DEPARTMENTS OF MECHANICAL & ELECTRICAL ENGINEERING OCT 1930 University of California THEOEY OF EQUATIONS. AN ELEMENTAEY TREATISE ox THE THEORY OF EQUATIONS, AVITH A COLLECTION OF EXAMPLES. BY I. TODHUXTER, M.A. F.R.S., -'r- HOXOEARY FELLOW OF ST JOHN S COLLEGE, CAMBELDGE FIFTH EDITION MACMILLAN AND CO. 1882. [The Right of Translation is reserved.] TL ' ' •• * library ©ambritge : PRINTED BY C. J. CLAY, M.A. & SOK, 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, 808... 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 vievv' 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 ehmination 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 Coiirs cVAlgehre 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 Encyclop(c- 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 Lady^s and Gentleman s Diary for the 3'ears 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 18C0. I. TODHUNTER St John's College, March, 1880. CONTENTS. I. n. in. IV. _— V. YI. ^t:i. vni. IX. X. XI. Xll. xin. XIV. XV. XVI. XVII. xvm. XIX. XX. XXI. xxn. xxin. XXIV. XXV. XXVI. xxvn. xxvm. XXIX. PAGE 1 15 22 31 40 Introduction . . . On the Existence of a Eoot Properties of Equations . . . . . Transformation of Equations ...... Descartes's Eule of Signs On Equal Eoots .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 Lagi-ange's Method of Approximation .... 135 Newton's Method of Approximation with Fourier's Additions 142 Homer's Method 150 XXXI. Symmetrical Functions of the Eoots Applications of Symmetrical Functions Sums of the Powers of the Eoots Elimination ..... Expansion of a Function in Series . Miscellaneous Theorems Cauchy's Theorem .... Kewton's Eule and Sylvester's Theorem Eemoval of Terms from an Equation Introduction to Determinants Properties of Determinants Applications of Determinants Trigonometrical Formulas Examples ...... Answers 165 174 181 193 206 214 231 236 251 256 266 .284 296 308 326 > 4 « > U THEOEY OF EQUATIONS. I. INTRODUCTIOK 1. The reader can easily obtain a general idea of tlie object of the following treatise by a reference to the theory of quad- ratic equations which he is supposed to have already studied. /The equation ax^ + bx-h c — has two roots, namely, ' 2a ' h and with respect to these roots, we know that their sum is — a c and their product is - ; that is, their sum is equal to the coeffi- a he cient of the second term of the equation x' + - x-{- — = 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 • INTRODUCTION. •' '' ' Any Algebraical expression which contains x may be called 'a. function of x, and may be denoted by f{x). Any quantity which substituted for x in f{x) makes f{x) yanish, is called a root of the equation y*(x) = 0. An expression of the form aa;" + 6a;""' + cx""^ + . . . + Z;a; + ?, where ?i is a positive integer, and the coefficients a, 6, c . . . ^', ^, do not involve x, is called a rational integral function of x of the ^i**" degree ; 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 ti* 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 ic, 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 ic" has a coefficient which is not unity. If we do not wish to suppose the coefficient of a" to be unity, we may conveniently denote it by p^ ; then the equation takes the form p,a;"+;7,a;"-'+;?,a:'-'+ ... +jp„_,a;' +p,._j;c + ;^„ = 0. The term jp„ 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 / 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 onr investigations ; for example, such equations as tan x - 6'= 0, INTRODUCTION. 3 nr xlogx~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 f(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 2^ ,i, Pi, V-i-, •••/'„) in the equation ;?X + pia;" "» + ^^^r""' + • • ■ + Pn-^a?' + p,_^x +p,^ = 0. In the quadratic equation ax^ + hx+ c = we are able to solve the equation without knowing what particular numbers are de- noted by a, 6, c ; all we require to know is that a, 6, c are some numbers independent of x. If we have to solve the equation a:^-12a;+15 = 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 seekino- 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. But with respect to what may be called the arithmetical solu- tion of equations in which the coefficients are given numbers, more success has been obtained. Thus, for example, although 1—2 / 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 degi'ee 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 byyfa;) the