negative coefficient. 92, By particular artifices we may frequently obtain a smaller supei'ior limit than the general rules supply. Consider the first example of the preceding Article. Here we have to find a superior limit of the positive roots of /(x) = 0, where f {x) may be written thus, x\x'-53) + 8x' (^^-y) + ^^(^-^) - now if X be equal to 4, or to any greater number, the expressions within the brackets are all positive, and so f{x) is positive. Thus 4 is a superior limit of the positive roots of the equation /{x) = 0. Again, consider the second example of the preceding Article. Here we may write f{x) thus, x^af-5x-lS) + 2x' + x-70; now by the aid of Art. 87 we see that x^ — 5x — lS is positive if 05=13 + 1 or any greater number, and obviously 2^3^ + a; -70 is positive when ic=14 or any greater number. Thus 14 is a superior Hmit of the positive roots of the equation /(x) = 0. LIMITS OF THE BOOTS OF AN EQUATION. 61 93. "We may now easily find an inferior limit of the positive roots of an equation, that is a number which is not greater than any of the positive roots. For transform the proposed equation into one whose roots are the reciprocals of the roots of the pro- posed equation, and then the reciprocal of the superior limit of the positive roots of the transformed equation will be an inferior limit of the positive roots of the proposed equation. Thus sup- pose the proposed equation to be ' put - for Xj and multiply by y" and divide by p^^, so that the ' transformed equation is •^ p„ p/ p/ p. Let a superior limit of the positive roots of this equation be found by one of the preceding Articles, and denote it by Z; then -= is an inferior limit of the positive roots of the proposed equa- tion. Suppose that we use Art. 87; let — denote that coefficient which is numerically the greatest of the negative coefficients of the transformed equation; then 1-— is a superior limit of the positive roots of the transformed equation, and therefore Pn-Pr is an inferior limit of the positive roots of the proposed equation. Here p^ is in fact the numerically greatest among those coefficients of the proposed equation which have the contrary sign to the sign of p^. For example, in the first equation of Art. 91 we have ^^=-18 I p^=56; thug the positive roots. _ 1 o IS and p^=56; thus — — — — , that is ^- , is an inferior limit of ^. 62 LIMITS OF THE ROOTS OF AN EQUATION. 94. We will now explain another method of determining a superior limit to the positive roots of an equation ; this method is called Newton^s Method. Let/(£c) = denote the equation which is to be considered; put h + y for x and expand f{h + y) by Art. 10. Thus the equa- tion becomes f{h) + 2//' (A) + 1 /"(h) + ... + ^/»(A) = 0. Now suppose h positive and of such a value that f{h), /'(h), f"{h), ./"(^) are all positive; then no positive value of y can satisfy the above equation. But y = x-li^ and as y cannot be positive, X cannot be greater than li; thus h is a superior limit of the positive roots of the equation f{x) = 0. We may observe that if the proposed equation is in its simplest form/"{7i) is neces- sarily positive, being equal to |?*. 95. For example, take the equation x> £c'^ + a;' -4a;' -6a:' -700a; 4- 500 = 0. Here / {h) = h' + h'- W- 6h'- 700h + 500, f'{h) = 5h'+ 4:h'-12h'-l2h - 700, If (h) = 10h'+ 6h'-l2h- 6, i/"'(/») = 10A"+a-4, It is convenient to begin with the last function of h and ascend regularly. Any positive value of h makes /'"' (7i) positive; 7i=l makes f"'{h) positive; A = 2 makes f"{h) positive; 7t=4 makes f'iji) positive; 7^-= 5 makes y(7i) positive. Then it will be found that 7i= 5 makes all the functions of h positive; and there- fore 5 is a superior limit of the positive roots of the proposed equation. It must be observed, that when according to the method here given we begin with the last function and increase the value LIMITS OF THE BOOTS OF AN EQUATION. 63 of h suitably as we ascend to the other functions, we shall not require ever to re-examine the sign of those functions of h which we have passed. For suppose, for example, w^e have ascertained that a certain value a when put for h renders all the functions of A positive up io f"{Ji). Then put a greater value for h, say a + hy and since /"(« + h) =f"{a) + If" {a) + ~/""(<^) + - and all the terms on the right-hand side are positive by sup- position, f"{a + h) is positive also. Hence in the preceding ex- ample, when it was found that h=5 rendered f (h) positive, it was unnecessary to try whether this value of h rendered the other functions of h positive, because the method of proceeding ensured this result. 96. To find the limits of the negative roots of an equation f{x) = we put — 1/ for x, and then find the limits of the positive roots of the transformed equation in y; then these limits, with their signs changed, will be limits of the negative roots of the proposed equation. Take, for example, the equation x'- 7x*- 15ir'4- 3^-4- 4a; -f 48 = 0; put — y for X and we obtain I 48 By Art. 90 we Lave ^ — - — - + 1, that is 5, as a superior 48 limit of the positive roots, and by Art. 93 we have -r^ — = as ^ ' • 48 -f- 7 an inferior limit of the positive roots. Thus the negative roots 48 of the proposed equation must lie between — 5 and — -^ . ^ 97. Having thus shewn how limits may be found between which all the real positive roots of an equation must lie, and limits between which all the real negative roots of an equation ust lie, we proceed to give some theorems with respect to the I 64 LIMITS OF THE ROOTS OF AN EQUATION. situation of the roots taken singly or in groups. It will be seen hereafter that the complete investigation of this part of the subject is involved in Sturm^s Theorem. 98. If we substitute successively for :k. in f(x) two quantities which include between them an odd number of roots of the equation f(x) = 0, we shall obtain results with contrary signs; if we sub- stitute successively two quantities which include between them no root or an even number of roots we shall obtain results with tlie same sign. Suppose A. and fx two quantities of which A is the greater; let a,bfCy..., k, be all the real roots of the equation /(ic) - which lie between X and [i; by Art. 43 we have f(x) -{x—a){x — b){x-c)...{x — lc)\^ (x), where \l/{x) is a function formed of the product of quadratic factors which can never change their sign, and of real factors which cannot change their sign while x lies between A and /x. Substitute successively A. and /t for £c; thus f{\) = {\-a){\-b){\-c)...{X^k)^{\\ /W = (^-«)(/^-^)(/^-^)---(/^-'^)'/'(/A Now all the factors X-a, X-b, \- c,... X^ k, are positive, and all the factors fi-a, fx-b, fx- c,... fi-k, are negative; and ij/ (X) and ij/ (ju) have the same sign. Therefore f(X) and f(fji) have the same sign or contrary signs, according as the number of the roots a, 6, c,..., Jc, is even or odd. 99. Hence conversely, if two quantities when substituted for X in f(x) give results with contrary signs an odd number of the roots of the equation f(^x) = must lie between the two quantities ; if they give results with the same sign either no root or an even number of roots must lie between the two quantities. This result includes that of Art. 19 as a particular case. 100. It is to be observed that the demonstration in Art. 98 does not require the roots a, 6, c,...,k, to be all unequal; only LIMITS OF THE ROOTS OF AN EQUATION. 65 it must be remembered that a root repeated m times is to be counted as m roots. We see that if /(A.) and /(/x) be of the same sign, either no root of the equation /(x) = lies between \ and /x, or else an even number of roots. Now in the preceding Articles of the present Chapter an argument of the following kind has been sometimes used ; the value fi or any greater value of x makes y (a;) positive, therefore /x is a superior limit of the positive roots of the equation y(a:) = 0. It must be observed that by the Avords makes /(x) positive, we mean makes /(x) a positive quantity/ and not zero. For example, iif{x) = (a;- 4)" (a; - 1), then if x is greater than unity /{x) cannot become negative; but we must not infer that unity is a superior limit of the positive roots, for 4 is a root. If then we only know that /(x) cannot become negative for any value of x greater than //,, we cannot infer that there is no root greater than fi; but we may infer that there is either no root or else a root or roots each repeated an even number of times. 101. We shall now investigate an important theorem which furnishes relations between the roots of the equation f(x)=0 and the roots of the equation /' (jc) = 0, where /'(cc) is the first derived function of f{x). The theorem is sometimes called by the name of Bolle, who first used it. -^ 102. A real root of the equation f ' (x) = lies between every facent two of the real roots of the equation f (x) = 0. Let the real roots of the equation f{x) = arranged in de- scending order of algebraical magnitude be denoted by a, h, c,...k. Let {x). N"ow put successively a,h,c,...,h, for x; the last term on the right- hand side of the identity vanishes in every case, and therefore the sign of /'{a) is the same as the sign of {a — h){a-c)...(a- k)^ the sign oif'{h) is the same as the sign of {b - a)(b — c) ... (b — k), the sign oif'{G) is the same as the sign of (c — a)(c — 6) ... (c — k), and so on; and these signs are alternately positive and negative, for the first expression has 7io negative factor, the second expressioD has 07ie negative factor, the thii^d expression has two negative factors, and so on. Hence by Art. 99 an odd number of the roots of the equation y (a:) = lies between every adjacent two of the roots of the equation /(x) = 0. 103. The demonstration of the preceding Article implies that the roots a, b, c,... k, are all unequal. Suppose however that the root a is repeated r times, that the root b is repeated s times, that the root c is repeated t times, and so on. We shall have /{x) = {x-ay{x-by(x-cY...'{x). LIMITS OF THE BOOTS OF AN EQUATION. 67 Let fi{x) denote the greatest common measure oi f{x) and f'{x), that is, let /, {x) = {x- ay-'{x - by-\x - c)*-\ . . Then •0^ = <^(x) ir{x-b){x-c)... ■hs{x-a){x-c)... + ...> + {x-a){x-h){x-G)...(fi'{x). Call this expression F (x) ; then as before we see that the equa- tion F (x) =0 has an odd number of roots between a and b, an odd number between b and c, and so on. And since we have f'{x)=f^{x)Fix), whenever F {x) vanishes so also does f'{x). Thus an odd number of the roots of the equation /' (x) = lies between every adjacent two unequal roots of the equation/* (a?) = 0. With respect to the equal roots of the equation /(x) = 0, we know that the root a which is repeated r times in the equation f(x) = is repeated r — 1 times in the equation /' (a?) = ; simi- larly the root b which is repeated s times in the equation f(x) = is repeated s—1 times in the equation f'{x) = 0: and so on. It will be convenient for us to imagine that the r roots equal to a of the equation fix) = include r — 1 intervals, in each of which a root a occurs of the equation /'(cc) = 0; and similarly for the other repeated roots. With this conception we may regard the enunciation of Art. 102 as holding universally, whe- ther the roots of the equation f{x) = are all unequal or not. 104. No more than one root of the equation /(x) = can lie between any adjacent two of the roots of the equation /' (a?) = 0. For if there could be more than one there would be a root or roots of the equation /' (03) = comprised between them, and so the two roots of the equation f'{x) = which were by supposition adjacent would not be adjacent. And similarly the equation /(x) = cannot have more than one root greater than the greatest root of the equation /' (x) = 0, or more than one root less than the least root of the equation r{x)^o. If the equation f(x) = has all its roots real, so also has the equation /'(cc) = 0; for the latter equation is of a degree lower 5-2 68 LIMITS OF THE ROOTS OF AN EQUATION. than the former by unity, and a root of the latter equation exists between each adjacent two of the roots of the former equation. And generally if the equation f{x) = has m real roots the equa- tion/' (a;) = has certainly m - 1 real roots, and may have more. 105. Since /"(a;) is the first derived function oif'{x), the equation f"(x) = has an odd number of roots between every two adjacent roots of the equation y (oj) = 0. Thus if the equa- tion f(x)=0 has m real roots, the equation /'(x) = has at least m — 1 real roots, and the equation /" (x) = has at least m - 2 real roots. Proceeding in this way we arrive at the result that if the equation /(x) = has m real roots, the equation f''(x) = has at least m-r real roots. Hence if the equation /"{x) = has jx imaginary roots, the equation /(a:) = has at least /x imaginary roots. For if the equa- tion /{x) = had less than /x imaginary roots it would have more than n — fi real roots, supposing n the degree of the equation ; thus the equation /'■(oc) = would have more than n— fi-r real roots, and as this equation is of the degree n — r it could not have so many as /x imaginary roots, which is contrary to the supposition. For example, let /(x) = x"{l— x)". The equation f(^x)=0 has all its roots real, namely, n equal to zero, and n equal to unity. Hence the equation /" (x) = will have all its n roots real and all lying between and 1; this equation is K^ 0-1 ^^+1 , n{n-l)(n+l){n + 2) 106. From Art. 105 we may deduce the following simple test, which will often indicate the existence of imaginary roots in an equation. Let p,_i, Pr, and p,^i he tJie coefficients of three consecutive terms in f(x), then if -pj^ is less than Pr_iPp+i there must he a pair of imaginary roots in the equation f (x) = 0. LIMITS OF THE ROOTS OF AN EQUATION. 69 Take the {n-r- Vf derived function of f{x) and equate it to zero; thus |r+l 1.2 '■ ' '^'^ Put - for X, and multiply by 2/'"^\ and divide by p^^^ \n-r- 1 ; thus y V ^ 1.2 V ^ — If the roots of this equation are all real the sum of their squares is positive; and therefore, by Art. 47, (n—rYp^ {n — r + l)(n — r)p_^ is positive. Therefore P\^x Pr^^ I 2- X xi n-r+1 p^ IS greater than ^_^ ^.-li^..,., d a fortiori p^ is greater than ^^_jjo^^.j. If then this condition does not hold there must be a pair of imaginary roots in the derived equation, and therefore also in the original equation. See also Art. 331. y 107. If we know all the real roots of the equation f'{x) = we can determine how many real roots the equation f{x) = has. For let the roots of the equation f'{x) = be a, ft y, ..., k, arranged in descending order of algebraical magnitude. Substi- tute for X in f{x) successively a, /?, y, ..., k, and observe the signs of the results. Then one root or no root of the equation f{x) = lies between any adjacent two substituted values, according as the corresponding results have contrary signs or the same sign. This follows from Arts. 98 and 104. 70 LIMITS OF THE ROOTS OF AN EQUATION. The equation f(x) = has one root algebraically greater than a, or none, according as /(a) is negative or positive ; and it has one root algebraically less than k if tlie equation be of an even degree and/(K) be negative, or if the equation be of an odd degree and /(k) be positive, otherwise not. See Arts. 98 and 104. Hence the number of real roots of the equation f(x) = will be the same as the number of changes of sign in the series ob- tained by substituting +a) , a, /?, y,... k, — oo , for aj in f{x) suc- cessively. If however y(£c) vanishes when any of the substitutions are made, it indicates that the equation f{x) = has equal roots, and the number of these may be discovered by Chap. vi. 108. As an example we will investigate the conditions that the equation x^—qx + r = may have all its roots possible, sup- posing q a positive quantity. Here /'(x) = 3x^—q, so that the roots of the equation /'(ic) = are =i= /(|j;leta = -f- /(|) Then /W = .(|/-.(|)^.^-2(|)^. First suppose l-j greater than (|) j then if r be positive /(a) andy(y8) are both positive, and the equation y (a:;) = has only one real root, which is algebraically less than /3j if r be negative /(a) and /(/5) are both negative, and the equation f{x) - has only one real root, which is greater than a. Next suppose l-j less thanf^j ; then /(a) is negative and /(P) is positive, and the equation /(^c) = has three real roots, namely one greater than a, one between a and (3, and one algebrai- cally less than ^. LIMITS OF THE ROOTS OF AN EQUATION". 71 109. A method of discovering the situation of the real roots of an equation was indicated by Waring, and reproduced by Lagrange, which we shall now explain ; it is called Waring^ s Method of separating the Roots. Let us suppose that tTie equal roots of an equation, if it has any, have been discovered and the corresponding factors removed, so that we have to deal with an equation which has only unequal roots. Let f{x) = represent this equation. Suppose h to be a quantity which is less than the difference of any two roots, and let s be a superior limit to the positive roots. Substitute for X iD.f(x) successively s, s—k, s — 2k, s—3k,... and so on down to a quantity which is algebraically less than the least root which the equation can have; and observe the series of the signs of the results. Then when a change of sign occurs one root exists between the two corresponding substituted values, and when there is a continuation of sign no root exists in that interval. For since k is less than the difference of any two of the roots we are sure that more than one root cannot occur in each in- terval. We have then to consider how the quantity k may be de- termined. Suppose that the equation has been formed which has for its roots the squares of the differences of the roots of the proposed equation, and that an inferior limit of the positive roots of this equation has been found ; denote this by 8. Then JS is a suitable value for k. " We have already in Art. 60 given an example of the con- struction of an equation which has for its roots the squares of the differences of the roots of a proposed equation, and we shall hereafter consider the question generally: see Chapter xx. It will then be found that on account of the complexity of the result obtained, Waring's method of separating the roots of a proposed equation is generally useless in practice for equations of a degree higher than the third, although theoretically it attains its proposed object. 72 LIMITS OF THE ROOTS OF AN EQUATION. f^ 110. As an example of Waring's method take the equation a;3_3i^-4a;+13 = 0. By Art. 60 the equation which has for its roots the squares of the differences of the roots of the proposed equation is 2/'-42/+441y-49 = 0. Put y = 1 ; thus 49;^'- 441;^*+ 42« - 1 = 0, that is, i^^{z-^) + 42 (^ - A) = ; thus 9 is a superior limit to the values of z, and therefore ^ is an inferior limit to the values of y. Hence /^ , that is, ^ , is less than the difference of any two roots of the proposed equa- tion. Now 4 + 1, that is 5, is a superior limit of the positive roots of the proposed equation, by Art. 87. And - (1 + sfli) is nume- rically a superior limit to the negative roots, by Arts. 96 and 89. Thus all the roots of the proposed equation lie between 5 and - 5. By substituting, in succession for x the values 5, 5-|^, 5--|,... it will be found that one root lies between 3 and 2f , one root > between 2f and 2 J, and one root between — 2 and — 2^. v3.- 111. We will conclude this Chapter with a proposition which may serve as an example of some of the principles already esta- blished. In the equation f{x) = 0, where f(x) = p^x"" + jt?jCc"~' -t- ... + x-r, if q is the numerical value of the numerically greatest coefficient, and r is positive and less than - — — - , there is a real positive root less than 2r. When X is zero f{x) is negative. Now a positive value of x will make f{x) positive, a fortiori^ if it make a;-r-5'(a;"+a;""^+ ... +a;^ + a;^) 1 -OJ""^ positive, that is, if it make x-r-qx^ — positi^'e. J. — X COMMENSURABLE ROOTS. 73 Hence a fortiori f{x) is positive if x is less than unity and {\- x){x - r) - qx^ is positive. Now put 2r for x in the last ex- pression and it becomes r { 1 — 2r — 4gr}, and this is positive be- cause by supposition r (2 -f- 4^) is less than unity. Thus /(xc) is positive when ic= 2r ; and f(x) is negative when a;= ; therefore a root of the equation f{x) = lies between and 2r. In like manner if the last term in f(x) is r instead of — r and r is positive and less than - — — the equation /(x) = has a root between and — 2r. YIII. COMMENSURABLE ROOTS. 112. By a commensurable root is meant a root which can be expressed exactly in a finite form, whole or fractional ; so that it involves no irrational quantities. We shall now shew that when the coefficients of an equation are rational numbers, whole or frac- tional, the commensurable roots of the equation can easily be found. We have seen in Art. 53 that if the coefficients of an equation are rational but not all integers, we can transform the equation into another which has all its coefficients integers and the coeffi- cient of its first term unity. We may therefore confine ourselves to equations of the latter form ; and we shall first shew that equa- tions of that form cannot have rational fractional roots. 113. Tf the coefficients of an equation are whole numbers, and the coefficient of its first term unity^ the equation cannot have a rational fractional root. Let the equation be ^"^ +p,x''-' +p^x''-' + ... ^p^_^x^ +p^_^x+p^ = 0, and if possible suppose it to have a rational fractional root which in its lowest terms is expressed by j- . Substitute this value for x, and multiply all through by 6""^ ; thus 74 COMMENSUBABLE BOOTS. and therefore The last result is impossible because the right-hand member of the equation is an integer^ and the left-hand member is not an integer. Therefore - cannot be a root of the proposed equation. 114. Thus we are only concerned with the investigation of integral commensurable roots, and we shall now explain the method by which they may be found. The method is sometimes called the Method of divisors^ and sometimes Newton's Method. Let the equation be x" + p.x''-' + p^x""-" + . . . ■¥p^_^x^ + p^_^x -vp^ = 0, and suppose a an integral root. Then substituting and writing the terms in the reverse order we have and therefore by division by a Pn a + p„., +p^.,a + ... -i-^X +P,a"-' + a""-' = 0. Hence — must be an integer; denote it by g'j and divide again by a ; thus 5^1 Pn-l , ^ I 1 ^ ^n— 4 I ^ ^n-3 , ^n-2 a Hence ^ — ^-^^^ must be an integer; denote it by q^ and divide again by a, and we shall find that — — ^-^^^ must be an integer. Proceeding in this way after dividing n times by a we shall arrive at a result denoted by ^^^ — ^ -^- 1 = 0. COMMENSURABLE ROOTS. 7o Hence the following conditions are necessary in order that the integer a may be a root of the equation /(a?) = 0. The last term of the equation must be divisible by a. Add to the quotient thus obtained the coefficient of £c in the equation; the sum must be divisible by a. Add to the quotient thus ob- tained the coefficient of x^ in the equation; the sum must be divisible by a. Proceed in this way until n — \ divisions have been effected, add to the quotient the coefficient of £c"~^ ; the sum must be divisible by a and the quotient must be -1. If at any step the required condition is not satisfied the inte- ger a is not a root. 115. We have in the preceding Article found the conditions which are necessary in order that the integer a may be a root of the equation y(cc) = ; it is easy to see that if the last of these con- ditions is satisfied the integer a is a root. For that last con- dition may be expressed thus ; a'' a"-' a"-^ a;' a and if this is true we see by multiplying by a" that a is a root of /(x) = 0. In order then to find all the commensurable roots of an equa- tion we have only to determine all the divisors of the last term, and try whether they satisfy the conditions of Art. 114. The labour will often be lessened by first finding positive and negative limits of the roots, because of course no integer need be tried which does not fall within these limits. 116. For an example take the equation Here 1 + 10 is a superior limit of the positive roots, by Art. 87 ; and by writing —yiovx we obtain the equation y'' + 3y(y-|) + 10 = 0, 76 COMMENSURABLE EOOTS. for which 3 is a superior limit of the positive roots. Hence all the roots of the proposed equation lie between 1 1 and - 3. The divisors of -10 which fall between these limits are 10, 5, 2, 1, -1, - 2 ; and we proceed to try if any of these numbers are roots. + 10 - 1 - 9 + 5 + 2 + 1 - 1 -2 - 2 - 5 -10 10 5 -10 -13 -18 2 -3 - 2 -18 - 2 - 5 -21 - 5 - 1 -21 + 5 In the first line all the divisors of the last term are written which it is necessary to try, and beneath each divisor the results are placed which arise from carrying on the trial with that divisor. Thus taking the divisor 10, we first divide the last term -10 by it, and set down the quotient — 1 ; then we add this to the coef- ficient of X which is — 8, and set down the sum — 9 ; this is not divisible by 10, so that 10 is not a root. With respect to 5 all the conditions are fulfilled, so that 5 is a root. With respect to + 2 and — 2 we arrive at points where exact division is not possible, so that these numbers are not roots. With respect to + 1 and —1 the final condition is not satisfied, so that these numbers are not roots. Thus the only commensurable root is 5 ; and denoting tlie equation hj/(x) = 0, we know that x— 5 is a factor oi/(x). The other factor will be found to be x'^+ 2x+ 2. For another example take the equation x' + 5x* + x^-Ux'- 20x - 16 - 0. It will be found that the commensurable roots are 2, - 2, and -4. 117. It is usual to omit +1 and -1 from the divisors to be tried, as it is simpler to test whether these values are roots by substituting them for x in the given equation. COMMENSURABLE ROOTS. 77 If any powers of x are missing from the proposed equation they should be supposed to be introduced with zero coefficients ; see Art. 51. When we have ascertained by the method here exemplified that certain nuiiibers a, 5, c,..., are the only commensurable roots of an equation y(cc) = 0, it still remains to determine whether any of these roots are repeated. We may divide f {x) by the product (x-a){x — })){^-c)... and denoting the quotient by ^{x) we may apply the method to the equation <^ {x) = 0, and thus determine whether any of the quantities a, b, c,... are roots of this equation. Proceeding in this way we shall determine the repeated roots of the equation /{x) = 0, and how often each root is repeated. Or we may apply the test of equal roots found in Chapter vi. to the equation /{x) = 0. 118. Suppose that instead of taking an equation, with unitT/ for the coefficient of the first term, as in Art. 114, we take an equation with any integer p^ for the coefficient of the first term. The only difierence in the resulting conditions is that the last quotient must be -p^ and not — 1. Suppose for example 2a;'-12a;^+13a;-15 = 0. 15 "re IT- + 1 is a superior limit of the positive roots by Art. 87, and there is no negative root by Art. 24, and by trial we see that 1 is not a root; thus the only divisors of the last term to be used are 5 and 3. The process being arranged as before we have TISi I -3 -5 10 8 2 -10 - 2 ^V — i^/ ^pThus 5 is a root, for all the conditions are satisfied, the last quotient being - 2; and 3 is not a root, because 8 is not divisible by 3. 78 COMMENSURABLE ROOTS. It must be remembered that if the coefficient of the first term is not unity the equation may have a commensurable fractional root; see Art. 113. 119. The number of divisors of the last term which it is necessary to try may sometimes be diminished by the following principle. Suppose a a root of the equation f{x) = ; for cc put m + y, then a - w is a value of y which satisfies the equation firii + y) = 0. The term independent of y in this equation is/(m), and all the coefficients of y are integers, if the coefficients in /(x) are integers and m also an integer; see Art. 10. Thus if a be an integer a — m is an integer and must therefore divide /(m) by Art. 114. Thus any integer a which divides the last term of /{x) is to be rejected if a — m does not divide /(m). Here m may be any integer positive or negative ; the values + 1 and — 1 are advantageous from the ease with which f{m) can then be calculated. Take for example the second equation given in Art. 116; here 4 divides the last term, but 4+1 does not divide /(- 1) which is — 9; thus 4 cannot be a root of the proposed equation. Again, take the example x^ - '20x^ + 164:X - iOO = 0. This equation has no negative root by Art. 24; and by writing it in the form x^(x — 20) + 164: (x— ~-t-] , we see that 20 is a superior limit of the positive roots. The positive divisors of the last term which are less than 20 are 2, 4, 5, 8, 10, and 16. Of these 5, 8, and 10 are not roots; for /(I) = -255, and this is not divisible by 5 — 1, or by 8 - 1, or by 10 - 1. Thus the only divisors of the last term which remain for trial are 2, 4, and 16; it will be found that 4 is a root. 120. As an example of a rational fractional root, consider the equation 4:X*-llx^+7x-6 =0, that is, OF THE DEPRESSION OF EQUATIONS. 79 First, put a; == ^ , in order to transform the equation into one with integral coefficients; see Art. 53. Thus 2/*-ll2/'+142/-24 = 0, that is, 2/'+ V- ll2/'+ 1%- 24 = 0. By Arts. 89 and 96 all the roots of this equation must lie between 1 + ^24 and —{1 +^24:); and we see by trial that +1 and — 1 are not roots. Thus the only divisors of the last term to be tried are 4, 3, 2, - 2, - 3, - 4. Also /(I) = - 20, and this is not divisible by 4 — 1 or by - 2 - 1 j thus the numbers 4 and — 2 may be rejected. The process being arranged as before we have 3 2 -3 - 4 -8 -12 8 6 6 2 22 20 2 1 - 5 -9 -10 -16 -3 - 5 4 -3 4 -1 - 1 y 3 Thus 3 and - 4 are roots; and since ic = f , we have ^ and - 2 as roots of the original equation. IX. OF THE DEPRESSION OF EQUATIONS. 121. In the present Chapter we shall shew how the solution of an equation may be made to depend upon the solution of an equation of lower degree, in certain cases where known relations subsist among the roots; this process is called the depression of equations. 122. When two equations have a root o^ roots in commonf it is required to determine the root or roots. 80 OF THE DEPRESSION OF EQUATIONS. Suppose the equations f{x) = and F{x) = to have a com- mon root a] then y (a;) and F{x) have the common factor x — a. Hence the greatest common measure of f{x) and F (x) must have x — a as a factor. Similarly every factor common to f{x) and F {x) will be a factor of their greatest common measure, and no other factors will occur in the greatest common measure. Hence, if we find the greatest common measure oi f{x) and F{x), and equate it to zero, the roots of this equation will coincide with the required roots which are common to the equations /(£c) = Oandi^(.T) = 0. p If any factor is repeated in f{x) and F {x) it will also be repeated in their greatest common measure. 123. Suppose, for example, we have the two equations x^-^Zx'-bx^-Qx~d>=.0 and x'-^ x^- ^x'+ lOaj - 8 = 0. The greatest common measure of the expressions which form the left-hand members of these equations is £c^ -f 2aj — 8 ; and ' if this be put equal to zero we obtain a? = — 4, or x = 2. Thus 2 and — 4 are the roots common to the two equations. 124. Suppose we know that there exists between a and h, two roots of the equation f{x) = 0, the relation pa + qb — r; it is required to determine these roots. Since a and 6 are roots of the equation f{x) = 0, we have f{a) = 0, and f{h) = ; but 6 = ^-11^ , therefore / (^JlP^\ = q. Thus a is a common root of the equations /(a;) = and/( — ~\ = 0. Hence a may be found by the preceding Article. Thus a is known and then h from the relation pa + qh = r. Hence /(oc) may be divided by the product of the factors x-a and x-b; and if the quotient be equated to zero we obtain an equation for determining the remaining roots of the equation f(x) = 0. OF THE DEPRESSION OF EQUATIONS. 81 125. Suppose, for example, that we have the equation x*-7x^ + Ux'-7x+lO = (1), and that it is known that two of its roots a and b are connected by the relation b = 2a+l. Substitute Sx + 1 for ic in (1) ; thus {2x + iy-7{2x+iy+U{2x+iy-7(2x + l) + 10 = 0, that is 1 6x' -24:X^ -IQx' -4:X + 8 = Q, or 4x'-6x^-ix'-x + 2 = (2). The greatest common measure of the left-hand members of (1) and (2) will be found to be a; -2. Thus a = 2, and therefore 6 = 5; that is, 2 and 5 are two of the roots of the proposed equa- tion. Then it will be found that sd'-7x^+Ux'-7x + lO = {x-2){x-5){x'+l), so that the other roots are =*= aJ{-1). ^ 126. It may happen that another pair of roots a and (3 is subject to the same condition pa + q/S-r. In this case the ex- pressions f(x) and /( — ^ J will have for their greatest common measure an expression of the second degree in x which will in- volve the factors x~a and x - a. If the roots a and b are both repeated in the equation /(a?) = 0, the factor x-a will be repeated in the greatest common measure of/(x) and /(^) 127. Generally suppose that two roots a and b of the equa- tion /(«)=: are connected by the relation b = {a). Then the equations /(cc) = and /{<^(a;)} = have a common root, namely a, and we may determine this common root by Art. 122. 128. There is a case in which the method of Arts. 124 and 126 does not assist us in solving a proposed equation. Suppose, T. E. 6 82 OF THE DEPEESSION OF EQUATIONS. for example, we have an equation f{x) = 0, and it is known that the roots of this equation occur in pairs, and that each pair of roots a and h satisfies the relation a + 6 = 2r. Then accordinir to Art. 124 we should proceed to investigate the common roots of the equations f{x) = and /{2r — x) = 0. But these equations will be found to coincide completely ; for by supposition /(a) = 0, that is,/(2r-5) = 0, and/{6) = 0, that is, /(2r - a) = 0, so that the' roots a and b are common to the two equations. Similarly every other pair of roots is common to the two equations, and so the two equations must coincide. yf^-^ 129. There are various ways in which we may depress the equation in the case considered in the preceding Article; we will explain two of them as they furnish exercises on the subject of the present Chapter. I. We may proceed thus. Assume a — b=2z, so that we have simultaneously /(a)=0, a + h=^2r, a-h = 2z. From the second and third of these equations a = z+r. Substitute in the first equation, so that y*(;2; + r) = 0. From this equation values of z must be found, and then corresponding values of a and h. It is easy to shew that the equation f{r + z) = () only involves even powers of z^ and so if we regard z^ as the unknown quantity the degi-ee of this equation will be half the degree of the proposed equation. For let a and h be one pair of roots of the proposed equation, a and ^ another pair, and so on ; then /(x) = {x-a)(x — h)(x-a)(x~l3)... f{z + r)-{z + r-a){z-\-r — h){z + r — a)(z + r- jS)... .(„ili_.)(„fi-'->)(,.^jS_.)(„i±_^-,).. that is, f{z + r) involves only even powers of z OF THE DEPRESSION OF EQUATIONS. 83 In fact, as no distinction in theory exists between the roots a and 6, it might have been expected that an equation which should be constructed to have --^— for a root would also have as a root ; and such is the case. II. We may also proceed thus. Assume z = ah. Then " {x — a)(x-h)=x^ — {a + b)x + ah=x^- 2rx + z. Hence if z be suitably determined, x^—2rx + z will be a factor of f{x). Perform the process of dividing /{x) by x^-2rx+z until the remainder takes the form Fx + Qj where F and Q are functions of z, but do not contain x. Hence the necessary and sufficient conditions for x^ — 2rx + z being a factor of f{x) are F=0 and ^ = 0. Find by Art. 122 a value of z which will satisfy both these equations ; then find a and b from a + b = 2r and z = ab. 130. Suppose we know that between three roots a, b, c of the equation /{x) = 0, the relation pa + qb + re = s exists ; it is required to determine these roots. Since a, b, and c are roots of the equation /{x) = 0, we have /(a) = 0, /{J) = 0, /(c) = 0. Thus /(a) = 0,/(6)=0,/(l:ii^«*V0. Suppose b eliminated between the last two equations ; we thus obtain an equation which we may denote by <^ {a) = 0. Thus the equations f{x) = 0, and <^ (x) = have a common root a, and this may be found by Art. 122. 131. We will here give a few miscellaneous examples con- nected with the subject of the present Chapter. 6—2 84 OF THE DEPRESSION OF EQUATIONS. (1) It is required to determine the roots of the equation aj" + ^^a;"-' + i^gX""' + . . . + p„ ■= 0, which are all in arithmetical progression. Denote them by «, a + 6, a + 26, By Art. 47, -p^ =a+{a + h) + {a + 2h)-\- ... + {a-¥n- 16), p;- 2p^ = a' + {a + by +{a+ 26)*+ ... (a + n^ibf. r™ . n(n — V)^ That IS, -p^ = na-\--^ — -b, p^- 2p^ = na?^- n{n-l)ab + ^r b' ; see Algebra, Chapter xxx. By squaring the first result and subtracting it from n times the second we obtain / IX . o n^n'-l)b' {n-l)p^'-2np^= ^^ ; thus 6 is known, and then a can be found. (2) The equation x^h 3x^-12x--4:8x-64: = has two roots which are equal in magnitude and of opposite signs ; find them. Here the equation obtained by changing the sign of x will have a root in common with the proposed equation. That is, the proposed equation has a root in common with the equation x'-Sx^-12x^+4:Sx-U = 0. Then by Art. 122 we may proceed to find the greatest common measure of the left-hand members of these equations. Or thus ; by subtraction, 6x^-96x = 0; therefore either x = 0, or else x^=16. The former does not give a root ; the latter gives ic = ± 4 ; and + 4 and — 4 are roots of the proposed equation. EECIPROCAL EQUATIONS. 85 (3) The equation 3x*- 19a;'+ 9x'- 19a; + 6 = has two roots the product of which is 2; find them. 2 Suppose 7/ to denote one root ; then ^ is another; hence 32/^-19?/' + 92/^- 19?/ + 6 = (1); that is, 6y*- 38/+ 3Qy'- 152^/ + 48 = 0, or 3^-192/'+ 18?/'- 762/+ 24 = 0.../ (2). The greatest common measure of the left-hand ^members of (1) and (2) is 3/- 19?/ + 6; and putting this equal to zero we obtain 2/ = ^, or ?/ = 6. Thus ^ and 6 are the required roots. X. RECIPROCAL EQUATIONS. 132. A reciprocal equation is one which is not changed when the unknown quantity is changed into its reciprocal. Hence if a be a root of such an equation, the reciprocal of a, that is, - , is also a root. We shall see that the solution of a ci reciprocal equation may be made to depend on the solution of an equation of not higher than half the degree of the proposed equation. We shall first determine the relations which must hold among the coefficients of an equation in order that it may be a reciprocal equation, and shall then shew how the equation may be depressed and so rendered easier of solution. 133. To find tJie conditions that a proposed equation may he a reciprocal equation. Let the equation be a;"+jp,a;'-^+^^a;'-'+ •.. + p^.^x"" ■¥p^_,x+p^ = 0...{l). Change x into • , then multiply by cc" and divide by ^„, and X 86 RECIPROCAL EQUATIONS. re-arrange the terms; tlius we have a;"+^^a;"-^+^^^a;"-'+...+^2cc'+^aj+— = 0...(2). In order that (2) may coincide with (1), the coefficients of the same powers of x must be coincident ; thus ^' P.' ^' P.'^-' P.' ^"- P.' ^' P,.'- from the last equation we have p„^=l, therefore p^ = + l, or -1, and this gives rise to two classes of reciprocal equations. I. Suppose 2^„= 1 ; then we obtain Pl = Pn-iy P2 = P„-2-> "'Pn-2=P2^ Pn-X= Pi' Thus an equation is a reciprocal equation when the coefficients of the terms equidistant from the first and last are equal. II. Suppose p^= — l; then we obtain Pl-=-Pn-l^ P2=-Pn~2^ '■'Pn-2=-P2^ Vn-,= -P,' In this case if the equation is of an even degree, we have among the above series of conditions p^ = ~p^^ where m = \n, and this is impossible unless p^ = 0. Thus an equation is a reciprocal equation when the coefficients of terms equidistant from the beginning and end are equal in magnitude and of contrary signs ; with the condition that if the equation is of an even degree the coefficient of the middle term is zero. 1 34. A reciprocal equation of the first class of an odd degree has a root — 1, as is obvious by inspection. Thus if /[x) = denote the equation, f{x) is divisible by a:+ 1; see Art. 6. Let <^ (x) be the quotient, then {x) = will be a reciprocal equation of an even degree with its last term positive. A reciprocal equation of the second class of an odd degree has a root + 1, as is obvious by inspection. Thus if y(a;) = denote the equation, /(x) is divisible by a? — 1 ; see Art. 6. Let cf> (x) be the quotient, then <^{x) = () will be a reciprocal equation of an even degree with its last term positive. REciPBocAL equations; 87 A reciprocal equation of the second class of an even degree has a root + 1, and a root — 1. as is obvious by inspection. Thus, if f{x)-0 denote the equation, f{x) is divisible by x^—\', see Art. 36. Let <^{x) be the quotient, then <^{x) = will be a reciprocal equation of an even degree with its last term positive. 135. The statements made in the preceding Article respect- ing the results of certain divisions will probably be admitted as obvious. But it is easy to give formal proofs. Consider the last case, that of a reciprocal equation of the second class of an even degree. Suppose /(cc) = to represent the equation; then we know that /(a:) is such that /(a;) = — cc^/f - j , and we know that /(a;) is divisible by a;'- 1; we wish to prove that the quotient is a function which has the coefficients of the terms equidistant from the first and last equal. • - • We have f{x) = — x'*f [-) t 1 " x'- 1 ^ w therefore „ . .,-,.,.,- 1 - 4 X' And this shews the truth of the statement, since , is what we 1-4 X" 1 . fix) obtain when we chancre x into - in — „---! . 136. It follows from Art. 134 that any reciprocal equation is either of an even degree with its last term positive, or may be depressed to this form. We may then consider this as the standard form of a reciprocal equation, and we shall now shew that such an equation may be depressed to one of half its degree. The fact that a reciprocal equation could be thus depressed was noticed by De Moivre in 1718: see his Doctrine of Chances, first edition, page 113. 88 KECIPROCAL EQUATIONS. 137. It is required to depress a reciprocal equation which is of an even degree with its last term positive. Let the equation be a3""+^^a;^'"~'+ j92a;"""^+ ... +J02aj^+;?ja; + 1 = 0. Divide by a;"* and collect the terms in pairs which are equidistant from the beginning and end; thus i*pi---^)-p^{--^~)-- = o. X Now assume x + - = y: then X nr^ + — - 01^ - 9, X +^2-y ^, and generally, ^'*' + ^^= («''+^) ('« + ^) " (*''"' + ^ ' so that we can express a;'"*'* + -^ as a rational function of ?/ of X the degree ^ + 1. Hence by substitution in the above equation we obtain an equation in y of the degree m. Then from each value of y we deduce two corresponding values of x from the equa- tion x^-yx+1 = 0. 138. Tlie general relation in the preceding Article may be thus expressed ; This shews that we may regard the quantities ,,1 si 3 1 U/"r f X "T 2 ) X n' jjj... as forming a recurring series in which the scale of relation is 1 -?/+!; see Algebra, Chapter xlix. We shall hereafter give in Chapter xxi. a general expression for x^ + —'ni terms of y. BINOMIAL EQUATIONS. 80 139. For an example of a reciprocal equation take the equation Here +1 and —1 are roots by inspection; and we can therefore divide the left-hand member by a;^-l. Thus we obtain 2x'' + x^-nx'+x+2 = 0; therefore x + -^ + 7:{x + -) -^ -^ = 0. x^ 2\ xj 2 Put £c + -=2/; t^s or f^l-'^^O; , therefore y = ^ or - 3. XT 15 1 _ Hence x + -= -, or x + - = — o ; x 2 X therefore x=^2 or ^ or - (- 3 ± ^5). 140, The following equation may be transformed into a re- ciprocal equation : + ... +^^c"-V +p^c"'-'x + 0"* = 0. For assume x = z ^c, and divide by c"* ; we thus obtain a reciprocal equation in z of the standard form. XI. BINOMIAL EQUATIONS. 141. An equation of the form x'*-A = where ^ is a known quantity is called a binomial equation. The roots of this equation are all different because the first 90 BINOMIAL EQUATIONS. derived function of o^~A is wcc""^, and no "v Qi^-A and nd(^~^ vanish simultaneously; see Art. 75. 142. If a;"- -4 = we have x = 'lJA\ that is, x is equal to an w**" root of A. But the equation a;"— ^=0 has n roots by Art. 33, and these roots are all different by Art. 141. Hence we obtain the following important result, any algebraical quantity has n different n*"" roots. By an algebraical quantity here we mean either a real quantity, or an imaginary quantity of the form p + q J—1. 143. Let a denote one of the n^^ roots of any quantity A, so that a"=^. Then in the equation x"-A = assume x = ay, so that a"y''-A = ; therefore ?/"— 1 = 0. Hence y = l/l, that is, y is equal to an n*'^ root of unit}'". And x = ay=a^l; but x = ^A; therefore jA = alJ\. Thus all the n*^ roots of any algebraical quantity Tnay be found by rtivltiplying any one of them in siccces- sion by the values of the n"" roots of unity. 144. Let us now suppose that -4 is a real positive quantity, and that we have to solve the equation cc"-^ = and the equation 'x^+A = 0. Let a be the arithmetical value of the n*^ root of A, which may always be obtained, at least approximately, by the aid of the Binomial Theorem ; see Algebra, Chapter xxxvi. Assume x = ay, then the proposed equations become respectively ^/"-l = 0, and 2/" + 1 = 0. These equations can both be solved by the aid of Trigonometry ; see Trigonometry, Chapter xxiii. We shall however now consider these equations without using the Trigono- metrical expressions ; and although we are not able to solve them generally by means of algebraical expressions, we shall be able to prove important results respecting them. 145. If a be any root of the equation x"-l=0, then a™ is also a root, where m is any integer, positive or negative. For (a")" = a*"" = (a")'" = r = 1. BINOMIAL EQUATIONS. 91 146. If o. he any root of the equation x° + l=0, then a™ is also a root, where m is any odd integer positive or negative. For (a"*)" = a'"" = (a")'" = (1 1)"'= - 1, if m be odd. 147. //* m he prime to n, the equations x™- 1 = and x° - 1 = have no common root except unity. Let p and q be two integers wbicli satisfy the relation pm -q7i = l; such integers can always be found by Algebra j see Algehra, Chapter xlvi. And suppose that a is a common root of the two equations. Then »'"= 1, therefore a^=\', and a"=l, therefore (x'"=l. Hence, by division, a^""'"= 1; that is a = 1. 148. If n is a prime numher, and a any root of the equation x" — 1 = 0, except unity, then all the roots of the equation will he famished hy the series a, a^, a^,...a". For these quantities are all roots by Art. 145. We have there- fore only to shew that no two of them are equal. If possible, suppose a'' = a*j then ar~* = l; and thus the equations aj" - 1 = and ic*""* — 1 — have a common root which is not unity. But this is impossible by Art. 147, since r-s is less than n and therefore prime to it. 149. If Tfc is not a prime number, and a is any root of the equation £c"— 1 = 0, it is true by Art. 145 that any power of a is also a root ; but it is not necessarily true that the successive powers of a will furnish all the roots. Suppose for example that n^pq; and let a be a root of the equation cc'' — 1 = j then a is also a root of the equation a^"— 1 = 0, and so is any power of a. But we can- not obtain more than p different values by taking powers of a; for aP"*"^ = a*" X a = a, a^"^^ = a^ x a^ = a', and so on. Thus the powers of a will not furnish all the roots of the equation a;"— 1 = 0. If n be not a prime number it is still true that some of the roots of the equation a;"-! = have the property of furnishing all the roots by their successive powers. This we shall shew from the Trigonometrical expressions for the roots. J 92 BINOMIAL EQUATIONS. For let r be any integer ; then 2r7r / — - . Irrr cos V J —\ sm n ^ n is a root ; denote it by a. Suppose r prime to n, then the succes- sive powers of a will furnish all the roots. Eor let s and t be two integers, neitlier of which exceeds n ; then a' and a' will not be equal. For , 2sr7r ,—- . 2sr7r a' = cos +J-1 sm , 2tr7r ,— - . 2tr7r a* = cos + J- 1 sm : n n and in order that these should be equal and - — must either n n be equal or differ by a multiple of four right angles. See Flane Trigonometry^ Art. 93. Thus {s ~t)r must be an mteo^er : n ° •' but this is impossible since r is prime to n and s ~ ^ is less than n. 150. The solution of the equation x" — 1 = where n is the* product of different prime numbers can he made to depend upon the solution of equations of a similar form having for the index of x the different prime factors ofn. Suppose, for example, that n is the product of three prime factors m, jo, q. Let a be a root of the equation a?'" — 1 = 0, let ^ be a root of the equation aj^ - 1 = 0, let y be a root of the equation 02^—1 = 0; these roots being all supposed different from unity. Then the roots of the equation a;"-l = will be the terms of the product (l + a + aV...+a"*-^)(l + /5 + /3^+...+/5''-^)(l + y + y=+...+/-^). First, any term of this product is a root. For suppose a'/3*y* to denote such a term; then {afi'y*y=l, since a''"= 1, /5'"- 1, and BINOMIAL EQUATIONS. 93 y = l. Secondly, no two terms of this product are equal. For, if possible, suppose a*" ;S* / = aP /3<^ / j then a'-'^ = Z^*^-' y^"'. The quantity on the left-hand side is a root of the equation x""- 1=0, and the quantity on the right-hand side is a root of the equation x^- 1 = 0; but since m is prime to pq it is impossible that these equations can have any common root except unity. Similarly we may proceed when n has more than three prime factors. 151. Next suppose that the prime factors of ?z occur more than once in n; for example, let n = fJL.ir.K, where /x, tt, k are respectively any powers of the prime numbers 7?i, p, and q. Then it will still be true that if we obtain the /x roots of the equation x'* - 1 = 0, the tt roots of the equation cc'^ - 1 = 0, and the k roots of the equation a;* - 1 = 0, and take every possible product of these roots, one from each system, we shall obtain all the roots of the equation aj" — 1 = 0. But, by Art. 149, the roots of each system cannot necessarily be represented by the powers of one root taken arbitrarily. Similarly we may proceed when n involves more than three different primes. ^_ ^ 162. It is usual to add one more proposition respecting the equation a;"- 1 = when n is a. power of a prime; and we will give it here although it is of little practical importance. Suppose, for example, that n = m^ where m is a prime number. Let a be a root of the equation a;"* - 1 = 0, let ^8 be a root of the equation a;'"- a = 0, and let y be a root of the equation «;'"-/? = 0. Then the roots of the equation cc"- 1 = will be the terms of the product First, any term of the product is a root. For suppose a^'fi'Y to denote such a term; then (a»";S'7')" = a'""/3*''y"= 1. Secondly, no two terms of this product are equal. For, if possible, suppose arp^y'^a^^^'f', thus o} = a\ where l=r+ — + — 75 and X = p + — + — r, . m 71V ' m m 94 BINOMIAL EQUATIONS. Therefore a^"^= 1, therefore a"— 1 = 0, where v = m^{l — \). But m^{J, — X) = m^{r — p) + m(s — a) + 1 — t, and this is prime to m, and therefore to m^ ; and therefore the equations cc" — 1 = and a*'— 1 = cannot have a common root different from unity. ^ 153. The preceding Article is of little practical importance, because the operations ' which it involves cannot be generally effected. Suppose that we can solve the equation a;"* — 1 = 0, and so find a; then all the quantities 1, a, a^, ...a^'^, are roots of the equation a;" — 1 = 0; so that we thus obtain m roots. But to find f3 we have to solve the equation a;"*— a = 0, that is, we have to find ^a where a^H^l; and there is no algebraical method of effecting this generally. Thus, for example, when we have solved the equations a;^- 1 = and a;^ — 1 = we can immediately/ form all the solutions of the equation a;^^- 1 = by Art. 150. But we cannot practically solve the equations a;'- 1 = or aj^^- 1 = by the method of Art. 152; we can only obtain three roots of the former equation and five roots of the latter equation. 154. We will now indicate the methods by which we can practically solve the equations x" — 1 =0'and a;" + 1 = 0, when n is not too great. We may observe however that if n be any power of 2 these equations may be solved by the process given in Algebra for extracting the square root of a binomial surd, repeated as often as is necessary; see Art. 28. If n =pm, where p = 2'', assume a^= y, thus the equations a;"— 1 = and aj"+ 1 = become respectively 7/"*— 1 = and 2/"*+ 1 = 0. Then if y can be found we can deduce x by the process of extracting the square root repeated r times. 155. In the equation a;"— 1=0 suppose that n is an odd number, and let n = 2m + 1 . The equation a;^""*"^ — 1 = has only one real root, namely + 1 ; for it has no negative root, and if x be made equal to any other quantity than unity x^""^^ will not be equal to unity; thus the equation has only one real root. Divide x^'"'^^- 1 BINOMIAL EQUATIONS. 95 byoj— Ij thus we reduce the equation to be solved to the fol- lowing, This is a reciprocal equation, and its solution can be made to depend upon the solution of an equation of the degree m. 156. In the equation a;"— 1 =0 suppose that n is an even number, and let n = 2m. The only real roots of the equation are + 1 and — 1 ; and we may divide x""" - 1 by the product of a; — 1 and x+1, that is, by a;*— 1. Thus we reduce the equation to be solved to the following, x"^-' + x'"'-''+.,.+x'+l = 0. This is a recijorocal equation, and its solution can be made to depend upon the solution of an equation of the degree m — 1. The equation aj^"* -1 = may also be conveniently treated by writing it thus, (a:;'"— l)(£c"*+ 1) = 0, and so resolving it into the equations a;'" — 1 = and aj"* + 1 = 0. Or we may adopt the method given in Art. 154. 157. In the equation 33"+ 1=0, suppose that n is an odcb number, and let w = 2m+l. The equation a;^'"'*'^ + 1 = has only one real root, namely — 1 ; and we may divide a;^"*"^^ + 1 by a;+l, and thus reduce the equation to be solved to the following. X 2m— 1 , ^m— 3 + a;'''"-'-...+aj'-a;+l = this is a reciprocal equation, and its solution can be made to depend upon the solution of an equation of the degree m. If n is an odd number in the equation a;" + 1 = 0, and we change x into — aj, we obtain a;"— 1 = Oj so we may if we please solve the latter equation, and then change the signs of the roots, and thus obtain the solution of the former equation. 158. In the equation a;"+l = 0, suppose that n is an even number; then the equation has no real root. The equation is a reciprocal equation, and its solution may be made to depend 96 BINOMIAL EQUATIONS. upon the solution of an equation of half the degree. Or the equation may be treated by the method given in Art. 154. 159. Thus in the four preceding Articles we have shewn how the solution of the proposed equations can be made to depend vipon the solution of other equations which are not of higher degrees than half the degrees of the proposed equations. In each case we remove the factors which correspond to the real roots and then put x+ - = z, and obtain an equation in z. Now it may be observed that this equation in z will have all its roots real. For suppose that a + pj — 1 denotes one of the imaginary values of Xy then the corresponding value of z is a-f^7^ + ^^^ .— , thatis, a + /?V^4-"^j^^— , and this is a real quantity, namely, 2a, provided that a^ + ^-=^ 1. We shall shew that a^ + ^^ is = 1. Since a-\- p J —\ is a root of the proposed equation a;" =f 1 ^ 0, by Ai-t. 41, a- pj -I is also a root. Thus (a + ^7^)" = ±l, and (a -/3 y^)" = ± 1 ; hence by multiplication (a^ + py = 1 ; therefore a^ + yS^ = ± 1, and since a^ + /3- is necessarily positive it must be equal to + 1. 160. We will now consider some examples of the equations £c" + 1 = and a;" - 1 = 0. (1) ic'- 1 = j this gives {x- 1) (a;'+ a; + 1) = 0. Hence the roots are 1 and ^ ; these values are then the three cube roots of + 1. By changing their signs we shall obtain the three cube roots of -1, or in other words the roots of the equation a;^ + 1 = 0. BINOMIAL EQUATIONS. 97 (2) a;*+l = 0. T\itx + - = z; wegetz'-2 = 0. Thus is = ^J2. Therefore x'+ 1 - {x' + xJ2 + l){x'-xJ2+l); and the solution can be completed by finding the roots of two quadratic equations. (3) x'- 1 = 0. This gives (x - l)(x*+ x^+ x'+ a; + 1) = 0. 1 1 X^ X Hence we have to solve x^ + —^ + x + - + 1 = 0, that is z'+z-l = 0. Thusz 'A Therefore aj'^- 1 = (oj - 1) ('^^+ o^i:^ + 1 Va^^+ ^ ^^ + l) ; and the solution can be completed by finding the roots of two quadratic equations. The roots with their signs changed will be roots of the equation £c^+ 1 = 0. 161. If we attempt to solve the equation a;^-l = 0, we ob- tain an equation of the tliird degree in z) and if we attempt to solve the equation £c^- 1 =0 we obtain an equation of ^q fourth degree in z. We shall in the next two Chapters shew how to solve equations of the third and fourth degrees ; it will however be found that the methods of solution are of little practical value when the equations to be solved have all their roots real, which is the case we have here to consider, by Art. 159. 162. In an equation of the form a;"" +^;a;" +3' = 0, we can by the solution of a quadratic equation find the values of a", and then the method of the present Chapter may be applied to find the values of x. -i We will close this Chapter by a proposition respecting the number of values of the product of two surd quantities. T. E. 7 98 BINOMIAL EQUATIONS. "^ 163. Suppose A and £ any two algebraical quantities, and m and n any positive integers. Then "^A has m different values, and J^B has n different values by Art. 142. Hence the product of "^A and ^y^ cannot have more than mn different, values; and we shall shew that it cannot have so many -values unless m and n are prime to each other. This we shall shew by proving the following proposition ; the number of different values of the product of "^A and IJB is equal to the least common multiple of m and n. Let a be one value of liJA; then all the values of "^A are included in a^l. Let h be one of the values of H^B] then all the values of ^B are included in h^\. Hence all the values of the product are included in a6 x yi x ;yi; and therefore the number of the different values of the product is the same as the number of the different values of ^1 x ^1. Let r be the least common multiple of m and n; then (;;yi x ^1)''= 1. Thus !;yi X ^1 is equal to an r"" root of unity, and therefore cannot have more than r different values. We have however still to shew that "l^l x Ijl really has r different values. Let p be the greatest common measure of m and ?i, and let m=pjx, and n=pv. Let a denote a value of ^1, and p a value of ^/l ; then ^ 1 x ijl may be written thus ^a x UfB, or Ijap. ISTow a/5 has /x x v values, and as each p^^ root of a^ has p values we have in all pixv values, that is r values. And these values are all different. For let a denote another of the yu, values, and P' another of the v values, and suppose if possible that ^ a P' = ^/ ap -J raise both sides to the p^^ power, then a'/8' = aj8; therefore — = 7^ . The left-hand member is a root of the equation a ^ of- — 1 = 0, and the right-hand member is a root of the equation a;^ — 1 = ; and these equations can have no common root except ' unity by Art. 147. Thus there are fiv different values of a/3, and r different values of ^1 x ^1. ^ 164. The essential part of the preceding Article is sometimes ,-.-.. "''*''* Tn •{- n treated thus. We have T/l x Z/1 = 1'"'* . and if be reduced ^ ^ 7nn Y CUBIC EQUATIONS. 99 to its lowest terms, the numerator will be' an integer and the m+n 1 denominator will, be r ; thus 1 "*" = l*" which has r different values. This method however is unsatisfactory, because the ordinary theory of surds in Algebra is only proved there for the arithmetical values of the surds, and thus does not furnish the 1 1 m+n relation 1™ x 1"= 1 ""* , in the sense in which this relation is here required. XII. CUBIC EQUATIONS. O ' '' 165. It is unnecessary to say anything on the solution of quadratic equations because that subject is fully considered in treatises on Algebra. We propose in the present Chapter to give the solution of equations of the third degree which are also called cubic equations. It appears from Art. 56 ^ that any proposed equation can always be transformed into another equation without the second term. As the roots of a cubic equation without the second term are more simple expressions than the roots of a complete cubic equation, we shall suppose that the cubic equation which we have to solve is without the second term. The process which we shall now give is usually called Cardan^s solution of a cubic equation. 166. To solve the equation x^ + qx + r = 0. Assume x^y + z, so that y and z are two quantities which are at present unknown. Substitute for x in the given equation; thus {y + zY + q{y + z) + r = 0, that is, y^ + z^+ {Syz + q){y + z) + r = 0. Now we have made only one assumption with respect to the two quantities y and z, namely that their sum is the value of a root of the proposed equation. We are therefore at liberty 7-2 1^ •lOO CUBIC EQUATIONS. to make another assumption ; suppose then that 3yz + q = 0. Thus we have Substitute for z in terms of y; thus K<-& + r = 0, thatis f + rf-^ = 0. Hence 3^ = _| =. ^(^ + g , and .3_,_^_r^yg^|!^_ Also x = y + z; it will lead to the same result in the value of X whether we adopt the upper sign or lower sign in the values of y^ and z^; for distinctness suppose the upper sign taken. Therefore "{-iV(i4)}'*fiV(j4)f- Thus the expression for x is the sum of two cube roots, and as every quantity has three cube roots, we must examine which cube roots are to be used in the present case. Let then by Art. 160, the three cube roots of 1 are 1, a, and a'. Let m denote one of the cube roots of--+ /(■j + ^jj then the other cube roots are ma and ma^; let n denote one of the cube roots of —-r — x / \T '^ 97) '^ then the other cube roots are na and na^. If we could ascribe to each of the cube roots which occur in the expression for x any one of its three values, we should obtain on the whole nine values of x. But a cubic equation can only have three roots, so that we are led to con- clude that only three values will be admissible for x. And CUBIC EQUATIONS. 101 in fact the process of solution requires that yz = — -^i and it is this condition which determines the admissible values of the cube roots. Suppose that m and n are so taken as to satisfy the condition mn = — ^ j thus we can have y = m and z = n as ad- o missible values. Then we can also have y = am and z = a^n\ and we can also have y — cLm and z = a.n\ for in these two cases we have the relation yz = — \ satisfied. No other pair of values however is admissible; for instance, if we suppose y = m and z = an^ we get yz = — -^ and not - ^ , and any other pair of values except those which we have admitted will make yz = — ~ o or = — ^ instead of - 1^ . 167. For example, suppose a;^ + 6a; — 20 = 0. Here q = Q and r = ~20; thus aj = (10 + 7108)3 + (10-7108)3. ' -- By numerical work it may be ascertained that (10 + 7108)3 = 2-732 ..., and (10-7l08)^ = --732..., so that we may presume that a; = 2 is a root, and this will be found the case on trial. Instead of expressing the other two roots by the method of the preceding Article it will be preferable to depress the equation to a quadratic. Since 2 is a root of the proposed equa- tion we know that x^ + 6x— 20 is divisible by a; — 2, and we find that a;' + 6a; - 20 = (aj - 2) (a^ + 2a; + 1 0) ; therefore the other two roots of the proposed equation may be found by solving the equation a;'+2a;+10 = 0; thus these roots are -l=fc7-^» thatis-l±37^. 102 CUBIC EQUATIONS. In the preceding example we may verify hy trial that (10 + 7108)^ = 1 + V3 and (10-7l08)3 = 1 - ^3, and so find the root 2 without any numerical extraction of roots. There is however no algebraical process by which we can universally obtain the cube root of an expression of the form a + Jb in a finite form; see Algebra, Art. 310. We may apply the binomial theorem to find the value of (a + Jb)^ in an infinite series ; in this case in order to obtain a convergent series, we must expand in ascending powers ol ^Jb or of a, according as ^/ft is less or greater than a; see Algebra, Chapters xxxvi. and xl. 168. We have seen in Art.. 166, that although apparently values are furnished for x only three are really admissible. may see a reason for the occurrence of the nine values. For the relation yz = — ^ was assumed, but this was transformed into 2/V = — ^ in the process; and the latter relation would not be changed if q were changed into qa or into qa^. Thus, in solving the equation a?^ + g'a; + r = 0, we really found nine solutions, three belonging to this equation, three to the equation x^ + qax + r = 0, and three to the equation x^ + qa^x + r = 0. 169. Let us now consider more particularly the form of the roots of the proposed cubic equation. We will assume that q and r denote real quantities. The expressions for y^ and z^ may be either real or imaginary. First suppose that these expressions are real. We may then suppose that m and n denote respectively the arithmetical values of the cube roots of y^ and z^. The proposed cubic equation has in this case one root which is certainly real, namely m + n; the other two roots are ma + na^ and ma^ + na. By substituting for a its value these roots become respectively CUBIC EQUATIONS. 103 and --^{m^n)--{m-n)J-'^, and these roots are imaginary unless m = n. When m = n the cubic equation has two equal roots each being equal to — m or - n. The condition which is necessary and sufficient to ensure m — n, that is, y^ = z^, is that — + ^ = 0. Conversely, if the roots of the cubic equation are all real and unequal the expressions for y^ and z^ must be imaginary. Next suppose that the expressions for y^ and z^ are imaginary; r^ q^ . that is, suppose that j + ^ is a negative quantity. We know from Art. 142 that y^ and ^ will each have cube roots of a cer- tain form. We may therefore suppose that m = ix-¥v J -\, and as z^ only differs from y^ in the sign of the radical, we can take n=fx-vj—l. In this case the roots of the proposed cubic equation are all real, namely, {fjL + v J- i)a + {fi-v J -l)a^j that is - /a - v ^^3, and (^ + vj- 1) a^+ (ja - v J- 1) a, that is - //, + v ^3. 170. It will now be seen that Cardan's solution of a cubic equation is of little practical use when the roots of the proposed equation are real and unequal. For in this case the expressions for y^ and z^ are imaginary; and although we know that cube roots of these expressions exist, there is no arithmetical method of obtain- ing them, and no algebraical method of obtaining them exactly. We have the roots in thi^ case exhibited in a form which is alge- 104 CUBIC EQUATIONS. braically correct, but arithmetically of little value. For example, take the equation 03^-1003-4=0. Here r = - 4 and q = -15. Hence we obtain x = (2 + J^UT)^ + (2 - J^Ul)i; that is, a; = (2 + 11 ^-1)^ + (2 - 11 J^)K Now here we have no obvious mode of extracting the cube roots. It may be verified by trial that (2 + 11731)^ = 2 + 7^1, and (2-ll7Tl)3 = 2-7^. Thus x = 2 + J^ + 2-J^l = 4. Hence 4 is a root. The other roots can then be found by the method of Art. 169; or we may proceed thus, a;* - 15aj - 4 =-- (a; - 4) {x' + 4aj + 1). "We have therefore to solve the equation (C^ + 4cc + 1 =0; the roots are — 2 =t JZ. Again, consider the equation a;^ - 3 ^^a; - 2 = 0. Here r = - 2 and ^ = - 3^2. Thus aj=:(l + 7rT)i+(i_7n)i It may be verified by trial that A^+V ^} 2^2 24/2 ^ ' ^ ^ ^. 2 «/2 2 4/2 "^ Thus ^~\2^2 ^2^2 ^ ^\2l72" 2f/2 ^~^ 1/2 2^2 ^ \2V2 24/2 -^ ^r 72 The other roots may then be found; they are CUBIC EQUATIONS. 105 171. The case in which the three roots of a cubic equation are real and unequal is sometimes called the irreducible case, and sometimes it is said that Cardan's solution /aiYs in this case; these expressions are used to indicate the fact that the roots are in this case presented to us in a form which is very inconvenient for arithmetical purposes. We may however use the binomial theorem in order to ap- proximate to the cube root of an expression of the form x> + ^ J~ 1- For if q be numerically less than p we can expand (p + q J~ 1)^ in a converging series proceeding according to ascending powers of q J-l; see Algebra, Chapter xxxvi. We can thus obtain approxi- mately {p + qj-l)^ in the form F+Qj^; and then (p-qj^)^ will have an approximate value P—^^— 1; and the sum of the tv/o cube roots will be 2F. But if q be numerically greater than p we may proceed thus; p + qj^ = j~l{q-pj^l); hence (p + qj^l)^ = U^)kq-pJ'^f- Now ->/- 1 is a cube root of ^- 1 as we find by trial, so that we have {p -^qj-^f^=^- J -^{q-pJ^)K And we can expand (q — pj—l)'^ in a converging series pro- ceeding according to ascending powers of pj~l; and thus we may find as before the sum of the cube roots of ^ + qj— 1 and The case in which p = q is really involved in the second example of the preceding Article. It may be observed that by means of De Moivre's theorem, we can express the cube root of any quantity p+qj —^ ina form involving Trigonometrical functions. 172. It appears from the preceding Articles that the cubic equ^ation x^+ qx + r=^0 may always be solved by Cardan's process 106 CUBIC EQUATIONS. "without any difficulty when g' is a positive quantity, and also when 3' is a negative quantity provided q^ is numerically less than — — ; and in these cases two of the roots are imaginary. If q^ is a negative quantity and numerically greater than —j- , Cardan's solution is inconvenient, and in this case all the roots are real. 21 r^ If q^ be negative and numerically equal to —r-^ so that — + ^ = 0, the proposed- cubic equation has two of its roots equal by Art. 60. We have by Art. 166 in this case m-n= / --r\ and the three roots are 2 m, - m, and - m. In eveiy case where one root of a cubic equation has been found we can, if we please, depress the equation to a quadratic, and so find the other two roots, instead of finding the other two roots by the process of the preceding Articles. 173. We will briefly indicate the results which are obtained in the solution of a complete cubic equation. Let the equation be assume x = z — , then we obtain , - a z^ + qz + r = 0, ^c W d She 2b' where q = o ^ , r = ^ + — -r . ^ a a a a a Hence by Cardan's method ^=(-2W4-'l7) n-2-V4^27) ' The condition which must hold if there are equal roots is L4.I- 4 27 = 0: CUBIC EQUATIOITS. 107 that is (26' - ^ahc + a'dy + 4 (ac - hj = 0.' It will be found by common Algebraical work that this can be put in the form {ad - hey -4:{b'- ac) (c' - hd) = 0. 174. Some cubic equations in which the coefficients have special values may be solved without using Cardan's method. For example, suppose This may be written that is, x^-(^a--j + 3\x-(a--]i = 0; and now we see that one root is given hy x = a — . Again, suppose we have the complete cubic equation x^ + ax^ + hx + c = 0, and that the relation 3ac = b^ holds among the coefficients. The proposed equation may be written — x^ = ax^ +bx + c, therefore - 3abx' = Sha^x^ + db^ax + 6", therefore (a^ - 3ab)x*=a^x^ + Sba'x'+ Wax + b^= {ax + bf, therefore xja^ - 3ab = ax + b, b therefore x = ija^- 'dab - a • 175. A process is given in the Trigonometry^ Chapter xvii. by which we may obtain the roots of a cubic equation in the irreducible case, by the aid of the Trigonometrical Tables. This 108 CUBIC EQUATIONS. is a matter of very little practical value, but we will shew how the Trigonometrical Tables may also be used for examples which do not belong to the irreducible case. Suppose x^+qx + r = 0; then -(-iV?^)'*(-iV?4)'- Q^ r^ If 2' is positive, assume ^= -j tan^^; then we get =(-^»)'{(»D'-(-'"l)'}- If q is negative, and 4:q^ niimerically less than 27r^, assume q^ r' ^ = - J sin' ^ j then we get ..(-5*5„..)'.(-|-|„.,)' 176. An important cubic equation occurs in many mathe- matical investigations, and it may be noticed here although not connected with the special subject of this Chapter. We propose to shew that the roots of the equation f(x) = are all real, where /(a;) denotes {x - a){x - b){x -c)- a"{x -a)- h'\x -h)- d\x -c)- 2a'h'c', The equation may be written thus, {x - a) ^{x - h){x - c) - a"\ - h'\x - b) + c'\x - c) + 2a'b'c\ = 0. CUBIC EQUATIONS. 109 K Let li and h denote the roots of the quadratic equation (aj-6)(a;-c)-«'" = 0, and suppose h not less than h. Then by solving the quadratic equation it will be seen that h is greater than h or c, and that h is less than h or c. Substitute successively + (X) ^ h, k^ — od for X in f{x) ; the results will be respectively + 00 ,-{b'J{h-h) + c'J{h-c)^ , i^'J{h-h)-c'J{c-k)^ Thus the equation /(cc) = has three real roots, one greater than /fc, one between h and ^, and one less than k. 177. There are two cases which require further examination as they are not provided for by this demonstration, (1) that in which h = k^ (2) that in which A or ^ is a root of the cubic equation. (1) Suppose h = k. Since the roots of the quadratic equa- tion are equal we shall obtain the condition (6 — c)^+ 4a'^= 0; therefore h = c and a'= 0. Hence it will be found that c is a root of the cubic equation; and on dividing f{x) hj x — c and equating the quotient to zero we obtain a quadratic equation which has real roots. (2) Suppose that h or ^ is a root of the cubic equation; for example, suppose that h is. Then the process of Art. 176 shews that the cubic equation has also a real root less than k; thus it has two real roots, and the third root must therefore also be real. Similarly if ^ be a root of the cubic equation, it has a real root greater than h; aud thus the third root must also be real. 178. "We may investigate the condition which must hold in order that h or k may be a root of the cubic equation. Suppose that X is a root of the quadratic equation and also of the cubic equation. Since X is a root of the quadratic equation, we have (X-6)(A-c)-«'^ = (1); I 110 CUBIC EQUATIONS. and since \ is also supposed to be a root of the cubic equation, "we obtain b"{\-b) + c''{\-c) + 2a'b'c' = (2). From (1) and (2) we deduce b"(\ -b) + c"{\ -c) + 2b'c'J{X.-b){k-c) = 0, that is, |6V(X -b) + c'J{X - c)X = ; therefore b" {X - b) = c'\X - c) (3). From (2) and (3) we obtain ^-^ = —Y\ ^~^ = ""^ (^)^ and therefore b — -77- = c ;- (5). Hence the relation (5) must hold among the coefficients of the cubic equation in order that one of the roots of the quadratic equation may also be a root of the cubic equation. Conversely, if (5) holds we may give to X the single value determined by (4), and then both (1) and (2) will be satisfied; and thus the quadratic equation and the cubic equation will have a common root. In obtaining (4) and (5) we assume that neither b' nor c vanishes. Suppose that b' vanishes; then from (3) either c' vanishes or X=c. If X = c then from (1) it follows that a' must vanish. 179. Let us now investigate the conditions in order that the cubic equation may have equal roots. If neither h nor A; is a root of the cubic equation, the demon- stration in Art. 176 shews that the roots of the cubic equation are unequal. But the process of Art. 176 may be conducted so as to use either of the quadratic equations {x-c){x-a)-b"' = 0, or{x-a){x-b)-c" = 0, instead of the quadratic equation (x-b){x-c)- a' = 0. CUBIC EQUATIONS. Jll Hence tlie cubic equation cannot have equal roots unless it lias a root in common with any one of these quadratic equations. Hence from equation (5) we obtain the following as necessary conditions for the existence of equal roots of the cubic equation, Vg' , da! a'l/ a - = b--rr = c-—r . a h c Converselyj if these conditions hold the cubic equation has equal roots. For denote these equal quantities by r, so that h'c' , cV ah' a = r+ —p . o = r + -rr , c = r + — - ; a c substitute for a, b, c in the cubic equation, and it becomes SO that the root r occurs twice, and the other root is h'c' c'a! olV r + ^ +-77- + — r . a c This assumes that a', 6', and c' are all different from zero. Suppose now that one of these quantities vanishes, say a'. Then from the quadratic equation (£c-5)(ic-c)-a'' = it follows that x must be equal to c or h. Suppose x, = c\ then from the other quadratic equations we see that 6'=0 and (c-«){c-6)-c'^ = 0. If a', y and g all vanish then in order that there may be equal roots, two of the three a, 6, c must be equal ; if they are all equal the cubic equation reduces to (a; - of = 0. I 112 BIQUADRATIC EQUATIONS. XIII. BIQUADEATIC EQUATIONS. 180. "We shall now proceed to explain some methods for the solution of equations of the fourth degree, which are also called biquadratic equations. We suppose the biquadratic equation which is to be solved to be deprived of its second term, for a reason already given; see Art. 165. The first solution which we shall give is called Descartes^s Solution. 181. To solve the equation x^ + qd(? + roj + s = 0. Assume x^ ■\- qx^ -^ rx + s = {x^ + ex +/) {x^ -ex + g); we have then to shew that the quantities e, /, and g can be found. Multiply together the factors on the right-hand side, and equate the coefficients of the several powers of x to those on the left-hand side; thus g+f-e' = q, e(g-/)^r, g/=s; T that is, g +/- q^e\ g -/- - , gf= s. e Find g and / in terms of e from the first two of these equa- tions, and substitute in the third; thus (j+,vr)(^ +,=„-) = 4.. From this equation by reduction we obtain e^ + 2qe*+{q^-4:s)e'-r' = 0. This may be considered as a cubic equation for finding e^, and it will certainly have one real positive root by Art. 20. When e^ is known we can find e, and then g and /become known. Thus the expression x'^ + qx^ + rx + s is resolved into the product of two real quadratic factors, and we can obtain the four roots of the proposed biquadratic equation by solving the two quadratic equations X- + ex +/= 0, x^-ex + g=^ 0. BIQUADKATIC EQUATIONS. 113 182. It will be observed that in one of tbe two assumed quadratic factors we introduced the term ex^ and in the other quadratic factor the term —ex; and the reason for this is that there is no term involving £c^ in the expression which we wish to resolve into quadratic factors. Now e is equal to the sum of the two roots of the second quadratic equation given at the end of the preceding Article, so that e is equal to the sum of two of the roots of the proposed biquadratic equation. Out of the four roots of a biquadratic equation two roots can be selected A Q in ^—^ ways, that is, in 6 ways ; and thus we see the reason why the equation in e should be of the sixth degree. But as the sum of the four roots of the biquadratic equation is zero by Art. 45, the sum of any two roots is equal in magnitude and opposite in sign to the sum of the remaining two roots; and thus we see the reason why the equation in e only involves even powers of e, so that the values of e^ can be found by the solution of a cubic equation. We may observe that when we have found e^ we can give either sign to the value of e, which we obtain by extracting the square root ; for by changing the sign of e we merely interchange the values of / and ^, and this has no influence on the results which are obtained by solving the biquadratic equation. 183. Suppose, for example, that a;*— lOaj^^ 20aj- 16 =0, Here g' = — 10, r = — 20, s = ~16. The cubic equation in e^ becomes e®-20e''+ 164e^- 400 = 0, and a root of this is e^=4; see Art. 119. Thus e = 2 ; then f= 2, and ^ = — 8 ; therefore X*- \W- 20ic- 16 = {sc' + 2x + 2){x'-2x-^). The four roots of the proposed biquadratic equation will be found to be 4, -2, -1 + /^, and ^l- J^, 184. Thus it appears that the solution of a biquadratic equa- tion can be effected if we can obtain one root of a certain auxiliary cubic equation. It becomes therefore a point of importance to ascertain when this cubic equation falls under the irreducible T. E. 8 (. 114 BIQUADRATIC EQUATIONS. case; see Art. 171. This gives occasion for tlie following pro- ])Osition. The auxiliary cubic equation will not fall under the irreducible case when the biquadratic equation has two real roots a7id two imaginary roots. For suppose the imaginary roots of the biquadratic equation to be denoted by a + /? J- 1 and a-p J- 1 ; then since the sum of the four roots is zero, the two real roots will be of the forms -a + y and — a - y. By taking the sum of every pair of these roots we obtain the expressions ^2a,^(y+/3j^), and ±(y-)S^^). Thus the three values of e^ will be (2a)^, {y + Pj-lY, and {y-Ps/-^Y'> if 7 is not zero two of these values of e^ are imaginary, and if y is zero the values of e^ are all real, but two of them are equal ; thus the cubic equation in e^ wiU not faU under the irreducible case. ^ • 185. If the roots of the biquadratic equation are all real the roots of the auxiliary cubic equation will be all real. If the roots of the biquadratic equation are all imaginary they will be of the forms a ± j8 J— 1 and — a ± y J— 1 . By taking the sum of every pair of these roots we obtain the expressions ±2a, =t(/8 + y)^/-l, and =«=(/? -y)^-lj thus the values of e^ are 4a^, -{/S + yY, and — (y8 — yYf and so are all real. Hence if the biquadratic equation has its roots all real or all imaginary, the auxiliary cubic equation will in general fall under the irreducible case ; we say in general, because it may happen that the cubic equation has two of its roots equal, and then it does not fall under the irreducible case. 186. We have in the two preceding Articles shewn what will be the forms of the roots of the auxiliary cubic equation cor- responding to the various forms of the roots of the proposed biquadratic equation. We will now state conversely what will be the foi-ms of the roots of the proposed biquadratic equation cor- responding to the various forms of the roots of the auxiliary cubic equation. Since the last term of the cubic equation is BIQUADEATIC EQUATIONS. 115 negative, there must be one positive root ; and as tlie product of the roots is positive, by Art. 45, the only cases which can occur are, (1) all the roots positive, (2) one positive root and two nega- tive roots, (3) one positive root and two imaginary roots. The following results follow from Arts. 184 and 185. (1) If the cubic equation has all its roots positive, the roots of the biquadratic equation are all real. ' 1/ (2) If the cubic equation has one positive root and two negative roots, the biquadratic equation has two real roots and two imaginary roots, or else four imaginary roots. (3) If the cubic equation has one positive root and two ima- ginary roots, the biquadratic equation has two real roots and two imaginary roots. ' '' 187. The four roots of the biquadratic equation can be ex- pressed very simply in terms of the three roots of the auxiliary cubic equation. Let a^, /3', y* denote the three values of e^ ob- tained from the cubic equation eV25e" + (g'-4s)e'-r*=0. Then by Art. 45 we have r^ = ^P'y\ and -2q = a? + jS" + y\ Thus we may put r = a^y, and take a as a value of e j therefore x^ + ex +/= x^ ■¥ ax + -^(q -{• a^ — ) = (k" 4- aa; -}- i (a'- ^'- y^- 2/?y). By solving the equation x^ + ex +f= we shall therefore obtain ^ = ^{-<^-P-y), or ^=2^-^ + P + i)' Similarly, by putting a;^ - ea; + ^ = we shall obtain a' = 2(«-^ + y)» or x = -^{a + P-y), Thus the four roots of the biquadratic equation are l{-a-P-y), l(_a + ^+y), \{a-p^y), \( ••• 5'm-i> denote the successive quotients which arise in performing the operations indicated; then we have the following relations, From these relations we can draw three inferences. (1) The last of the functions /^(ic) is not zero; for by supposi- tion it is independent of x and if it were zerof(x) and/^(£c) would have a common measure, and then the equation f(x) = would have equal roots by Art. 75, and this is contrary to the hypothesis. (2) Two consecutive auxiliary functions cannot vanish simul- taneously; for if they could all the succeeding auxiliaiy functions would vanish including f^(x)j and this is impossible by (1). (3) When any auxiliary function vanishes the two adjacent functions have contrary signs. Suppose for example thsLtf(x) = ; then from the third of the above system of relations we have Now no alteration can be made in the sign of any one of Sturm's functions except when x passes through a value which makes that function vanish; and we shall now prove that when X passes through a value which makes f(x) vanish one change stukm's theoeem. 