expression then the value of this expression when x = a may be denoted by /{a). We will shew how the numerical value of /(a) may be most easily calculated, supposing that the coefiicients of /(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 First obtain p/j, ; add^:>^, this gives i^^+i^j multiply by a, this gives ^^'^a" + 2^^^ > add ^2 , this gives p^a' + p^a + p, ; multiply by o,, this gives 2>^ • We may arrange the process in the following way; j)^(i p^d^ + p^a p^a^ + p^ar + p.xt 2\a -f i\ p.a' +p,a +2\ ViP^ -^ 1\^' ■^Pf'+Ps We may proceed in the same way whatever may be the degree of/(x). For example, required the numerical value of 2>x^ -2x"-6x+7 when a; = 3. 3-2 0-5+7 + 9 4-21 +63 +174 + 7 +21 +58 +181 Thus tlie result is 181. INTRODUCTION. 5 6. If any rational integral function of x vanishes when X = a, the function is divisible by :k.- el. Let f{x) denote the function; then we have given that f(a) = 0, and we have to prove that /(a:) 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 H the re- mainder if there be one. Then f{x) = Q (x- a) + B. Tn this identity put a for x ; since ^ is a 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=ahj supposition. Thus J? vanishes when x = a ; but E does not contain £C, so that if it vanishes when a; = a it always vanishes. That is, E = and x-a divides /(ic). 7. The above demonstration is important and instructive; we may however 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" +P,X'*-' +JPX"' + • •• +Pn-,''' +Pn-X^ +Pn^ then since /(a) = we have /(a;) =f{x) -f{a) = p^ (a:" - a") +p^ {x"-' - a""') +p, (aj""' - a""') + . . . +p,_, {x - a). Now the terms aj"-a", a""* -a""*, ... are all divisible hj x-a (see Algebra, Art. 483). By performing the division we obtain for the quotient p^ ix"-' + ax"-« + aV-' + . . . + a^'-^'x + «""') + p^ (a;"-' + aa;"-' + a V"* + . . . + a"-*) 4- ... We may rearrange the quotient thus : ^oa^""' + {P q2=ciq,+p^, q^==m2-^Pzy ; that is, each new coefficient is found hy multiplying the preceding new coefficient hy a and then adding the corresponding old coeffi,- deiit. It will be observed that these new coefficients are succes- sively determined by the process of Art. 5. 8. If 'K — B, 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 f{x) is divided hj x — a, then f(x) = Q {x — a). In this identity put a for x, then Q is not infinite, and therefore Q{x-a) vanishes. Thus /(re) vanishes when x = a, and therefore a is a root of the equation/ (a:) = 0. 9. To find tlie remainder when any rational integral function of -s.is divided hy 's. — q, wliere c is any constant. Lefc/(a;) denote any rational integral function of x^ and divide fix) by a:-c until the remainder is independent of ajj let ^ denote the quotient and R the remainder. Then f{x)=Q{x-c) + B. In this identity put c for ic, then Q is not infinite, and therefore Qix-c) vanishes ; thus /(c) = R. That is, E is equal to /(c) when ic = c, but R does not contain cc, so that R is equal to /(c) always. For example ; if 3a3* — Scc^ - 5aj + 7 is divided by a; — 3, the quotient is 3a;^+ 7a;^ + 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 4-160 + 620 + 10 + 40 + 155+627 Thus the quotient is 3a;' + 10a;^+ 40a; + 155, and the remainder is 627. This process is a particular case of Synthetic Division; see Algehra, Chapter LViil. INTRODUCTION. 7 10. Let fix) be any rational integral function of x^ and suppose x + y put for x ; then we propose to arrange f{x + y) according to powers of y, and to determine tlie coefficients of the different powers. Let f{x) =^0^" +;?!«""' +^30;""'' + ... +p^_^x +p^ ; then /(x+y)=p,{x+yy+p^{x+yy-'+p^{x+yy-' + ... + p„_^{x+y)+p^. Expand (x + yY, (x+yy~^, ••• bj the Binomial Theorem, and arrange the whole result according to powers of y; we thus obtain for f{x-ry) the following series : p^x''+p,x''-'+p^x''-'+ ... +p,_,x+p^ + ?/Wa;" ' + {n-l)p^x'' ' + {n~2)p,x''-^ -^ ... + pX f;^\^{n-\)p,x^-'-^{n-l){n-2)p^x'^-'+,..^2p^,)^ 7;' + + + ... The first line of this series is obviously /(a). We shall denote the coefficient of y by /'(x), the coefficient of ~-^ by f"{x)j 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 p will be denoted byy''(a;). Hence /(^ + y) =/W +2//'(^) + i^/"(^)+S-^"'W + - ...+^/'(x) + ...+^/"(x). 8 INTRODUCTION. By inspection it will be seen that the functions f{x), f'{x), f"{x), f"'{x), ...f"(x) are connected by the following general law: in order to obtain f"*^{x) we multiply each term in /"(a:) by the exponent of a: in that term and then diminish the exponent by unity. 