123 of sign is lost by Sturm's functions, and that no change of sign is lost or gained in consequence of x passing through a value which makes one of the auxiliary functions vanish. I. Suppose c a root of the equation y (a?) = 0, so that y(c) = 0. Let h be a positive quantity. Now f(c — h) may be expanded in powers of h by Art. 10, and h may be taken so small that the sign of the whole series shall be the same as the sign of the first term that does not vanish, hjjAxt. 14; that is^ the sign of/{c- A) will be the same as the sign ^- h/^(c) since /(c) = 0. The sign of yj (c — h) will be the same as the sign of/ (c) when h is taken small enough. Thus if x = c-h and h is taken small enough, /(x) and / (x) have contrary signs. Similarly, it may be shewn that if x = G + h and h is taken small enough, /(aj) and/(cc) have the same sign. Thus as X increases through a root of the equation /(a;) = 0, Sturm's functions lose one change of sign. II. Let c now denote a value of x which makes one of the auxiliary functions vanish, for example, f^{x), so that/.(c) = 0. Then /._,(c) and /.+ ,(c) have contrary signs, and thus just before x = c and also just after x = c, the three terms/_j(rc), /^{x), /+i(^) t. : will present one permanence of sign and one change of sign; for if ( fr-A^) 9'^d/(fl3) have the same sign,/(.'^;) a,ndf^^^{x) have contrary signs, and vice versa. Thus Sturm's functions neither lose nor gain a change of sign when x passes through a value which makes one of the auxiliary functions vanish. No value of x can make two consecutive functions simul- taneously vanish. If two or more vanish simultaneously which are not consecutive, then, if /(x) be one of them, it follows by I. that a change of sign is lost as x increases through that value, and iff{x) be not one of them it follows by II. that no change of sign is lost. . Thus we have proved that as x increases, Sturm's functions never [ lose a change of sign except when x passes through a root of the equation f(x) == 0, and never gain a change of sign. Hence the i / 124 STURM'S THEOREM. number of changes of sign lost as x increases from any value a to a greater value ^, is equal to the number of the roots of the equa- tion /(a;) = which lie between a and yS. 195. "We have shewn that no alteration occurs in the number of the changes of sign in Sturm's functions in consequence of x passing through a value which makes one of the auxiliary functions vanish; but alterations may take place, and in general do take place, with respect to the order in which the signs + and — are distributed among the series of functions. Suppose, for example, that a and h are two roots of the equation f{x)=0 and that a is less than h ; then f{x) and /, {x) have contrary signs just he/ore x = a and have the same sign, just after x = a. "Now just be/ore x=b the signs of f(x) and f^{x) are again contrary. In fact the equa- tion fj^x) = has one root between x=a and x = b, and so f^{x) must pass from positive to negative or vice versa between x = a and x = b. This transition of f^ (x) from positive to negative or vice versa between a and b, cannot alter the whole number of changes of sign in the series of Sturm's functions, as we have proved, but it does modify the distribution of the signs + and - among the series, and thus renders it possible after a change has been lost as X increases through a, for another change to be lost as x increases through b. The present Article adds nothing to the proof of Sturm's theorem ; but is merely intended to assist a student in the diffi- culty which is often felt as to how the changes of sign are lost. ""^ 196. In counting the number of changes of sign in the series of Sturm's functions, it may happen that the value of x which we are considering makes one of the auxiliary functions vanish. Then it is indifferent whether we ascribe the positive sign or the negative sign to the vanishing function, since the signs of the functions which precede and follow it are necessarily contrary. 197. In order to find the whole number of real roots of an equation fix) = 0, we may first put — oo for x and then + oo for x in Sturm's functions; the excess of the number of changes of sign in stuem's theorem. 125 the first case over the number of changes of sign in the second case is the whole number of real roots. When x is made equal to + 00 or — 00 the sign of any one of the functions will be the same as the sign of the highest power of x in that function. 198. Let n denote the degree oi/{x); then the number of the auxiliary functions/j(aj),/2(£c),...will in general also be n; because each remainder is generally of one degree lower than the preced- ing remainder. We will suppose that the number of auxiliary functions is the same as the degree of /(x), and we will suppose that the highest power of x inf{x) has a positive coefBcient. (1) If the first terms in all the auxiliary functions have posi- tive coefficients all the roots of the equation /(a;) = are real. For all Sturm's functions will then be positive when a;= + oo , and they will be alternately positive and Negative when a; = — oo ; thus n changes of sign are lost as x passes from — oo to + oo . (2) If the coefficients of the first terms are not all positive, there will be a pair of imaginary roots for every change of sigti in the series formed of these coefficients. For suppose that in this series of coefficients there are m changes of sign and n — m con- tinuations of sign. Then when a; = + oo there .are m changes of sign and n — 7)% continuations of sign in Sturm's functions. Now change X from +00 to - 00 j then the changes of sign are replaced by con- tinuations of sign and the continuations of sign by changes of sign, so that for a; = - 00 there are n — m changes of sign. The excess of the number of changes of sign when a; = — oo over the number when a; = + 00 is therefore n - 2m; thus there are n — 2m real roots of the equation y (a;) = 0, and therefore 2m imaginary roots. Hence in order that an equation may have all its roots real, it is necessary and sufficient that the coefficients of the first terms in all the auxiliary functions should be of the same sign. 199. Suppose that among the auxiliary functions we find one, as f^{x), which cannot change its sign; then we may disregard all the functions which follow it, and count only the number of changes of sign in the series /(x), f^ (x), J\{x), . . ./. {x). For in the original 126 Sturm's theorem. demonstration of Sturm's theorem the necessary property of the last auxiliary function is that it should not vanish, and as f^{x) cannot vanish, the demonstration will hold for the series This remark is of practical importance, because the labour attending the formation of Sturm's functions is considerable in examples of equations of high degrees, and thus it is usefiil to have a rule which sometimes relieves us from the necessity of forming the entire series of functions. "^ 200. Suppose {x) to be a function which has no factor in common with /(a;), and suppose that <^{x) and /^{x) take the same sign when any root of the equation f{x) = is substituted for X in them. Then we may use <^ {x) instead of /j {x) and deduce the remaining auxiliary functions from f{x) and {x) instead of from /(£c) and/j(a3). For on recurring to the demonstration of Sturm's theorem it will be seen that with this new set of functions the two fundamental properties are still true, namely, that no change of sign is lost owing to the vanishing of any auxiliary function, and that a change of sign is lost when /(a:) vanishes. 201. We have hitherto supposed that the equation to be treated by Sturm's method is cleared of equal roots; we shall now shew that this limitation is unnecessary, and that the theorem will always give the number of distinct roots between assigned limits, no regard being had to the repetition of any roots. Suppose for example that the root a occurs p times and the root h occurs q times in the equation f{x) = 0. Let f{x) = {x-ay{x-h)\x-c){x-d),., then f^{x) = {x-ay-\x-hy-'{p{x-h){x-c){x-d).., + q{x — a){x — c){x — d)... } Sturm's theorem. 127 Thus {x - aY~^ (x - hy~'^ is the greatest common measure of /(x) and y,(a3), and this expression will divide all the auxiliary f unc- tions jf^ (a;), /g(aj) ,.../„ (a:) which are formed as in Art. 194. Now let if/ (x) = {x — a){x - b) (x — c) {x — d) . . . and (x) is not the first derived function of if/(x), for that would be what (x) would become if ^9 = 1 and q=l; but cf> (x) has the same sign as the first derived function of 1/^(0;), when we make x = a, or b, or c,... Hence, by Art. 200, we may determine the situation of the real roots of the equation i}/(x) = by taking if/{x) and <^ (x) as the first two of Sturm's functions and forming the rest from them. But the series of Sturm's functions formed from f(x) and /^ (x) only difiers from the series formed from i}/(x) and cfi(x) by reason of the additional factor (x — af~^ {x — by~^ in every term of the series. Thus when any value is ascribed to x, the signs of the terms in the former series will all be the same as those of the latter, or all contrary; and thus the number of changes of sign will be the same. Hence by examining the series of Sturm's functions formed from f(x) and /^ (x) we can ascertain how many of the roots of the equation ij/(x) = lie between assigned limits, that is, how many distinct and separate roots of the equation /(a;) = lie be- tween those limits. Thus we need not apply the test for equal roots before we apply Sturm's method; in fact, in calculating Sturm's functions we shall be warned of equal roots if they exist by the fact that the last remainder will be zero. A 128 STURM'S THEOREM. 202. We may observe tliat in the operation by "which all the auxiliary functions after the first are found, we may always mul- tiply or divide the divisors or dividends by any positive number we please, as in the operation of finding the greatest common mea- sure; for the auxiliary functions thus only become multiplied or divided by positive numbers, so that their signs remain un- changed. We may by Sturm's theorem determine the number of real roots of any proposed equation. Then, by substituting successive integers for x in the series of Sturm's functions, we can determine between what consecutive integers the roots lie; or if it is found that more than one root lies between two assigned integers, we can substitute 'for x successively fractions which lie between those integers, until we at last determine intervals between which the roots lie singly. 203. We will now take some examples. Suppose /{x) = x^- Sx'- 4aj + 13 = 0. Here /i (x) = dx^- 6x - 4,- f^{x) = 2x-5, The roots of the equation are all real by Art. 198. The following is the series of signs corresponding to the values of x indicated. /i{*) /.(^) M'^) /(^) + 1 + 2 4- 3 + Here there are two changes of sign when x = 2, and none when a; = 3 ; thus there are two positive roots between 2 and 3, and no other positive roots. STURM'S THEOREM. 129 It will be found that when x = -3, the succession of signs is - + - +, and when x = - 2 it is + + - +, so that one change of sign is lost in proceeding from — 3 to — 2, and therefore the negative root lies between — 2 and - 3. To separate the two roots which lie be- tween 2 and 3 we should substitute for x some number or numbers lying between 2 and 3. Suppose, for example, we put x = 2^; then the succession of signs is + , and thus we have only one change of sign, whether we consider the to carry the sign + or — . Thus a change of sign is lost in proceeding from 2 to 2^, and therefore one root lies between 2 and 2^ ; hence the other root lies between 2J and 3. Again, suppose /(a;) =«;''- 6x^+ 5x^+ 14a; - 4 = 0. Here /^ (x) = 2x'^- 9x^+ 5x + 7, omitting a factor 2, f^(x) = 17x'-57x-p, f^{x)=\62x-457, In this example it will be found that the calculation of f^(x) is somewhat complicated; it is sufficient for our purpose however to know the sigriy and thus when we ascertain that it is positive we need not calculate it exactly, but merely put downy^(a3) = +. The roots of the equation are all real by Art. 1 98. . The following is the series of signs corresponding to the values of X indicated. • -2 + - + - + - 1 - - + - . + _.+ __.+ 1 + 4- . - - , + 2 +•---.+ 3 + - - - + 4 + + + + + There is one change of sign lost between — 2 and - I, one be- tween and 1, and two between 3 and 4. T. E. ' 9 A 130 Fourier's theorem. If we put 3 J for aj the succession of signs is - + + +, and thus* there is only one change of sign, so that one root of the equation lies between 3^ and 3 J; therefore another root lies between 3^ and 4. Again, suppose/(aj) = 2a;*-13a;^+ lOic- 49 = 0. Here f^ {x) - 4aj^- 13a3 + 5, omitting a factor 2, /,(«) = 13a;'- 15aj+ 98. It is easy to see that the roots of the equa,tion fjix) = are imaginary, that is, fj^x) cannot vanish for any real value of x ; therefore by Art. 199 we need not obtain any more of Sturm's functions in this example. When aj = — oo the succession of signs is + - +, and when cc = + oo the succession of signs is + + + ; thu& the equation has two real roots and two imaginary roots, ^ne of the real roots is positive and the other negative by Art. 21. XV. FOUHIER'S THEOREM. 204. Sturm's theorem constitutes the complete solution of a problem which has engaged the attention of many of the most eminent mathematicians during the last two hundred years; this theorem was published in the volume of Memoires presentes i pa/r des Savants Etrangers, Paris, 1835. \ Among those who attempted the solution of the problem before Sturm two are deserving of especial notice, Budan and Fourier; the methods of these two mathematicians start from a theorem which English writers usually call Fourier's theorem, and which Fi'ench writers connect with the name of Budan as well as with that of Fourier. Fourier's work on equations was published in 1831 after the death of, the author; Budan published a work on the subject in 1807. There is evidence however that Fourier had given the theorem in a course of lectures delivered before the • publication of Budan's work. We will now enunciate and prove the theorem. Fourier's theorem. 131 205. Fourier's Theorem. Let f(x) be an algebraical function of the 71* degree; let /(a;), f^{x),...fjx) be the successive derived functions of f{x). Let a be any quantity and /? another which is algebraically greater; then the number of the real roots of the equation /(cc) = between a and ^, cannot be greater than the excess of the number of the changes of sign in the series f{x)^f^(x),fjx),...f^{x), when a3 = a, over the number of the changes of sign when x = p. We shall call the whole seriea/{x),/^{x\f^{x)f...f^(x)j Fov/rier's functions. No alteration can occur in the sign of any one of Fourier's functions except when x passes through a value which makes that function vanish. We shall now have four cases to consider. I. Suppose when x = c that f{x) vanishes and that f{x) does not vanish. Put c — h for x where h is a positive quantity; then h may be taken so small that the sign of f{c — h) is the same as that of -hf(c), and the sign of f^(c-h) the same as that of /^(c); see Art. 14. Thus if x = c-h and h is taken small enough, /(«) aiidy*j (x) have contrary signs. Similarly it may be shewn that if x-c-^-h and h is taken small enough, /(cc) and f^{x) have the same sign. Thus as X increases through a value c, which is an unrepeated root of the equation f{x) = 0, Fourier's functions lose one change of sign. II. Suppose when x — c that /"(cc) vanishes and also the de- rived f unctions /j(cc),/2 (a?),... up tof_J^x), and that/.(ir) does not vanish. Put c — h for x where A is a positive quantity; then h may be taken so small that the signs of the series of terms f{c-h), /,(c-/»), Uc-h), ./,.,(c-A), fXo-h) shall be respectively the same as the signs of the series of terms (-j^rm, (-*)-■/,(«), {-hrvxc),...-hfXc),fxo); 9—2 132 FOURIER'S THEOREM. see Arts. 10 and 14. Thus ii x = c-h and h is taken small enough, the first r + 1 of Fourier's functions present r changes of sign. Similarly it may be shewn that if x = c + h and h is taken small enough, the first r + 1 of. Fourier's functions present no change of sign. Thus as X increases through a value c which is a root of the equation f(x) = repeated r times, Fourier's functions lose r changes of sign. III. Suppose when x=c that one of the derived functions vanishes, but neither of the two adjacent functions; thus letf^(x) vanish when x^c but neither f^_^{x) nor /^+i(£c). Then if h is taken small enough, when x = c — h the signs of the three terms fr-M)i fri^)> fr+ii^)' ^^'® rcspcctivoly the same as the signs of fr-i{(^)) -hfr+ii^)i/r+i(^)f ^^^ whcn X -^ G + k thc signs are the same as the signs of X_^(c), A/,^.,(c), f^^{c). Thus if f_^(c) and /^^c) have the same sign, Fourier's functions lose two changes of sign as X increases through c, and if /^_^{g) and /^^^{c) have contrary signs Fourier's functions neither gain nor lose a change of sign. IV. Suppose when x = c that several successive derived func- tions vanish; for example, suppose when x = c that the 7n func- tions fXx), f^^{x),...f^^_^{x) vanish, and that /.^ (a?) and /^,„(a;) do not vanish. By proceeding as before, and supposing h taken small enough and positive, we shall obtain the foUowiag results with respect to the ?7i + 2 terms, f_,(x), /.(a?),. ••/+«.-! (a^), /r+mH- (1) Let m be even. If /^_^(c) and/.+^(c) have the same sign, the terms present m changes of sign when x = c—hf and no change of sign when x = c + h. If /,_i(c) and f+J.c) have contrary/ signs, the terms present m+ 1 changes of sign when x = c — h, and one change of sign when x = c + h. Thus in both cases Fourier's func- tions lose m changes of sign as x increases through c. (2) Let m be odd. If f_^(c) and f^Jc) have the same sign the terms present m + \ changes of sign when a; = c — A, and no change of sign when x = c + h. Thus Fourier's functions lose m + 1 Fourier's theorem. 138 changes of sign as x increases through c. Ify^_j(c) and/^^^(c) have contrary signs, the terms present m changes of sign when x = c — h, and one change of sign when x = c + h. Thus Fourier's functions lose m — 1 changes of sign as x increases through c. Thus on the whole Fourier's functions never gain a change of sign, but they do lose one change of sign when x increases through a root of the equation /(a:;) = 0; and thus the theorem is proved. 206. It will be observed that the demonstration of Art. 205 gives us something more than the enunciation to which for sim- plicity we confined ourselves. For it appears that whenever an alteration occurs in the number of the changes of sign of Fourier's functions, except by reason of the variable increasiug through a root of the given equation, an even number of changes of sign is lost. Thus on the whole we have the following result if we sub- stitute successively a number a and a greater number y8 in Fourier's functions. (1) Suppose that Fourier's functions lose no change of sign; then no root of the given equation lies between a and fi. (2) Suppose that Fourier's functions lose an odd number of changes of sign; then we are certain that some odd number of roots lies between a and j8, but cannot tell what odd number, ex- cept when only one change of sign is lost, and then we are certain of on,e root. (3) Suppose that Fourier's functions lose an even number of changes of sign; then we can only infer that there is either no root or else some even number of roots between a and jS. 207. The advantage of Fourier's theorem is that it can be easily applied, because the successive derived functions of a given function can be immediately formed. The disadvantage of. the theorem is that it may require an almost unlimited number of trials. For if two roots are very nearly equal, it would require very minute subdivision of an interval in which they were con- jectured to lie, in order to distinguish them from two imaginary 134 FOURIER'S THEOREM. roots. It would be necessary to aj^ply the test for equal roots before begiimmg Fourier's process, as otherwise an even number of repeated roots might remain undiscovered. 208. Budan and Fourier both gave methods for examining a doubtful interval more closely in order to discover whether roots of the proposed equation were or were not situated in the interval. But it is unnecessary to explain these methods since Sturm's theorem attains the proposed object with simplicity and certainty. 209. It may be shewn that Descartes's rule of signs is included in Fourier's Theorem. Suppose that f{x) = is a comphte equation. If we put a; = in Fourier's functions the signs are the same as the signs in the expression f{x) taken from right to left; and if we put x=co in Fourier's functions the signs are all positive. Hence, by Fourier's theorem, the equation /(x) = cannot have more positive roots than f(x) has changes of sign. If the proposed equation be not complete, we may suppose the absent terms supplied with zero coefficients, and such signs may be ascribed to these coefficients as to make Fourier's functions have the same number of changes of sign when these terms are counted as when they are omitted. The part of the rule of signs which relates to the negative roots can be deduced from that part of it which refers to the posi- tive roots; see Art. 63. 210. Fourier's theorem also includes the rule given by New- ton for finding a superior limit to the positive roots of an equa- tion ; see Art. 94. For if /(x) = be the equation, Newton's method directs us to find h such that when x = h Fourier's func- tions are all positive; and then by Fourier's theorem no roots of the proposed equation exist between x = h and a? = + oo . lagra-Nge's method of appi^oximation. 135 XYI. LAGEANGE'S METHOD OF APPEOXIMATIOlSr. 211. We have already shewn how the commensurable roots of an equation may be found; we shall now consider how the approximate numerical values of the real incommensurable roots may be calculated. By Sturm's theorem we can always determine how many roots lie within a given interval, and we may then divide that interval into smaller intervals within which the roots lie singly. Suppose then that we know that an equation has one root and only one between two given quantities a and ^, and we wish to approxi- mate to the value of this root. If we substitute any quantity y which is intermediate between a and P for x in f{x), we shall know by the sign of f{y) whether the root lies between a and y or between y and p. Suppose it to lie between a and y] then we may substitute for x a quantity S which lies between a and y, and we shall know by the sign of /(S) whether the root lies between a and 8 or between 8 and y. This process may be continued to any extent, and we may approximate as closely as we please to the numerical value of the root; for by each operation we can thus halve the interval within which the root must lie. The operation here described would however be very laborious and methods have been proposed for attaining the required result, with less calculation. We shall first explain Lagrange's method. 212. Let/*(aj) = be an equation which is known to have one root, and only one, between two consecutive positive integers a and a + 1. Put x = a + -. and substitute this value of x in the y proposed equation; thus /( a + - j = 0. If we clear this equation of fractions, we obtain an equation in y of the same degree as the original equation in x ; denote it by {y) = 0. This equation in y has only one positive integral root, since the original equation in x 136 Lagrange's method of approximation. has only one root between a and a+\. We may then determine the consecutive integers between which the value of y must lie, by substituting in {y) positive; therefore the required value of y must lie between 10 and 11. Assume y=\0+-', then z {lO) = W- 10 . 10^- 6 . 10 - 1 = - 61, <^'(10)= 3.10^-20.10-6 =94, icf>"{lO)= 3.10-10 =20, and the equation in 2; is - 612;^+ 9 4^;^+ 20^; +1 = 0, that is, 6U'-94;s'-20;2;-l=0, say if;{z)=--0. Here z = 2 makes if/(z) positive, so that the required value of z must lie between 1 and 2. Assume z = l+-: then u f (1) = 61 . r- 94 . r- 20 . 1 - 1 = - 54, i/r'(l)= 183.1^-188.1-20 =-25, Ji/a"(1)= 183.1-94 =89, and the equation in «* is - 54?*^- 25u^+ 89u + 61=0, that is, 54?^^+ 25^^-89^-61=0. This equation shews that the value of u must lie between 1 and 2 ; and we may proceed as before. Hence x=2 + r- 10+ 1 + 1+&C. LAGRANGE'S METHOD OF APPROXIMATION. 139 The convergents corresponding to this continued fraction are J , Y^, YT , ^ry, See Algebra, Chapter xliv. The difference 1 lU 11 ZL 4:4: . 1 between -—■ and the real value of the root is less than „., ,„-, — rrrr , 21 21(21 + 11) that is, less than ■^=^ . b7 J For another example take the equation cc^ — 7a; + 7 = 0. By Art. 108 this equation has all its roots real; and by Sturm's theorem it maybe shewn that one root lies between 1 and IJ, and that another root lies between IJ and 2, Therefore if we put cc = -^ and form an equation in x' this equation will have one root between 2 and 3, and one root between 3 and 4. The equa- tion in a;' is CD - 7 1- + 7 = 0, that is, x'^~ 2Sx'+ 56 = 0. The root which lies between 2 and 3 will be found to be 1 1+ 1 2 + &C. The root which lies between 3 and 4 will be found to be 1 1 1 +&C. The roots of the original equation will be obtained by taking half of each of these values. Or we may apply Lagrange's method to the original equation without any preliminary transformation. Assume a; = 1 + - ; thus fl^-J-l fl+-\ + r = 0. This will give y^- iy' + 3y+ I = 0, say (j>{y) = 0. Here <^(1) is positive, ^(2) is negative, and <^(3) is positive; thus one value of y must lie between 1 and 2, and the 140 LAGRANGE'S METHOD OF APPROXIMATION. other between 2 and 3. Then we may put y =1 + - in order to continue the approximation to the first root, and y = 2 + -m z order to continue the approximation to the second root. The equation a;^— 7a5+7 = has one negative root; we may- find it by changing x into —x and calculating the positive root of the resulting equation, that is of the equation (-xY-1{-x) + 1 = Q. Or since the sum of the three roots of the equation a;^— 7a;+7 = is zero, when two of the roots are calculated approximately the third can be immediately found approximately. 216. If in following Lagrange's method we arrive at an equa- tion which has an integer for a root, we obtain a finite continued fraction as a root of the original equation, that is, we obtain a com- rtiensurahle fractional root. This of course cannot occur if we have previously determined all the commensurable roots both whole and fractional of any proposed equation, and removed the corresponding factors by division. 217. It may happen that in following Lagrange's method we arrive at an equation which is identical with one of those which preceded it; in this case the quotients of the continued fraction ■ recur, so that the continued fraction is a periodic continued frac- tion and its value can be found by solving a quadratic equation; see Algebra, Chapter xlv. The roots of this quadratic equation will involve a quadratic surd, and both of the roots will be roots of the proposed equation by Art. 44. 218. We will here give the general process which has been exemplified in Art. 215 in the second method of treating the equation cc^- 7a3 + 7 = 0. The object in view, is to apply Lagrange's method of approximation when a proposed equation has more than one root between consecutive integers. Let f{x) = be the pro- posed equation; form the avjxiliary functions f^{x),f^{x), f^{x),... Lagrange's method of approximation. 141 which occur in Sturm's theorem, stopping when one is obtained which is positive for all values of x; see Art. 199. Suppose that more than one root of the proposed equation lies between the consecutive integers a and a+1. Put a + - for a? in the functions /{x),f^{x)^f^(x),..., and denote what they become respectively by F(y), F^(y), F^(y),... If in the latter series of functions we sub- stitute successively any two numbers, as b and 6 + 1, the difference of the numbers of the changes of sign in the two cases will give us the number of roots of the equation F(y) = which lie between b and b + 1. For the results which we obtain by substituting b and 6 + 1 in F(y), F^{y), FJ^y),...^ are the same as those we should obtain by substituting respectively « + r a-nd a + - — - in the series f{x),f^{x),/^{x),...'j and therefore the difference of the numbers of the changes of sign must be equal to the number of the roots of the equation f(x) = which lie between a + j- and a + 7— y , that is, to the number of the roots of the equation F(y)=0 which lie between b and b + 1. If then we find that more than one value of y lies between the consecutive integers b and 6+1, we substitute 6 + - for y z in the series F{y), F^{y), FJ^y),...; then, by giving two consecutive integral values successively to z and substituting them we can determine whether more than one value of z lies between two consecutive integers. o We proceed in this way until we obtain an equation which has only one root between consecutive integers; and after that we need not pay any regard to Sturm's functions but continue the calcula- tion for this particular root by the method of Art. 212. Thus we are able to separate the roots and can calculate them without any omissions. As we do not require to know the values, but only the signs of F(y), F^{y), F„{y), ^.., we may in all cases multiply these 142 NEWTON'S METHOD OF APPROXIMATION. functions by such powers of y as will clear them of fractions; for y is supposed to be a positive quantity, and therefore any power of y is positive. Thus, for example, instead of F{y), that is, instead of 1 , we may use A-% 2/'/(«) + 2/"-'/'(») + ^V"(») + - + j^/"(»). supposing that/(a3) is of the degree n. XVn. NEWTON'S METHOD OF APPROXIMATION WITH FOURIER'S ADDITIONS. 219. We shall now explain Newton's method of approxima- tion to the numerical value of a root of an equation. Let f(x) = be an equation which has a root between certain limits a and jB the difference of which is a small fraction; let c be a quantity between a and p which is assumed as a first approxi- mation to the required root, and let c + h denote the exact value of the root, so that h is a small fraction which is to be determined. Thus/(c + h) = 0, that is, by Art. 10, /(c) + hf{c) -^ 1^2^(0) + ^/-(c) + ... + |/«(c) = 0. Now since h is supposed to be a small fraction A^, h^,... will be small compared with A; if we neglect the squares and higher powers of A in the above equation we obtain / (c) -f- A/"' (c) = 0; thus Supposing then that we thus obtain an approximation to the f{c) value of h, we have c — ,, ' as a new approximation to the root / {^) of the proposed equation. Denote this new approximation by Cj , fie ) and then proceeding as before we obtain c^ - VtVK as a new ap- proximation; and so on. We shall presently examine more closely the conditions which , Newton's method of approximation. 148 must hold in order that this method may be safely applied. It is of course obvious that such examination is necessary, since the pro- cess is not universally applicable; for if /'(c) is small compared with /(c) the supposed approximate value of h is not a small fraction as it should be. 220. As an example of Newton's method we will take the equation which Newton himself selected, namely, x^~2x-6 =0, say fix) = 0. Here as = 2 makes /(x) negative, and x = S makes /(x) positive, so that a root of the equation f(x) = lies between 2 and 3. Again, aj= 2J makes /{x) positive, so that the root lies between 2 and 2^; also a3 = 2-2 makes /(«) positive; thus the root cannot differ from 2*1 by so much as •!. Suppose then c = 2-l; then thus c, = 2*0946 nearly. Then for a new approximation we have c, -p^ = G^ - -.00004852 nearly = 2-09455148 nearly. 221. This process is very simple in theory and not difficult in practice; but it is not of certain success unless some precautions are taken which we shall presently explain. For suppose that c is an approximate value of the root, and that c^ = c- jrj-r , we are not sure without further investigation that c^ is nearer than c to the real value of the root. In the preceding example, after we had ascertained that there was a root between 2 and 2*2, we assumed 2-1 as a first approximation and deduced 2 '0946 as a new approximation. But we are not sure as yet that 2-0946 is nearer to the root than 2*2; if however we put 2-1 for x we find that f{x) is positive, and thus the required root must lie between 2 and 2*1, and now we know that 2-0946 is nearer than 2-2 to this root. But we do not know even now that 2-0946 is nearer to the root than 2-1. If we put 2-0946 for x we find 144 NEWTON'S METHOD OF APPROXIMATION that f{x) is positive, and this shews that the root lies between 2-0946 and 2; thus 2*0946 is nearer to the root than 2-1. 222. Fourier has given a rule by which we are saved the trouble of such repeated examinations as we have exemplified in the preceding Article; this rule guarantees the success of New- ton's method when certain conditions are satisfied. Fourier's supplement to Newton's method depends upon a property of the first derived function of a given function, which we will now prove. 223. If a and h are any two quantities, some quantity \ inter- mediate between a and h exists, such that m-f{o) = {h-a)f'{\). For let F{x) denote f{x) -f(a) - 1^ |/(6) -/(a)\ ; then F{x) vanishes when x = a and also when x = b. Therefore by Art. 102 the equation F'{x) = has a root between a and b. And, by Art. 1 1, F'{x} =/'(x) — 1 ) hence some quantity X intermediate be- tween a and h must exist, such that /'(^) ~^}, — "LjlJ. = O; there- fore /(*)-/(«) = (6 -a)/'(X). 224. Suppose that h is greater than a-, then f(b) is algebrai- cally greater or less than f{a) according as /'(X) is positive or negative. Ji f'{x) is positive between x = a and a3 = 6, then /'(A.) is necessarily positive, and if /' (a?) is negative between x-a and x=h, then /'(A) is necessarily negative. Hence we have the following result; iif'{x) is constantly posi- tive through any interval, f{x) increases with x through that interval, and ii f {x) is constantly negative, f{x) decreases as x increases through that interval. By the increase or decrease of f{x) we mean algebraical increase or decrease. We may however state our result thus; \i f'{x) retain the same sign through any interval, then as x increases through that interval f{x) increases numerically when it has the same sign as /'(x), and decreases numerically when it has the contrary sign. WITH Fourier's additions. 145 225. We shall now enunciate and prove Fourier's rule. Let f{x) = be an equation which has one root and only one between a and p', and suppose that the equation /'{x) = has no root between a and p, and also that the equation /"(x) = has no root between a and /?; then Newton's method of approximation will certainly be successful if it be begun and continued from that limit for which y (a?) and /"(oo) have the same sign. It follows from our suppositions that /(x) changes sign once and only once between a and ^8, and that /'(x) and /^'{x) do not change sign between a and p. We will suppose /3 - a to be positive. (1) Suppose that f{x) and /"(x) have the same sign when x=a. Take a for the first approximation ; then Newton's second approximation is a - "^ttt- • I^et a + h denote the true value of the root ; then /(a + h) = 0. Now by Art. 223, we have /{oi+h)-/{a)=h/'(k), where X lies between a and a + h, thus h = - •{// , and the true value of the root is a - ^7^ . We have / W J {^) then to shew that a - 'i,Y^ is nearer than a to the true value of the root. Since h is necessarily a positive quantity, /(a) and /'(A) are of contrary signs, and /(a) is of the same sign as f'\o), and therefore /'(A) and /"(a) are of contrary signs. Hence/' (x) decreases numerically as x increases between a and P, by Art. 224, so that y*'(A) is numerically less than/' (a) j therefore — ^, , is a positive quantity which is numerically less than the positive quantity - ^^ . This shews that Newton's second approxima- tion is nearer to the true value of the root than the first approx- imation. Let a^ = a - "0^^ ^ *^^^^ /('^i) ^^^ /''("i) ^^^® *^® ^^^® ^^o^» >/ \ ) and the approximation can be continued from a,. T. E. 10 146 Newton's method of approximation (2) Suppose that /(a;) and /"(a;) have the same sign when x=^p. Take P for the first approximation, then Newton's second approximation is P- j-rnl:- I^et P + h denote the true value of the root; then /{/3 + h) = 0. Now, by Art. 223, we have f{P + h)-f(/3) = h/'{\)j where X lies between ^ and /3 + h; thus h = - ^i^T7v-\ • ^^ "hsLve then to shew that /? - jftttt is nearer than p to the true value of the root. Since h is necessarily a negative quantity, /{P) and /'(A.) are of the same sign, and/{/S) is of the same sign asy"(/5), and thereforey*'(A) £ind/"{p) are of the same sign. Hence /'(a?) increases numerically as x increases between a and Pj by Art. 224:, so that /'(A) is numerically less thsmf (p). Therefore , ^ is a positive quantity which is numerically less than the positive quantity j^yfy. • -^^^^ shews that Newton's second approximation is nearer to the true value of the root than the first approximation. Let P, = P- P^ ; then /(^J and f"{P,) have the same sign, and the approximation can be continued from p^. 226. The preceding Article shews that the conditions given by Fourier are sufficient to ensure the success of Newton's method of approximation. When these conditions are satisfied, and the approximation is begun and continued from that limit for which f{x) and f"{x) have the same sign, we obtain a succession of values, which continuously increase up to the real value of the root or diminish down to it, according as the limit from which we start is less or greater than the true value of the root We will now briefly shew that Fourier's conditions are necessary. If we start with an assumed value c, Newton's second ap- f(c) proximation corrects this by adding — "^/Vt, while the true value of the root would be obtained by adding — -tttA • Hence the WITH foueier's additions. 147 permanence of sign oi f'{x) is necessary in order that we may be sure that f'{c) and f'Q^ have the same sign; if these quantities do not have the same sign the Newtonian correction has the wi'ong sign, and Newton's second approximation is further from the true value of the root than the first approximation. The permanence of sign oi f'\x) is necessary in order to en- sure that f'{\) is numerically less than f {c). If this is not the case the Newtonian correction is numerically greater than the true correction, and thus, supposing the correction to be of the right sign, the true value of the root lies between Newton's first and second approximations. In this case Newton's second approxi- mation may be nearer to the true value of the root than the first approximation, but is not necessarily so. 227. In the example of Art. 220, it may be shewn that the equation f{x) =0 has only one root between 2 and 2*1, and that the equations f'{x) = and f"{x) = have no roots between these limits; also /(ic) and /''(cc) are both positive when £c = 2*l. Hence the Newtonian approximation will certainly succeed if it be begun and continued from the limit 2'1. For another example take the equation a;^ — 7a3 + 7 = 0, say f{x) = 0. It may be shewn by trial that the equation has one root between 1'3 and 1*4; the equations /'(a;) = 0, and/"(a;) = 0, have no roots between these limits; also y (a;) and /"''(a?) are both positive when a; = 1 "3. Hence the Newtonian approxima- tion will certainly succeed if it be begun and continued from the limit 1*3. 228. We will now shew how to estimate the rapidity of the approximation. Suppose c to be the approximate value of the root which has been obtained at any stage of the process; then fie) the true value of the root is c - ^./ ' , so that the numerical value / v) fie) of the error at this sta^e is ■.;;/, . which we will denote by r. e next approximate value will be c - jftt-} ^^^ ^^'^ *^® ^^" 10—2 i 148 Newton's method of ArPROXiMAxioN nierical value of the error is "].} \ - ^^^ , that is, r ~ ^ - . / W J \9) J (c) And by Art. 223, we have /' (c) -/' W = (c - ^) /" W, where [i lies between c and A; thus the error is — ^^ — ,.;•{ ^^A . Now X / (c) lies between c and the real value of the root, so that c — A, is less r^ f"(n\ than r : hence the error is less than •;, , \^ . Let the greatest value which f"(x) can take between the limits considered be divided by the least value which f {x) can take, and denote the quotient by q\ then the error is a fortiori less than qr^. For example, in Art. 220, the root lies between 2 aud 2 1. Thus to find q we divide the value of 6a; when x = 2'1 by the value of 3oc^- 2 when x=2; therefore q=l-26; and as q is nearly unity, the number of exact decimal places in the approximation will be nearly doubled at each step. 229. The student who is acquainted with the elements of the application of the Differential Calculus to the theory of curves, will find it easy to illustrate geometrically Fourier's rule for con- ducting Newton's approximation. Suppose FQE to be a part of the curve determined by the equation y=f{x). Then we may be supposed to know OJ/and ON^ and to require the value of OQ) that is, we require to know the point where the curve cuts the axis of x. At the point P it is obvious that f{x) is negative if Oy be the positive direction of the axis of y\ and f"{x) is also negative at P, since the curve at F is convex to the axis of x. Draw the ■fl \ tangent TTj let OM=a, then MT =-~~, as is known by the *^ \ / Differential Calculus; so that, starting from M the ISTewtonian approximation proceeds to T. And as T falls between M and Q it is obvious that the method succeeds in this case, and that the approximation can be continued from T. WITH FOURIER'S ADDITIONS. 149 A At the point R we have /(a;) positive and/"(£c) negative. Draw the tangent RS-, then, starting from N the Newtonian ap- proximation proceeds to S, and *S' and N are on oj^posite sides of Q. Moreover there is no security that ^*S' is less than QN, and there is no security that the approximation can be continued from S. Thus the approximation cannot be safely begun from N. The student may easily illustrate by figures the condition that f'{x) and f"{x) should retain an unchanged sign between the limits considered. If however, in any example, we know that NS is less than NM we may start from N^ as the point aS' will then fall between Q and J/, and the approximation can be continued from S. 'i,jn[ is less than /S- a. Messenger of Mathematics^ Vol. iii. page 40. Let ON = P; then we may start from N if 150 HORNER'S METHOD. # XVIII. HORNER'S METHOD. 230. "We shall now explain the method of approximating to the numerical value of a root of an equation which was in- vented by the late W. G. Horner. Foy the history of this part of the subject we refer to a memoir by Professor De Morgan in the Companion to the Almanac for 1839. Lety(a;) = be any equation ; then y (a + a;) = is an equation the roots of which are less by a than the roots of the first equation. The equation f{a + x)^0 becomes when developed f(a) +^/'(a) -.x^-g + «=' 7^-^+ - .+ ^'"•^ = 0- Now the essential part of Horner's method consists of a pro- cess by which the coefficients of the last equation may be system- atically and economically calculated ; we have already - observed that such a process will be useful ; see Arts. 11, 54, and 214. 231. Suppose, for example, that f{x) = Ax' + Bx^ ■¥ Cx^ + Dx^ + Ex + F ; then f{a) =Aa' + Ba*+ Ca^ + Da' + Ea + F, f (a) =5Aa' + iBa' + 3(7<+ 2Dc^ + E, 1 f"(a) = lO^a' + QBd" + 2>Ca + D, ^, 0^ i/""'(a)=x' (1) We may calculate /(a) in the manner explained in Art. 5, thus; / hoener's method. 151 A =1, Aa + B =F say, Fa + C=Aa^ + Ba + Q = § say, Qa+B=Aa' + Ba' + Ca+D = R ssij, JRa +11 = Aa^ + Ba^ + Ca' + Da + U = S say, Sa + F= Aa' + Bo" + Ca^ +Da^ + Fa + F =f{q). Here each line is obtained by multiplying the preceding lino by a, and adding on in succession the terms B^ G, D, F, F. (2) We may now calculate /'(a) in the same way as /(a) was calculated, using A, P, Q, F, S in the same way as A, B, C, F, F, F were used; ^ziJuJ-^Jii A - A, Aa + F=2Aa + B = Tseij, Ta+Q= 3Aa' + 2Ba + = CTsay, Ua + F = 4:Aa^ + 3Ba' + 2Ca + F= F say, Va + S = 5Aa' + 4:Ba^ + 3Ca' + 2Fa + F =/'{a). (3) We may now calculate \f"{cL) in the same way as /(a) and/'(cfc) were calculated, using A, T^ U, F; A =A, Aa + T = 3Aa + B=Wssij, Wa+U= QAa^ + 3Ba + C = Xssij, I- Xa + F= lOAa' + eBa' + 3Ca + F = lf\a). I (4) We may now calculate -r^f"'{a) in the same way, using A =A, Aa+ W= 4:Aa + B^ Y say, Ta+X= lOAa' + iBa + (7= i/'"(«). 1^ 152 Horner's method. (5) We may now calculate 17 /""(^) ^ ^^® same way, using A and Yj A =A, Aa+Y^5Aa + B = ^f'"\ay (6) Lastly, ^ = ^ /-'(«). The above process may be conveniently arranged thus ; An Fa Qa Ba Sa Aa' Ta Ua Ya IT, F 7>1. Aa W Wa X Xa Aa Ya f Aa * \\r"i-)~ The quantity under any horizontal line, is obtained by adding the two quantities immediately over the line. We have thus shewn Horner's process of forming the coeffi- cients of the- equation f{a + x) = when the equation is of the fifth degree ; we will hereafter prove that this process is applicable whatever may be the degree of the equation. We will give a numerical illustration of the process and then explain the use of the process in approximating to the root of an equation. For a numerical illustration suppose a = 2 and f(x). = Zx\-x^ + ix' + 5x-K ■ If nHORNERS METHOa v ^ 153 a f -1 + 4 + 5 -8 (Jr ^ 6 12 2^ , 5^ 114 ./-) 6" 11 26 57 106^ 6 24 70 192. 12 35 . 9.6 249' f ^^K . 6 36 142 18 71 238^, :t/^ (^ ' 6 48 -^^ " '.■ 24 ¥? , a III 6 30 r.n^ Thus/(2 + a;) = 3a;'+30«* + 119a;'+238aj ^+ 249a; +106.. 232. Suppose, for example, that we have an equation with a root lying between 300 and 400 ; form a second equation the roots of which are less than those of the first equation by 300, so that the second equation has a root lying between and 100. By trial let the greatest multiple of 10 which is contained in this root be found; suppose it to be 70; form a third equation the roots of which are less than those of the second equation by 70, so that the third equation has a root between and 10. By trial let the greatest integer which is contained in this root be found; suppose it to be 2; form a fourth equation the roots of which are less than those of the third equation by 2, so that the fourth equation has a root lying between and 1. By trial let the greatest number of tenths which is contained in this root be found; suppose- it to be 8 tenths; form a fifth equation the roots of which are less than those of the fourth equation by '8, so that the fifth equation has a root lying between and -1. By trial let the greatest number of hundredths which is contained in this root be found; suppose it to be 7 hundredths. Now suppose that -07 is exactly a root of the fifth equation; it follows that 372-87 is exactly a root of the first equation. Next suppose that '07 is not exactly a root of the fifth equa- tion; then it follows that an equation exists the roots of which are less than those of the first equation by 372-87, and which 154 Horner's method. has a root lying between and '01. Thus the first equation has a root which lies between 372-87 and 372-88. Thus we see how by a series of operations of the kind given in Art. 231, we either arrive at the exact value of the root of an equation, or we may approximate to it as closely as we please. 233. In the preceding Article we have stated that certain numbers must be found hy trial; we shall now shew that we can easily guide ourselves in these trials. Let f{x) = be the proposed equation, and suppose that by one or mare- operations we have derived the equation which has its roots less than those of the proposed equation by c, that is, suppose we have formed the equation f{G + x)=-0, and suppose that this last equation has a small root. Then c is an approximate value of a root of f ( A the original equation; hence by the pi-eceding Chapter c — ^, > will be in general a nearer approximation to that root. Thus — jr)-:] is an approximate value of the number which we want in ordei- to continue the operation. 234. Example. Let f(x) = 2x^ - 473a:^ - 234a; - 711-. It will be found by trial that/(200) is negative and/ (300) positive, so that the equation /(x) = has a root between 200 and 300. We proceed to diminish the roots by 200. 2 -473 -234 -711 {20.(1 400 -14600 -2966800. -73 -14834 -2967511 . 400, 65400 327 50566 400 727 ,\ Hence the equation which has its roots less than those of f[x) = by 200 is 2ic'+ 727ic'+ 50566aj- 2967511 = 0; so that /(200) = - 2967511 and /' (200) = 50566. HORNER'S METHOD. 155 Hence - ^/,^ ^ is more tlian 50. We then proceed to climi- nish the roots of the equation just given by 50. 2 727 50566 -2967511(^50 100- 41350 4595800 827 91916 , 1628289 We thus find that 50 is too large a number, for we have /(250) = 1628289 a positive quantity, while /(200) is negative; _so that the root we are seeking is less than 250. In £act, in guiding ourselves in the manner explained in Art. 233 we are Jiable to select too large a number for trial, especially in the early part of the operation; a similar failure occurs sometimes in the -ordinary process of extracting the square root of a number. We then try 40. '2 727 50566 -2967511 (^40 ^ _^ 32280 3313840 • 807 82846 346329 ms 40 is also, too large, for/(240) is positive. We then try 30. 727 50566 -2967511 (30 2225280 • -742231 60 787 23610 74176 60 25410 847 99586 60 "9^7 Thus /(230)= - 742231 a negative quantity, so that 30 is the right number. Hence the equation which has its roots less than those of [a:) = by 230 is 2ic^+ 907ic'+ 99586a; - 742231 = 0. [Here/' (230) = 99586 so that - 4L^ -7 approximately. 156 HORNER S METHOD. We proceed then to diminish the roots of the equation just given by 7. 2 907 99586 -742231 (J 14 6447 742231 921 106033 ' This shews that /{237) = 0j so that 237 is a root of the original equation. The whole operation is usually exhibited thus; -473 -234 -711 {237 400 -14600 -2966800 -73 -14834/ -2967511*) 400 65400 2225280 327 U " 50566*- -7422311^ 400/ 23610 742231 727* 74176 60 25410 787f 99586t 60 1 6447 847 r 106033 60/ 907t 14 _^j 921 Here the mark *. shews wliere the first part of the operation ends, and the mark f shews where the second part of the operation ends. 235. We will now take an example of an equation which has no commensurable root. Let f{x) -x^- 3x^— 2x + 5. It will be found by trial that /(3) is negative and /'(4) positive, so that tlie equation /(x) = has' a root between 3 and 4. The following will be the operation for approximating to this root as far as three places of decimals. hoener's method. 157 ■ 3 -2 5 (^3-128. 3 -6 -2 -I*-. 3 ^ •761 3 T* - •239t 3 •61 7-61 •167128 ■ 6* --071872^, •l -62 8-23t. •068273152 61 - -003598848 •1 -1264 6-2 8-3564 •1 •1268* 6-3t •02 8-4832^^^ •050944 6-32 8-534144 •02 •051008 6-34 8-585152 •02 6-36J •008 6-368 •008 6-376 •008 6-384 Here to find the second figure of the root we have — ~ , so & 7 ' that *1 is the nearest number to be tried ; to find the third figure — -239 of the root we have — ^Toq" > ^^ *^^* *^^ ^^ *^^® nearest number to be tried; to find the fourth figure of the root we have 158 HORNER S METHOD. •071872 , so that -008 is the nearest number to be tried. In ^ 8-4832 all these cases the number suggested is found to be correct. 236. As another example take the equation given in the pre- ceding Article, and approximate to the root which it has between 1 and 2. The operation is usually exhibited thus ; 1 3 1 -2 — 2 -4 -1 5 (1-2016 -4 2 1 1000* -992 1 1 -500* 4 '80000001 -4879399 00* 2 -496 8 3120601000+ -2927060904 2 2 -48800001 601 193540096 4 2 -4879399 602 600t 1 -487879700+ 36216 601 1 -487843484 36252 602 1 6030+ 6 6036 6 6042 6 - 487807232 6048 Horner's method. 159 The difference between this arrangement and that in Art. 235 arises from the fact that it is usual in practice to omit the decimal points, just as they are omitted in the process for extracting the square roots of numbers approximately. The following rule with respect to the decimal part of the root will be sufficient. When all the whole figures of the root have been found and the decimal part of the root is about to appear, annex one cipher to the right of the first working column, two ciphers to the right of the second working column, three ciphers to the right of the third working column, and so on if there are more than three workinor columns : then proceed completely through one stage of the operation as if the new figure of the root were a whole number. Then annex ciphers again as before. It will be observed that after the 2 in the root the next fiomre considered as an integer would be approximately given by 8000 in..-,, . . 1 . . -, Toonn ' ^ •^^ than unity ; so a cipher is put m the root and we annex another cipher to the first working column, two more to the second working column, and three more to the third, and proceed as before. The ciphers will serve to distin- guish the several stages of the operation, so that the marks'^! J may be omitted. It is obvious that in all the preceding examples the first work- ing column might have been shortened by performing in the head the easy work which occurs, and putting down only the results, but we have thought it clearer to exhibit the whole for the student. 237. After a certain number of figures in the root have been found correctly, an additional number may be obtained by a con- tracted operation. We will exemplify this by calculating the positive root of the equation x^ + 3x^ - 2x - 5 = 0. We will first perform the operation at full until five decimal places of the root have been determined. 60 horner's method. 3 -2 -5 (1-33005 1 • 4 2 4 2 -3000 1 5 2667 5 700 - 333000 1 189 332337 "60 889 - 663000000000 3 198 564352475125 63 108700 -*9864752487igi 3 2079 66 110779 3 2088 690 112867000000 3 3495025 693 112870495025 3 3495050 696 1128739900^ 3 699U00 5 * 699005 • 5 699010 5 699015 Tlie rule for contracting the operation is the following ; strike off at every step one figure from the right of the last column but one, two figures from the right of the last column but two, and so on. We will now resume the example just considered and apply this contracted process. HORNERS METHOD. 161 699015 112873990075 55921 -98647524875 (^1-33005873 90299639432 11287454929 55921 - 8347885443^ 7901261018 11287510850* 489 -4466244251 338625624 1128751574 489 - 107998801 1128752063t 2 112875208 2 112875210 At the point where the full operation terminated we have 8 sug- gested for the next figure; we then reject 5 from the end of the last working column but one, and 15 from the end of the last working column but^wo. The first step in carrying on the work is to multiply 6990 by 8, and put the product in the next working column; the product is considered to be 55921, because we con- ceive 69901 multiplied by 8 and the last figure struck off", and so 55921 is nearer than 55920 to the true value. Then we add 55921 to 11287399007; the figure in the units' place of the sum we take to be 9 by allowing for the rejected 5. Thejmark ^ indi- cates where the first stage of the contracted operation finishes. Kow strike off" from the end of the second working column and 90 from the end of the first working column, so that the first work- ing column is reduced to 69. The next figure of the root is 7, and this stage of the operation finishes where the mark f is put. Strike off" 3 from the end of the second working column and 69 from the end of the first working column. The first working column now disappears, but still exercises a slight influence be- cause the next figure in the root is 3, and when 69 is multiplied by 3 and two figures rejected there remains a 2. T. E. 11 162 HORNER S METHOD. Only two working columns are now left; the remainder of the work coincides with the ordinary process of contracted division, and it will supply eight more figures in the root. 1287521,0 -107998801 (1-3300587395679825 101587689 1128752,1 -6411112 5643761 112875,2 -767351 677251 11287,5 -90100 79013 1128,7 -11087 10158 112,8 -929 902 11,2 -27 22 1,1 -5 The approximation may be relied upon very nearly up to the last figure. For if the whole operation were performed at full, the last working column would present a large number of figures on the right-hand side of those here exhibited, but those which are here exhibited would retain their places without alteration except perhaps the exchange in some lines of the last figure for another difiering from it by unity. 238. The root found in the preceding Article is the numeri- cal value of the negative root of the equation x^— 3x^- 2a; + 5 = 0. Hence the sum of the roots found in Arts. 235 and 236 should exceed the root found in Art. 237 by 3; because the sum of the three roots of the equation with their proper signs is 3. This will be found to hold approximately; and the student may exercise himself in carrying on the approximations to the two hoenek's method. 163 positive roots to more places of decimals than we have given, in order to verify more clearly the connexion between the sum of those two roots and the root calculated in Art. 237. 239. Various suggestions have been offered with the view of saving labour in the use of Horner's method. With respect to such suggestions we may quote the following remarks which occur in connexion with one of them. "But considering that the process is onfe which no person will very often perform, we doubt whether to recommend even this abridgment. All such simplifica- tions tend to make the computer lose sight of the uniformity of method which runs through the whole; and we have always found them, in rules which only occur now and then, afford greater as- sistance in forgetting the method than in abbreviating it." Fenny Gyclopcedia^ article Involution. 240. In Art. 231 it was stated that it would be proved that Horner's method of forming the equation f{a + x)=0 is uni- versally true. We will now consider this point. Let f{x) =p^x''+p^x''-'-hp^x'"'+ ... +p^^_^x+p^^, for X put y + a, and suppose thaty(a;) then becomes we have to prove that 5'„, 2'„_i, ... 5'ij S'o? ^^^ found correctly by Horner's process. It is obvious that g„=^^. Since y = x — a the following expressions are identically equal, p^x''+p^x"-' + p^x"-'+ ... +p„_,x+p^, and q^ {x - a)"+ q^ {x - «)""' + q^ {x - aY~^+ . . . + q^_^ (a; - a) + q^. Therefore q^^ is the remainder that would be found on dividing f{x) hy x-a; also the quotient arising from this division must be identically equal to 11—2 164 Horner's method. Then again q^_^ is the remainder that would be found on di- viding* the last expression by a; — a; also the quotient arising from this division must be identically equal to %{^ - »)""'+ ^, {^ - «)""'+ ^.(^ - «)""'+ • • • + 9,.-,' Then again q^_^ is the remainder that would be found on di- viding the last expression hy x-a; also the quotient arising from this division must be identically equal to q,{x-ay-'+q^{x-ay-'+q^{x-ar-'+...+q,^_^; and so on. Thus q^, q„_^f q,^_^, ^n-z^ ••• ^^® *^® successive remainders which occur in dividing, first f{x) hj x-a, then the quotient by x-a, then the new quotient by a; — a; and so on. And we see by Arts. 5, 7, and 9 that Horner's process determines these succes- sive remainders. 241. We have thus sufficiently discussed the subject of the approximate values of the real roots of equations. There is no easy practical method of calculating the imaginary roots of equations at present known; but theoretically this may be made to depend on what has been already given. For suppose a + hJ—\ is an imaginary root of an equation f{x) = ; then since the real and imaginary parts of/ (a + h J— I) must separately vanish, we obtain two results, which we may denote by P = and Q = i), as in Art. 41. Here F and Q will be functions of a and h, and if we eliminate a or 5 from the equations F = and ^ = 0, we obtain a single equation involving one unknown quantity ; and we require real values of this unknown quantity. Hence we can determine the imaginary roots of a given equation if we can form a certain other equation and determine its real roots. We shall hereafter shew how to form the equation which results by eliminating one of two unknown quantities from two given equations. We shall in Chapter xxi. explain another method which has been used for calculating the imaginary roots of equations. The student may also consult Dr Rutherford's essay on the Complete Solution of Numerical Equations. SYMMETKICAL FUNCTIONS OF THE ROOTS. 165 XIX. SYMMETRICAL FUNCTIONS OF THE BOOTS. 242. A function of two or more quantities is said to be a symmetrical function of those quantities if the function is not altered when any two of the quantities are interchanged. Thus, for example, a^ + lf + c^ is a symmetrical function of the three quantities a, h c; so also is ab + bG+ ca; for each of these functions is unaltered when we interchange a and b, or a and c, or b and c. 243. The coefficients of an equation are syminetrical func- tions of the roots of the equation. For by Art. 45, if the equation be £c" -hp^x''~^ +i^2«^"~^ + ... = 0, we have -/>j = the sum of the roots, p^ = the sum of the products of the roots taken two at a time, and so on ; and it is manifest that the functions of the roots which occur here are symmetrical functions. The object of the present Chapter is to shew that every rational symmetrical function of the roots of an equation can be expressed in terms of the coefficients of that equation. We shall begin with proving Newton's theorem for the sums of the powers of the roots of an equation. 244. Let /(^) denote cc" +jc>ja;""^ +;:>2aj""^-f ... +/>„, and let a, 6, c, d,... denote the roots of the equation f{x) = 0. Let S^ = a + b + c -h d + ..., S^ = a'+b'+c' + d'+..., S^ = a^ + b'+c^+d'+..., and so on ; thus S^ is the sum of the roots, S^ is the sum of the squares of the roots, S^ is the sum of the cubes of the roots, and in general S^ is the sum of the m^^ powers of the roots. 166 SYMMETRICAL FUNCTIONS OF THE ROOTS. By Art. 74 we have "^ "" ' X — a X — x — G The divisions indicated on the right-hand side of this identity can all be exactly performed by Art. 7 j and we have fix) fix) and similar expressions hold for "^—j^ , , . . . By addition then we obtain fix) = 72^"-' + (.S', + np^) x''-^ + (S^ +p^S, + np^'x""-^ -f . . . Also f'{x) = nx"-' + {n- l)p,x"-' + {n - 2)p^^x"-^ + ... ■\-{n-m)p^x"— ' + .... Equate the coefficients of the same powers of x in the identity ; thus Si + np^ = (')i-l)p^ or S^+p^ = Oy S^ +p,S^ + np,= (n - 2} p^ or S^ +p^S^ + 2p^ --= 0, and generally <^r ^n. +PA-1 +P2^m-2 + • • • +Pm-lS. + ^^Pm = ^' In this general result m is supposed to be less than n. By means of the general result we can express the sum of the m^ powers of the roots in terms of the coefficients and the sums of inferior powers of the roots ; and thus by repeated operations we may express the sum of the m^^ powers of the roots in terms of the coefficients only. SYMMETRICAL FUNCTIONS OF THE ROOTS. 167 Next suppose that m is not restricted to be less than n. Multiply the given equation f{x) = by aj"'"'*; thus aj"'~"/(cc) = 0, that is, x"" + p^x"^"" + p^x"^'"" + . . . +px'^~'' = 0. Substitute for x successively a, 6, c, ... and add the results ; thus By this theorem we can express the sum of the m**" powers of the roots of an equation in terms of the coefficients and the sums of inferior powers of the roots when m is not less than n ; and thus by repeated operations we may express the sum of the m^^ powers of the roots in terms of the coefficients. Practically the following is a very convenient method. We have ^ ' x — a x-o x-c therefore xf'ix) XXX f{x) X — a X— b x — c '('-r*('-r*('-r- 'S', s^ S^ X x^ x^ Thus, if we actually divide af(x) hj /(x), the coefficients of the terms in order will be n, S^, S^, The division may be advantageously performed in the manner explained in the Algebra^ Chapter lviii. 245. To find the sum of the negative powers of the roots of the equation /{x) = 0, we may put — for x and find the sum of the corresponding positive powers of the roots of the transformed equa- tion in 2/. 168 SYMMETRICAL FUNCTIONS OF THE EOOTS. Or we may make m successively equal to ?i - 1, ?^ - 2, 9^ - 3, . . . in the result of the preceding Article; and thus obtain suc- cessively >S'_j , a9_2, aS'_3,. . . . 246. The general problem of finding the value of any rational symmetrical function of certain quantities may be reduced to the problem of finding the value of certain simple functions, as we shall now shew. Any rational symmetrical function which is not integral will be the quotient of one rational symmetrical integral function by another ; so that only integral functions need be considered. Any rational symmetrical integral function which is not homogeneous will be the sum of two or more rational symmetrical integral func- tions which are homogeneous; so that only homogeneous functions need be considered., A homogeneous function may consist of different parts in which although the sum of the exponents remains the same, the exponents themselves are different; in such a case the homogeneous function is the sum of two or more homogeneous functions of the same degree in which the exponents are the same for all the terms. Hence it follows that we need only consider such rational symmetrical functions as are integral and homogeneous, and in which the exponents are the same for all the terms. 247. Let (X, h, c, d,... denote the roots of a given equation. By Art. 244 we can express in terms of the coefficients the value of a'^ + h"' + c"' + d'"+ ... This function may be said to be of the Ji7'st order, since each term involves only one of the roots. A function may be said to be of the second order when each term involves two of the roots, as a"'b^ + a'"c^ + h"'c^ + ... Here every permutation is to be formed of the roots taken two at a time, and the exponent m placed over the first root and p over SYMMETRICAL FUNCTIONS OF THE ROOTS. 169 the second. We shall denote this function by '^oTh^, as it is the sum of all the terms which can be formed like a""}/. A function may be said to be of the tJiird order when each term involves three of the roots, as arh^d'+ oTc^'d' + a'^h^d' + . . . Here every permutation is to be formed of the roots taken three at a time, and the exponent m placed over the first root, p over the second, and q over the third. We shall denote this function by '^oTh^'c', as it is the sum of all the terms which can be formed like Similarly we may have functions of the fourth and higher orders, and may use a similar notation to represent them. Since we have shewn how to express the function denoted by S^^ in terms of the coefficients of the equation it will be sufficient to shew that any of the functions we have to consider can be expressed in terms of such functions as S . 248. To find the value of the symmetrical function of the second order Sa^b''. We have /S',„ = a'" + 6'" + c'"+ .,., S^, = a'' + Jf + (f + .... By multiplication we obtain SJ^ = a'""P + h'^^v + c'+P + . . . that is, SJ^ = S^^^ + %a-lf, and therefore ^a'^lf = SJ^ - S^^^. This supposes that m and p are unequal. If we suppose p equal to m the terms in "^a""!)^ become equal two and two, so that this sum may be expressed thus, 2S (a^)""; and therefore 170 SYMMETRICAL FUNCTIONS OF THE ROOTS. 249. To find the value of the symmetrical function of the third order 2a™bPc**. We have '^a'^b' = a""!)' + h^'c^ + a'^c^^ . . . , By multiplication we obtain S^a'^y = a"'^'b'' + JT'-^'c'' + c'"-'W + . . . + a'"h^^'' + h'^c^^'^ + c'^a^^' + . . . + a'^lfc'+... The terms on the right-hand side form three sets, which in our notation are denoted by ^a'^^'^l/, ^a^^'^lT, ^a'^h^'c''; thus S.^a^'b^ = ^a"'^^b' + ^a^^^"^ + S^r^V. Substitute for ta'"b^, ^a'^^'^b^, and ^d'^'^b'^ their values from Ai't, 248, and we obtain We have supposed m, p, q all unequal. Suppose, however, that m=p; then, as in Art. 248, we have 2S (a5) V = SIS, - S^S, - 2S^,,S^ + 2S,„^,. If m =p = q, the sura "S.aTW reduces to 2 . 35 {abc)'; thus The method of this and the preceding Article may be con- tinued to any extent, and thus a function of any order like Sa"^ and '^a"'b^c'^ may be expressed in terms of the coefficients. Hence by Art. 246, the object proposed in tlie present Chapter can be attained. 250. We have shewn how the function denoted by aS',^ can be expressed in terms of the coefficients; and thus of course the sum of any number of such functions as S^ can be so expressed. The following method will, however, be generally more advan- tageous in such a case. If (f>{x) denote any rational integral function of x, it is required to express in terms of the coefficients the sum (a) + <^ (6) + ^ (c) + ... SYMMETRICAL FUNCTIONS OF THE ROOTS. I7l ^^r 1 r(x) 1 1 1 We have -ft-^ = + , + + ...; / {x) X — a X — X — c therefore ^^f^ ^ *M ^ ^ . M ^ ... J {x) x — a X — X —c ct>(x)-{a) , (b) , cf>(x)-(c) , = f- — , 1 f- . . . x—a x—o x—c ^ {c) , _ X—a x—b x—c In this identity the integral parts and the fractional parts will be separately equal; also such expressions as — are in- X — a tegral by Art. 7. Let {x)f'{x) be divided hj /(x), the process being carried on until the remainder is an integral function of x of lower degree thany(aj); let li be this remainder. Then by considering the fractional parts of the identity we have E _Ma) , <^(^) , He) ^_^ y (£fc) x — ax — h x — c Multiply up; then i?- a"-* U{a) + 4>{h) +{c) + ...| + terms involving lower powers of x than a;"~\ Thus (fi{a) + (f>{b) + {c) + ... is equal to the coefficient of x"~^ in I?. 251. As an example of the formulae of this Chapter suppose it required to find the sums of the powers of the roots of the equation x* -x^ — 7x^ + x+ 6 = 0. ^\---pA-2p^=l + U=15, ^« = -i^A-M- 3^^3 = 15 + 7-3 = 19, ^^* = -^A-^A-M-4i'.= 19 + 105- 1-24= 99, '^5 = -pA-pAs-PsS.-P,^,= 99+133-15- 6 = 211, 'S'6 = -M5-i^A-M-J^A=211 + 693-19-90 = 795, and so on. 172 SYMMETRICAL FUNCTIONS OF THE ROOTS. Put - for X in the given equation; then Thus for the sums of negative powers of the roots of the original equation we have and so on. These results may be easily verified, as the original equation has been constructed so as to have for its roots - 2, - 1, 1, 3. L^- Again, suppose we require the values of S^ , S^, S^ and S^ in the biquadratic equation x*-irpx^+ qx^+ rx + s = 0. S\+p=^ 0, therefore S^ = -p, aS; +pS^ + 2q = 0, therefore aS^ =/ - 2q, S\ + 2jS^ + qS^ + 3r = 0, therefore S^ = -p(p^- 2q) +pq-3r = — p^ + 3pq - 37'j S,+pS^ + qS^ + rS, + 4:3 = 0, therefore S^ = -p{-p^ + 3pq -3r)-q {p^ - 2q) + rp-is = // - 4:p^q + 4rp + 2q^ - is. As another example, let a, p, y, S denote the four roots of the biquadratic equation a;* + px^ + qx^ + rx + s = 0; letA = l{al3 + yS), 5 = i(ay + ;88), C = ^(a8 + /3y); and let it be required to find the value of the following sym- metrical functions of the roots of the biquadratic equation, (1) A+JB + C, (2) AB + BC+CA, (3) ABC. SYMMETEICAL FUNCTIONS OF THE ROOTS. 173 (1) A+B+C=haP + ay+aS+l3y + pS+yS)=l , (2) AB + BC-^CA = ^ {a'l3y + a'yS +•.•) = T ^<^'Py = ^ (^S^^S^ - S,' - 2S^S^ + 2S^) j by the metliod of Art. 249. Then the values of S^, S^, S^ and S^ may be substituted which have already been obtained, and the value of - ^a^/3y will be known. Or we may proceed thus, And a^yS = 5, and 2 ^ =^ - 4, by Art. 48 ; therefore AB + BC + CA = - {j^r - is). (3) ABC = \ {a'Pyh + . . . + a^^V' ^"')=\ ^^'/^^^ + \ ^""'^V- The values of these two symmetrical functions may be found by the methods of the present Chapter directly ; or we may ab- breviate those methods thus, 2a'/3yS = a/?y82a' = S (/ - 2g), ct \_s s J for to find 2-^ we have only to obtain the sum of the squares of the roots of the equation in y which is formed by writing - for x. Thus ABG = ^{f^fs-iqs). The values of the functions o£ A, B, G which have been found may be verified; for A, B, C, by Art. 189, are the roots of the cubic equation in m in Art. 188. 174 APPLICATIONS OF SYMMETRICAL FUNCTIONS. XX. APPLICATIONS OF SYMMETEICAL FUNCTIONS. 252. In the present Chapter we shall give two applications of the theory of symmetrical functions of the roots of an equation ; the first application will consist in forming the equation which has for its roots the squares of the differences of the roots of a given equation, and the second application will be to prove an important theorem in elimination. 253. To form the equation which has for its roots the squares of the differences of the roots of a given equation. Suppose the given equation to be of the rC^ degree, and denote its roots by a, b, c, . . . . Then the roots of the required equation will be (a — by, {a — cf, . . .(6 — c)^. . . ; the number of these is the same as the number of combinations of n things taken 2 at a time, that is, -n{n — V); and this number will therefore denote the degree of the required equation. Put m for -n(n — 1), and suppose that the required equation is denoted by £c'" + g^cc"*-' + g,aj"*~^+ . . . + g,„ = 0. Also let s^ denote the sum of the r^^ powers of the roots of this equation. We have only to determine s^, s^,...s^, and then the coefficients of the required equation will be found in succession by the formulse of Art. 244, namely, s^ + q^ = 0, s^ + q^s^^ + 2q^ =^ 0, and so on. Let c}>{x) = {x-ay' + {x-by'' + {x-cy'+ ..., then 2s^ = (l>{a) + {b) + {c)+ ... Now let aS'p S^, S^,... denote the sums. of the powers of the roots of the given equation ; thus <}>{x) = naf'- 2rS^x^-' + ^'' ^^'''^'^ S,x''-' --.., + S^^. APPLICATIONS OF SYMMETRICAL FUNCTIONS. 175 Put for X in succession a, h, c,... and add ; thus The terms on the right-hand side which are equidistant from the beginning and the end are equal; therefore by rearranging and dividing by 2 we obtain s, = nS,^ - 2rSS_ + M?!!^ s.S, 1.2 ..4(_l)^M2>-l)...(r.l)^^._ Now S^, S^,... can be expressed in terms of the coefficients of the given equation; thus s^ can be found, and then finally the coefficients of the required equation. //' • 254. The last term of the required equation, namely that denoted by q^ in the preceding Article, may be calculated in another way. Let the given equation be denoted hjf{x) --=^ 0, so that /(x) = (x — a){x — h){x-c)... Then f '{x)=^ {x-h){x- c) ... + (x- a)(x-c) ... + ... ; thus f'{a) = {a-h){a-c)..., f\h) = {h-.a){h-c)... Hence g^=/'(a)/X&) /'(c).... Now let a, p, y,... be the roots of the equation/'(cc) = 0; then /'(x) = n{x-a){x-l3){x-y)...; therefore / ' (a) f (b) f (c) . . . = 7i"(a-a)(a-/5)(a-y)...(6-a){6-/5)...(c-a) But («-a)(6-a){c-a)... = (-l)''/(a)..., (a-^)(6-^)(c-/5)... = (-l)y(^)..., and so on ; 176 APPLICATIONS OF SYMMETRICAL FUNCTIONS, thus f{a)f{h)f{c)... = n- (-ir^"-V(a)/(/S)/(7) ... = ^"/(a)/(^)/(7)-.., for (-1)"'"-''=!. Now y(a)y*(/8)y(y)... is a symmetrical function of the roots of the derived equation/' (cc) = 0, and may therefore be calculated. 255. In Art. 109 we have explained one use which we may make of the equation whose roots are the squares of the differences of the roots of a proposed equation ; namely, we may thus determine the situation of the real roots of the proposed equation. But Sturm's theorem now answers this purpose more readily. However the equation which has for its roots the squares of the dififerences of the roots of a proposed equation will sometimes on inspection give information respecting the number of imaginary roots in the proposed equation ; for it is obvious that if this new equation can have negative roots the proposed equation must have imaginary roots ; and if the new equation has no negative roots the proposed equation has no imaginary roots. Also if the new equation has imaginary roots the proposed equation must have imagirary roots ; it will not however follow that if the new equation has no imaginary roots the proposed equation has none. For example, the proposed equation might be a biquadratic equation with roots ^\J—\ and ^fjij—l; in this case the new equation will only have real negative roots. It will be convenient to give the product of the squares of the differences of the roots in algebraical equations of the second, third, and fourth degrees. (1) ax" + 2bx + c = 0. rvu 1 ^ • iib'^-ac) Ihe product is ^ . (2) x^ +px" + qx + r = 0. By Art. 60 the product is APPLICATIONS OF SYMMETRICAL FUNCTIONS. 177 If the equation be aoi? + 2>bx^ + Zcx + c? =^ 0, this becomes - ^ |(26' - 3a6c + a^d^ + 4 («c - 5')^ j , or more symmetrically -^f{ad- hey -4:{b'- ac) (c' - hcl) \ . (3) x* + qx^-i-rx + s=0. By Art. 187 the product is where a", jS', y^ are the roots of a certain cubic. Hence the product is - ^ |(27r^ - 72^5 + 2qy - 4 {q' +12sf \ . If the equation be ax^ + 4:bx^ + Qcx^ + idx + e = 0, this becomes by Art. 187, -^^Uae-ibd+^cy-21{ad' + eh'+c^-ace-2hcdy\. 256. We shall now shew how to eliminate one of the unknown quantities irom two equations containing two unknown quantities, by the theory of symmetrical functions. Let the equations be and q^p(P + 5'j^""^ + S'g^""^ + ...+ q^^ = 0. The coefficients joi^,^:> J, p^i---, 2'oj 3'i) 5'2>--- ^^^ supposed rational integral functions of a quantity y, and x is to be eliminated. T. E. 12 178 APPLICATIONS OF SYMMETRICAL FUNCTIONS. Suppose that from the first of these equations the values of x could be found in terms of ij ; let these values be denoted by a^ hj c, Substitute them in the second equation, and we obtain m equations for determining y^ namely qy + q^-' + q^-' + . . . + ^„ = 0, so that all admissible values of y are contained among the roots of these equations. And conversely any root of any one of these equations is an admissible value of y. For suppose, for example, that the first of these equations has a root /?, and suppose, when /? is put for y in a, that the value is a; then a; = a, 2/ = /8 will satisfy the two original equations. For these values obviously satisfy the second equation ; and the first equation is satisfied by x = a, whatever y may be, and is therefore satisfied when we take x = a and give to y itl a the value jS. Hence it follows that by multiplying together the left-hand members of the above equations in y and equating the product to zero we obtain the final equation in y. Now in this product no alteration is made by interchanging any two of the quantities a, b, c, ..., so that the product is a symmetrical function of these quantities, and the value of it can therefore be expressed in terms of the coefficients Poy Pi^ P2^--' ^^ the first equation. Thus we shall finally obtain a rational integral equation in y, and this equation has for its roots all the admissible values of y and no others. " 257. For a particular example, suppose that the first equa- tion is a cubic in x, and the second a quadratic in x, so that we have to eliminate x from the equations where the coefficients are supposed functions of y. Here with the notation of the preceding Article we have APPLICATIONS OF SYMMETRICAL FUNCTIONS. 179 {q^a^ + q,a + q,) (qj)' + qfi + ^J (q^c' + q^c + q^) = 0, that is, q.2 + q^ahc + q^a%^c' + q^q^a?lr + q^q^a-h^c + q^q^tah + ?,<7/:S« + 5'o?2'^a' + qoqi'^a'bo + q,q,q,^a'b = 0. Also ahc = ~^\ a'b'c'^Ky ^a'b'c = abc^ab = -^:$ab = - ^--^^ 2 7 Vo Po Po Po Po ta-bc^abc^ci^^-^^, Po ^cirb = abc^ "" = - ^ (Ilil-^ _ 3 V by Art. 48. And by substituting these values we shall obtain the equation which results from the elimination of x. 258. If ive eliminate one unknown quantity between two equations of the degrees m and n respectively, the degree of the residting equation ivill not exceed mn. Let the equations be p,X- + p^X-^ +p^T"'-' + 4-^„. = 0, q^x" + q^x""^ + q^^x""'^ -f- + 9',. = ; the coeflScients in these equations are supposed to be functions of y. Moreover it is now supposed that the sum of the exponents of X and y in the same term is never greater than m in the first equation, and never greater than n in the second equation ; so that pp and qp may be of the degree p in y, but not higher. 12—2 180 APPLICATIONS OF SYMMETRICAL FUNCTIONS. Now suppose that x is eliminated by the method of Art. 257; the first member of the final equation in 2/ then consists of a series of terms, each of which is the product of m factors, and is of the form q a""*" x $,6""* x g'^c""* x ... And as we know that the series of terms forms a symmetrical function of a, 6, c, ..., the aggregate of the terms with the exponents just indicated will be Now the degree of qj^cit--- is not higher than r + s + ^+..., so that we have only to shew that the degree of %a"~''b"~'c'*~\.. is not higher than n-r + n-s + n-t+ ..., and then it will follow ^tt that the degree of the product is not higher than mn. The re- quired result follows from two observations. (1) From the formulae of Art. 244, it can be shewn that Sp does not involve higher powers of y than yp. (2) From the process of Arts. 248 and 249, it will follow that the value of '^a^bf^C ... will involve powers and products of JS^, S.^, AS3, ... /S'a+^+v+...; and in each term the sum of the subscript letters attached to the symbol S is X + ijt. + v+ ... Hence we conclude that in the final equation in y no power of y higher than y""" will occur. 259. The preceding Article gives the limit which the degree of the final equation in y cannot surpass; it may however in particular cases fall short of this limit. The theorem may be extended and the following general result obtained; if between any number of equations involving the same number of unknown quantities all those quantities are eliminated except one, the degree of the final equation cannot exceed the product of the degrees of the original equations. See Serret's Cours d'Algehre Superieure. SUMS OF THE POWERS OF THE ROOTS. 181 XXI. SUMS OF THE POWERS OF THE EOOTS. 260. By Newton's method, which is explained in Art. 244, the sums of the powers of the roots of an equation may be found successively; we shall now explain a method by which the sum for any assigned integral power of the roots of an equation may be obtained independently. Let a, &, c,... denote the roots of an equation fix) = 0, so that we have / (x) = {x — a) (x - b) (x - c). .. ; and suppose the equation of the n*'^ degree. Then fix) (. a\(. h X \ xj \ xj \ xj Take the logarithm of both sides, and then expand the loga- rithms on the right-hand side; thus log'^-^r^ = - - (a + ^ + c + ...) ° X x^ ' 1 ^ Thus on the riofht-hand side the coefficient of — is — '" ; X m S 1 hence we have -^ = the coefficient of -- in the expansion of m X ^ 1 f(^) •IT n - log -\ in descending powers of x. X This supposes m positive; if the sum for any negative integral power is required we can change x into - and find the sum for the corresponding positive power of the roots of the equation in y. 182 SUMS OF THE POWERS OF THE ROOTS. 2G1. For example, find the sum of the m^^ powers of the roots of the equation x^ -px 4- g' = 0. XX 2\x x-J 3 \x x^J m \x x^J The complete coefficient of —^ may be obtained by selection X fix) from the various terms in the value of -I02: -V^in which this ° x^ power of X can occur; these terms written in the reverse order are m \x X J ■ m-l\x icV 'm - 2 \x wi-2 The coefficient of — ;;, is therefore -1 1. ^ ^^m-2 1.2 ^ ^ - Thus S,=p'^- mp'^-'q + ^ ^ ^^ p m—4. _^,_ m(m-r-l) (m-2r + l) ^ Suppose q—1, then the quadi'atic equation is a reciprocal equation, and its roots are of the form a and-: see Art. 133. a Thus we have a + - = v, and also SUMS OF THE POWERS OF THE ROOTS. 183 , ^, 77i(m-r- l)...(m- 2r + 1) „ ,, + i-iy-^ ^-r^ ^p-^^4- ... We have thus obtained a general expression for a"" + -- in terms of powers of a + -; see Art. 138. Again, suppose q = — l; then the roots of the quadratic are of 1 / 1\'" the form a and — : thus we get an expression for »'"+(--) in terms of powers oi a — . a 262. Again, let it be required to find the sum of the 7/i*^ powers of the roots of the equation a;" - 1 = 0. fix) , 1 , f(x) 111 1 Here the coefficient of — is zero unless m is a multiple of n, and then the coefficient is — ; so that ^„= unless w is a multiple of 01, and then *S'„^= n. This result is often useful, and we will give three applications of it in the following three Articles. 263. We will shew how to find the sum of selected terms of a given series. Suppose that the sum of a^ + a^x + a^x^ -v ... ad infinitwin is known, and denote it by <^{x) : and let it be required to find the sum of the series a^cc"* + o^,„+„a3'"^" + a^+2„a5'""^^" + «c? infinitum. Let a, ^, y, . . , denote the rif°- roots of unity, that is, the n roots of the equation a;"- 1 = 0. Multiply both sides of the given identity by a"~"*, and then change x into ojx) thus 184« SUMS OF THE POWERS OF THE ROOTS. Similarly, /-'"(y:c) = «„/-"' + ay-'''-'x+ a^f-"'^'x'+ .., and so on. Add together the n identities which can thus be formed ; then on the right-hand side we obtain n times the required series, by Art. 262; thus = I |a"-> (ax) + r- (I3x) + /--<^ (yx) + ...