11. Let us suppose, for example, that/(x) is of the fourth degree; let /{x)=p^x* + p,x^+p^x' + p^x+p^. Tlien /' (x) ^ ip^^^+ ^p,x'+ 2p^x +p^, r(x) = 4:.3p^x'^3.2p,x + 2p^, /'"{x) = A.3.2p^x+3.2p,, f""{x) = ^.3.2.p,; If we suppose numerical values assigned to p^, p^, p.,, p^, p^, and ic, we may calculate separately /(x), f'{x), ... by the method of Art. 5 ; we shall however hereafter, in explaining Horner's method of solvin<^ equations, shew how these calculations may be most conveniently and systematically conducted. Por another example suppose that/(x) =p {x^ cy. Then f{x) ^ p L" + ncx''-' + '^ ^^ ~^^^ cV-^+ ... + nc-*a;+cj ; therefore f'{x) =p [nx"-^+n(n-l)cx'-'+ '^ ^''~^]^^ " ^^ cV-^4- ... + ,10"-] ; that is f\x)=pn{x^cy-': similarly f"{x)=pn{n-\)ix-¥ cy-', f"'(x) =pn {n - 1) (n - 2) {x + cy-\ and so on. Suppose that (^ (a:) and \p{x) are two rational integral functions of X, and that f{x) = {x) + i}/ (x) ; then it is easily seen that /'{x) = it>'{x) + yp\x), and /'\x) = <^'\x) + ^"{x), and so on. -2 INTRODUCTION. 9 12. If we write the series for/{x + y), beginning with the highest power of y, we shall have ( , _- (n-l)(n-2) 3 n{n-\){n-1) ^ „_3 + \P^ + (^ - 2) p,x + ^ J\^ -^ p^x' + -^ ^ ^ p^x^)^ 2/ + ... + ... ■¥f{x). This may be seen from the form already given for f{x + y)y or by expanding separately every term in f{x + y), and arranging ac- cording to descending powers of y. 13. The function f [x) is called the Jirst derived function of y(a;), the f unction /" (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 f(x + y) in powers of y is an example of Taylor's Theorem. Moreover, it must be observed that/"(x) is deduced fromy^(aj) in precisely the same way asy'(a;) is deduced from /(a;). Thus f"{x) is the first derived function oif'{x), and/"'(cc) is the second derived function of /'(x), and so on. Hence by the preceding Article we have 2/' Similarly /"(x + y) =r{x) + yf'\x) + /^f""(^) + - \r- 2 ^ ' \ n - And so on. 10 INTRODUCTION. 14. In any rational integral function of tl 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 ^0^" +^^a;"~' +^20;""^+ ... +^„_2cc^ +^„_ja3 + ^, be any ra- tional integral function of x; suppose for example that the r^^ term p^_ x"""^^ occurs; that is, suppose ^^_j not zero. Let q denote the numerical value of the greatest of the coeificients p^, Pr+i, '--Pn- The sum of all the terms which follow the /** term cannot exceed ^n-r+l_ J q{x"~' + x"~''~^ + ...+x+l), that is, 5' j — . The ratio of the r'** term to this is ^^^.-r^'_ ^x » t^iat is, i^-AA__^^ . 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 ic = - , then we obtain the series 2/"" {Po+p.y+p^f + ••• + Pn-iV"'' + py}' 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 terms Pi^ + p y+ ...+p^_^y''~^ 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. IG. Let/(£c) be any rational integral function of x, aiidf{a)-, fil>), the values olf{x) corresponding to the values a aud h of x- INTRODUCTION. 11 then as x changes from a to h the function /{x) will change from /(a) to f{b), and will pass through every intermediate value. Let any value c be ascribed to x, and let /(c) be the corre- sponding value of /(x) ; let c + h be another value which may be ascribed to x ; then by taking h small enough /{c + h) may be made to differ as little as we please from y(c). For f{c + h) =f(c) + h/'{c) + j^/"(c) + ... + ~r-\o) + 1/' (e). Then, by Art. 14, by taking h small enough, the first term of the series hf'{c), - — ^/"(c), r^f"'{c), ... 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 y(c + h) -f{c) can be made as small as we please by taking h small enough. This shews that as 0} changes, f{x) changes gradually, so that \i f{x) takes any value for an assigned value of x, 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 icitkout 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 f{b), or always decreases fi-om f{a) to f(h); it may be sometimes increasing and sometimes decreasing. What we assert is, that it passes without any sudden change of value, from the value /(a) to the value /(h). 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 iu the succession of values which /(.r) 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 byy, so that ?