}. As an example we will find the sum of x^ x' . . x+ rj + ■rj^ + ... ad infinitum. Here w = 1, ti = 3, {oo) = e*. Thus the required sum = i{aV(aa.) + /3^^(i8iB) + /<^ (yx)| . Now a=l, ^= 2 » 7 = Y • Hence {l3x) = e ^ ^ {yx) = e 2(^cos -^--V-lsm-y-^. And finally the required sum is i.-Je-1(eos^-#-V3sin£f^). SUMS OF THE POWERS OF THE ROOTS. 185 264. Again, by means of Art. 262 we can prove the folio wing theorem ; the expression {x + y)" -x" -y" is divisible by x^+xy+y^ if n be an odd positive integer not divisible by 3, and it is divisible by (of + xy + y^Y if w be a positive integer of the form 6w + 1. Let 1, a, j8, be the three cube roots of unity, that is, the three roots of the equation a;^ — 1 = 0. Then the product of these roots is 1, that is, a/? = l, by Art. 45; and 1 + a*" + /S"* = 0, provided m be not a multiple of 3, by Art. 262. Thus ^ + xy + 2/^ = (a; - a?/) {x — py). Hence {x + yY - a;" - y" is divisible by x^ + xy + 'if provided it vanishes when x = ay, and when x = l3y; and it is divisible by (x^ + xy + i/Y, provided its derived function also vanishes when x = ay, and when x = py : this derived function, by Arts. 11, 13, is 71 (a; + ?/)""' -m;"-'. When x = ay we have (a, + 2/)" _ a:" - 2/" = 2/" {(1 + a)" - a" - l| = 2/» |(- iS)"- a"- 1| , and this vanishes when n is an odd integer which is not divisible by 3. Also, when x = ay^ n{x + yy-'-nx''-'=ny"-' |(l+a)"-^-a'-^| = 7i/-^|(-i8)"-*-a-'j ; this vanishes if n — 1 is an even integer and a multiple of 3, because a^=l, and j8^=l. And if oi-l is an even integer and a multiple of 3, it follows that n is an odd integer and not divisible by 3, so that (x + y)" — x" — y" also vanishes. The same results would be obtained by putting py for x. Comptes Rendus Yol. ix. p. 360. 186 SUMS OF THE POWERS OF THE ROOTS. 265. The last application we shall make of Art. 262 is to pi'ove the following theorem. Let S denote the sum of the series n-3 (n -4:){n- 5) (n - 5)(n - (j){n-7) 3 "... Then S=- ii. w is an odd positive integer divisible by 3, /S' = if n is an odd positive integer not divisible by 3, S = — if ?^ is an even positive integer divisible by 3, n . 2 S=- iin is an even positive integer not divisible by 3. In Art. 261 put xy for q and x + y ior p, so that S^ = x" + y" ; thus, if n is a positive integer, (x + yf-x^-f^ nxy {x + y) Ux + 2/)""' - ^ xy {x + y)"-' Let 1, a, /3, denote the three cube roots of unity ; put x = ay, then the right-hand member of (1) becomes „a(lH.a)y»{(l + a)"--^a(l+ar-+(^^ri|izi).=(U„)w_...}. But a/3 = 1, and therefore /3^ = a^^ = a ; also a + ^ + 1 = 0, so that — y8 = a + 1 ; thus a = (a + 1)^ Hence the right-hand member of (1) reduces to /-. ^n „(. ^-3 (n-i)(7i-5) ) ,z(l+a)y|l— ^-f^ g ^-...j, that is n{-pyy"S. SUMS OF THE POWERS OF THE ROOTS. 187 Also when x = ay the left-hand member of (1) becomes 2/"|(l + a)"-a"-lj, that is, 2/" |(-/8)" - a" - l| . Therefore {-py-ar-l=n{- /3yS (2). If n is an odd positive integer divisible by 3, the left-hand member of (2) is equal to - 3 by Art. 262; therefore -3 = -nl3''S = -nS ; 3 therefore S=- . n If n is an odd positive integer not divisible by 3, the left-hand member of (2) is zero by Art. 262 j therefore S -^. If n is an even positive integer divisible by 3, the left-hand member of (2) is —1, and the right-hand member is nS\ therefore 8 = . If n is an even positive integer not divisible by 3, the left-hand member of (2) is ^"-a"-!, that is 2jS", since a" -f- j8" + 1 - ; thus 2Q'' = nB"S, and therefore S = ^ . It is to be observed that the series denoted by S consists of a finite number of terms ; in fact if n-2m or 2m + 1 there are m terms in the series. Crelle's MatJiematical JouvTial, Yol. xx. p. 321. This Article serves to illustrate the present subject : but we may observe that the result can be obtained more simply by another method. It is known, see Plane Trigonometry, Chapter xx, that 2 cos nO = (2 cos ef-n (2 cos Oy-' + Vl^^z3 (2 cos Oy-' ~ ...+(-l)-"("-'-^)("-'--^)-("-^'-+^)(2coser'-+... Put ^ = o ; hence, transposing and dividing by n, we obtain o ..l(l-2co.f). 188 SUMS OF THE POWERS OF THE ROOTS. 266. As another example of the theorem of Art. 260 we will shew how to express a;" + 2/" + (- cu - y)" in terms of x^ + xy + if, and xy {x + y). Let a^x^ -^xy + y^, b^xy(x + y), and put z (or - X - y. Then x + y + z-^0^ xy + yz + zx = xy—(x + yy= — a, xyz = — h; thus X, y, and z are the roots of the cubic equation f-at + h=0; and therefore - (x" + 2/" + «") is equal to the coefficient of — in the 7Z 3 + expansion of - log ( 1 - t^ + 73 ) . Now -log (1-1+^3) 1 / b\ 1 / bV 1 / bV We can then expand la — ) , (a — ),... and collect the coeffi- cient of any assigned power of — . If n be an even number we thus obtain a formula for (x + yy + x^' + y"; and if n be an odd number for (x+yY-x^'-y^ The following are special cases : (x + yY -xJ -y'' = la'b = 1{x^ + xy->r yyxy {x + y), (aj + 2/) V ic« + 2/' = 2aV 8a5' = 2 (x^ + a;?/ + f) {{x' + xy + yj + ixY {x + y)'}. i SUMS OF THE POWERS OF THE ROOTS. 189 The general formulae may be easily obtained by putting 2m and 2m -r 1 for n. Thus it will be found that {x + yy-^x^-+y'- _ar . m - 2 ^,.3^, ^ (m - 3) (m - 4) (m - 5) ^,_,^, 2//1 m 1 . 2 14 (m-r-l)(m-r-2)...(m-3r + l) ^^-3.; .._^ . \Ir ' and that 2m + 1 \3 _^ (m-r-l)(m-r-2)...(m-3r) ,_, ^_,^, |2r+l 2G7. It has been proposed to make use of the values of the sums of the powers of the roots of an equation in order to ap- proximate to a root of the equation; we will give an account of this method drawn from Murphy's Treatise on the Theory of Algebraical Equations. Let a, b, c,... denote the roots of an equation; suppose them all real and a numerically the greatest. We have ^... ^n^+l^jm + I^^n^-M + ... 1 a"" ' + 6"* + c"* + , + ... 1 + ©"*©"-^' Thus if 771 be taken large enough the right-hand member can be made to approach as near as we please to a, that is, to the value of the numerically greatest root. 268. We may now examine how far the result of the pre- ceding Article is modified by the presence of imaginary roots. Let p + yJ—\ and jS — yJ — l be a pair of conjugate imaginary roots ; their sum is 2/3 and their product is (3^ + y^, which is the square of their modulus; see Algebra, Chap, xxv. 190 SUMS OF THE POWERS OF THE ROOTS. Now p^yj:-i=i,(^^Tj'lA), Assume - = cos 6, and - = sin ^, so that tan 6 =^ and /x^= P^+y^; thus fjL is the modulus. Then the conjugate roots may be put in the form tJi,{cosO^J^ sinO); and hy De Moivre's theorem the sum of the 7n}^ powers of the two roots is 2/a"* cos mO. Thus if the numerical value of the greatest real root be greater o than the greatest modulus of the imaginary roots, —^ will tend to a limit as m is indefinitely increased, namely, to the numerically greatest root ; but if there is a modulus of the imaginary roots greater than the numerically greatest root, there will be no S limiting value of — ^^ . Example. x^-2x-5 = 0. Here the series aS'^ , S^, S^, is 0, 4, 15, 8, 50, 91, 140, 432, 735, 1564, 3630, 6803, 15080, 31756, 64175, 138912, 287130, 598699, By dividing each term by the preceding, we observe a tendency to a limit a little greater than 2, so that we may conclude that there is a real root a little greater than 2. The example however is not a very favourable one for the method; for since the product of all the roots is 5, and the real root is rather greater than 2, the product of the other two root.-s is nearly 2 '5. These two roots are imaginary by Art. 172, and as their modulus is the square root of their product, the modulus is greater than I'O; thus the modulus is not very small compared with the real root, and so the expression —~- approaches slowly towards its limit. 269. We may obtain the product of the two numerically greatest roots in certain cases, by a method similar to that in Art. 267. SUMS OF THE POWERS OF THE ROOTS. 191 For ' S^ = or + }r + c'^+..., m + 2 Therefore SJ^,^^,-S\^, = a-h-{a-hf-^a-G''\a-cy + bV{b-cy+.„ We will denote this by u^, so that ..=<«>-.)■ {wf(s)-.5(^j....}. Hence by proceeding as in Arts. 267 and 268 we may obtain the following results. (1) If all the roots are real -'"^ can be brought as near as we please to the product of the two numerically greatest roots by increasing m sufiiciently. . (2) If there are real roots numerically greater than the modulus of any imaginary root, there is a limiting value of ^ , namely, the product of the two greatest of these roots. (3) If there be one or more moduli greater than the numori- cally greatest real root there is a limiting value of -'"— , namely, u the square of the greatest of these moduli, that is, the product of the corresponding imaginary roots. (4) Thus the only case in which "'"^^ can fail to have a limit u is when there is one real root, and only one, numerically greater than the greatest modulus of the imaginary roots. In this case that real root can be found by Art. 267. 270. We may also obtain in certain cases the sum of two roots of an equation by a similar method. 192 SUMS OF THE POWERS OF THE ROOTS. From the values of /?„, S^^^y S^^^y and S^^^, we shall obtain we will denote this by v„. Then u^ having the meaning assigned in the preceding Article, we shall find that there is a limit of -^ in the cases named in the preceding Article, and that this limit is the sum of the numerically greatest roots, or the sum of the two imaginary roots with the greatest modulus. 271. Thus in cases (1), (2), and (3) of Art. 209 we can get the product of two roots by Art. 269 and their sum by Art. 270; and in cases (1) and (2) w^e can get the sum of two roots by Art. 270 and the greater of them by Art. 267. 272. Example, x' +x^ + 4:X^ - 4:X+1 = 0. Here we obtain the following values : forks',, /^„...-l, -7, 23,-3,-116, 227, 202,-1571,...; for w,, u^,... - 72, - 508, - 2677, - U137, -74961, - 397421,... ; for v^, ^,,...164, 881, 4873, 25726, 136382,... Here no definite limit is obtained by dividing each term in the series 8^,8^,... by its predecessor; we are therefore sure of the existence of imaginary roots. By dividing each term of the series Wj, Wg,... by its predecessor, we obtain quotients which indicate 5 -301... as the value of the product of two roots. By dividing each term of the series v^, v^,... by the corresponding term of the series u^, ^^2'••• ^^ obtain quotients which indicate — 1*8 19... as the sum of these two roots. From these values we can obtain approximate values of two imaginary roots. Since the sum of all the four roots of the equation is — 1, and their product is 1, the sum of the remaining two roots is '819... and their product — ; these two roots are therefore also D'OKJi. .. . imaginary. ELIMINATION. 193 Thus we shall find in this example that the modulus of the first pair of imaginary roots is about five times as great as the mo- dulus of the other pair. Hence with the notation of Art. 269 we shall find that in taking u^ = a'"b"' (a — by and neglecting the other terms, the error is about — of the whole quantity; and hence we o can judge of the accuracy of our result. For example ; we have given above the values of u^ as far as u^ and u^, so that we can depend upon having found the product of the roots with an error /l\th of only about f —5 j part of the whole. XXII. ELIMINATION. 273. Suppose that we have to solve two simultaneous equa- tions involving two unknown quantities ; there are certain cases in which the solution can be readily effected. Suppose that x and y denote the unknown quantities; then if one of the equations involves x"" and no other power of x, we can immediately find x"' from this equation in terms of y and substitute it in the other equation ; we shall thus obtain an equation involving y only, and the roots of this equation may be found exactly or approximately by methods already explained. Again suppose that the equations are represented hj A = and B = 0, and that A and B can be readily decomposed into factors ; suppose for example that A = UU'U" and B^VV\ Then all the solutions of the proposed equations are obtained by solving the simultaneous equations U = and F= 0, t^=Oand F' = 0, U'=^0 and F=0, ?7'=.0and F'^O, ^"=Oand F= 0, ^'' = Oand F' = 0, Thus the solution of the proposed equations is made to depend upon the solution of other equations of lower degrees. It may happen that one of the factors of A is identical with one of the factors of B ; for example, suppose that U and F are iden- T. E. 13 104 ELIMINATION. tical. Then any values of x and y which satisfy the equation ^=.0 will satisfy the simultaneous equations ^ = and ^ = 0. Thus if U involves both x and ?/, we can assign any value we please to one of the unknown quantities and determine the cor- responding value of the other, and so obtain as many solutions as we please. If U involves only one of the unknown quantities we can satisfy the equations A-^ and ^ = 0, by giving to that un- known quantity a value deduced from the equation U^^^ and any value we please to the other unknown quantity. 274. We have already shewn how by the aid of the theory of symmetrical functions we can eliminate one of the unknown quan- tities from two equations, and so obtain a final equation which involves only the other unknown quantity. We are now about to explain another method of performing the elimination, which depends on the process of finding the greatest common measure of two algebraical expressions. 275. Let the two simultaneous equations be denoted by f^{x, 2/) = 2i\idi f^{x, y)^0. Suppose that x^a and y = /3 are values which satisfy these equations ; then the equationsyj (x, fS) = Had f^{Xj P) = are satisfied by the value x = a. Hence yj(a7, )3) and yi('^) /^) must have a common measure; this common measure must be such that when equated to zero it furnishes the value a, and also any other value or values by which in conjunction with v/ = ^ the proposed equations are satisfied. Suppose then that we arrange /^(x, y) andy^(£c, y) according to descending powers of x, and proceed in the usual way to find their greatest common measure, carrying on the operation until we arrive at a remainder which is a function of y only, say ^ (?/). Then no values of y will be admissible except such as make {y) = ^; for unless <^(2/) vanishes /^ (a:, y) and /^fe v) have no common measure and therefore do not vanish simultaneously. It is not however true conversely that every value of y which makes ^{y) vanish is necessarily admissible. For it may happen that in the process the coefiicients of some of the powers of x are ELIMINATION. 195 fractions involving y in their denominators; and a value of y which satisfies the equation (^y) = may make some of these denominators vanish, and thus introduce infinite or indeterminate quantities. Suppose, for example, that we have Then if q is an integral expression it will not be rendered infinite by any finite value of y, and any value of y which makes <^ (?/) vanish, combined with the corresponding value of x deduced from the equation /^(a;, y) — 0, will makeyj(cc, y) vanish. But if q is a fraction, involving y in its denominator, q may be infinite when <^ (v/) vanishes, andy,(.x, y) will not necessarily vanish when (fi(^y) =0 -dnd/^^x, y) = 0. The same exception may occur when we carry on the process in the usual way, and introduce factors which are not functions of x in order to avoid fractional coefficients. Suppose, for example, that we multiply yj(.^, y) by a quantity c in order to avoid the fractional coefiicients which are functions of y ; and suppose we now have ^/i{^,y) = 9A(^^ y) + ^(yy If we find y from the equation <^ (y) = 0, and then x from the equation ^2(3?, y)=0, the values so obtained must necessarily make (l/\{x, y) vanish; but it does not follow that ^j (a;, y) vanishes, for it may be that the value of y which has been taken makes c vanish. Hence we require a rule which shall point out the admissible solutions, and to this rule we shall now proceed. We shall suppose that in finding the gi-eatest common measure the usual precautions are taken to avoid fractional coefficients. "We may assume that in the equations which we shall denote by ^4 —0 and i/ = 0, neither A nor B contains any factor which is a function of y only; for such a factor can be separately considered and all the solutions found which depend on it. The method we are about to explain is due to MM. Labatie and Sarrus ; we shall give it from I the Algebra of MM. Mayer and Choquet. 276. Let the two simultaneous equations be denoted by and JB=0; we will suppose that neither A nor B has a 13—2 196 ELIMINATION. factor which is a function of y only, and that B is not of a higher degree in x than A. Let c denote the factor by which A must be multiplied in order that it may be divisible by B\ let g be the quotient and rR the remainder, where r is a function of y only. Let Cj denote the factor by which B must be multiplied in order that it may be divisible by R ; let q^ be the quotient and rj^^ the remainder, where r^ is a function of y only. Proceed in this way, and suppose, for example, that at the fourth division we have a remainder which does not contain x, and which we may denote by rj. Thus we shall have the following identities : c A = qB -\- rR , cfi =q^R +r^R^, c,R =q,R, + rfi^, (1) Let d be the greatest common measure of c and r, let d^ be the greatest common measure of —~ and r^, let d^ be the greatest cc c common measure of -~~ and r^, let d^ be the greatest common del, measure of //, ^ and r„. We shall now prove that the solutions dd^d^ ^ of the equations -4 = and ^ = will be obtained by solving the following systems : - = and^ =0, (t 5- = and^ =0, I ^ = and i?j = 0, ^ = andi? =0; d^ (2) that is, we shall shew in the first place that all the solutions obtained from (2) do satisfy the equations ^ = and B~ 0, and in ELIMINATION. 197 the second place that all the values of x and y which satisfy the equations ^ = and ^ = are included among the solutions obtained from the system (2). Divide both members of the first identity (1) by d; thus 5^ = l^^5^ (^)- Now, by hypothesis, -^ and - are both integral functions of y; thus -y- is also an integral function; but by hypothesis B has no factor which is a function of y only, and therefore d must divide q. The identity (3) shews that the values of x and y which satisfy 7* C C 7* the equations -; = and ^ = make -. A vanish : but - and -: by ^ d d ' d d ^ hypothesis have no common factor, and therefore these values make A vanish. Hence all the solutions of the equations -^ = and i? — satisfy the equations ^ = and B — 0. Again, multiply both members of the identity (3) by Cj, and substitute for c^B its equivalent obtained from the second of the identities (1); thus cc The expression - — -r-^-^ is integral, for r and q are divisible by Ob CC^ d; moreover this expression is divisible by d^, for d^ divides -^ and rj and does not divide R. Di\ide by c/, ; then, for shortness, putting M for ^ and M^ for ^ . , , we have d dd ,^A^M,R,^MR,... (4). 198 ELIMINATION. ft Multiply both members of the second of the identities (1) by - ; thus Since d^ will divide ^ and r^, it will divide -^ ^ ', but R is not divisible by d^ and therefore -|-' must be. Divide by d^ ; then, C CO for shortness, putting i\r for - and N^ for -^j , we have ^5 = if.i?.-^i? (5). The identities (4) and (5) shew that all the values of x and v/ 7' CC CC which make -y and E vanish, make -tt-^ ^^^^ tt ^ vanish: but d^ dd^ dd^ CC 7* ~r and -7^ have no common factor, and therefore all the solu- dd^ «! T tions of the equations -7 = and R—0 satisfy the equations ^ = andi? = 0. Again, multiply both members of the identity (4) by c^, and substitute for c^R its equivalent from the third of the identities (1); thus ccxc, . / ,, cjr. By hypothesis d^ divides the first member of this identity, and (c v \ q^M^ + -~ MjR^, but i?i is not divisible by d^; therefore q^^i + -^- M is divisible by d^. Denote the quotient by M^-, thus J^^^ = ^A+^i/A (6). ELIMINATION. 199 Multiply both members of the identity (5) by Cg, and substitute for c^R its equivalent from the third of the identities (1); thus cc lid f^B--{,^Nyf^N)E^.r^N^R^, We may prove as before that the coefficient of R^ is divisible by - 0, nor of T . T the equation y=0, but is a root of the equation y=0; since -^ y does not vanish when y = P, it follows from (11) that the values x = a and y = P make R^ vanish, and so they satisfy the equa- tions ^? - and i?= 0. Next suppose that P is not a root of any of the equations T T V T . -^0, -r = 0, v = 0, but is a root of the equation -r = 0: since -j-j -f ^^^^ Jiot vanish when y = p, it follows from (12) that the 1 2 values x = a and y = P make Ji^ vanish, and so they satisfy the equations -y =- and i?^= 0. This proves the second part of the proposition. T T V T The equation -: - ^ -? -5 = which jnves all the admissible dd^d^d^ values of y may be called t\iQ final equation in y. 202 ELIMINATION. 277. Examples. (1) x'+'iyx'+{Zy'-y-^\)x + y'-y-+2y = 0, x^+2yx+y'^-y = 0. Here we have x-¥2y for tlie first remainder, so that r = 1, and y~—y for the second remainder, which is independent of x. The T only solutions are those furnished by -^ = and E = 0, that is, by y^*- y = and x + 2y = 0, (2) «'+ 22/a;'+ 2y{y-2)x + y'-4:=^0, a^+ 2yx + 2y'- 5y + 2 = 0. The first remainder here is (y — 2)[x + y + 2) ; so that r = y-2 and Ii=x + y + 2'j the second remainder is y^— 5y+ 6, which is 7' independent of x. The solutions are those furnished by - = aud-J5 = 0, that is, hy y~2^0 and x^+27/x + 2y'- 5y + 2=^0; and those furnished by y = and ^ = 0, that is, hy y'—5y + 6 = and x + y+2^0. The final equation in 2/ is (?/ - 2) (2/^- 5y + 6) = 0. (3) x^ + 3?/x'- 3a;' + 2,y^x - 6ya; - aj + 2/^- 3?/' - 7/ + 3 = 0, a;'- 32/a;'+ 3x'+ 3?/'£c - 6?/a; - a; - 2/' + 3^/' + y - 3 - 0. The first remainder is 2(?/- l)(3a;^ + 2/*— 2y- 3); the second remainder is 8(y^—2y)x; the third remainder is y'^-2y— 3. The solutions are those furnished by y- 1 = 0, and x""- 2>yx'+ 2>x^+ '6ifx -6yx-x-y^-\-^y^+ y-Z= % by y'- 2y = 0, and 3a;'+ y^-2y-^ = 0, and by y^— 2?/ - 3 = 0, and a? = 0. The^wa^ equation in. y is {y - \) (y^ - 2y) {y^— 2y—^)- 0. ELIMINATION. 203 (4) {y-2)x^-2x+5y-2 = 0, yx'— 5x'4- iy = 0. Here we multiply the left-hand member of the first expression by y to render the division possible without introducing fractional coefficients. Thus c = y. The first remainder is (3?/— 10) a; + y^+ 6?/. In order to carry on the division we now multiply yx^— 5a; + 4?/ hy 3?/ — 1 0, and perform the following operation : (3y - 1 0) .'« + y^- + 6 ?/| (32/ - 1 0) yx'' - {?>y - 1 0) 5cc + (3?/ - 1 0) iy^yx {?>y-lO)yx'+{y~+Qy)yx - (2/^ + 6/ + 15y - 50) a; + 12/ - 407/ We may either regard the terms in the last line as forming the second remainder, or we may continue the operation of division as the remainder is not of a lower degree in x than the divisor; if we adopt the latter plan we must again multiply by 3?/ -10, which will give rise to the same remainder as if we had originally multi- plied by (3?/ - 10)^. Thus we continue the operation as follows: -(2/^+62/^4-15y/-50)(3y-10)a;+(122/^-402/)(3y-10)|-(2/^+62/^+157/-50) -(y+6?/'+15?/-50)(3?/-10)a:-(2/'+6/+15?/-50)(7/'+6?/) ' ?/'+122/*+87/-2002/'+100?/ We have here a remainder independent of x, which is the value of rj and d^ here =2/; so that the solutions are those furnished by y'+ 12/+ 87/- 200v/ + 100 = 0, and {^y -I0)x + /+ Gy = 0. 278. The following remarks may be made on the process of Art. 276. I. We may always take c such that c and r have no common factor. .For if d be the greatest common measure of c and r the c division of ^ ^ by 5 can be eflfected without introducing fractional coefficients, as appears from the identity (3); thus c is not the most 204 ELIMINATION. simple factor which can be used as a multiplier of A before divid- ing by B. Hence by choosing the most simple factor we can make Similarly we may take Cj, c^,..., such that c, and r^ shall have no common factor, and that c^ and r^ shall have no common factor, and so on. Hence on the whole we may take c, Cj, c^, Cg,... so that c?= 1, that d^ is the greatest common measure of c and r, , that d^ is the cc greatest common measure of -7^ and r^, that d.^ is the greatest com- ^\ cc c mon measure of -r^ and r , and so on. dA n. Suppose that the remainder independent of x which has been denoted by r^ is zero; then j?^ is a common measure of A and B. Hence the solutions of the equations A = and JB=0 consist, (1) of an infinite number of values of x and y which may be deduced from the single equation i?2=0, (2) of the finite number of values of x and ?/ which may be obtained by solving A B the equations yj- ^ ^ ^^^ "p" ^ ^' -^^^ since ^3= it follows from the identities (1) of Art.. 276 that R^ divides E and i?,. Divide the identities (3), (4), (5), (6), (7), (10), (11), (12) of Art. 276 by li^; we thus obtain new identities in which A, B, E, E^ and E^ are , ^ ^ A B E E - E„ _- - , . , . replaced by -rj- , -7- , -77 , -75^ and -^^ . Bv means of these identi- -^2 K K -^2 -^2 ties we can prove, as in Art. 276, that all the solutions of the A B equations -p- = and p- = ^ will be obtained by solving the 2 — 2 following systems: J = Oand| = 0. '^ = and^ = 0. d, E^ ELIMINATION. 205 For example, suppose a;'+ yx^- {y''+ \)x + y-y^= 0, and re'- yx'- (y'+ 6y + 9)x + y^+ Qy^+ 9y = 0. Here the first division gives 2Ujx^+ {3y + 4)a; - (/+ 3y'+ iy) r for the remainder, so that we may take R = yx^+ (3?/ + 4)^;- (7/'+ 3?/"+ 4v/). To perform the second division multiply the dividend by y, and after one step in the division multiply again by y in order to continue the division. We then obtain 8 (t/^ + 3?/ + 2) (x - ?/) for the remainder rJR^. Divide R by x-y and the quotient is yx + y^+?>y + 4, and there is no remainder. Thus the solutions of the proposed equations consist, (1) of an infinite number of values of x and y which may be deduced from the single equation ic-?/ = 0, (2) of the finite number of values of x and y which may be obtained by solving the equations ?/'+3?/ + 2 = and ya? + 2/*+ 3?/+ 4= 0. III. The demonstration in Art. 276 implicitly supposes that the values of x and y are finite; it is however possible to have infinite solutions of an equation. Suppose for example that (y — 1) a;" - 2cc -4- ?/" = ; then so long as y is not equal to unity the two values of x furnished by this quadratic equation are finite. If y approaches indefinitely near to unity one value of x increases indefinitely; see Algebra, Chapter xxii. Thus when ?/ = 1 we may say that x has an infinite value. We have not included such infinite values of x and y in our investigations in Art. 276; these can be easily discovered indepen- dently. If, for example, we wish to ascertain whether an infinite value of X is admissible, we may put — for x, then clear of frac- tions, and suppose a;'=0; we have now two equations in y, and if they have a common root or roots, such root or roots combined with an infinite value of x may be said to satisfy the proposed equations. 206 EXPANSION OF A FUNCTION IN SERIES. XXIII. EXPANSION OF A FUKCTION IN SERIES. 279. Suppose we have an equation connecting two unknown quantities x and y. If we could solve the equation so as to obtain the values of y in terms of x, we might expand each value of y in a series proceeding according to powers of x. We are now about to explain a method for effecting these expansions of the values of y in series, .without having previously obtained the values of y in finite terms. The method in its complete form is due to Lagi^ange; it was suggested by a process given by Newton which is called NewtonJs Parallelogram. For the history of the method, and for full infor- mation respecting it, the student may refer to Memoirs by Professor De Morgan in the first volume of the Quarterly Journal of Mathe- matics and in the ninth volume of the Cambridge Philosophical Transactions; from these memoirs the brief account of the method which we shall give has been derived. An account of Newton's Parallelogram will also be found in the translation of Newton's work on Lines of the Third Order by C. R. M. Talbot, published in 1861. 280. Let the equation be denoted by where A, B, ...K, ...Sj are all functions of x. We suppose a, jS, ...K, ...o" to be arranged in descending order of algebraical magnitude; and throughout the investigation such words as ^rea^er and less, greatest and least, are to have their algebraical meaning. Let A be of the degree a, that is, suppose x" the greatest power of X which occurs in A; let -5 be of the degree 6, , K of the degree A:, , aS' of the degree s. Our object now requires the solution of •the problem given in the next Article. 281. It is required to determine all the ways in which t can be taken so that two or more out of the following series of terms may be equal and greater than any of the rest : a + at, b + ptj k + Kt, s + (Tf. EXPANSION OF A FUNCTION IN SERIES. 207 Begin by supposing that ^ is + oo ; the first term is then greater than any of the others. As t diminishes each term diminishes, but each term diminishes more slowly than any of the terms which pre- cede it. Let t have that value for which a + at first becomes equal to one or more of the subsequent terms. This is found by taking the greatest value of t which can be obtained from the equations a + at — h + ^tj a + at = c + ytj ...a + at = k + Ktj ...a+ at = s + crt^ that is, the greatest value of t must be found from the set h — a c — a h — a s — a a — y8 ' a — y' a — k' a — tr * Let be the greatest of these values, if one is greater than a— K any of the others; or if several are equal and greater than any of the rest, let be the last of them : denote by t. tt-K a-K '' Let t continue to diminish from the value t until h-hKt first becomes equal to one or more of the similar subsequent terms. This value of t is found, as before, by taking the greatest value of t which can be obtained from the equations h + Kt = l-{-\tj k + Kt = m + fxt, k -h Kt = s + at, that is, the greatest value must be taken from the set l — k m — k s — k K — X' K — fl* k—(t' Let the greatest of these be selected, if one is greater than any of the others; or if several are equal and greater than any of the rest let the last of them be selected; let t denote the value of the selected term, which we will suppose to be . K—V Let t continue to diminish from the value t'; and proceed as before to find another value t" from the equations n + vt = p + TZitj n + vt = s + at. 208 EXPANSION OF A FUNCTION IN SERIES. This process must be continued until the term 5 + cr^ is used in obtaining a value of t. Thus we see how all the suitable values of t may be found. 282. Suppose now that A=x:'{a^ + A^, where a^ is indepen- dent of (T, and A^ vanishes ■\Vhen x is infinite; similarly let B = x^(f>^ + B^; and so on. Assume y =x*{u-[-U), where u is independent of x, and U vanishes when x is infinite. Substitute these values in the proposed equation involving x and y, thus ... +x^^+'^t{k^+iQ{u+uy+... + xs+<^t(s^+s^){io+ uy=o. Since this is to hold for all values of x it must hold when x is infinite; and this will not be the case if the highest power of x occurs in only one term. In other words, the sum of the coeffi- cients of the highest power of x must vanish. At this point the investigation of the preceding Article finds its application. By supposition r is the greatest admissible value of t, and we obtain for the part of the expression on the left-hand side of the above equation involving the highest power of £c, ■'d+ar |(<., +AXu+irr+... + {k, + JQ (u ^uyj. When X is infinite the coefficient of x'^'^"'^ must vanish; this gives the following equation for finding u, a^u- + + k^u" = 0. From this equation values of it must be obtained, and to each value of u corresponds a value of y in which the term involving the highest power of x is ux'^. In a similar way by considering the value t' we arrive at the following equation for determining u, h^ + -f- n^ = 0. From this equation values of u must be found, and to each value of u corresponds a value of y in which the term involving the highest power of a; is wa;'". EXPANSION OF A FUNCTION IN SERIES. 209 By proceeding in this way, we shall obtain the highest power of X in each value of y. Next use one of the pairs of corresponding values of t and u which have been determined; put y = x*(u+ U), and substitute this value of y in the original equation involving x and y. We thus obtain an equation connecting x and U and known quantities. We then apply the method to determine the highest power of x in the values of U, and thus we obtain the second terms in the expansions of the several values of y in series proceeding accord- ing to descending powers of x. And this process may be con- tinued to any extent we please. 283. There is nothing in the preceding method which re- quires the given exponents a, ^, ... the equa- tion 6i^'-2 = 0, so that i^=-i— -. Thus the first terni of U is - ._= or — j= . Therefore, as far as we have gone, we hav^e 287. The nature of the values of U may be seen by exartiin- ing the formation of the general equation in U. Let us first P^t x^u for 7/ and then change u into u+U, When we put x^u ior y the left-hand member of the proposed equation will take the j-orm where w^, w^, Wg, ... are supposed in descending order of magnitude. Denote this expression by (w); then the equation in TJ will be ^ (w + U) =0. We will suppose the exponents of y in the pi'o- posed equation positive integers. The equation in U may. ^^ written aU''+4>^_,U'-'+^^_,u-'-'+... + ^,ir+ = o, where 4*a stands for — a- y ^ - i "• Now if no special value were assigned to u, the coefficients of the several powers of U in the above equation would be functions of x, all of the same degree, namely n^ Thus by Art. 284 the values of IT would all commence with the zero power of X. But if u be such that Xi(^0 '-^ ^' *^^® function <^ is of a lower degree in x than the function ^ ; hence one of the values of U begins with a negative power of x, namely, with cc-("i~^2). And this is the value of U which we are seeking, because Xi(^) = ^ is the equation from which u is to beTound according to our process. If however the equation Xi(^) — ^ ^^^ equal roots, we obtain more than one suitable value of U. Suppose, for example, that the particular root which we have selected occurs /our times ; then ^ will be of the degree n^ in a;, while 3, t^g, j, <^, will only be of the degree n^. Hence, by Art. 284, there will be four suitable values of Cf, each commencing with x raised to the negative power --(?i^- 71^). We have here supposed that X2(^*) ^^^ ^^^ derived functions do not vanish for the value of u which is considered. 288. In what we have hitherto given we have investigated values of y proceeding according to descending powers of x. Thus if we illustrate our results by geometry, and suppose curves traced corresponding to the values of y in terms of x, the first term of the series which we have found for a value of y will exhibit the nature of the curve at a great distance from the origin. But the method may also be applied to find the values of ?/ proceeding according to ascending powers of x, so that the first term in a value of y will exhibit the nature of the curve close to the origin, when the curve passes through the origin. In order to apply the method to find the values of 7/ proceed- ing according to ascending powers of x we need only make the following changes. We must suppose a, /?, ... a arranged in as- cending order of algebraical magnitude; and A^ must vanish when X vanishes and not when x is infinite, so that a;" must be the lowest power of cc in ^ and not, as before, the highest power; a similar 214 • MISCELLANEOUS THEOREMS. change of meaning must be made in B^ and 6, and in the remain- ing similar quantities. Then when ^ is + oo the following quantities are in ascending order of magnitude, a + at, h + ^t, ... k + Kt, ... s + at As before, the greatest value of t is to be found from the equations a + at = h+pt, a + at= c + yt, ... a + ai=^k+ Kf, ... a + at = s + a-t. XXIV. MISCELLANEOUS THEOREMS. 289. In the present Chapter we shall collect some miscel- laneous theorems of interest and importance, which will exemplify many of the principles established in the preceding pages. To prove that the following equation has no imaginary roots, A' B' C K' , ^ + + + . . . + —: -X=0. x—a x—o x—G x—k If possible suppose that p + ^J—^ is a root; then p — q J— I is also a root. Substitute successively these values for x and sub- tract one result from the other; thus f A' B' C^ _ _K' \ '^\{p-ar + f'^{p-br + q''^(J^cy^q'-'-'''^{p~ky~+qi- and this is impossible unless q=0. Or we may prove the theorem thus. Denote the left-hand member of the proposed equation by (f>{x), and suppose a, b, c,...k, in ascending order of algebraical magnitude. When aj is a little greater than a the first term of c{>{x) is very large and posi- tive, and by taking x sufficiently near to a we may ensure a jjositive value for (x) (x -a){x- h) {x— c) ... {x-k) = 0. 290. Required the values of the w quantities x^, x^^ x^, ... x^ from the following n equations, cc^ + a?^ + cCg + . . . + £c,, = 0, a^x^ + a^x^ + a^x^ + . . . + ax^^ = 0, a^\ + aX + «3H+ • • • + «^>„ = ^> a" "aj, + a " ^x^ + « " ^x^ -\- ...-{■ a '' ^a: = 0, a^-'x^ + a^'^x^ + a.^~'x.^ + ... + «„"~'£c„ = h. Multiply these equations respectively by c^_j, c^_„, ...c^, Cj, 1, where c^_,, c^.g, ...c^, c^, are at present undetermined, and add the results. Assume c^.j, c^^_^^...c^^ c^, such that the coefficients of x^^ a?3,...cc^^, vanish; then x^{a;'-' + c^a;"' + c^a/'-^ + ... + c^.gCij + c„_j) = 5. From the assumption with respect to c^_j, c^_,, ...c^, c^, it follows that ^2) ^3? •••"„ ^^^ tli6 roots of the equation s"-' + c/- + c,."- + ... + c„.,. + c,,., = 0. Therefore the left-hand side of this equation is identically equal to {z-a^){z-a^)...{z-ay Hence substituting a, for z the equation which determines x may be put in the form x^{a^-a^){a^-a.^...{a^-a^) = h. Thus x^ is known; and the values of cc^, x^,...x^^, can be deduced by symmetry. 216 MISCELLANEOUS THEOREMS. 291. Eequired the values of the n quantities x^y^z^... from the following n equations, + 7-^+T + ... = 1, k^ — a k^ — h k^ — c X y z 7 + r^ +1 + ... = 1, X y z k^-a k —h k In n We may regard the n quantities k^, k^^...k^ as the roots of the single equation X 11 z k-d k—h k-c which is of the ?i*^ degree with respect to k. Assume k = a — t', it will follow that a-k^, cL — k^, a-k^,... are the values of the roots of the following equation in ^, 1 + -+ —^ ■¥- +... = 0. t t + — a t+ c — a Multiply by the product of the denominators so as to put this equation in the usual form ; thus r +^/-» +^/-^ + ... +^„ - 0, where the term independent of t, that is -4^, is x(b-a){c-a)... Therefore, by Art. 45, {a-k^){a-k^){a-k^)...={~iyx{b-a)(c-a)...y that is, x = - (^-K)i^-K)(<^-K)-" {a — b){a—.c)... From this expression the values of y, z,... may be deduced by symmetrical chauges in the letters a,h,c... Grunert's Archiv der Mathematik und PJiysik, Yol. xxiii. p. 235. MISCELLANEOUS THEOREMS. 217 292. To prove that the sum= of the products of the n quanti- ties c, c^, c V • • c"> taken m at a time is (c-l)(c«-l)...(c--l) ' ' Assume (x+c){x + c^)...{x + c'') = x''+pX~'+ "• +Pn-i^+K (!)• Then by Art. 4.5 we have to find the value of p^. In (1) change X X into - and multiply by c"; thus c (x + c^)(aj + c')...(aj + c""'')=a;" + fica3""'+ ... +^^_^c''~'aj + 2?„c"...(2). From (1) and (2) we obtain {x + c''-''){x" +^X"' + ••• +Pn-X^+2\) = {x + c)(x'' ^p^cx"'^ + ... +p^_^c"~^x+p^(f). Equate the coefficients of ic""'""*'^ in the two members of this iden- tity; thus therefore p„ =p^_^ c<^ ' ^^^' we can c(c"-l) And p^ = c + c^ + ... +c'*-- — ~ ; then by meaus of (3) c — J. determine successively p^) P^j P^i-"'} and thus we shall arrive at the required value for p^. 293. Let there be n quantities a, b, c,... ; let s^^ denote their sum, s^_^ the sum of any ti— 1 of them, and so on; and let >S^ denote («„)'- s{«,.,)' + sfe._/- ...+(- ir-'s(.,)'. Here '^{s^Y denotes the sum of such terms as {s^Y formed by taking all possible selections of m quantities out of the n quanti- ties a, b, c,... Then we shall shew that aS'= if r is less than n, 218 MISCELLANEOUS THEOREMS. and that >S^ is divisible by a6c... if r is equal to n or greater than n-y and in particular that /S^ = [w a5c. . . , when r = w, k + 1 and S = —,r— {a + h + c + ...)ahc...^ \vhenr = w4-l. We may separate S into two parts, one part in which a occurs in every term and another part in which a does not occur at all. We may write the former part thus, w- s.(v,r + 2,k J'- - + (- 1)"-'«'. and the latter part thus, where 3^ indicates certain of the terms formerly included under % and iSg indicates the remainder. Now suppose a = 0, then «S^ vanishes; for we have in this case .s,K.,r-s.(.„.3)'=o, . Similarly, we may prove that aS^ vanishes when 6 = 0, and when c = 0, and so on. Thus we conclude that S is in general divisible by each of the quantities a, b, c, . . . and therefore by their product. But the product will be of n dimensions, and therefore if S be of less than n dimensions it must be identically zero. And as S is of r dimensions it follows that aS' vanishes when r is less than n, and is divisible by ahc, when r is not less than n. When r = n we have therefore S=XahG...f where A. is some numerical quantity which is to be determined. To determine X. suppose that a, h, c,... are all equal to unity; then S becomes „-_„(„_l)"+!i^^(„_2)"_..., that is Iti, by Algebra, Chapter xxxix. MISCELLANEOUS THEOREMS. 219 Next, suppose r = n + l. Then aS' is divisible by ahc.,.; and as S is o£ n+1 dimensions, it must have a factor which is of one dimension and symmetrical with respect to a, 6, c...; this factor must therefore he a + b + c + ... Hence S^ iiahG...{a + 6 + c + ...), where /x is a numerical quan- tity which is to be determined. To determine /a suppose that «j 6, c,... are all equal to unity; then S becomes n"^' -n{n-\y^' + '^^'^~^^\ n-2y^' - .... . and this must equal iin. Hence by Algebra, Chapter xxxix. we have a = „ . /^ 294. Let [c\ denote c(c - l)(c— 2) ... (c -r+ 1), whatever c may be; then will [a + hi = [«]„ + n [al_fi + ?^M [„],__^[6]^ + . . . + [J],_. For suppose that a is a positive integer; then we know that this theorem is true for any positive integral value of b, for it follows by equating the coefficients of cc" in (1 -k-xY'^'' and in (1 + x)" x (1 + x^. Hence since this is true for more than n values of b it is iden- tically true by Art. 39; that is^ when a is a positive integer the theorem is true for all values of b. Then since it i^ true for any positive integral value of a, it is true for more than n values of a, and therefore by Art. 39 it is true for all values of a. Thus we are able to prove the proposed theorem, by assuming the Binomial Theorem for a positive integral index and also the Theorem of Art. 39. The theorem is sometimes called by the name of Yandermonde. The theorem is required in Euler's proof of the Binomial Theorem for any index, and as is well known, is there established by an appeal to the principle of the permanence of equivalent forms. 220 MISCELLANEOUS THEOREMS. 295. Let (f>(x) = be an equation which has a root a, so that we may suppose {x) = (x-a) \{/(x); then X =(i-3^w. Suppose that log ^-^ can be expanded in a series involving posi- X tive and negative powers of cc, and that log y\r (x) can be expanded in a series involving only positive powers of x ; then assuming the identity of the two members of the equation we obtain this result, — a = the coefficient of - in the expansion of log — ^ . X X 296. The theorem of the preceding Article is given by Murphy in his Theory of Equations and illustrated by examples ; see his pages 7 7... 82. The demonstration of the theorem is imperfect, since the infinite series may be divergent; but the theorem is of some importance. It had been noticed before Mur- phy drew attention to it ; see De Morgan's Differential and Inte- gral CalculuSj pages 328 and 644, and also the Philosophical Magazvne for June 1848, page 421; according to the latter work the theorem was given by Lagrange in 1768. It appears that the process furnishes the numerically least root of the equation to which it is applied; and some reason may be assigned for this, at least when all the roots are real. For suppose that the roots of the equation <^{x) = ^ are a, 5, c, . . . in ascending order of magnitude. Then (j>{x) = Aix - a){x -h){x - c) , where -d is a constant. . 0..^ ^)=.(i-«)(i-|)(i-3....... where ^ is a constant. MISCELLANEOUS THEOREMS. 221 Since a is the numerically least root of tlie equation ^ ioc) = 0, if X lies between a and b the expansions of iog(i-3, iog(i-g, iog(i-3 will all give convergent series; and hence we see that log -^-^ can be developed in the required form in a manner which is arithmeti- cally intelligible and true. Then as x can have any value between a and b we may, by a natural extension of the theory of indeter- minate coefficients, equate the coefficient of - in the expansion of X \o2 ljl^ to -a. X In the same way as the coefficient of - in the expansion of los: — - is seen to be - a, we see that the coefficient of -r is ; ° X X n thus we can determine the value of any assigned integral power of the numerically least root of the equation (x) = 0. 297. For example, required a root of the equation x" + cx-b = 0. = logc-5;-2« "3^; -..., b a;"-' b /, cc"\ where » = = — 1 - -- . ex c cx\ b J We have now to pick out the terms involving - ; we shall obtain X such a term from «, from «""^\ from s^""^^, and so on. Hence we shall find for the root the series b_ b^ ^ 2nb'"-' dn{3n-l)b'^-' 2.3 c^"^^'^'" 222 MISCELLANEOUS THEOREMS. 298. Let (fi{x)~0 be an equation of whicli a^, a^j ••'«,„> ^i'© roots, so that we may suppose '{a). This determines A; and similar values will be found for B,C,.,.K. ■ 300. Next suppose that j/^ (a;) is not of lower degree than {x). By common division we may obtain where F (x) and /{x) are integral functions of a?, and /{x) is of a f/x) lower degree than (£c). We may then decompose --r)—. into partial fractions in the manner shewn in the preceding Article. Since we have iHx)^{x)F{x)+/{x); it follows that ij/^x) and/(a;) have the same value when (x) vanishes. Hence the partial fractions corresponding to — ^-/ , when determin- ed by the method of Art. 299, can be found without previously dividing xj/^x) by {x); we must however not omit the part F(x) il/ (x) if we wish to obtain the complete value of -i-7-c . 224 MISCELLANEOUS THEOREMS. 301. Various Algebraical identities may be established by means of the principles of the preceding two Articles. For example, if n be any positive integer \n \ n I («+ l)(iC+2)...(£c + 7^+ 1) £c+l la;4-2 1.2 a;+3 x-\-n-\-\ For we may assume that the left-hand member can be put in the form A^ J„ A. A.^, £C+ 1 x+2 x+'6 aj + w + l' and then we may determine A^, A^,...A^^^: this is effected by multiplying both sides by (a; + l)(a;+2) ... (cc + w+l), and then substituting for x in succession the values - 1, — 2, ... Again, if w be any positive integer 1 n n{n — \) ^T~(^+l)(£c + 2)'^ (a; + l)(a; + 2)(a; + 3)^"* (oj + l) (a; + 2) ... (a3+7i+ 1) x + n-\-\' For we may assume that the left-hand member can be put in the form A. A^ A. A ^, — V + — S + — ^ + + — ''^H-; aj+l x-^^, £c + 3 x + n+l multiply both sides by (cc-H l)(a;+ 2) ... (a; + ?i+ 1) and then sub- MISCELLANEOUS THEOREMS. stitute for x in succession the values - 1, obtain Thus ^, = (1 - 1)" = 0, j^'%7\^^^ 1.2 (l-l)'-=0,\^^jpQ^^ and by proceeding thus we find that A^, A^,...A^^2iTQ bM zero, and that A_^, = 1. Again, if m be any positive integer (^ + l)(x-+2) 1-2/ m ( w — 1 ) (a;+l)(a; + 2}(£c+3) 1 -2// (a;+ l)(a; + 2)...(a; + 7;i + l) _1_ x + 1 my m{m—V) y^ i^T2"^ 1.2 ^4^3 (ryi (-i)y _ a5 + m+ 1 * This may be demonstrated in the way already exemplified by assuming that the left-hand member can be put in the form x+ 1 then we deduce 4- + x + )L a; + m + 1 A^ = - my {I -y -\-y)"'-' ^- my, and so on. Or we may establish this result by the aid of the second ex- ample. For if we expand the left-hand member in powers of y, and compare the coefficients of y" in the two sides, we find them equal by the second example. 302. We have in Articles 299 and 300 given separately the decomposition of a rational fraction when its denominator has no repeated factorSy on account of the simplicity of the result; it T. E. 15 226 MISCELLANEOUS THEOREMS. is however only a particular case of the general investigation to "wbich we now proceed. Suppose that <^(£c) is a function of x which involves repeated factors ; for example, let (x) =p^ {x - ay(x - by {x - cy. ..{x- Jc), and let il/{x) be any other function of x. Then the expression "^^-r niay be resolved into the following parts. (1) Any factor x-k which is not repeated will give rise to a single term — , . on '^ K (2) The factor {x — cbf will give rise to the series of terms {x - ay {x - ay ^ {x - ay~^ ' x-a A similar series of teims will arise from each of the other repeated factors. (3) There will also be an integral expression if ij/{x) be not of a lower degree than <^(aj). For suppose <^(a;) = (»-«)'' x (a?); then we have identically, whatever A may be, {x) {x-ay <^{x) Now let A be determined by the equation i/^ (a) - ^x W = ^ 3 *^®^^ }l/{x)-Ax{x) vanishes when x=a, and is therefore divisible by a; - a. Therefore with this value of A we may put and therefore {x + y J —^) can be expressed in the form p + q J — I. A point whose co-ordinates are such that p and q simultaneously vanish, will be called a radical point. Describe any contour ABCD ; then the number of radical points which lie within this contour will be given by the following rule. Let a point move round this contour in the positive direction, and note how often - passes- through the value and changes its sign; suppose it to change k times from + to — , and I times from — to -f ; then tlie number of radical points which lie within the contour is r){^- 0- 2A It is to be observed that the contour is supposed to be so taken that no radical point lies on it ; also if any imaginary root of the equation <^ (2;) = is repeated two, or three, or more times, we consider that we have two, or three, or more radical points, although these points coincide. By movement in the positive direction we imply that a radius vector drawn from a fixed point within the contour to the moving point passes over a positive angle equal to four right angles, while the moviag poiat passes round the contour. The theorem is proved by first considering tiie case of an in- finitesimal contour, and then the case of a finite contour. cauchy's theorem. 233 309. Take any point G, which is not a radical point, within the contour, and describe an infinitesimal contour including G. Suppose that the moving point passes in the positive direction round this infinitesimal contour; we have then four cases to consider. (1) Suppose that neither p nor q vanishes within or on the contour. Here - does not change sign at all during the circuit ; so that the rule asserts that there is no radical point within the contour, and this is true because p and q do not vanish. (2) Suppose that q does not vanish within or on the contour, p but that p does. In this case — may change sign as the moving point passes through a position for which p vanishes. But at the end of the circuit p has resumed its original sign, and thus there must have been the same number of changes from + to — as from — to +. Hence h and I are equal, and the rule asserts that there is no radical point within the contour, and this is true because q does not vanish. (3) Suppose that p does not vanish within or on the contour, 7) but that a does. In this case - never vanishes, so that the rule asserts that there is no radical point within the contour, and this is true because p does not vanish. (4) Suppose that both p and q vanish within or on the con- tour. If they do not vanish simultaneously we may divide the space bounded by the contour into other spaces, for some of which 2) alone vanishes, and for others q alone vanishes; thus we obtain two or more contours instead of one, and these fall under the cases (2) and (3). We have then only to consider the case in which p and q vanish simultaneously, so that there is a radical point within or on the contour. And we may suppose the con- tour so small that there is only one distinct radical point within it, and none on it. 234 cauchy's theorem. Let a, h be the co-ordinates of this radical point ; and put x = a + r cos 0, and y=h-{-rsiu6', thus x + y J - l=a + b^-l+r (cos + jj^ sin 6\ = a + hj—l + v, say. Suppose now that the equation (z) = has the root a + h J-l repeated m times ; then {a-^h J —1 + v) takes the form cv"" ■>(■ c^v'^'^^ + c^^"^^ + ..., where c, c^, c^,... are certain imaginary expressions of the standard form; so that we may suppose c = h{cos a + ^ - 1 sin a), c^ = h^ (cos a^ + ^ - 1 sin a^), . . . Hence, by De Moivre's theorem we shall obtain p hcos{md + a) + h^r cos {(m + I) + a,} +h^r^ cos {(m + 2)6 + a^}-i- .. . q ~ h&iD.{mO + a) + 7^/ sin{(m + 1)6 +a^} + hy shi{{m + 2) + a^}+ . . . We may suppose r so small that the number of changes of sign IT) shall be unaffected by r ; that is, we may proceed as if - = cot (m^ + a). And as m6 increases from one multiple of tt to the next multiple of tt, there is always one passage through zero accom- panied by a change of sign from + to -. Thus we have k = 2m, and 1 = 0; so that - {k-l) = my according to the rule. 310. The theorem is thus proved for an infinitesimal contour; and we shall now consider the finite contour A BCD. Let the contour be divided into an indefinitely large number of infini- tesimal contours, these contours being so taken that no radical point falls on any of them. Then the number of radical points within ABGD can be found by making a point describe all these infinitesimal contours, and adding together the numbers furnished by the rule, which we may denote by ^ ^ (^ — l). But the same result will be obtained if we omit all the interior lines of division, and retain only the boundary ABCD. For each point on any interior line of division belongs to two contours, and is therefore cauchy's theoeem. 235 traversed by the describing point twice and in contrary directions; so that, if in one case there is a change in - from + to — , there is a change in the other case from — to +, and on the whole the number -^^{k — I) is unaffected. Hence the interior lines of division may be omitted, and the moving point constrained to describe the contour A BCD alone. Thus the theorem is proved. 311. We can now immediately deduce the theorem that an equation of the n^^ degree must have n roots. Suppose the contour ABGD to be a circle with the origin as centre and an indefinitely 7) large radius. The value of - will now depend only on the term in- volving the highest power of z in 4>{^)} ^^^ if ^^ suppose that term to be h (cos a + /y/— 1 sin a);s", we shall have - = cot {nO + a). Thus we shall obtain h = 2w, and ^ = ; so that -{k — l)=n. 312. We have drawn the figure in Art. 308 so that if from any point within the contour a radius vector is drawn in one direction it meets the contour in only one point. The figure however need not be so restricted; it may be such that a radius vector drawn in one direction may meet the contour any odd number of times. Hence as a point moves round the contour the radius vector drawn to the moving point from any fixed origin within the contour will not always revolve in the same direction. By the positive direc- tion of movement of the describing point we must understand that for which, although the vectorial angle may not be always increas- ing, yet on the whole the positive angle 2ir is gained in the circuit. The demonstration will not be affected by the admission of the kind of figure here contemplated; for the infinitesimal contours 2^6 NEWTON S RULE AND SYLVESTERS THEOREM. may still be supposed, if we please, ovals whicli have only one radius vector drawn in any definite direction from a fixed origin. Or if we do not adopt this restriction we must observe that at the end of Art. 309, as $ now does not always increase, there may be more values of for which — vanishes, than we contemplated; but if so, there will be exactly as many more changes from + to — as from - to +. 313. We have supposed throughout that there is no radical point on a contour considered. If there be, no change is made in our investigations except at the end of Art. 309; and here instead of having the range 27r for 6 we have only ir, so that m occurs instead of 2/?^ as the number of changes of sign. 314. Cauchy's Theorem is given in the Penny Cyclopcedia, Article Theory of Equations, in Mr De Morgan's Trigonometry and Double Algehra, and in Mr De Morgan's Memoir to which we have referred in Art. 32; from these sources the present account of it has been derived. XXVI. NEWTON'S RULE AND SYLVESTER'S THEOREM. 315. Newton enunciated a rule respecting the number of positive, of negative, and of imaginary roots in an equation, which remained without demonstration until the recent researches of Professor Sylvester, who has established a remarkable general theorem which includes Newton's rule as a particular case. The original sources of information on the subject are th.Q Philosophical Transactions for 1864, the publications of the London Mathema- tical Society, No. li., and the Philosophical Magazine for March, 1866; from these sources the exposition which we shall now give has been essentially derived. 310. We begin by enunciating in substance Newton's rule. NEWTON'S RULE AND SYLVESTER'S THEOREM. 237 Let/(£c) = be an algebraical equation of the n^^ degree; and suppose /./ X „ «-l ^(^— 1) «-2 J [x) = a^x + na^x H — ^ — -^ a^ + . . . + na^_ ^x-¥ a^-y then a^, a^, a^,...a^ may be termed the simple elements of/{x). Let a new series of quantities A^, A^^ A^,...A^ be formed in the following way : then A^j A^, A^,...A^ may be termed the quadratic elements oif{x). We shall call a^, a^^j a succession of simple elements, and yl^, A^^^ a succession of quadratic elements ; and we shall call an associated couple of successions. Now a succession may present either a permanence or a varia- tion of sign j and this will be termed for brevity a permanence or a variation. Thus in an associated couple of successions we shall have one of four cases ; two permanences, or two variations, or a superior permanence with an inferior variation, or a superior variation with an inferior permanence : these may be called respectively a double permanence, a double variation, a perma- nence-variation and a variation-permanence. The following is equivalent to Newton's complete rule : Write the whole series of quadratic elements off(x) under the whole series of simple elements in their natural order ; then : The number of double permanences in the associated series is a superior limit of the number of negative roots of the equation /W = o. The number of variation-permanences is a superior limit of the number of positive roots. From either of these two statements the other follows by changing the sign of x inf(x). 238 Newton's kule and Sylvester's theorem. It follows from these two statements that the whole number of real roots cannot exceed the number of permanences in the series of quadratic elements ; and therefore the number of imagi- nary roots cannot be less than the number of variations in the series of quadratic elements. It should be noticed that writers who have quoted Newton's rule seem always to have restricted themselves to that part which relates to the number of imaginary roots. 317. "We will illustrate Newton's rule by some examples. Suppose 2x^-13af+10x- 4:9 = 0. Here the series of simple and quadratic elements are 2. 0,-^, ^. -49, . 13 169 1199 .... *' T> 16-' -T2-' ^*^^- Thus whether we suppose the zero which forms the second of the simple elements to be positive or negative, we find that there is one double permanence, and one variation-permanence ; so there cannot be more than one positive root, and there cannot be more than one negative root : there are then certainly two imaginary roots. These results agree with those in Art. 203. In this example Descartes's rule would indicate that there cannot be more than three positive roots ; so that Newton's rule gives us fuller informa- tion than Descartes's. Next suppose x^ + x^-x^-x^ + x^ -x+l = 0. "We will write down the series of simple elements, and the signs of the quadratic elements : '6' 15' 20' 15' 6' ' + , +, +, +, -, -, +. Here there are two double permanences, and two variation- permanences; so that by Newton's rule there cannot be more than two positive roots, and there cannot be more than two nega- tive roots. Newton's kule and Sylvester's theorem. 239 Descartes's rule would indicate that there cannot be more than/our positive roots. In this example it may be shewn by Sturm's theorem that all the roots are imaginary; or we may obtain the same result thus : It is obvious that there can be no positive root greater than unity ; and the equation may be written in the forms x' + {l-x){l+x'-x') = 0, x'(x+iy{x-l) + x'-x+l = 0, which shew respectively that there is no positive root less than unity, and no negative root. Next suppose x^ - 12a;'' + 60x* + l2Saf + 4567a;- 89012 = 0. We will write down the series of simple elements, and the signs of the quadratic elements • 1, -2, 4, 0, 1\ 1^, -89012, +, 0, +, -, +, +, +. There is one double permanence whether we suppose the zero in the upper series to be positive or negative, and one variation- permanence if we suppose the zero in the lower series to be nega- tive; so that by Newton's rule there cannot be more than one positive root, and there cannot be more than one negative root. Descartes's rule would indicate that there cannot be more than three positive roots. In this example we know by Art. 21 that there is certainly one positive root and one negative root ; it will be found on trial that the former lies between 1 and 10, and the latter between — 1 and - 10. 318. The preceding examples shew that Newton's rule may often be applied with facility. It is obvious that it always tells us as much as Descartes's rule, and often tells us more. For with respect to positive roots, for example, Descartes's rule takes the number of variations in the series of simple elements, while Newton's rejects those variations which are unaccompanied by a permanence in the series of quadratic elements. 240 NEWTON'S EULE AND SYLVESTER'S THEOREM. 319. The following is Professor Sylvester's Theorem : Let f{x + X) be arranged according to powers of x ; let the series of simple elements and the series of quadratic elements be formed, and let the number of double permanences be called the number of double permanences due to X, and be denoted by cr (X). In like manner let the number of double permanences for f(x + ^) be called the number of double permanences due to /a, and be denoted by c7(/i.). Suppose /a greater than A; then cr(ju} —m{X) is either equal to the number of roots of the equa- tion /(x) = between X and yu,, or surpasses that number by some even integer. 320. Before demonstrating this theorem we will shew that it includes Newton's rule. Put for ft and — x for X. We have m {-cc ) = ; for when X is — oc , the simple elements oi /(x + X) are alternately positive and negative, so that there can be no double permanences. Thus ^{0) = 'UT (0) - CT (- oc ). Therefore, by the above theorem, ztr (0) is either equal to the number of roots of the equation y (a?) = between — x and 0, or surpasses that number by some even integer. This establishes the first part of Newton's rule, from which the other parts follow. 321. The simple elements off{x + X) are rw i r-'w 1.2 /-(x) i.^awm [n. ' n \ri-l ' n{7i-\) \n-2_' n'^ W»/W It will make no change in sign if we multiply every element by I w ; thus the series becomes /-W,/"-'W, 1.2/"-'W, |3/"-=W, !»-!/' (A), \n/{X). In like manner by omitting the square of |r — 1 from the /'' quadratic element we obtain the series where G^(\) stands for {f'{X)Y - yj"-' (X)/'^' (X), , ^. n~r+l y, denoting . Newton's rule and Sylvester's theorem. 241 We have then to determine the laws of the change in the number of double permanences in the associated series f'{t),r-'{t),r-\t), /'(<},/(«), GMG.-A'-)'Gn-M, GAt),G((), as t increases. Ko change can take place except when t passes through a value which makes one or more terms vanish in either or both of the series of elements. 322. It will be necessary to investigate the value of the de- rived function of a quadratic element; let G^^{t) denote the quadra- tic element, and G'^{t) its derived function. To obtain G\^{t) we must suppose G^^{t +h) to be expanded in powers of A, and take the coeflBcient of h. The coefficient of ^ is (2 - r J/" (<)/"■*' («) - r„/"- (OZ-^W- Kow it is easily seen thafe /m+X thus G'„{t)=^r{t)r»{t)-yj"'-\t)r*\t) 7»n+l I = :^/" W/""(0 - ^j|/""W +^^5 <^»(0 /"W +1777 '^"+1 W + TS+rrrt G„{t). T. E. 16 242 Newton's rule and Sylvester's theorem. 323. Suppose that a single term in the series of simple ele- ments intermediate between the first and the last vanishes when Let h be an indefinitely small quantity ; this will be the mean- ing of h throughout the investigation : then/*'(c + /i) has the sign of /i/''"*'*(c). Thus the associated terms /'^'(c + h), fic + h), /'-'(c + A). have the same signs as r\c), hf'^^c), /'-'(c), {r"{o)}\ -f'*'{c)f'-\c), {/'-(«)}'. Ify*''"*'^(c) andy'"~^(c) have the same sign, the terms here con- sidered have no double permanence. If/'"^^(c) and /"^^c) have contrary signs, there is one double permanence whether we suppose h negative or positive. Thus no change is made in the number of double permanences when t increases through the value c. 324. Suppose that a single term in tl^e series of quadratic elements intei-mediate between the first and last vanishes when t = c, say G^{c) = 0. Since G^g) = it follows that/'"^(c) and/'""^^(c) have the same sign. Thus the associated terras f'»{c + h), f'{c + h), /-(c + A), Gr„{c + h), e,{c + h), G,_,(c + h), have the same signs as . /'"{<=), /'(c), /-'(c), G.lA"), ''{c\ A^(^(c),...7.'^(c), /--(c). We proceed to investigate the signs of the quadratic elements : ^r+.(c) = ] /'"*"' (^) [ > which is positive, 0^_^(c) = \ f^^(c) [■ , which is positive, IG— 2 244 Newton's rule and Sylvester's theoreal expand in powers of h and take the term which involves the lowest power of h : thus we obtain so that the sign is the same as that of -/^-(c)c/>(c)A-\ We shall now shew that the other quadratic elements which we have to consider are positive. For let IX stand for r + s - m ; then by expanding and picking out tlie term which involves the lowest power of h we obtain the sign of this is the same as the sign of that is as the sign of r+s-w+1 n—m+l r + 8 — m that is as the sign of 1 r + 8 — m n — m Now r + s is not greater than n so that the sign is never negative; the case in which r + s = n will require further examination. In this case /'(c) = 0, /-'(o) = 0,.../"-'(c) = 0; and as/''(^) is of 7i-r dimensions in t it follows that all the roots of/'(^) = are equal to c. Thus /■"(«) is of the form G{t-cy-' where (7 is a constant. Newton's rule and Sylvester's theorem. 245 Then f-' it) = C{n-r){t- cy-"-', f^'{t) = C{n-r){n-r-\){t-cy-''-'; and thus it will be found that G^^^{t) is identically zero. And in like manner it will be found that G-^^^{t)^ ^r+sW? ••• ^n-iW ^^'^ all identically zero. We will adopt the convention that these quadratic elements which are identically zero shall be supposed to have the positive sign ; and thus the case in which r-^s = n will lead to the same results as that in which r + s is less than n. Thus finally the signs of the terms of the associated serie? which we have to consider are the same as the signs of <^(c), h{c\ A'<^(c), /.'(^(c), /^-'(c), + , +, + +, -h'-'4>{c)r-\c), +. We can now ascertain the number of double permanences; the following results will be easily obtained : Suppose s even, and <^ (c) and f'~^ (c) of the same sign ; when h is negative there is one double permanence, and when h is posi- tive there are s - 1 : thus s-2 double permanences are gained when t increases through the value c. Suppose s even, and <^(c) and /"^(c) of contrary signs; when h is negative there is no double permanence, and when h is posi- tive there are s : thus s are gained. Suppose s odd, and <^(c) and /"""'(c) of the same sign; when h is negative there is no double permanence, and when h is positive there are s — l: thus s — 1 are gained. Suppose s odd, and <^(c) and f'~\c) of contrary signs; when h is negative there is one double permanence, and when h is positive there are s : thus s—l are gained. Hence an even number of double permanences is gained when t increases through the value c. 246 NEWTON'S RULE AND SYLVESTER'S THEOREM. 326. Suppose that several consecutive terms of the series of quadratic elements vanish when t = Cj say &,..M = 0, G^,Uo) = 0. e,{c) = 0. Thus we suppose s consecutive terms to vanish, and as G^{c) is a constant which cannot vanish, r + s cannot be greater than n: we suppose that r is not zero. In consequence of the vanishing of the s consecutive quadratic elements, we have the following conditions holding among the simple elements comprised between /,+,{c) and f^_^{c), both in- clusive : /,+,(c), /r+s-i{c), /,+,_4(c),... are aU of the same sign; /,+,_i(c),/,+,_3(c), /+._s(c),... are all of the same sign. If the terms in the second set have the contrary sign to those in the first set there is no double permanence when t = c + h, whether we suppose h positive or negative. We have then only to consider the case in which the terms in the two sets have all the same sign. Let G^^^[t) and G^^{t) be any two consecutive quadratic elements comprised between 6^^+,(0 and G^^), both inclusive: then G^^_^^{c + h) and G^^(c + h) shall have contrary signs when h is negative and the same sign when h is positive. Por by Art. 322, Put c + h for t, and expand in powers of h. Suppose that in G^^ {c + h) the term which involves the lowest r) power of ^ is — A'', so that R is the value of the p^^ differential P coefficient of G^{c) with respect to c. Then the term which r> involves the lowest power of h in G\X'^ + h) will be r~Z\^^'"'' Newton's eule and Sylvester's theorem. 247 Hence from the above equation the term which involves the lowest power of A in G^_^_^{c + h) will be Ifm + W \^) J?Tp-l 7>yEi ■ Hence finally G^^(c + h) has the sign of Rh^ and G^^^ (c + h) has the sign of RltJ''^ ; so that G^^_^_^{G + h) and G^X^ + h) have con- tiary signs when h is negative, and have the same sign when h is positive. Thus the simple elements which we have to consider have all the same sign; and the quadratic elements comprised between ^'r+X^ + ^0 ^^^ ^r{^'*'^0> ^^^^^ inclusive, have alternate signs when h is negative and have the same sign when h is positive. "We can now determine the number of double permanences; the ibllowing results will be easily obtained : Suppose s even, and G^^g{c) and G^_^{c) of the same sign; when h is negative there is one double permanence, and when h is posi- tive there are s + 1 : thus s double permanences are gained when t increases through the value c. Suppose s even, and G^+^{c) and G^_y^{c) of contrary signs; when h is negative there is no double permanence, and when It is posi- tive there are s: thus s are gained. Suppose s odd, and G^^.^{c) and G^_^{c) of the same sign; when h is negative there is no double permanence, and when ?i is positive til ere are s + 1 : thus s + 1 are gained. Suppose s odd, and G^_^_^{c) and G^_^(c) of contrary signs; when h is negative there is one double permanence, and when h is posi- tive there are s: thus s — 1 are gained. Hence an even number of double permanences is gained when t increases through the value c. 327. We now consider what takes place when an extreme term vanishes. 248 NEWTON'S RULE AND SYLVESTER'S THEOREM. /''(^) and G''{t) are constants, and can never vanish; and G{t) is essentially positive. Suppose however that /-yc) = 0, /*-'(c) = 0,... /(c) = 0, so that c is a root repeated s times of the equation fix) ~ 0. Then, as in Art. 325, the last s+ 1 terms of the associated series will have the same signs when t=c + A, as /'(c), //'(c), hT{c), A'-y'(c), hT{ch 3 s + , 2-y,_„ 2-'/*-2 ^^TJ"'^^' ■*"• Here when h is negative there are no double permanences, and when h is positive there are s: thus s double permanences are gained when t increases through the value c. 328. This completes the demonstration of the theorem. The general result is that the number of double permanences belonging to the associated series is increased by at least as many units as there are real roots, equal or unequal, passed over as t increases from one specific value to another ; and the excess, if any, of such number over the number of real roots will be an even number. Thus, with the notation of Art. 319, we know that the num- ber of real roots between A. and fi cannot exceed 'n7{fjt) — -cr (A). If we know that some of the double permanences gaioed arise from the vanishing of any of the elements except /(^) we can of coui'se make a corresponding reduction in the extreme num))er of real roots. Thus, for example, suppose that s double permanences are gained in the manner considered in Art. 325, then the number of real roots between A and /x, is not greater than -cr (/a) — -cr (A) — s. 329. Some extension may be given to the theorem by ascrib- ing another value to y^. The principal property of y^ which is required in the preceding investigation is that used in Art. 322, namely. NEWTON'S RULE AND SYLVESTER'S THEOREM. 249 We may then examine what form of y^ will satisfy this equa- tion. We will solve" the equation, though the process will require from the student a knowledge of the elements of the Calculus of Finite Differences. Put Ir^—'y thus 2-^ = ^- therefore u^^^ - ^u^^^ + u^ = 0. The solution of this equation is where A and B. are constants. Hence y^= \ /^~ _ C+r-l " C + r * where G stands for ^ . n The student who is not acquainted with the Calculus of Finite Differences may easily verify that this value of y^ satisfies the relation 1 7.+ 1 We have also to satisfy the condition that y^ shall be positive, and also the condition assumed in Art. 325, that y^ shall be positive ; these conditions will be satisfied if G be any positive quantity, and also if G be negative provided it does not lie between and — n. 330. Professor Sylvester observes that his theorem bears the same relation to Newton's rule which Fourier's theorem bears to Descartes's rule. Fourier's theorem may be stated thus : 2-r,' 550 Newton's rule and Sylvester's theorem. Forni the simple elements corresponding to/(a; + X) and to /{x + fx). Let jo(X) and J9(/a) denote the corresponding numbers of permanences of sign; and suppose ft greater than A, Then p{fx) -jo(>v) is either equal to the number of roots of the equation f{x) = between X and /x, or surpasses that number by some even integer. 331. We have given in Art. 106 a simple proposition which resembles a special case of Newton's rule; and it is easy to extend the pioposition so as to convert the resemblance into a coinci- dence. For take the equation there obtained, this equation has at least as many imaginary roots as any of its derived equations. Take the (r- 1)^ derived equation, which is (r+\)r 3 r(n-r)p^ (n-r+l)(n-r)p^ , ^ --.—«- y + -^ -^ y + ^ ^ — -T —- ^ 0. This equation has imaginary roots if {n-T)rp^ - {n-T ^-Y) {t ^■\)ip^_^p^^^ is negative ; and hence in this case the original equation has imaginary roots. It will be found that the above condition is equivalent to having one of the quadratic elements negative; and as the first and last quadratic elements are positive, there must be at least two variations in the quadratic elements and therefore at least two imaginary roots. See Art. 316. This special case of Newton's rule, and only this, had been established before Professor Sylvester's investigations. 332. If we consider the intrinsic beauty of the theorem which has now been expounded, the interest which belongs to the rule associated with the great name of Newton, and the long lapse of years during Avhich the reason and extent of that rule remained REMOVAL OF TERMS FROM AN EQUATION. 251 UD discovered by mathematicians, among whom Maclaurin, Waring, and Euler are explicitly included, we must regard Professor Sylvester's investigations as among the most important contri- butions made to the Theory of Equations in modern times, justly to be ranked with those of Fourier, Sturm, and Cauchy. XXYII. REMOVAL OF TERMS FROM AN EQUATION. 333. We have already in Art. 56 shewn how to transform an equation into another which shall want an assigned term. We shall now consider this subject more generally, and shew how theoretically any number of terms may be removed. The method of transformation which we shall explain is called by the name of its inventor Tschirnhausen. 334. Suppose we have the equation £c" + />j.'r"~^ +^'2^''"^+ +/'„_i^ + i\ = (!)• Assume y = a^ + a^x + a^x''-^ + a^x"" (2), where m is an integer less than n, and a^,, a,, ... a„ are constants at present undetermined. We propose to eliminate x, and thus form an equation in terms of y. Since there are as many values of 2/ as of a; the equation in y will be of the degree n. The elimination may be effected thus : raise the equation (2) to the powers denoted by 2, 3, ...w; and by means of (1) de- press the exponents of x, so that none of them shall exceed w- 1, in the following way, a;" = _^^a;'-^-^^a;'-2_ -Vn-v^-V.^ x''^'=-p,x'' -p.^x''-' - -Pn-i^^-Pn^> substitute for .t" its value from the preceding line, and we have o:""^^ expressed in terms of x"'^ and lower powers of x; then mul- 252 KEMOVAL OF TERMS FROM AN EQUATION. tiply by x and substitute as before, and we have a;""^* so expressed ; aaid so on. Thus we shall obtain results of the following form : y'=bQ + b^x + h^x^+ ... +b^ 2/" k^ + k.x + kx^ + ... + k i 2 n (3). Here h^, ^j, ... &„_i are integral homogeneous functions of the second degree of the undetermined quantities a^, a^,...a^; also c„, Cj,...c^_j are integral homogeneous functions of the thio'd degi'ee of a^, a^, ...»„; and so on. Let 5j, 5jj, ^3, ... denote the sums of the first, second, third, ... powers of the roots of (1) ; and let S^, S^, tS^y ... denote the sums of the first, second, third, ... powers of the roots of the required equation in y. Then from (2) and (3) we have ^. = ^K + K^ + K^ + '"+ K-i^n-r > 'S'3 = ?^C„ + CjS, +c/^ + S^^ = nh^ + k^s^ + k^s^ + . . . + ^^_ j5„_, . (4). Thus, as the sums of the powers of the roots of the equation in y are known we can construct the equation; see Ai*t. 244. Or we may proceed thus : from equations (3) we can obtain the values of £c, x^, ... ic"~^ in terms of the powers of y ; then by substituting in (2) we have the required equation in y. This method has the advantage of giving x as a rational function of y, and thus the value of each root of (1) will be known as soon as the equation in y is solved. 335. We may now take the hitherto undetermined quanti- ties a^, a^y ...a^ so as to make some terms disappear from the equation in y. For example, suppose we wish to make the coeffi- cients of the m terms which succeed the first disappear; it will be sufficient to put s=o, s.-o, ^„=0. REMOVAL OF TERMS FROM AN EQUATION. 253 But from equations (4) we see that aS'i is of the first degree with respect to a©* »i> ••• ^^> *^^* '^2 is of the second degree, S^ of the third degree, and so on. Hence by Art. 259 the determi- nation of these quantities a^^, «,,... «„,, of which one may be assumed arbitrarily, will depend on the solution of an equation of the degree Im. 336. "We shall make an application of the preceding method which is of especial interest in connexion with equations of the fifth degree : a preliminary proposition will be required, which we shall now give. 337. An integral homogeneous /unction of the second degree of n variables can he expressed as the sum of the squares of v linea/r functions^ the number v not being greater than n. Let V be an integral homogeneous function of the second degree of the n variables ajj, x^,...x^. If 71 = 1, the function contains only one variable, so that it is of the form f^x^, that is, {x^slPT' Suppose that n is greater than 1, and that V involves the square of one of the variables, say x^ ; then by arranging in powers of cc^ we obtain V=Px,^ + 2Qx^ + R, where ^ is a constant, § is a linear function of the n-\ varia- bles, ajjj, £^3, ...a;^, and i? is an integral homogeneous function of the second degree of these n—\ variables. Put ^,=^. + |, ^. = -K-|; thus F=(X,V/3r+F„ and Fj is an integral homogeneous function of the second degree of 71 - 1 variables at most. Next suppose that V does not contain the square of any of the variables j then, arranging V with respect to two variables x^ and ajjj, we have 254 REMOVAL OF TERMS FROM AN EQUATION. V = jSx^x^ + Qx^ + Ex^ + S Put x. = l(».. + x, + ^), • thus F=^(x.'-x/)+ r,=(x, V/8r + (x.V^)'+ r,. Here Xj and X^ are linear functions which may involve the n variables aj^, a;^, ... a;^; and F^ is an integral homogeneous function of the second degree which involves at most w — 2 variables. Thus in the first case the function F which involves n vari- ables is made the sum of a certain square and of Fj, where Fi involves only n-\ variables at most ; and in the second case F is made the sum of two squares and of F^, where F^ involves only w — 2 variables at most. Then by continuing the process on Fj or Fg we can finally express F as the sum of not more than n squares. 338. Let there be an equation a" +;?^a;"~' +^^03""^ + ... +jp^ = 0. Assume y = % + «i^ + cOg^^ "^ ^a^ "^ *4*** Let the equation in y obtained by eliminating x be de- noted by 2/" + ^y + qy + . . . + g, = 0. Now from Art. 334 it will follow that q^, q^, 5-3, ...are re- spectively of the first, second, third, . . . degrees with respect to the quantities a„, a^, a^, a^, a^. Suppose then that we wish to make the second, third, and fourth terms of the equation in y disappear. We put y.-O, q^=0, ^3=0. REMOVAL OF TERMS FROM AN EQUATION. 255 The first of these equations is of the first degree. Suppose we obtain «„ from it in terms of a^, a^, c^a, a^, and substitute in the second and third equations; and then denote these equations bv Here q\ is an integral homogeneous function of the second degree with respect to a,, a^, a^, a^ ; and q'^ is an integral homo- geneous function of the third degree. By Art. 337 the equation §''2 = may be put in the form where yj g, h, and k are linear functions. This equation will be satisfied by putting these two equations are linear. Suppose we deduce from them the values of a^ and a^ in terms of a^ ajid a^ and substitute in the equation q\ = ] and then denote this equation by Here q'\ is an integral homogeneous function of the third degree with respect to a^ and a^. One of these quantities may be taken arbitrarily, and the other can then be found by the solution of a cubic equation. If we wish to make the second, third, and fifth terms disappear from the equation in y the process will be similar but the final equation will be of the fourth degree. 339. If with the transformation of Tschirnhausen we com- bine that of changing the unknown quantity into its reciprocal we can by the aid of a single equation of the third or fourth degree remove from an equation the three terms which precede the last, or the two terms which precede the last, together with the fifth from the end. 340. Thus we see that the general equation of the fifth degree can always be reduced to any one of the following forms : x^ +px->rq = Oj x'^ ■¥px^ + q = Oj x^ ^px^ ■¥q = Oi x^ -^-px* + q^O, 256 REMOVAL OF TERMS FROM AN EQUATION. 311. The foregoing Articles of the present Chapter have been derived from Serret's Cours d'Algebre Superieure, / The reduction of the equation of the fifth degree to the form of the preceding Article was given by Mr Jerrard; it appears from a paper by Mr Harley in the Quarterly Journal of Mathe- matics, Yol. VI., that the result had been previously obtained by E. S. Brinp^, a Swedish mathematician. *o» Mr Jerrard considered that the algebraical solution of equa- tions of the fifth degree could be effected ; his proposed method formed the subject of an enquiry by Sir W. R. Hamilton in the Reports of the British Association, Vol. vi. Most mathema- ticians admit that Abel has demonstrated the impossibility of tlie algebraical solution of equations of a higher degree than the fourth. An abstract of Sir W. E,. Hamilton's exposition of Abel's argument will be found in the Quarterly Journal of Mathematics, Vol. v. A simpler demonstration due to Wantzel will be found in Seri'et's Cours d'Algebre Superieure. An Essay on the Resolution of Algebraical Equations by the late Judge Hargreave has been recently published; the results arrived at are to some extent at variance with those of Abel and Sir W. E. Hamilton. XXVIII. INTRODUCTION TO DETERMINANTS. 