/ = /(cc) 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, b. good idea is obtained of the necessary consecutiveness in the values assumed by fix) between the values y (a) andy(6). It must be observed that we do not restrict a, h,/(a), /{b), to be positive quantities ; and by values intermediate between /{a) and /(b) we mean intermediate in the algebraical sense; that is, any quantity z is intermediate between /(a) and /(b) which makes z —/(a) andy* (6) — z of the same sign. 19. If two numbers substituted for n in a rational integral expression f (x) give results with contrary signSy one root at least of the equation f (x) = lies between those values of x. Let a and b denote the two numbers ; then y (a) andy (5) have contrary signs. By Art. 16, as x changes gradually from a to 6, the expression/ (a:) passes without any interruption of value from f{a) to f(b); but since y* (a) and y (6) are of contrary signs the value zero lies between them, so that f{x) must be equal to zero for some value of x between a and b ; that is, there is a root of the equation y(a;) = between a and b. "We do not say that there is only one root. And w^e do not say that ii f{a) and f(b) are of the same sign there will be no root of the equation y(a:) = between a and b. 20. An equation of an odd degree lias at least one real root Let the equation be denoted ^J f{x) = 0, where f{x) =p^'' + p^x""-' + ... ^p,_{c +pl, and n is an odd number. When X is large enough the first term of fix), namely pjxT, will be larger than the sum of all the rest by Art. 14, and there- fore the sijjn of f{x) will be the same as the sign of 7?„rc". Thus, by taking x large enough, the sign of f{x) can be made the same INTRODUCTION. 13 as the sign of 2>o "^^^^^ ^ is positive, and the contrary to that of 2> when X is negative. Since then f{x) changes its sign as o: passes from a suitable negative value to a suitable positive value, there must be some intermediate value of x which makes f [x) vanish; that is, there must be some real root of the equation y (a;) = 0. "We maj determine whether this root is positive or negative. For when we put zero for x the sign of f{x) is the same as that of p^. Thus if p^ and p have the same sign^ so that — is positive, there will certainlj be a negative root of the equation y (a?) = 0; and if p„ and p^ have contrary signs, so that — is negative, there will certainly be a positive root of the equation y*(a:) = 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 a;, it will have a real root of the sign contraiy to that of the last term. 21. An equation of an even degree which is in its simplest form, and has its last term negative, lias at least two real roots of contrary signs. 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 value 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) = has certainly one negative root and one positive root. 22. If 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 j)Ositive root; we will shew that it has 07ily one positive root. Let f{x) =;^X +2^^"'' +P,^"~"' + .-. +Pn-v^ + A- Suppose the coefficients ^g, ^9^, ...jy^ all positive, and the remaining 14 INTRO DUCTIOX. coefficients negative; let p^^^ = -P^^^, ^r+2 = -^r+2) '"Pn^-^n- Then we may write /(x) thus, The expression p^pf +p^x''~^ + p.^""'^ + ... + p^ increases as x in- creases, unless r = 0, and then it remains constant; the expression P P P . . . — •^+ —^4-... + -;^ diminishes as x increases. Thus as x in- XX' X creases from zero onwards, the two expressions cannot be equal more than once. Th at is, f{x) = has only 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 only 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 fix) never changes its sign within a certain range of values for x there is no root of the equation fix) = within that range of values for x. Take for example x^ — Qx+ 9 ; this expression never changes its sign, and yet it vanishes when a; = 3 : the expression is equal to [x — of. But if the equation y*(a:) = has no root within an assigned range of values for x we are sure that y (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: ox THE EXISTENCE OF A ROOT. 15 (1 ) If the coefficients in f{x) are all positive, tlie equation f{x) = has no positive root. (2) If all the coefficients of the even powers of a; infix) have one sign, and all the coefficients of the odd powers of x the contrary sign, the equation y (re) =0 has no negative root. (3) Ify(^) involves only even powers of x and the coefficients are all of the same sign, the equation /(a;) = has no real root. It is supposed in this case that there is a term independent of a;. (4) If /(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 ma2/ in some cases have imaginary roots ; see Algebra, Chapter xxv. And in fact we shall now proceed to shew that imaginary roots must 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 + h J— 1, where a and h are real ; such an expression as a + 6 ^— 1, where a and h 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+h J - 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. 