342. We now propose to give some account of the theory of determinants, a branch of Mathematics of comparatively recent origin, but already of great and rapidly increasing importance. In the present Chapter we shall consider some particular examples and illustrations which will enable the student to form a conception of the nature and properties of determinants ; in the next Chapter we shall 4emonstrate the principal general theorems of the subject, and in the following Chapter we shall give some applications to the theor}'- of equations. INTRODUCTION TO DETERMINANTS. 257 Consider the simultaneous equations from these equations we obtain V'» + 6,?/ = Ca; afi^ y= (X',b„ — ajb. The common denominator cbfi^-(^fi^ is called the determhiant of the four quantities a,, h^, a^, b^, and is denoted by the following symbol, The numerators of the values of x and y are also determinants ; and we may exhibit the values of x and y thus, «u bi »2. ^'^ I 343. The determinants here considered are all said to be of the second order, because they consist of terms each of which is the product of two quantities. Th^ quantities a^, b,, a^, b^ which occur in the determinant cifi^-aj)^ are called the constituents of the determinant; the products afi^ and ajj^ are called the ele- ments of that determinant. Thus a determinant of the second order consists of two elements involving four constituents. In the symbol used to denote this determinant the constituents are arranged in a square forming two rows or two columns, 344, We shall now indicate some properties of determinants of the second order. Since we have b\^K it follows that the determinant is not altered by changing the rows into columns. T. E. 17 258 INTRODUCTION TO DETERMINANTS. 345. The following identities may be easily verified. a,, 5i K «l %^ K K % «2» h K % ^i» ^ K «x Thus in the determinant, if the two rows or the two columns are interchanged, the sign of the determinant is altered, but not its value; if both these interchanges are made, the determinant is unaltered. 346. We have JP«,» \ «^i» K pa^, p\ «^i» K v%^ K =p "^2^ K f «2» K =p %^ K Thus if each constituent in one row or in one column is multiplied by a given quantity, the determinant is multiplied by that quantity. 347. We have = 0, ttj, «1 I Q Thus if two rows or two columns are identical the determinant vanishes. 348. It may be proved by developing the determinants that «2, h 1, ^1 1.' W ., bj «<,', bj Thus the determinant, each of whose constituents is the sum of two terms, is equivalent to the four determinants which can be formed by taking instead of each column one of its partial columns. As a special case, suppose h a„ b; INTRODUCTION TO DETERMINANTS. 349. By Art. 348 we have 259 a,a, + bAi + hA + hA , = + bA, ^2 + a,a„ bA2 + ^2A, a2a2 = < x,a^ a,, a, + PA b„ b, ,aA a,, b, a„ b. ,As 6„a b„a 1 2 by Art. 346. By Art. 347 the first two of the four determinants just written vanish. And by Art. 345 5„ a. «!, b^ »2) h Thus we have left (aA-Atts) refore a„ 5i that is X a,, b. I i8. A ^3, 62 «1, ^1 »2> ^2 ai^i + ^iA? ^ittg + ^iA a^a^ + bAii a^ct^ + bAi Thus the product of two determinants of the second order is a determinant of the second order. As a particular case, suppose the constituents a, , ^1 , Og , A *o be respectively equal to the constituents a^, b^, a^, b^; then we tind that the square of the determinant ^2, 6s is equal to the determinant IB. l<+^^ «l«2 + ^^2> »2' + V 17—2 260 INTRODUCTION TO DETERMINANTS. 350. We will now proceed to determinants of the third order. Consider the simultaneous equations a^x + h^y + c^z = d^ , a^x + h^y + c^z = c?„ a^x + h^ + c^z = d^] from these equations we obtain and similar expressions for the values of y and z. The denominator of the value of x is called a determinant of the third order, involving the nine constituents a^, b^, c^, a^, b^, c^, a , b , c^; the determinant consists of six elements, each element being the product of three constituents. This determinant is de- noted by the following symbol, — ' Since the value of this determinant is-1 we may express it in terms of determinants of the second or Cl K <^2 Tlie numerator of the value of x is also a determinant of the third order; we have only to change a^, a^i % i^*^ ^i> ^2» <^3 respectively in the symbolical expressions already given for the denominator, and we obtain symbolical expressions for the numerator. We shall now see that determinants of the third order have the same jiropei-ties as determinants of the second order. INTRODUCTION TO DETERMINANTS. 261 351. Suppose a^ = 1, a2 = 0, and ^3 = 0; then we have 1, h^, Ci, 0, &2> <^2J 0, &3» ^3 J Thus the determinant of the third order reduces in this case to a determinant of the second order. The values of hy and c^ have no influence on the value of this determinant, and we may if we please suppose them zero. Hence we see that when we have any relation holding among determinants of the third order we can deduce the correspond- ing relation for determinants of the second order by supposing certain constituents to vanish. 352. It may be shewn by developing the determinants that «1 ^2 J ^2 + «2 ^3, C3 + <^3 Ky C, K Cz b,y C, hi ^2 = aj K h + ^ (^2, ^3 + <'l «2, «3 ^2 J ^3 ^2, ^3 h, h = h, h, h that is, Thus the determinant is not altered by changing the rows into columns. «i, 6,, <^X «'2, &2» ^2 = «^3, ^3. <^3 353. The following identities may be easily verified, by expressing the determinants of the third order in terms of deter- minants of the second order and developing : I ^2, 62J ^2 «3J hi ^3 h, a„ c. ^2i ^2i ^2 = hi «3> ^3 = h Thus if two columns are interchanged the sign of the determi- nant is altered but not its value, and therefore if this operation is 262 INTRODUCTION TO DETERMINANTS. performed twice the determinant is unaltered. Hence, by Art. 352, if two rows are interchanged the sign of the determinant is altered but not its value, and therefore if this operation is performed twice the determinant is unaltered. Hence too ib follows that if two columns are interchanged and also two rows the determinant is unaltered ; so that «!, ^, Cj h,, a^, c^ ^2, 5^, c^ = b,, a„ c. «35 ^3' ^3 hy «^3> Cg 354. As in Article 346 we may prove that if every consti- tuent in one row or in one column is multiplied by a given quantity the determinant is multiplied by that quantity. 355. It is easy to shew that <^2> Ki ^2 = and a3> h^ K «!, Sj, C a^y h^, c. = 0. Thus if two rows or two columns are identical the determinant vanishes. 356. It is easy to see that the determinant ttg + ag' + aa", 63, C3 1 is equivalent to the sum of the three determinants a^y h„ Ci «2> ^2> <^2 ^3> h, ^3 ) <', Ki «2 <^3'i ^3> C3 ^3 J ^ 3> ^3 and a similar result would be obtained if each constituent in the first column consisted of the sum of four terms, or of the sum of five terms, and so on. Again, if each of the constituents 61, h^y 63 is replaced by three terms, each of the above three determinants becomes equivalent to the sum of three determinants; and so on. INTRODUCTION TO DETERMINANTS. 263 In this way tlie following determinant may be seen to be equiva- lent to the sum of 27 determinants : a^ + K Cs u The result is that the six determinants which do remain constitute f, (ftra - P.y.) + «. (/^sT. - ^,Ta) + «3 (/J.y. - ^.7,) that is, Hence we see that the product of two determinants of the third order can be exhibited as a determinant of the third order. If we suppose a^, b^, ... respectively equal to a^, ft,... we obtain a determinant of the third order which is equivalent to the square of a determinant of the third order. ^» A> 7i X S> ^2> 72 %, ft' 73 '1 '2 '3 358. We have now given sufficient examples of the nature and properties of determinants to enable the student to form a conception of the subject. We might have confined ourselves to determinants of the third order, because by Art. 351 the pro- perties of determinants of the second order can be immediately derived from the corresponding properties of determinants of the third order, but the method we have adopted will be of service to the beginner. In the next Chapter we shall give general demon- strations applicable to determinants of any order. It will be observed that we introduce the subject of determi- nants by considering the forms obtained in solving certain simul- taneous equations. The student thus may see at once that the expressions called determinants do naturally present themselves in mathematics. It is however more convenient in treating the general theory to give an independent definition of a determinant, and this we shall do in the next Chapter. It will prepare the student for that definition if we here consider the determinant of the third order in this new light. INTRODUCTION TO DETERMINANTS. 265 359. The value of the determinant i» ^' 2» 2' 2 «3> ^3' <^2 The first element here is ct,J)^c^, which is the product of con- stituents situated diagonally in the square symbol denoting the determinant. The other elements may all be deduced from the first element in a way which we shall now explain. The suffixes 1, 2, 3 are to be attached to the letters a, b, c in all the different ways in which permutations can be made of these suffixes ; and the sign + or — is to be prefixed to any element according as it can be deduced from the first element by an even number or an odd number of mutual interchanges of two suffixes. Thus the second element given above is oi}>.f^\ this can be derived from the first element by interchanging the suffixes 2 and 3, and so according to the rule it is to have the sign — prefixed. The third element is ^2^3^i ^ *^^^^ ^^^ ^® derived from the second element by interchang- ing the suffixes 2 and 1, and therefore it can be derived from the first element by two interchanges of two suffixes, and so according to the rule it is to have the sign + prefixed. Similarly the remain- ing elements with their proper signs may be determined. V 360. The following examples are particular cases of determi- nants of the third order, which the student may verify : .(1) (2) • (3) I a, h, g K h, f = 9, L c 1, ^1, 2/i 1, ^2» y. \ «^3. V J = ahc — of' — hg^ — ch" + 2/gh. = ^1^2 - ^^Jx + ^2?y3 - ^32/2 + ^^X - ^12/3 1, a, + a^, a^a^ 1, ^ + ^2, ¥2 1» ^1 + ^2. ^1^2 (^-^.)(^-0(^-^2) + (^2-^)(^2-^l)fe-«l)- 266 PROPERTIES OF DETERMINANTS. (4) 1, y, -P -y, 1, a A - a, 1 \+o^ + P' + y\ h > XXIX. PROPEETIES OF DETERMINANTS. 361. Let there be n symbols «Xj, a^, ...a^; then one of these symbols will be called higher than another when it has a greater suffix, so that for example a.^ is higher than a^ or a^ , a^ is higher than ^3 or a^ or a^ , and so on. Now suppose that permutations are formed of these symbols ; then whenever in a permutation the higher of two symbols pre- cedes the other there is said to be a disarrangement. Thus, for example, in the permutation a^cb^aj^^ there are four disarrange- ments, namely a^a^^ a^a^, <^^^t and a^a^. 362. The permutations of the symbols a^, a^^...a^ may be divided into two classes, those in which there is an even number of disarrangements and those in which there is an odd number. 363. When in any 'permutation two symbols interchange their places while the others remain unchanged the number of disarrange- ments is increased or diminished by an odd number. Let g and k denote two symbols of which k is the higher. Let A denote the group of symbols before g and k, let B denote the group between g and k, and let G denote the group after g and k ; so that the permutations which we have to compare may be denoted by AgBkO and AkBgC. Then the difference of the numbers of the disarrangements depends upon the symbols which constitute the groups gBk and kBg. Let B consist of /5 symbols and suppose that P^ of them are higher than g and p^ of them higher than k. Then in the group gBh, besides the disarrange- ments in B itself, there are /3 - /?, + yS^ disarrangements ; for g is higher than j^ - /?, of the symbols in B, and there are P^ symbols PROPERTIES OF DETERMINANTS. 267 in B higher than k. In the group hBg, besides the disarrange- ments in B itself, there are ft- ^2 + fi^ + 1 disarrangements ; for k is higher than ft - P^ symbols in B^ and there are ft^ symbols in B higher than g, and k is higher than g. Therefore the difference of the numbers of the disarrangements is that is, 2 (/^i - p^ + 1; thus this difference is an odd number. 364. By repeated interchanges of two symbols all the permu- tations of a set of n symbols taken all together can be deduced from a given permutation. In this mode of deriving the permu- tations we shall, by Art. 363, obtain alternately permutations with an even number of disarrangements and permutations with an odd number of disarrangements. The whole number of the permu- tations of a set of symbols taken all together is an even number ; hence it follows that there are as many permutations with an even number of disarrangements as there are with an odd number of disarrangements. 365. Let there be n^ quantities arranged in the form of a 1.1 J «i. luare, thus a, 3 > '-''1, 1 ^2, 3 > ^2 Here for any quantity a^^ the first suffix, r, indicates the row, and the second suffix, h, indicates the column in which the quantity ^r.k appears. The above symbol is used to denote the determinant of the n^ quantities occurring in it ; these quantities are called constituents of the determinant. The value of the determinant is found by taking the aggregate of a certain number of elements, each element being the product of n constituents. The first element is the pro- duct of the constituents a^ ^ , a^ 2 > ^3. 3 j • • • *» .« 5 which lie in the 268 PROPEETIES OF DETERMINANTS. diagonal drawn from the upper left-hand corner of the square to the opposite corner; we shall call this diagonal the diagonal of the square, for we shall only have occasion to refer to this diagonal. All the other elements are to be derived from the first element ^1.1^2. 2^8.3 •••^n.n ^7 permutatlous of the second suffixes, the first suffixes being left unchanged. The sign + or - is to be prefixed to each element of the determinant according as it is or is not of the same class as the first element, the class being determined by the number of disarrangements in the permutations of the second suffixes ; see Art. 362. 366. The above detei-minant is said to be of the n^^ order because each element is the product of n constituents. The num- ber of elements is the same as the number of the permutations of n things taken all together, that is \n] half of these elements will have the sign + prefixed, and half of them the sign — prefixed. It will be seen from the mode of formation of the elements, that each element involves one and only one constituent out of each row or each column in the symbol which denotes the detenuinant. 367. Instead of the above symbol for the determinant, it is sometimes denoted by 2=^<^i,i0^2,2<^3.3 ••• ^n, n^ ^^^^ is? ^^^ fi^st ele- ment is written and the symbol S =fc put before it to indicate the aggregate of elements which can be derived from the first element by suitable permutations and adjustment of the signs + and — . The constituents of the determinant may be denoted in various ways; tlius sometimes (■i, k) is used instead of a^ j, and in this case we must remember that (^, k) and (k, i) in general denote difierent quantities. In examples of determinants of low orders, we may find it convenient to avoid double suffixes, and use the same letter for all the constituents in one column, distinguishing the con- stituents by single sufiixes ; this notation was adopted in the pre- ceding Chapter. 368. The other elements of a determinant are derived from the first element by permutations of the second suffixes while the PROPERTIES OF DETERMINANTS. 2(59 first suffixes remain unclianged ; these elements may however be derived in a different way, namely, by permutations of the first suffixes while the second suffixes remain unchanged. For suppose that a, p, y,...v represents a certain permutation of the n numbers 1, 2, 3,...n; then a^ a «2, /g <^3, v • • • ^n. v i^ ^^^ element of the determi- nant which arises from the first element by changing the second suffixes 1, 2,...n, into a, /3, y,...v, respectively. This element may however also be derived from the first element ^i, 1 0^2.2- ••^n.« ^^ the second suffixes are left unchanged and the first suffixes are suitably changed, namely, a to 1, j8to 2, y to 3,...v to n. In these two modes of derivation there is the same number of interchanges of two suffixes, and therefore the same sign is obtained to prefix to the element by the rule in Art. 365. 369. The value of a determinant is not altered if the successive roivs are changed into successive columns ; that is fti- , . Oj- ...... a-. a, , . a.^ „ • . . . . a„ (ii ,. a ....a '^l, 2> <^2. 2J-- ^n. 2 ^1, n> ^2, n) ^n,n For it is obvious from Art. 365, that the elements in these deter- minants are of equal value; and they have the same signs, as appears from Art. 368. 370. If two roivs or two columns are interchanged, the sign of the determhiant is changed. t For let R denote the given determinant, R' that which arises from the interchange. Then the elements in R and R' are the same as to value, and we have only to examine their signs. The first element in R' can be derived from the first element in R by interchanging two of the second suffixes, and thus these elements have contrary signs in the two determinants. Then an element 270 PROPERTIES OF DETERMINANTS. in R' which arises from the first element in 2i' by m interchanges of the second suffixes will be deducible from the first element in R by m + 1 interchanges, and therefore it will appear in R and R! with contrary signs prefixed. 371. If two rows or two columns are identical , the determinant vanishes. For by interchanging two row^s or two columns, a determinant is changed froQi i? to -^ by Art. 370. But if two rows or two columns are identical, the interchange of these rows or columns can have no influence on the determinant, so that R = — R; and therefore R — 0. 372. When all the constituents excejit one of a row or of a column vanish^ the determinant reduces to the product of that con- stituent and of a determinant of the next inferior order. Consider, for example, the determinant «1> *., «i. d, «2, K, e^, d. «3, h, "3. d. 0, 0, ".. By three successive interchanges of single rows we can bring the row which contains c^ to be the highest row ; and by two suc- cessive interchanges of single columns we can bring the column which contains c^ to be the first column. Thus, by Art. 370, «n h. Ci, d. «„ K <^2, d.. «3> h. C3, d. 0, 0, ^V ^4' 0, •0, ^15 a,, K d. ^2, «2, K d. C3, «3, K d. The first element of the determinant on the right-hand side is c^a,&2^3, and the other elements are to be derived from this by )>ermutations of the suffixes. But c^ is the only constituent with the suffix 4 which is not zero, and thus c^ will be a factor of every PROPERTIES OF DETERMINANTS. 271 element which does not vanish, and the other factor will be de- ducible from ap^d^ by permutations of the suffixes 1, 2, 3. Thus the original determinant reduces to (-!)'«. This mode of demonstration applies, whatever may be the order of the proposed determinant. The negative sign which arises in this example from (-1)^ may if we please be removed by interchanging two rows or two columns in the determinant of the third order. »l. K d, a„ K d. «3. h, ds 373. The top row of a determinant of the n^^ order can be brought to the bottom hy n-1 successive interchanges of two rows; and similarly, the first column can be brought to the end hy n—1 successive interchanges of successive columns. Each of these is called a cyclical interchange, and it is sometimes conve- nient to effect any proposed interchange of rows or columns by a series of cyclical interchanges, for the sake of greater symmetry in the arrangement of rows and columns. In tlie preceding example we may bring c^ to the place which we want it to occupy by per- forming three successive cyclical interchanges of rows and two successive cyclical interchanges of columns. Thus we obtain for the original determinant the following forms successively : (-1)= (-1)- «2. K Ci., d. »3. h, «a. d. 0, 0, h^ Czi ^3 (-1)' 0, 0, c,, »!> hi Cli ^1 <^2t hy ^2> ^2 > c„ 0, 0, (-^y\ Cj, d^f «!, 6, ^2, ^2, a^, h C3, c?3, a^,b^ > 0, 0, c,, »l> ^, c,, c?i «2> hi <^2) ^2 ^a* Ky ^3? ^a ^2, «2> ^2 272 PROPERTIES OF DETERMINANTS. 374. A determinant can always he expressed in the form of a determinant of any higher order. For example, by Art. 373, «n K Ci 1, 0, 0, 1, 0, 0, 0, <*2> ^2J ^2 = A »i, ^, Cj = /., 1, 0, 0, «3, K, ^3 y, a^, &2, c^ 8, ag, 63, C3 P, 7) »2> ^2» <^2 0-, 8, C^3, &3, C, where /?, y, 8, /x, v, p, o-, are any quantities. Similarly, we may carry on this process to any extent. 375. Let i and h denote any two suffixes out of the set 1, 2, ...ii; let R denote the determinant 2=tai. i«2, 2 ...«„„ ; and let Ai^ j^ denote the coefficient of a^ ^ in i?. Then each of the expressions and ai.iA* + «2. A *+•••+ ^n.i^«.i> is equal to R or to 0, according as i and h are equal or unequal. For every element of R contains as a factor one out of the constituents a^j, a^.^, a^g, ... a,, „, which form the i^ row. And since ^^^ ^ denotes the coefficient of a^ ^ in R, we have ^ = ai. A i + ^f.2^i.2 +••• + »». n^«.«- Similarly we have ^ = «i.Ai + «2. A<+---+^>t.Ai- In the first of these expressions for 7?, put a^, i"«;fc. u ^i. 2 = ^jfe,2J ••• ^^^ so on; thus we obtain the value of a determi- nant with two rows identical, which is zero by Art. 371. Similarly, in the second expression for R put «i. i = »i, ^ > *s,« = ^2. *j ••• and so on j thus we obtain the value of a determi- nant with two columns identical, which is zero by Art. 371. PKOPERTIES OF DETERMINANTS. 273 376. If every constituent in one row or one column is multv plied by a given quantity^ the determinant is multiplied by that quantity. For R = a^^A^.^^+ a.^A..^-\- ... +ai.„^i,„; and if every term in the i*^ row is multiplied by p we must put pa^^ for a^ i, pa^^ for a. 2, and so on; thus we obtain p times the former result for the new determinant. Similarly, we may prove the theorem in the case in which all the constituents of a column are multiplied by a given quantity. 377. If each of tlie co7istituents in one row or one column is the sum of m termSy the determinant can be considered as the sum of m determinants. Suppose, for example, that each constituent of the i^^ row is the sum of m terms j and suppose that «i.i = 7-^1 + ^1 + ^1 + ••• Then A' = a. i.4,i + a,.2^,,2+ +(ii,n^i,n ^ PAi.l+ P2A2+ + PuAn + Mi.l+ ^2^i.2+ + 9n^i.n + M,.i+ r,A,,+ + r„^., + Thus i? may be considered as the sum of m determinants which have for their i^^ row^s respectively P,^ P2^ Pny ?u (?2' 5'„» ^'ij ^> ^„v T. E. 18 274 PROPERTIES OF DETERMINANTS. 378. We shall now shew how the coefficient of a. ^ in a de- terminant can be itself exhibited as a determinant. In order to obtain those elements of a determinant which involve a certain constituent a^J^, and those alone, we may suppose all the consti- tuents in the i^ row to be zero, except a^ ^ ; then putting 1 for a. ^ we shall obtain the required coefficient. In this way we get ^i,*= «i.i> «!.*> 0, ... 0, 1, 0, ... a. %,k+li Thus -4 . t is here exhibited as a determinant of the n^^ order. We may, without influencing the value of A^^^, put for each constituent in the k^^ column except that which is 1. By Art. 372, or by Art. 373, we may exhibit -4^ ^ as a determinant of the (n-lf^ order. Thus, adopting the method of Art. 373, we make i— 1 cyclical changes in the rows and ^-^1 cyclical changes in the columns. Therefore ■^i,* — €^ ^t+i,i+i> •• "i+i. ., «^, •+1,1' i+l.i-1 ... a„ ... a. a,i, ... a^ ^i-i.k+if ••• ^. where €= (- 1)«-^+*-^k«-i)^(_ ly (i+i)(«-l) 379. By the aid of Arts. 375 and 378 we can express any de- terminant of the n^^ order as the aggregate of n terms, each of which is the product of one constituent and of a determinant of the (n~iy^ order; the determinants of the (n—iy^ order may themselves be similarly treated ; and the process continued to any extent. For example, PROPERTIES OF DETERMINANTS. 275 «,j ^i, Cj, d^ = ^1 ^2» ^2' ^2 -^ c^^ d,, 0^, + Cj ^., «2, ^2 -d. «2, ^2, C, a^, b^c'^, d^ K ^3' ^3 ^3' ^3' «3 ^3' ^3» K «3. ^3' ^3 «3' K ^3' ^3 ^.« ^.. < ^4. ^4' «4 d„ a^, \ «4' ^4' ^4 «4' K ^.. K + ^2!«3» ^3 + c/„ + a„ ^3' <^ "'{ {'■ «3' ^3 i+«2 K ^3 + ^2 ^3' »3 ^4, 64 ! 64, ^4 C^4, (^4 K % ^\ «3' «3 + ^2 «3. ^3 64, C4 C4, ^4 «4, &4 C4, c/4 1 i } 380. We now proceed to an important part of the subject, that which relates to the multiplication of determinants. Let there be two given sets of symbols, namely, "Li.* and n.p) K., From these let a third set of symbols be formed, Xni ese symbols being determined by the general relation Let R denote the determinant ^'^c.^xC^^,., c^.^. We shall now prove the following results : 18—2 276 PROPERTIES OF DETERMINANTS. (1) Suppose p less than w; then i? = 0. (2) Suppose p = n] then R is equal to the product of the two detenninants which consist of the two given sets of symbols in the order they occupy. (3) Suppose p greater than n ; then H is equal to the sum of a set of products of pairs of determinants, each pair of determinants being formed by taking any n columns out of the first given set of symbols for one determinant, and the corresponding n columns out of the other given set of symbols for the other determinant. The first element of B is c^ iCg 2 ••• c„ „, and the value of this is where in the first factor ^ denotes a summation with respect to r, in the second factor 2 denotes a summation with respect to s, in the third factor 2 denotes a summation with respect to t, and so on ; and all these summations extend from 1 to p, both inclusive. Thus the product may be obtained by taking the sum of the values of the expression where r, s, ^, . . . take all integral values from 1 to p. We may denote this sum by 2,..,,,...Kr«2..«3.« ••• KAsht"-)' The other elements of E are derived from the first element by permutations of the second suffixes and prefixing the proper sign. Now from the general value of c,^ ^ it follows that by changing the second suffixes of the symbol c no change is made in the suffixes of the symbol a, but the first suffixes of the symbol b are changed, and these alone. Hence "we obtain a result which we may denote thus, ^^X.».t....Kr^2„%f"^^KAAf-)' PROPERTIES OF DETERMINANTS. 277 Here ^^bj^^^h-^^b^^... constitutes a determinant of the n^^ order, wliich is formed from the second given set of symbols by taking certain columns, and the ^ refers to changes of the first suffixes j see Art. 368. We shall denote this determinant by Q. Now, in the first place, suppose p less than n. The suffixes Ty s, tj ... are n in number, and none of them can exceed p ; hence it follows that there must be always two or more of them which have the same value. Thus Q always vanishes, by Art. 371 ; and therefore H vanishes. Secondly, suppose p = n. Then the system of suffixes r, s, f,... can be a permutation of the n symbols 1, 2j...n; and they can be nothing else without making Q vanish. And by taking in succession difierent permutations the sign of Q will change, but not its value, by Art. 370. Thus the value of E reduces to the product of the determinant formed out of the second given set of symbols, into the sum of all the elements denoted by ^=*=<^i,i<^2,2 ••• ^«,n» where 1$ refers to changes of the second suffixes. Therefore when p = n, R «1.1, • •«^l.n X Jm, . -h. ««.l;- ••^„.n J.,„ • -K. Lastly, suppose p greater than n. Then the system of suffixes r,s,t,... can be any combinatioi^ of n numbers that can be formed out of the^ numbers 1, 2,...p; and the number of such combina- [p tions is , — ■ . Let P denote what Q becomes by chanmnor b into a. Hence, as in the second case, we shall obtain FQ for one term in i?, which arises from the selection of a definite combina- \p tion out of the , — ■ possible combinations. Therefore when \n\p-n ^ p is greater than n we have H = %PQ, where S refers to the sum- \E mation of r-; terms arising from all the possible combina- m \ p — n ° tions. 278 PROPERTIES OF DETERMINANTS. 381. By the second case of the preceding Article we see that the product of two determinants of the order n can be exhibited as a determinant of the same order. Similarly, the product of three determinants of the order n can be exhibited as a determi- nant of the order n ; for we can first exhibit the product of two of them as a new determinant of the order n, and then the product of this new determinant and the third of the original determi- nants can be exhibited as a determinant of the order n. Thus we see that the product of any number of determinants which are all of the same order can be exhibited as a determinant of that order. Hence generally the product of any number of determinants of any orders can be exhibited as a determinant of the same order as that of the determinant of the highest order among the factors. For by Art. 374, all the other determinants may be made to be of the same order as that which is of the highest order; and then the product of these determinants of the same order can be ex- hibited as a determinant of that order. 382. minants Suppose we wish to form the product of the two deter- and By Art. 369 we may change the successive rows into successive columns in either or both of these determinants. Thus, if we denote the product by ^1,1 J ••• ^l,n PROPERTIES OF DETERMINANTS. 27.9 we may form the new constituents in four ways, for we may adopt either of the following laws throughout, or C^* = «a.i&*.l + «^2. A,2+ ••• + (^n,An> or Ci.k = ai.Ak + <^2,Alc+'"+(^n,iKk' 383. Let A. J. denote the coefficient of ct. ^ in a determinant E. The system of symbols ■^2, 15 -^2, 2) •••-^2, n AAA is called the reciprocal of the system of symbols ^1, U ^1, 2>'*'^l, n "2, U "^2, 2J ...tto 384. The determinant of a system which is the reciprocal of a proposed system of n* symbols is the (^ — 1)*^ power of the determi- nant of the proposed system. If we multiply the determinants and we obtain for the product I 280 PROPERTIES OF DETERMINANTS. where c,. * = ^i, i o^*. a + ^f, 2«*. 2 + . . . + ^ . ^a,^ „. Hence, by Art. 375, the constituents of the last determinant have the value -S or according as i and k are equal or unequal. Thus this determinant reduces to its first element Cj^iCg 2...c„ „, that is, to R". Therefore therefore 385. Suppose we have a determinant of the n^ order, and in the square symbol denoting it suppose m columns and m rows destroyed; the remaining symbols may then be supposed moved close up so as to form a new square symbol which is a determinant of the order n — m. This determinant is called a partial determi- nant or a minor determinant, with respect to the original determi- nant. The symbols common to the w rows and columns will form a square symbol which is a determinant of the order m. This is also a partial determinant or minor determinant. The two partial or minor determinants are said to be complemeiitary to each other. 386. Let i? denote a' determinant of the order n. A partial determinant of the reciprocal system of the order m is numerically equal to the product of H""'^ into the complementary of the corre- sponding partial determinant of the original system. Let /, g,.'.ry s,... denote one permutation of the n numbers 1, 2,...n'j and let ^, h,...u, r,... denote another permutation. And suppose fi g^... and ^, ^, . . . to be groups of w numbers each, while r, », . . . and w, v, . . . are groups oin-m numbers each. Thus g.kv is a partial determinant of the reciprocal system of the order m ; we shall denote it by S. mOPERTIES OF DETERMINANTS. 281 Now ^/,i» *^/.*' •••^/.«> ^/.t. y.fj «..i> ^..*5 • -'%,u, %..•> ••• €i2 where € is + 1 or - 1 according as the permutations f, g,...r, s,... and i, k, ...Uj V, ... belong to the same class or to different classes. We now propose to obtain the product of these two deter- minants. The determinant S may be raised to the order n by inserting additional constituents; see Art. 374. Thus we may put for S the following determinant, ■^,.i< A,"-- ■^,.., '*.,.,- ■K. £ where the constituents denoted by the letter B with suffixes are all supposed zero, except those standing in the diagonal which are all supposed equal to unity. Now form the product of S and ei?, which will be a new deter- minant of the order n. Let the constituents of this new determi- nant be denoted by the letter c with two suffixes, the first of which indicates as usual the row and the second the column. By Art. 382 there are four ways by which we may determine the constituents in the product of aS' and €R; we shall select the first of these, according to which c^^ ^ is obtained by multiplying respec- tively the constituents in the p^^ row of aS' by those in the q^^ row f ci?. Thus A f.i'^f.i ^f.i^'^f.i + ...+^,_„ay^„+^,^,ay^,+ ^/.i«^.i + A.»^ir.*+ ••• +^/.«»^.«+^/.A,»+ ••• 282 PROPERTIES OF DETERMINANTS. Therefore by Art. 375, we have 0,1, C2 2,...c„,^ all equal to R, while all the other constituents in the first m rows of the determi- nant which is the product of S and tR are zero. For the first term in the (m + 1)* row, we have because all the symbols denoted by B with suffixes which occur here are zero except B^^, and that is unity. For the second term in the (m + 1)*^ row we have similarly Proceeding in this way, we find that the (w+1)*^ row in the product of aS' and ci? is the same as the (m + 1)"" column in Ui. Similarly, the (m + 2)*^ row in the product is the same as the (m + 2)*^ column in ^R. ' The determinant then which is equivalent to S^R reduces by Art. 372 to the product of R"" and the following determinant of the {n - rrif order, Thus S=€Rr- <^r.ui »r.«J a. 387. The following examples may be verified by the student. In examples (4), (5), and (G), we have determinants of which the constituents are themselves determinants. (1) 0, a, A y. a, 0, 7i> ^, A r,» 0, «, y» P. «i» a'a.'+^'/J,'+/y,'-2aa./3y3,-2ao.yy.-2/3y3.yy. PROPERTIES OF DETERMINANTS. 283 (2) 0, a, ft y r,-^., = (aa^-^ft+yy,r (3) 0, a, ^, 7 a, 0, y,, /?, A -7i) ^J «i ■y, -ft,-a^, ^ ?4 + 0' {a' + IS' + y'+ af + P^' + y;) + (aa,-^ft + yy,r (4) c, 5' ?, » 1 ^, a /,^ = a a, h, g 9, « a, h K h, f /,^ h,b ^./j c (5) g,a f.c a, h Kg h, h gj = h ay Kg KKf (6) hj /,<= h,b f,o k,g 9 J /,« 0,9 9, a h,g 9, a. f,h h,b 9, a- a, h 9,/ f,h Kb a, Kg square of K K f g,f, c (7) a, h c, d ^v^x bj c a, d (Tj, d^ c, a b, d = 0. 284 APPLICATIONS OF DETERMINANTS. XXX. APPLICATIONS OF DETERMINANTS. 388. Suppose we have to find the values of n unknown quan- tities iCj, x^^...x^^ from the following n simple equations Let R denote the determinant S=*=a^na2,2 •••»„.„? and let A.^^ denote the coefficient of a^^i in R. Then the values of the unknown quantities will be given by the formula Rx, = u,A,^ , + u.,A^ *+...+ w,.^„. *, where h may have any value between 1 and n both inclusive. For let the given equations be multiplied respectively by ■^1*) ^%ki-'-^n,k'} ^^^ a^^ *^6 results. The coefficient of x^ is then which is equal to R by Art. 375. The coefficient of x. is which is zero by Art. 375. We may write the formula which gives x,^ thus Rx^ = S, where S is also a determinant, namely the determinant which is obtained from R by removing the k^ column of R and substituting ■ for it the column formed oiu^, u^^...u^, 389. Suppose that the determinant R vanishes; then the values of the unknown quantities become infinite. This indicates that the given equations are inconsistent; see Algebra, Chapter xv. APPLICATIONS OF DETERMINANTS. 285 390. Suppose that u^, u^j ... u^ vanish, and that E also vanishes. The method of Art. 388 gives for the unknown quan- tities the indeterminate form ^ . In this case we may take n—1 of the given equations, and these will be sufficient to determine the ratios of ?* - 1 of the unknown quantities to the remaining unknown quantity. These ratios can however be at once assigned : we shall have X, : X,: x^ : ...=A^, : A^^ : A.^^ : ... where i is any integer not greater than n. For since i?= 0, we have by Art. 375, for all integral values of i and k between 1 and w, -^^^ %. 1 ^ f. 1 + «*. 2 ^ .. 3 + «*. 3 ^ i. 3 + = ; and thus when x^, x^, x^,... are taken in the ratios assigned above, we have a*.ia^i + a*.3^2 + «*.3»^3+ = ^• By taking n-l of the given equations, and supposing «j, u^,...u^ all zero, we shall obtain in general a single definite^ value for the ratio of each of ?^ — 1 of the unknown quantities to the remaining unknown quantity. Hence it follows that when ^=^0 the ratios ^i.I ' ^i.2 - A are independent of i. 391. If w,, u^^,...u^ all vanish, and E does not vanish, the system of equations in Art. 388 has no solutions, except we suppose a?!, .x^,...x^ all zero. The condition i? = is thus necessary in order that the unknown quantities may have values which are not zero. 392. For example, in order that the equations a^x + \y + c^z = 0. »!, K Cl 0^2, f>., <^2 «^3, K Cz 286 APPLICATIONS OF DETERMINANTS, may admit of solutions which are not zero we must have = 0. If this condition is satisfied the equations may be satisfied by X '. y \ z ov X : y '. z or X : y : z These three forms of solution coincide by Art. 390.. 393. From the given equations in Art. 388 we have deduced U,A,^ 1 + ^2^2. 1 + ^3^3, , + ...+ w,^„,i = Bx„ . U^A,^ 2 + U^A, 2 + t^3^3 2 + • • • + ^H A. 2 = ^^2 J h, «. : 1^2, »2 : ttj, h K, Cj !C3, «^3 ^a. h > h, "s : Cg, ^3 : ^3, h K «I <^l'> a, a„ h » h. «J: <^iy a, : a„ h K, c= ^2, ^2 ttj. h u,A^^^ + u^A^^ + u^A^^+ ...+u^A^^^=Rx^. Let p denote the determinant ^±^i i-42_2---Ai,«^ 3,nd let a^ ^ denote the coefficient of ^^^ in p. "We may from the above equa- tions find the values of «^j, u^,...u^; and by proceeding as in Art. 388 we shall obtain the general result pu, = i2 i^x,a^ 1 + x,a,^ 2 + ... + x^aA . By comparing this result with the given equation in Art. 388^ we have, since the values of w^ must be identical, P APPLICATIONS OF DETERMINANTS. 287 But p^E"-' by Art. 384 ; thus 394. We now proceed to apply determinants to another problem, that of forming the product of all the differences of given quantities. Let 01 quantities be denoted by a^, a^,..,a^. Let P denote the product of the differences obtained by subtracting each of these n quantities from all those which follow it, so that ^ = K - «l) («3 - «l) • • • {% - «l) («3 - %) k -%)'- («n - «„-l)- Then F may be exhibited as a determinant of the order n. For consider the determinant ., a^, a/, ...a/ 1, This determinant is a rational integral function of the quantities a^, a^,... a^; and it vanishes when any two of these quantities are equal, by Art. 371. It is therefore divisible by the product which we have denoted by P. Also both the determinant and the product P are of the degree —\ — ^ in powers and products of a^, ttg, ...a^; therefore the quotient when the determinant is divided by P is some number. And this number must be unity, as we see by comparing the first element of the determinant with the product of the first terms of the binomial factors of which P is composed. 395. The above determinant of the w*^ order consists of ^ terms. The product P prior to simplification and cancelling would w(n-l) involve a much larger number of terms, namely, 2 '■^ . Thus the determinant is an advantageous form for the product on account of the saving in terms. 288 APPLICATIONS OF DETERMINANTS. 396. We have F' 1, a , a*,... a"-' 1, aj, a^^,...a^ 1, ttg, a/,... a/~^ 12 n— 1 , a , a ,... a Now the product of these determinants can be exhibited as a single determinant; adopting the last of the four methods given in Art. 382, we have °i? *'2> •••*'«-i ^here a;+a/+...+a/. 397. Suppose, for example, that a^ , a^, . . . a^ are the roots of an equation of the n^^ degree ; then P^ is the product of the squares of the differences of the roots. Thus the product of the squares of the differences of all the roots of an equation can be exhibited as a determinant, the constituents of which are known in terms of the coefficients of the given equation, for s^ can be expressed in terms of the coefficients. 398. Suppose we have to find the values of the n unknown quantities a^ x^, ... x^ from the equations jrj + a?2 + 0^3+ ... +93^ = 1, «^i«i + ^2«2 + a^3«3 + • • • + ^„«„ = ^» The values of the unknown quantities will be determined by the formula APPLICATIONS OF DETERMINANTS. For by Art. 388 we have Iix^=S, where i?= 1, 1, 1, ...1 289 1, ...1, 1, 1, ...1 and^: n-1 „n-l yn-1 „n-l „ n-l Now let the i*^ column in ^ be placed first, and the i* column in She placed first; see Art. 373. Then let the two determinants be changed into products of differences by Art. 394; and by can- celling: common factors in the numerator and denominator we obtain the value of ccj in the form assigned above. As a verification we observe that if ai=t the equations are obviously satisfied by supposing x^= 1, and all the other unknown quantities zero, 399. The method of determinants may also be used to obtain the resulting equation when certain quantities are eliminated i'rom given equations. Suppose we have to eliminate x from the equations /(x) = and [x) = 0, where /{x) = %+ a^x + ajjif + a,oi?^ ^ (x) = b^ + h^x + \x*. We may proceed thus ^f{x) -a^ + a^x + a^ + a^a? + 0, xf{x) = + a^x + a^T^ + a^ + a^*^ {x) = 6^ + h^x + l^x} + + 0, x^ (ic) = + h^x + hx^ + h^^ + 0, a'<^( £c) = + + l^x^ + &,x' + h^x\ 19 290 APPLICATION'S OF DETERMINANTS. Let R= %, a,, a^, a^, I ^ f %, «i) «^2» «3 I ^0. ^, ^„ 0, I 0, ^'o. K K 0, 0, Jo, ^, h\] then since by supposition y (a;) = and ^ (cc) = 0, and therefore also xf{x), X(j> (x), and x'^ (x) are all zero, it follows by Art. 391 that j^ = is the necessary relation which must hold among the coeffi- cients of /{x) and <^ (x). 400. We have given a particular example in the preceding Article, as the general investigation to which ^Ve now proceed will thus be more intelligible. Let /(x) = % + a^x + a^a^ + . . . + a„x'"= 0, (x) = b^ + h^x + h^x^ + ... + bx"" = 0; and suppose we have to eliminate x between these equations. We have /{x) = a^ + a^x + a^x^ + . . . + a ^o;'", xf{x) = a^x + a^x" +...+ a^^_^x^ + aj^^\ x-'f{x)^ cj> {x) = h^ + h^x + h^x' +... + hx\ x4> (x) = \x + h^x^ + ... + ^..^a;" + hx""^', x'^-'{x) = hx"'-' + h.x'"+... Let E denote the determinant of the order m + n which has for ita first n rows «o> »,» «2» •••»«.» 0> ^, ^,-- 0, a„ a^,...a^_^, a^, 0, 0,... 0, 0, «„,... a^_^, «,„_,, «,„, 0,... APPLICATIONS OF DETERMINANTS. 291 and for its next m rows 6^, \, &„...&., 0, 0, 0,... 0, \, ^,...6„_,, 6„, 0, 0,... 0, 0, 6o,...&„_3, 6„_,, 6„, 0,... then i?:= is the necessary relation among the coefficients in order that fix) and <^ {x) may simultaneously vanish. The relation i? = has been called the resultant or the elimi- nant of the proposed equations /(a;) = and K, 0, Jo, 6., ...5, -K 0, 0, 0, «o. 0,0,... K Therefore by evaluating the determinant on the left-hand side, and rearranging that on the right-hand side, we obtain ^'^ir K 0, 0, 0, 0, 0, K 0, K K-, r— 2' r 402. We will now give some applications of the theory of determinants which occur in a case of the transformation of func- tions by linear substitutions. Let there be any function of the n independent variables x^, x^, ...x^; and let these variables be expressed in terms of n new independent variables 2/i> 2/2? •••2^n ^7 means of the following n linear equations : (1); Xn= »«.l2/x + an.22/2 + ••• +«„.„2/„ J then by substituting the values of cCj, x^, ...x^, the assigned func- tion becomes a function of 2/,, 2/2) •••y„* Suppose now that we impose the condition that < + <+... + r«; = y/ + y/+...+3,; (2); then certain relations will hold among the coefficients of the linear equations (1); these relations we shall now demonstrate. APPLIOATIONS OF DETERMINANTS. 293 T. For every value of i and k between 1 and n inclusive <^i\i + <.i+ + ^2,2> ••• *2.n a«,i> «^„.2>--- »« Denote the proposed determinant by E : then JR^, by Art. 380, is equal to the determinant '1, If ^1, 2 > ... c. . c rhere <^i.*=i«i.l»*.l+ai.2»i.3 + "-+ »«.n«*.«' 294 APPLICATIONS OF DETERMINANTS. Thus, by (5), we have c,i=0 when i is not equal to Jc, and c^^=l. Thus the latter determinant reduces to its first element, that is, to unity : therefore JS* = 1. V. Let ulj^jfc have the same meaning as in Art. 388: then we obtain from (1) Hence, comparing this result with (4), we have A. "^'^ £ ,(6). YI. The following partial determinants which can be formed of the constituents of H are numerically equal: ^m+2,m + li <^m+2.7n + 2> '•• ^m+2. : and ^2, 1) ^2,2J ••• ^2,OT Denote the first determinant by P, and the second hj Qj then by Art. 386, -^l.l> -^1,2 > ••• -^1,. ^2.1» •^2,2> ••• ^2,1 -^m,l> ■^m.2> ••• -^m. is numerically equal to E"'~^F. And by Art. 376 and equation (6), -^2,l> -^2,2> ••• -^2,, <„ ^ »i,l> ■'*«t,2» . ^. APPLICATIONS OF DETERMINANTS. 295 Hence, since ^^ = 1, we see that P and Q are numerically equal. 403. "VVe will finish with some examples. (1) Shew that 1 a, by c = a' + b^ + c'-dabG, c, a, b b, c, a Shew that a + h + c must be a factor of this determinant. (2) Shew that a, h, c, d = a* -b' + c* ' d*- 2a'c' + 2b'd' d, a, b, c - 4:a'bd + Wac - ic'bd + Ad'ac. c, d, a, h b, c, d, a . Shew that a + b + c + d must be a factor of this determinant. (3) Let there be a determinant of the order w + 1 in which all the constituents are equal to unity except those which form the diagonal series, and these are 1, 1 + a^, 1 +a^, ...1+a^: the value of this determinant is a, a ...a . 1 2 n For if any one of the quantities a^, a^,...a^ vanishes the determinant vanishes, because it then has' two rows identical ; thus the determinant is divisible by a^a^ ••• ^„« And the quotient of this division must be unity, as we see by considering the first element of the determinant. (4) Let there be a determinant of the order n in which alf the constituents are unity except those which form the diagonal series, and these are l+a^, 1 + a^,... 1 + a^: the value of this determinant is IFoT if any one of the quantities a,, a^, ... a^ vanishes the determinant reduces to a case of the preceding example ; and the term a^a^ • • • ^m ^^ found by considering the first element of the determinant. Quarterly Journal of Mathematics, Vol. I. page 364. ( 296 ) XXXI. TPJGONOMETRICAL FORMULA. 404. "We will give in the present Chapter a few propositions which bring the Theory of Equations into connexion with Trigo- nometry. 405. In Art. 272 of the Plane Trigonometry we have an expression for tan^i^ in powers of tan^, supposing n to be a positive integer. Suppose now that tan nO is given, and we require tan 6. Clear of fractions, and thus we obtain the following equation of the n^ degree for determining tan 6', . .L n{n-\)^ ,. n{n-\)(n-2){n-Z), ,. \ tan w^ U ^ — ^- tan^d + — ^^ '^ '^ tan'^ -. . .J = n tan^ ^^ ^ ' tan'^ \^ ^ ^(n-l)(.-2)(^-3)(n-4) ^^^,^_ [5 Now the value of tan nB is not changed if we put instead of any one of the following angles : n n n n Hence we infer that the roots of (1) are tan^, tan(^ + -), tan(^+ — ) tan{^+ ir] . \ nj' \ nj \ n J Let S denote the sum, and P the product, of the n quantities just expressed ; then, by the aid of Art. 45, we may deduce from (1) values for S and P : but for this purpose we shall have to consider separately two cases. TRIGONOMETEICAL FORMULA. 297 I. Suppose n even. Then (1) becomes tan 71^ h -'^-^^ tan*^ + + (- if tan"6>| = 7itan^- ''^''"^^^''"^^ an^^+... + 7i(-l)'^tan'-^^. In this case in order to put our equation in the standard form, that is, with unity for the coefficient of the highest power of the n unknown quantity, we must divide by (— 1)^ tan nO. Thus we obtain n-a S^ n(--l) _^ — = _7icotn^, P=ilir = (-lf ...(2). (- 1)"^ tan nO (- 1)^ II. Suppose n odd. Then (1) becomes tanw^ |l - ^^"""^^ tan^^+ ... +n{- l)"'^\an'-^^| = ^taii^- ^^^"|)<^-"^) tan-g + ... + (- l^-^Un-f?. In this case in order to put our equation in the standard form, n-l we must divide by (—1) /* . Thus we obtain n-l S = nt3inne, P = (-l)« tanw^ (3). 406. Again, take equation (1) of Art. 405, and multiply by cot"^; thus we have tan nO \coV6 ^r — ^ cot" ^6 + -^ ^-\-, — ^-^ ^ cot" *$-. . . V .-1/. n(n-l)(n-2) ^..-n = n cof"'^ ^ p^-^ cot* .^6 If ^ n{n-l)(n-2)(n-Z){n-i) ^^^...^ 15 298 TRIGONOMETRICAL FORMULAE. Divide by tan nO, and we obtain an equation in the standard form for determining cot^ when tanw^ is given. Hence, proceed- ing as in Art. 404, we have cot ^ + cot (^ + -j+COtf^ + — j+ + COt (O ^^^^^TTJ = n cot nd (4). 407. From equations (3) and (4) we see that, if n be any odd integer, the product of -ftan ^ + tan ( ^ + - j + + tan T^ + ''llL_ ^ jl into jcot ^+ cot (^ + -) + + cot (6 + ttH 408. Propositions like those of Arts. 405 and 406 may be easily deduced from other formulae of Trigonometry. We will By Art. 287 of the Plane Trigonometry we have, when n is even. . - w* . „. n^n'-2') . ,. ,^, cosn^ = l - tt; sm-^ + — ^-^ ^sm*^- (5). [2 ^ 14 Let cos nO = 0', then we may put for 6 any one of the follow- ing n values : IT Stt Stt n — \ 2n' 2n' In' "In Let 771 = ^ , and a; = cosec^^; then dividing (5) by sin"^, we get "-^ [2^ ^ [4 ^ The 771 values of x are cosec rr— , cosec ^r— , cosec — ;r — tt : In zn 2n hence, by Art. 45, the sum. of these m quantities = Tn • TRIGONOMETEICAL FORMULA. 299 Thus, if n be an even integer, cosec 7— + cosec ^^ + + cosec -—: — 7? = --, 2n In 271 2 409. We see by Art. 142 tliat any algebraical quantity has n different 7^*^ roots. If then we have found an expression for the n^^ root of an algebraical quantity that expression must be susceptible of n different values, unless some restriction has been introduced in our reasoning by virtue of which this multiplicity of values has been excluded. In other words, if two expressions are asserted to be equal, one of them must in general admit of as many values as the other. Various Trigonometrical formulae involving expansions were given by some of the older mathematicians, as for instance by Euler and Lagrange, which were not in accordance with the principle here stated, and which have been shewn to be inaccurate by Poinsot in a memoir, published in 1825, entitled Becherches stir Vanalyse des sections angulaires. A memoir by Abel also treats on the same subject : see his Oeuvres Completes^ Vol. I. page 91. We will illustrate the point by considering one case, and will follow Poinsot, though his method is not very rigorous : for a more elaborate investigation we refer to Abel. 410. Let it be required to investigate a series for (2cos^)'* in terms of cosines or sines of multiples of 0. The case in which tj- is a positive integer is treated in the Plane Trigonometry, Art. 280 ; we proceed to the more general proposition in which n is not restricted to be an integer, though it is assumed to be positive. First suppose cos 6 positive; and let p denote the arithmetical value of (2 cos Oy, Then we may put (2cos^)- = l> (1). Now 2 cos ^ = e*^ + e~'^, where t is used for J- 1 ; thus (2cos^)"=(e'^ + e-^«)'' 300 TRIGONOMETRICAL FORMULA. But e"'^ = cos nO + L sin. nOf e("-2)tfl = cos {n-2)e + L sin (» - 2) 0, and so on. Hence (2 cos $y = c + ts, where c stands for a certain series involving cosines, and s for a corresponding series involving sines. Again, l"p = (cos 2nfxTr + t sin 2n}nr) p, where ti denotes any integer. If then we were to equate this to c + 15 we should fall into the error against which we are warned in Art. 409. "We observe that (1) remains unchanged when 6 is increased by any even multiple of Stt. Let then c^ and s^ denote what c and s respectively be- come when in them 6 is changed to ^ + 2m7r. Then we may put c^ + f^^= (cos 2nfX7r + l sin 2nfnr) p (2). 411. If we suppose n an integer, we have c^ and s^ coin- ciding with c and s respectively. Then equating the real and imaginary parts of (2) we obtain c = p and s = 0. The former agrees with the result which is obtained and more closely discussed in the Plane Trigonometry^ Art. 280. 412. But we now suppose that n is not an integer. The first point to be established is that in equation (2) we must take m = fi. This point has sometimes been assumed; but Poinsot gives a reason for it in the following manner. Let us suppose to diminish without limit. Then it will be found that c^ = cos 2n7mr . TRIGONOMETRICAL FORMULAE. 801 The series within the brackets may be regarded as equal to 2", by the Binomial Theorem ; so that .c^ = 2" cos Inrmr = p cos ^nrmry s^ = 2" sin 2nmir = p sin 2/imir, Hence by (2) we get cos ^nrrnr + t sin 2n7mr = cos 2w/x,7r + t sin 2?i/i,7r ; from which we conclude that m = ft. 413. Thus, when n is not an integer, we have from (2) ^^^P-z^Z-:::z (3); '^ cos 2nmTr sin 2nm so that p may be expressed either in a series of cosines or in a series of sines. 414. If we put w = in the first of equations (3) we obtain p = c ; this coincides with Art. 280 of the Plane Trigonometry in form, and we see that it is true so long as cos 6 is positive. Again, putm = in the second of equations (3); then, since sin 2n7mr vanishes with m, it follows that s = so long as cos d is positive. 415. Let us now suppose that cos 9 is negative ; and let p de- note as before the arithmetical value of (2 cos 6)". Then we may put (2cos^)"-(-l)>. Also (-!)'• = cos (2/X + l)n7r + t sin (2/x,+ l)?27r. Hence instead of (2) we now obtain c^ + t5^ = {cos(2ft + 1) mT-¥ tsin(2/x+ \)mr] p (4). 302 TEIGONOMETRICAL FOEMULiE. 41 G. If we suppose n an integer, we have c,„ and s^ coinciding with c and s respectively. Then equating the real and imaginary parts of (4) we obtain c = — p and s == 0. The former agrees substantially with the result obtained in the Plane Trigonometry, Art. 280. 417.- But we now suppose that n is not an integer. We first shew, as in Art. 412, that m = fx; then, as in Art. 413, we have and p = — — -; — ^^^— TT (5). cos(2m> + l)mr ^ sin ('2m + I) nrr If we put m = in the first of equations (5) we obtain c p = . cos UTT This shews that when cos^ is negative the numerical value of (2 cos 9)" is not equal to c, but to c divided by cosqittj and this divisor is in general not equal to unity. Also, if we put m = in the second of equations (5) we obtain s •^ sm mr thus s is in general not zero. 418. Return to equation (3) of Art. 413; and let us deter- mine when p can be expressed by cosines only, and when by sines only. We may suppose that n is equal to some integer together with a proper fraction ; let this proper fraction in its lowest terms be r denoted by - ; then we shall not require to consider a value of m greater than s - 1. If p can be expressed by cosines only, it is obvious we must have sin 2nm7r =0 ; thus m = is one value of m, and if s be even, m = - ia another. TRIGONOMETRICAL FORMULiE. 803 If p can be expressed by sines only, it is obvious we must have cos 2nm7r = j therefore must be an odd multiple of ^ ; thus s 3* if r is odd and s sl multiple of 4 we may take wi = - or = — . 419. Again, take the equations (5) of Art. 417; and let us determine when p can be expressed by cosines only, and when by sines only. Use the same notation as before. If p can be expressed by cosines only, it is obvious we must have sin (2m +l)mjr=0 ; therefore sin (2m + 1) - tt = j thus, if *• is s odd, we can take 2m +l=s. If p can be expressed by sines only, it is obvious we must have * T COS (2m + 1) 7177= 0; therefore cos (2m + 1) -7r = ; thus, if r is odd, s and - an odd integer, we may take 2m + 1 = ^ or 2??i + 1 = -- . 420. Abel shews that the formulae here obtained for (2 cos Oy hold when n has any positive value ; and also when n has any negative value numerically less than unity, except for those values of 6 which make cos vanish. 421. We might investigate series for (2sin^)" in the same way as for (2 cos 6)" ; or we may deduce the results by putting TV -- 6 ioY 6 in the formulae already obtained. 422. In the Plane Trigonometry, Chapter xxiii., the ex- pression a;"" - 2cc" cos a + 1 is resolved into quadratic factors by a process which depends on De Moivre's Theorem, and which there- fore involves the use of the imaginary symbol J- 1. It has been 304 TRIGONOMETRICAL FORMULiE. lately shewn by mathematicians that the result can be obtained without the use of the -imaginary symbol ; we will here reproduce the process employed for this purpose by Professor Adams in Vol. XI. of the Transactions of the Cambridge Philosophical Society. 423. The relation between successive values of a"* +-- cor- responding to successive integral values of m is given by the formula when m = 1 this becomes An exactly similar relation holds between the successive values of 2 cos m^ ; thus 2 cos (m+\)e = {2 cos 6) (2 cos w^) - 2 cos (m - 1)^ ; when m = 1 this becomes 2 cos 26 = (2 cos 6) (2 cos 6) - 2. 'Now let v^y v^, v^^...%\ be a series of quantities the successive terms of which are connected by the same relation as that which we have just seen to hold for the successive values of a;'" + — and of 2 cos m.9 3 that is to say, let Also, as in those cases, let v^ = 2, but let v^ be any quantity whatever ; thus we have and so on. ^2 ~ '^1^1 ~ ^? ^a = ^i^B - "^1 = V^ — 3^1, TRIGONOMETRICAL FORMULA. 305 Hence it is obvious (1) that v^ is a definite integral function of t'j of n dimensions, and that the coefl&cient of Vj" is unity; (2) that a v^ = x + -, then i?„= ic" + — ; (3) that \iv^ = 2 cos 6, then v^^ = 2 cos nO, Hence 'y„ — 2 cos na will vanish when v^ is equal to any one of the following n quantities : 2 cos a, 2cos(a + /5), 2cos (a+ 2y8), ... 2cos(a + w- l,/5), where B is put for — . Therefore v -2 cos na ^ |v,- 2 cos aXiv, - 2 cos (a + /S)| |v, - 2 cos (a + 2/3)| ... ... lvi-2cos(a + w-l^)| . Now put X + - for tJ^; thus we obtain • £c" + — - 2 cos na = a; la; + -— 2 cos a I L + - - 2 cos (a + /5)| |ic + - - 2 cos (a + 2^)| . . . ... |ce + --2cos(a + 7i-ly8)|-. This gives the required resolution. Similarly if we put 2 cos for t\ we obtain 2 cos n9-2 cos wa = |2cos^-2cosa||2cos^-2cos(a+y8)j|2cos^-2cos(a + 2^)| ... ... |2cos(9-2cos(a + 7r^^)|. Hence we see that the two equations just found are particular cases of the general equation from which they have been derived; Vj being in the former case numerically not less than 2, and in the latter case numerically not greater than 2. Two special examples may be formed by taking first a; = 1 or ^ = 0, and then £c=- 1 or ^=7r. T. E. 20 30G TRIGONOMETRICAL FORMULA. 424. Various theorems may be obtained by the aid of the imaginary symbol, which can be verified if required by other methods. For example, we have by Plane Trigonometry, Art. 287, if n be an even integer, co8nO = l--^ sm'' 6 + — ^-, sin^^ _.-(n--2-)K-4-)^.^.^_ (^^^ and if 71 be an odd integer, . ^ . ^ n{n'-l) . ,. n(n'-l)(n'-3') . ,. ,^. sm ne = nsmO ^-rr, sm' + -^ r^ ' sin» 6-... 12). Now substitute x- ( ^ ) for sin 0, where t denotes ^ — 1 ; and for brevity put p for x . z Then from (1) we deduce, when n is even^ 1 (." + ^.) = 1 + 2?J2 y + —t]4pP* n>--2-')K-4-) ^ 2«[6 ^ ^ ''' and from (2) we deduce, -wlieii n is odd, 2V «V 2^^ 2'f3 ^ 2^|5 ^ ^' Thus we obtain the algebraical identities (3) and (4). These may be verified by the aid of Art. 244. For suppose /{xj^x^-px-l, so that the roots of/(£c) = are of the form 1 ' 1 z and — , and z = p. Then z z xf'ix) ^ x(2x~p) ^ ^ x'^l ^ 1 , A , ^\ A 1 ;^y^ f{x) af—px—l x^-px-l \ xy \ x^ x) =-04){(>-r*('4n*('-ir&'-}- TKIGONOMETRICAL FORMULAE. 307 First suppose n even; then by Art. 244 we have ;s" + ( ) , that is 2;" + — , equal to the coefficient of — in the expression just given. The terms involving odd powers of p will not furnish any part of the coefficient, since n is even. Now the coefficient of — in ( 1 + -^ ) ( 1 A is 1 + 1, that is 2. so \ SC J \ SC / The coefficient of— -in(l + -i)(l 5) '^is X \ X J \ X J X «'-'• ffi(i'')*(i-')i}''""|S- The coefficient of— infl+-;2)(l — j) "^is X \ X / \ X / X .("-ji„)C_ii'..)("-.,)(--i5..)).. "■"0(l-')iG*')S*^)H5-')6-')5S*')}^ And so on. Thus we obtain (3). Similarly by supposing n odd we obtain (4). 20—2 ( 308 ) EXAMPLES. I. 1. Find the quotient and the remainder when x' + Ix" 4- 3a;' +\lx^ + lOx-U is divided by cc - 4. 2. Expand (a + 5x)" in powers of x, and then obtain the^rs^ ^ derived function of {a + 6x)". "^ 3. Shew that the equation a;' + 3£c^ + fl3- 6 = has one root and oiily one between 1 and 2. II. 1. Find a root of the equation x^ = + J-l. 2. Find a root of the equation x^ = - J- 1. III. 1. Form the equation whose roots are 1, 1, 1, - 1, - 2. 2. Form the equation whose roots are 1 =±= ^ -^ ^^^ 2 =t^/-3. 3. Form the equation of the eighth degree one of whose roots 4. Solve the following six equations in each of which one root is given : (1) x^-x'+3x+5 = 0; 1-2^^. (2) x' + 4:x'' + 6x' + ^x + 5 = 0; J^. (3) x'' + x^-25x' + Ux+66 = 0- 3 + J^. (4) x*+2x''-4x'-ix+4: = 0; J2. (5) x''-2x^-5x'-6x+2 = 0; 2+^3. (6) x'-a^-8x'+2x'+2\x'-9x-H = 0; J2 + J^. EXAMPLES. 300 5. Solve the equation x^ - x^ -\- Sx^ - 9x - 15 =^ 0, one root being ^^3, and another 1 — 2^-1. 6. The equation x^—4:X^ + x + c^0 has one root =3; find c - (^ and the other roots. 7. Find the sum of the reciprocals of the roots, the sum of the squares of the roots, and the sum of the squares of the reci- procals of the roots of x^ - 6x^ + iOx^ +60a;^-aj-l=0. 8. The equation x* - 21ic' + UQx^ - 54:6x + 580 = 0, has roots of the form a, /?, a + ^ + (a — ;8) J— 1 j solve the equation. r^ 9. Find the sum of the cubes of the roots of a given equation. ^ 10. Form the equation the roots of which a, jS, y, 8, are 1 (l + V3 ± v/2V3) , and i (l - ^3 i 7^1^) ; 2 j^ 02 2 j^ 2 and thence shew that ^ + — + . . . = 0. a/j ay 11. If a, 6, c, ... are the roots of an equation, find the value of 12. Assuming that the arithmetic mean of any number of positive quantities is greater than their geometric mean, shew that if p^^ - 2p^ is less than np^^ the ■ equation has impossible roots. 13. If a, 6, c, ... are the roots of an equation in its simplest form, shew that 14. If a, 6, c,...are the roots of an equation in its simplest form, shew that p; - 1p,p., + 2/., - a%'' + aV + h\^ + . .. 310 EXAMPLES. IV. 1. Transform each of the following three equations into another the roots of which are formed by adding to the roots of the original equation the number assigned : (1) x'-3x*-x'+4: = 0', 1. (2) x' + x + l = 0; 3. (3) x'+ix^-x' + U^O; -3. 2. Transform each of the following four equations into another wanting the second term ; (1) x'-3x'+ix-4: = 0. (2) x^-6x'+l2x + l9 = 0. (3) a;'- 8a;' +5 = 0. (4) x'+5x'+3x^+ x'+x-l = 0. 3. Transform each of the following four equations into two others each wanting the third term : (1) x^^6x'+Sx-l=0. (2) a;''- 6^'+ 9^ -10 = 0. (3) x^-'8x''+l8af-l5x + U = 0. (4) x'-lSx'-60x'+x-2 = 0. 4. Transform the equation x^+ 2£c^+ - x+ -^ = into another with integral coefficients, and unity for the coefficient of the first term. 5. Kemove the second term and solve the equation a;=^-18a;'+ 157a;- 510 = 0. 6. Transform each of the following two equations into another whose roots are the squares of the differences of its roots; and discuss the nature of the roots : (1) a;^+7a;-l-0. (2) a;^-6x+6-0. 7. Transform a;*— 12a;^+ 12a; — 3 = into an equation whose roots shall be the reciprocals of those of the given equation ; and then diminish the roots of the transformed equation by unity. 8. Shew that the equation a;'* + a;^-8a;- 15 = has two real roots of contrary signs, and that it cannot have more real roots ; and that they lie between - 2 and 3. EXAMPLES. 311 9. The roots of the equation cc^ ■k-'poi? + ga: + r = are denoted by a, 6, c ; transform the equation into others which have the roots assigned in the following fourteen cases : (1) a\ h\ c\ (2) h + c, c + a, a + h, /3\ J_ _1_ J_. U\ ^ L L ^' h + c* c + a^ a+h' ^^ be* ca^ ab' (5) b'c\ c'a', a%\ (6) ^{ha), sj{kb), J{kc). (7) -(b^c-a), -{c+a-b), -{a + b-c). (8) b + c + ka, c + a + kb, a + b + kc. (9) _«^_ ^ ^ . ^ ^ b + G-a^ c + a-b* a + b-c (10) bc+-, ca + l, ab + -. (11) b'+c', c'+a\ a'+b\ a b . b c c a a b ,^ . ^^-^^ c^b' a'^c' b'^a' ^^^^ 6V ' cV ' a'b' c c a (14) b-c, c — b, c-a, a-c, a -by b-a. 10. The roots of the equation x^+qx + r^O are denoted by a, by c; transform the equation into others which have the roots assigned in the following two cases : 0) G~)'. c-f.y. (.4j)' (2) ba + aCy cb + ba, ac + cb. 11. If «., 5, c denote the roots of a;^ — 6x"+ llx-G =^0, form the equation whose roots are 1 1 1 52 + c^' c'^a'' a' + b'' 12. If ay b, c denote the roots of x"-2x''+ 2=0, form the equation whose roots are b^ + c^ c^ + a^ a^ + b^ ~~^ ' ~~W~ ' d" ' 1 3. Shew that the third term of the equation x^ +px^ + qx+r=0 cannot be removed if p* be less than 3q. 812 EXAMPLES. 14. Shew that the second and fourth terms of the equation can be removed by the same transformation if ^Pq = p^{^P2 — p^). 15. Solve the following two equations : (1) x'+4.x^+1x^+Qx-lO = 0. (2) x'+4.x^ + 2>x'-2x-Q = 0, 16. Shew that the equation £c^ + 4aj^+ 6a;+ 3 = does not admit of the second and third terms being remoA^ed by the same transformation, but that it does if multiplied by x. 17. Shew that it is possible to remove the second and third terms of an equation of the 'nP^ degree if n X (sum of squares of roots) = square of sum of roots. Y. 1. Shew that the equation x^-4:X^+3 = has at least two imaginary roots. ^. '^ 2. Shew that the equation x"^- 2x*+x^-l = has at least four imaginary roots. 3. What may be inferred respecting the roots of the follow- ing two equations 1 (1) x''-5x'+x'-x~l=0. (2) x^"-x"' + x'' + x + l = 0. VI. 1. Solve the following twenty equations, each of which has equal roots : (1) aj'-7.oc=+16ic-12 = 0. (2) x^-3x'-9x + 27 = 0. (3) a;^-a;^-8a; + 12»0. (4) x'-5x'-Sx + 4.S=^0. (7) a;'+8;:c^+ 20a: +16-0. (8) x'-lx + ~^0. ^ ' 2 lb EXAMPLES. 313 (9) a;^-llaj'+18a;-8 = 0. (10) x^-2x'-x'-4:x + 12 = 0. (11) X* - 7cc'+ Ux'+ 3aj - 18 = 0. (12) x'-4:x^- Qx' + 3Qx-27 = 0. (13) a;*+13:«^+33a;^ + 31aj+10 = 0. (14) 2x^ -Ux^+ 19^'- 6ic + 9 = 0. (15) a;^+16:c''+79aj^+126aj + 98 = 0. (16) 8^'+4ic'-18a;^+ll:r-2 = 0. (17) a;'-a;*-2x'+2;r'+cc-l = 0. (18) x'-2x''-6x^+4:x' + Ux + Q^0. ^ (19) aj'- 13^*+ 67a;'- 171a;'+ 216^- 108 = 0. (20) x'- 3x'+ 6a;'- 3a;' - 3a; + 2 = 0. 2. Find the condition tliat a;"— j9a;' + r = may have equal roots. 3. Shew that x* + qx^ + s = cannot have three equal roots. 4. If a;"+jOja;"~^+ ... +p,, = have two roots equal to a, shew that p^x"'^ + 2p^x"~^ + . .. + np^ = has a root equal to a. 5. If a;'4-g'a;'+ra;' + ^ = has two equal roots, prove that one of them will be a root of the quadratic ^ 5r^^3r 15 ~ ^• YII. 1. Find limits to the positive and negative roots of x'- 5x'+ x'+ 12a;'- 12a;' + 1 = 0. 2. Write x* - Sa;' + 12a;'+ 16a; - 39 = so as to shew that 6 is a superior limit of the positive roots. 3. Shew that the real roots of the following six equations lie between the limits respectively assigned : 314 EXAMPLES. (1) x*-x^+4:x'-3x + l = 0; i and 1. (2) a;" + a;'-10ic'-a;+15 = 0j -4 and 3. (3) x'+ 5x'+ x^-lQx'- 20x-l6 = ; - 5 and 3, by Art. 92. (4) («'- 26)(x'+5x + l) + Q0x = 0; -5 and 3. (5) (x'- 4:x - 2y- 43 = j - 2 and 6. (6) x'+xU x'- 25x-36 = 0; - 3 and 3, by Art. 92. 4. Find by Newton's method limits to the roots of the following five equations : (1) x'^-x^-dx'+Sx^Q^O. (2) x^-5x' + 6x-l = 0. (3) x*-x^+4.x' + x-4: = 0, (4) x''-5x^+naf-20=^0. (5) x'- 2x'- 3x'- 15x-3--= 0. 5. Prove that x^ + 5x*-20x^ -l9x-2 = has one root be- tween 2 and 3, but none greater than 3, and one root between — 5 and - 4, but none less than - 5. 6. Apply the method of Art. 102 to find the number and situation of the real roots of the following six equations : (1) a;' -12a; + 17 = 0. (2) a;' - 325C + 20 ::= 0. (3) x^-3x + 3 = 0. (4) 4xV9a;'-12£c + 2 = 0. (5) x'-a'x'+c'^O. (6) x'"'-px'+r=0. 7. Shew that the equation Sx* + 8x^ — 6x^ — 24a; + r = will have four real roots if r is less than —8 and greater than —13, and two real roots if r is greater than -8 and less than 19, and no real root if r is greater than 19. YIII. 1. Obtain the commensurable roots of the following twelve equations : (1) aj''- 106a; -420 = 0. (2) a;'- 9a;'4-22a;- 24 = 0. (3) a;"- 2a;' -25a; + 50 = 0. (4) 2a;'- 3a;' + 2a;- 3 = 0. EXAMPLES. 315 (5) 3x'-2x'-Qx + 4: = 0. (6) 3x' - 2Qx' + Six - 12 = 0. (7) a;*-2x'+8a;-16 = 0. (8) x''-af-Ux' + 16x-iS = 0. (9) x*-x^-x'+19x-4:2 = 0. (10) x^+8x^-7x'-i9x + 5Q = 0. (11) a;'-3£c*-9x'+2l£c''-10a; + 24 = 0. (12) a;«- 7^'+ llo;^- 7x'+ Ux'- 28a; + 40 = 0. 2. The coefficients of the equation f(x) = are all integers : shew that if/(0) and /(I) are both odd numbers the equation can have no integral roots. IX. 1. Solve the following four equations each of which has two roots of the form a, - a : (1) x^-2x^-2x'+8x-S = 0. (2) a;*+ 3ir'~-7£c'- 27a; -18-0. (3) x'+3x^ + 2x'+dx-3^0. (4) a;' + a'- 11a;-"- 9ic + 18 = 0. 2. Solve the following four equations in each of which the roots are in Arithmetical Progression : (1) x^-6x'+nx-6=^0. (2) a;^ - 9a;' + 23a; -15 = 0. (3) a;^-8a;^+14a;^+8a;-15 = 0. (4) a;*+4a;'-4a;'- 16a; = 0. 3. Solve the following six equations in Avhich certain con- ditions relative to the roots are given : (1) 3a;' - 2a;'- 27a; + 18 = ; product of two roots is 2. (2) X* - 3af — 6a; — 2 = ; product of two roots is - 1. (3) x^- 4a;' + 5a;'- 16a; + 4 = ; product of two roots is 1. (4) 2a;*-5a;' + lla;'-lla;+ 6 = ; product of two roots is 1. (5) X* - 45a;'— 40a; + 84 = ; difference of two roots is 3. (6) a;' - 7a;'' 4- 15a;' - 15a;' + 14a; - 8 = j one root double another. 4. Solve the following six equations in which the roots are of the forms respectively assigned : (1) a;'-10a;'+27a;-18 = 0; a, Sa, 6a. (2) a;*-10a;'+35a;'-50x+24 = 0; a + 1, a-1, 6 + 1,6-1. 316 EXAMPLES. (3) 6x*-4:Zx'+l07x'-10Sx+36 = 0: a,h,^, -. a (i) x' + 8a;* + 5a;"- 50a;'- 36a; + 72 = ; a, 2a, b, 2b, a + b. (5) x'-4:x'+lOx*-16x^+Ux'-l6x + o6 = 0; adt: J2^.Jb, ^Jc. (6) a;'-12a;*-2a;"+37a;'+10^-10 = 0; 1^ Ja, b^J2, ^^c. 5. Solve the following two pairs of equations, eacli pair having a root in common : (1) a;" -3a;' -16a; -12 = 0; a;"-7a;'+5a;+ 13 = 0. ■ (2) a;"-3a;'+lla;-9-0; a;"- 5a;'+lla;-7 - 0. 6. Solve a;' -7a;' + 36 = 0, and a;'-3a;' - 10a; + 24 = 0, the former of which has a root equal to three times one of the roots of the latter. 7. Solve the following two equations which have two roots in common : X*- 2x'-7x' + 26x-20^0; a;* + 4a;'- 2a;'- 12a; + 8 = 0. 8. Find in terms of m and a the roots of the equation a;* +pax^ + (m' + m) a^af + qa^x + a* = 0, which are in Geometrical Progression ; and determine p and q in terms of m and a. 1. Solve the following ten reciprocal equations : (1) a;*-2a;^+ 3a;'- 2a;+ 1 = 0. (2) a;* + 4a;' - 5a;' + 4a; + 1 = 0. (3) 2a;*- 5a;^+ 6a;'- 5a;+ 2 = 0. (4) a;*+4a;»-10a;'+ 4a; + l =0. (5) a;'-2a;*-19a;"-19a;'-2a;+l=0. (6) a;'-4a;*+a;'+a;'-4a;+l = 0. (7) 6a;'*-lla;*- 33a;'+ 33a;' + lla;- 6 = 0. (8) 2x' - 5x' + 4a;*- 4a;'+ 5a; - 2 = 0. (9) 8a;'-16a;*-25a;*-16a;'+8=0. (10) l + x'^a{l+xy. EXAMPLES. 317 2. Obtain roots of the following four equations, and depress the equations : (1) x''-2x'+x'+x^-2a^+l = 0. (2) jc^+ 2x^- 8x'- Ix"- 7x^- 8£cV 2.^• +1 = 0. (3) x^+2x'+^x^+2o(^-2x^-2>x^-2x^\=^0. (4) x''-\ = 0. 3. Exhibit the roots of x^+px^+ 1 = in the form a, 6, - , J-. a 4. If a, 6, c, . . . denote the roots of the recurring equation a;* +jpa5'*"^ + qx""'^ + . . . + g'a;^ + joa; + 1 = 0, 5. In the recurring equation af" — ^x'""^ + ... = 0, if the terms are alternately positive and negative and p less than 2n, the roots cannot be all real. XI. 1. Solve the following three equations : (1) aj«-l = 0. (2) a;«-l = 0. (3) aj'+l=0. 2. Shew that the factors of a^+h^+c^- 3ahc are of the form a + bi + ci^f where i^—l = 0. 3. Shew that the factors of a'{a'-4:hd - c')-b\b'- iac^d') + c" {c'- ihd-a') - d\d'- iac - h') are of the form a-\-hk + c¥+ dk^, where ^''- 1 = 0. XIT. 1. Solve the following eight equations : (1) a;'-3aj-2 = 0. (2) a;'- 9a; -28 = 0. (3) a:'-a;+6 = 0. (4) a;' + 3a; = |. (5) 3a;'-6a;''-2 = 0. (6) a;'- 15a;'- 33a; + 847 = 0. (7) a;^+6aa;''=36a^ (8) a;'-3 (a'+ 6')a; = 2a(a«-35'). 318 EXAMPLES. 2. Determine the relation between q and r necessary in order that the equation a;^ + g'a; + r = may be put into the form and hence solve the equation 8x^— 36aj + 27 = 0. 3. If the roots of the equation ^ + 'pQ(^ + g'o; + r = are in Geometrical Progression, r/>^= (f. Hence solve the equation a;'-a;'+ 2a;- 8 = 0. 4. If the roots of the equation a?" + g'a; + r = are diminished by 7i, shew that the transformed equation will have its roots in Geometrical Progression if h be such that 27rA^- ^(fK^- q^= 0. 5. If the roots of the equation x^-\- ?)px^+ 2>qx + r^0 are in Harmonical Progression, 2q^=r{^pq — r). 6. If the roots of the equation x^ + 2>p3(? + 2fqx + r = are in Harmonical Progression, the equation rx^ + 2q^x + qr = contains the greatest and least of them. 7. The impossible roots oix^ + qx-\-r = being put under the form a ± ^ ^^1, shew that ^' = 3a' + q. 8. If r, a+ JP, a — J/S, are the three roots of the equation x^ +p^oi:^ +p^ +JP3 = ^) of which r is real, and if x^ + m^a:;' + m^x = is the equation resulting from the diminution of all the roots by r, shew that a = — -^^ +T and ^ = — - (m^ + 3p^ -Pi)- 9. Reduce the equation x^+px^-¥qx-\-r = to the form y^—2>y + mF= 0, by assuming x = ai/ + b; and solve this equation by assuming y = z + -. Hence shew that if the original equation has equal roots, 4 (/- 3qf= (2/- 9pq + 27ry. 10. If the roots of the equation x^4-px^+qx + r = are in Harmonical Progression, so also are the roots of the equation (pq - r) 2/'- {p^- 2pq + 3r) y^+ (pq -3r)y-r= 0. EXAMPLES. 319 XIII. 1. Solve the following four equations : (1) x*+4:X^+3x'-Ux-S4:=^0. (2) x'- 6x'-8x-3 = 0. (3) a;*-12ic"+49^'-78a;+40 = 0. (4) x'- 2ax'+ {a'- W) x'+ 2ah'x - a'b'= 0. (Art. 192.) 2. If r^-p^s = the equation x*+px^+qx^+rx + s = may be solved as a quadratic. 3. If s and p are positive, and 27p* less than 2565, the roots of the equation x*+px^+ s-0 are all imaginary. 4. Assuming that the equation x^+ qx^+rx + s = has roots of the form a^/Sj-l, shew that the values of a and jS may be found by the equations 64a« + 32^a*+(4^^-16s)a^-r*=0, l3'=a'+^^ + ~, 2 4a XIV. 1. Apply Sturm's Theorem to determine the situation of tlie real roots of the following five equations in which the values of some of Sturm's functions are assigned : (1) a;^-4aj'-3a; + 23 = 0; /3{aj) = -491a; + 1371, /4(x) = -. (2) a^-4:X^+x'+(jx + 2 = 0; f^{x) =6x'-10x-7, f^{x) = x-\, (3) x'+x^ + x-l = 0; /^(a:) = 3:c^-12a; + 17, Art. 199. (4) x'-2x^+x^-8x + 6 = 0; f^{x) = 16af-2dx + 9, (5) x^+ bx"- 20a;'- 19x-2 = ; f^{x) = 20x'+ 60x'+ 36x - 9, f^{x)=:^96x'+lS7x+67, /, (a;) = 43651a; + 54571, /,{x) = +, 2. Apply Sturm's Theorem to shew that each of the follow- ing two equations has only one real root; and determine its situation : (1) a:'+6a;'+10ic-l=0. (2) a'- 6a;' + 8a; + 40 = 0. 820 EXAMPLES. 3. Determine the situation of the positive roots of the equation x^- 2x^+ ^x^-6x-l = 0, having given f^{x)=:^x{x-iy+\^x-\-Q. 4. Apply Sturm's Theorem to the following four equations : (1) x^+x^-2x-l = 0. (2) a;'- 4x'- 4a; +20 = 0, (3) tc*+2a;^-4aj+10 = 0. (4) aj"-a; + l«0. XY. 1. Shew that the equation ««_ 3a;^_ 24a;'+ 95a;'- 46a; - 101 = has all its real roots between —10 and 10, that it has one real root between —10 and —1, one between —1 and 0, no root between and 1, and one at least between 1 and 10. 2. Apply Fourier's Theorem to the equation a;*+3a;'+7a;'+10a; + l = 0. XYI. 1. Approximate by Lagrange's method to the positive root of the equation 3a;'' - 4a; - 1 = 0. 2. Approximate by Lagrange's method to the root of the equation x*-\- x^— 2a;'- 3a;- 3 = 0, which lies between 1 and 2. —-— XYli. 1. Apply Newton's method to calculate the root which is situated between the assigned limits in the following five equations : (1) a;^- 4a; -12 = 0; root between 2 and 3. (2) a;'- 4a;'- 7a; + 24 = ; root between 2 and 3. (3) a;'- 24a; + 44 = ; root between 3-2 and 3-3. (4) x^— 15a; - 5 = ; root between 4 and 4'1. (5) x*- 8a;V 1 2a;' + 8a; - 4 = ; root betweer. and 1. EXAMPLES. 321 2. Apply Newton's method to calculate a root of the follow- ing two equations ; (1) a;^+3a;-5 = 0. (2) a;'- 3x'- 3a: + 20 = 0. XVIII. 1. Apply Horner's method to calculate the root which is situated between the assigned limits in the following three equations : (1) x^+ 10af+ 6a; - 120 = ; root between 2 and 3. (2) x*- 2x^+ 21a;- 23 = ; root between 1 and 2. (3) X*- 5x^+ 3x'+ 35a; - 70 = ; root between 2 and 3. 2. Solve the equation x^— 17 = by Horner's method. 3. Calculate the real roots of the following four equations by Horner's method : (1) a;'4-a;-3 = 0. (2) a;' + 2a; - 20 = 0. (3) 3a;'+5a;-4:0 = O. (4) a;'+ 10a;^+ 8a;- 120 - 0. XIX. 1. Find the value of the following seven symmetrical func- tions of the roots a, 6, c of the equation a;^+^a;''+ qx + r = 0: (1) {a + h + ah){b + c + hc){c + a + ca). (2) {a + b-2c){b + c-2a)(a + c- 26). (3) :^{a + by{a + c). (4) ^{a + b-2c){b + c-2a). (7) (b-cY(c-ay{a-by. I 2. If a, bf c, d are the roots of the equation x*+px^+ qx^+rx + 8 = 0, find the value of 2 (a + b) {c + d). T. E. 21 322 EXAMPLES. 3. In the equation x^+p^x"*'^ + .,. +p^_^x+p^ = 0, suppos- ing the roots to he a, b, c, ... I find (1) 2a'b. (2) ^{a + h){a + c)...{a + r). 4. Form the equation the roots of which are the squares of the sums of every three roots of the equation a;* + px^ + rx + s = 0. Also form the equation the roots of which are the sums of the squares of every three roots of the same equation. 5. If the equation x"* + p^x"~^ +p^x''~^+ p^x"*'^^ ... +p^ = is transformed into another of which the roots are the sum of every pair of roots of the original equation, find the first three coefficients of the transformed equation. XX. 1. Transform the following three equations into others whose roots are the squares of the differences of their roots : (1) a;'-4a;+2 = 0. (2) x'+4:X+3 = 0. (3) a;^+l = 0. 2. Eliminate x from the equations ax'+hx + c = 0, ax^+h'x + c'=0. XXL 1. Find the sum of the assigned powers of the roots of the following five equations : (1) a;*-aj'-19a;'+ 49a;- 30 = 0; the cubes. (2) a;'- 3x^- 5x + l =0 ; the fourth powers. (3) x'- 2x^- 22x^- 2Sx'+ 72a; + 144 = ; the cubes. (4) 03*+ 2a; + 1 = 0; the inverse squares. (5) x^- a; - 1 = ; the sixth powers. EXAMPLES. 323 2. If a, 6, c,... are the roots of a?"- 1 =0, find %arh^. 3. If the sum of the r^ powers of the roots of the equation a;"+ a; + 1 = be expressed by S^, and the sum of the r**" powers of their reciprocals by 2^, prove that ^„_,->S, = l, and 2„_,-S,. = ^-2(-l)-. 4. In the equation a;"- a;^+ 1 = 0, find 5a""^, :§»""*, and 5a" ; supposing w greater than 3. 5. Find the sums of the r*^ and (2w)* powers of the roots of the equation x-''—px^+ 2'= 0, supposing n greater than r. XXIL 1. Solve the equations {y-l)x + y = J* 2. Solve the equations (^j-l)x'+y{y + l)x'+{3y'+ij-2)x + 27j = 0) (y-l)x'+y{y + l)x + 3y'-l=0 ' )' 3. Shew that the following equations have no solution : yx^-{y^-3y-l)x + y = x'-y'+3 = }■ XXIII. 1. Find the first term of each value of y when expanded in descending powers of x from the equation y*x - y V + Zyx^- y^x + hj-2x = 0. 2. Find the first term of each value of y when expanded in ascending powers of x from the equation x'^+x''+ x''y ~ a^y + 2a;y- xY+y' - Zxf+ x'Y' = 0. 324 MISCELLANEOUS EXAJMPLES. MISCELLANEOUS EXAMPLES. 1. If there be n quantities a, 5, c..., and if n functions of them be taken of the form (a; - 5) (aj - c) . . . shew that the sum of these functions is unity. 2. Remove the term which involves the cube of the unknown quantity from the equation a5'+5a;V200x'-lla; + 6 = 0, 3. Shew how to transform an equation which has both changes and continuations of signs (1) into one which has only continuations of sign, (2) into olae which has only changes of sign, 4. If j9 and q are positive, the equation a;^"-^a;^''+g = has four different real roots or none according as (—J is greater or less than ( j ; and it has two pairs of equal roots if (5y=(.4f- 5. If — ^„_,«"~', — i?„_^a;"~^ -j^„_,a^"~*> ••• are the negative terms of an equation of the rlf^ degree, then the greatest root of the equation will be less than the sum of the two greatest of the quantities (;?,_j^ {Vn-^\ (P«-,)'> - 6. If li be the last term of an equation of the ri^ degree whose roots are in geometrical progression, shew that - A;'* is a root, if n be odd. Shew that, in a similar manner, one root of an equation of an odd degree whose roots are either in arithmetical or harmonical progression may be found. 7. Find the greatest common measure of a' - a' - 3a: - 1 and «*- 6a;* + 7a;' + 7a;' -6a; -3. Solve the equation a;**- ^x*->f 7a;' + 7a;'- 6a; - 3 = 0. MISCELLANEOUS EXAMPLES. 325 8. Diminish by h the roots of the equation X* + q:*? + raj + s = ; give such a value to h that the roots of the transformed equation may be of the form a, — , &j x ' ^^^ shew how this equation may be solved. Example, x^— 2a3*+ 16a; + 1 = 0. 9. Shew by the process for extracting the square root of an algebraical expression that the equation a;*+joa;^+5'x^+ra; + s=-0 can be immediately reduced to quadratics if j^^s — ^.qs + r^ = 0, or if 'p^- 4:pq + 8r = 0. 3 10. Prove that the equation ic*+ ^ qx^+ rx + s = cannot have all its roots real if q^-\- r^ is positive. 11. If y(cc) be a rational integral function of x^ either y (a?) = or fix) - has certainly a real root. 12. Shew how to find the value of the semi-symmetrical function o^h + h^c + c^a of the roots of a cubic equation. 1 3. Let a, 5, c, . . . ^ denote the roots of the equation <^ {x) = 0, which is of the vl^ degree and in its simplest form, and suppose these roots all unequal : shew that the expression a' V c' le + T77TT + -rn-\ + ••• +T77TV ^'(«) <^'W ^{<^) '" *<^'W is equal to unity if r = n — 1, and is zero if r is zero or any positive integer less than n—1. Shew also, that if r = - 1 the expression = ^^r—^ — , . 14. If ' "^UV^H"^- ^' ^'2^ V2 • III. 7. -1; 36; 121. 8 a=5; p=2. 9. _jt>^3+ 32?j??2- 3^3. 10. x"- 2x^- 2^ + 1 = 0; then see Art. 48. 11. (p^'- 2p^) ^'"-~ ^f '^"-^ - n. 13. In the identity of Art. 45 substitute successively J -I and —J -I for x. IV. 5. The roots are 6, 6 ±77-1. 7. ^*-2y' + -^ = 0. 8. See Arts. 22 and 50. 15. Apply Example 14. YI. 1. (15) - 7 is a root. (I6) ^ is a root. (1?) The root 1 occurs three times. (18) The root -1 occurs three times. (19) 2 and 3 are roots. (20) The roots 1 and -1 are repeated. 3. Denote the root which is repeated by a, and the other by h ; then the left-hand member of the proposed equation must be identical with {x-aY{x-b) ; then we may equate coefficients. VII. 7. The roots of /' (cc) -= are - 2, -1, 1; use Art. 102. VIII. 1. (4) |. (6) |. IX. 2. (3) -1, 1, 3, 5. (4) -4,-2,0,2. 3. (1)3, |. (2)1=^^2. (3)2^^3. (4)1(3^^^7). (5) -2, 1. (6) 1, 2. 4. (l)a=]. (2)a = 3,6 = 2. (3)a = 2,6=3. {4>) a = l, h = -3, (5) a = l, 6 = -3, c = -2. (6) a = 3, 6 = -l, c = 5. ' 5. (1) -1. (2) 1. 6. The roots are 6 and 2. 7. The common roots are given by a;^+ 2a; - 4 = 0. ANSWERS. 327 8. Denote the roots by -^ , -^ , a/?, a^' ; equate tlieir product to a*, and the sum of the products of every pair to (m^+ m) a*. It may be shewn that p must be equal to q. XII. 1.(1)2. (2)4. (3) -2. (4)23-2-^. (5)1(^2^2 4-2^1 ^a (6) The root 1 1 occurs twice. (7) -^ 1 . (8) 2a. XIII. 1. (1)3,-2. (2) The root -1 is repeated. (3) Diminish the roots by 3, then the biquadratic can be solved. XIV. 1. (1) A root between 2 and 3, another between 3 and 4, and two impossible roots. (2) Two roots between and -1, and two between 2 and 3. XVIII. 1. (1) 2-833066480704857... (2) M57451508098991... (3) 2-64575131106459059 ... 2. 2-57128159065823535... 3. (1) 1-2134116627622296... (2)2-4695456501065939... (3) 2-13781194169747... (4) 2-76834546088879... XIX. 1. (1) (r-qy+p{r-q) + r. (2) 2p^-9pq + 27r, q^ + rp (3) -2p^ + pq-^r. (4) 9^-3/. (5) r —pq (6) p-'-^-^^^- (7) l{3g-p'){5pr-f)-l{P3-^ry. 2. 2q. 3. (1) ^Ps-PiP^- (2) If we denote the equation by /{x) = 0, the proposed expression following the symbol 2 becomes -^ — =^ , Hence the required sum is I {s.-. -pA-, +^A-3- ••• + (- i)>A,} • (3)^^. + n^_2„. (4) ^,_ fa'-y>- ,. 328 ANSWERS. 5. Let the transformed equation be a;'"+ 5',a;'"~' + q^^~^-\- q^x""^+ . . . = ; " then m = '—^^^ . Find the sums of the powers of the roots of 2 the transformed equation, and then the Coefficients by Art. 244. We shaUget q,= {n-\)p,', q,= ~ ^^^ " ^^ p'+ {n -2)p^; XXII. 1. The solutions are given by y^-2i/ = and {i/-l)x + y = 0. /^y 2. The solutions are given by y»-l = and (7/-l)a; + 2y = 0. /— 2 XXIII. 1. y = x^ ...) y = ^j2>x+ ...J 2/ = 3-^+--- 2. Six values of the form y = x^{u-h U), where u is to be determined from l-u^-u* + u^ = ; three values of the form y=:x~^ (u + U), where u is to be determined from 1 - 3u^= ; and _13 four values of the form y=x * {u + U)y where u is to be deter- mined from 3-u*=0. MISCELLAKEOUS EXAMPLES. 1. Call the sum {x) ; then shew that