16 ON THE EXISTENCE OF A ROOT. Thus, for example, the positive value of the square root of o^ + h' is called the modulus of each of the expressions a + bj-l and a — b 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 + bj^ and a +b' J-l is aa -bb' + {ab' + a'b) J-l, and the modulus of this is the positive value of the square root ot' (aa' - bb'Y + {ab' + a'bf, that is, of (a" + b') {a'' + b") ; that is, th modulus is the product of the moduli of the two given expressions. Also an imaginary expression a + bj—l is considered to vanisj when a and b vanish ; that is, an imaginary expression vanishei 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 exp>ressions 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 Algebra, 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 : cc" = l, cc" = -l, a" = + 7^, a" = -V^ (1) cc" = 1. It is obvious that ic = 1 is a root of this equation. (2) a;" = — 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 27?i ; we have then to shew that there is a solution of x'""^ — 1 ; this amounts to shewing that there is a solution of a;'"= ^yf— 1, ajid is therefore included in the next two cases. ON THE EXISTENCE OF A ROOT. 17 (3) x" = + J- 1. If n is an odd number it must be of one of the two forms 4m + 1 and im + 3 ; in the former case + J— 1 is a root, since {+J-iy"'^^ = + J-l, and in the latter case - J^ is a root, since (- J-IY""^^ = + J -I. If n is an even number suppose it equal to mp, where ni is an odd number, and p is some power of 2, say 2*. Put y = of, then the equation x""^ = + J^ may be written y"'- + ^/-l} and by what has been already shewn + J^ or - J- 1 is a suitable value of y, according as m is of the form 4r + l or 4:7'+ 3. We have then to find a value of x which will satisfy x^= + J-l or xf = -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 jS are real ; take the square root of a + l3j-l, which ^\dll 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^y = + J-l or =-^-1. (4) x" = - J^. 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:m + 1 or 4m + 3. If n be an even number suppose it equal to mp, where w?, is an odd number and 2^ = 2^, and proceed as before. 29. Every rational integral equation has a root real or imaginary. Let /(ir)=pX + Aa^""'+i^2^"~'+ ••• ^Pn-2^'^+Pn-i^-^Pni ^^ere the coefficients ^p^, p^, ..-p^^^i P^-ii Pn ^^^2/ ^^ either real or imaginary ; we have to shew that the equation y(ic) = has a root either real or imaginary. If any imaginary expression be substi- tuted for x in. fix), we shall obtain a result of the form U+Vj-\, 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 U^+ V^ is always a real positive quantity, if it cannot be zero there must be some value which is not greater than any other value, that is, there must T. E. 2 18 ox THE EXISTENCE OF A ROOT. be some value which cannot be diminished ; but we shall now prove that if U^ + V' have any value different from zero we ca7i diminish that value by a suitable change in the expression which is substi- tuted for x; so that it follows that U^+ V' must be capable of the value zero, that is, JJ and Y must vanish simultaneously. Suppose a particular value assigned to ?;, namely, a + hj—l; let f{x) then become F+Q J—1, wdiere F and Q are not both zero. Now put a + b J- l + h for x in/(a;) ; the value which/(a;) then takes may be found by first expanding /[x + h) in powers of h, and then putting a + b J- 1 for x. Suppose then h' ^„ n n n, f{x + h) = X + hX'+-^X"+ -^^PolVi where X, X', X", ... are functions of x; see Art. 10. Put a+b^f—i for X, then X becomes F + Q J—i, Some of the coefficients X', X", ... may vanish for this value of a;, but they cannot all vanish, since the last coefficient, which is that of r— , is p^\n. . \n - — Suppose /i"* the lowest power of h for which the coefficient does not vanish, and denote the coefficient of If" by R+Sj—1, so that R and S are not both zero. Thus when a + b J- 1 + A is substi- tuted for X the function /(x) becomes F+Qj^+{R + Sj^)h''' + ..., where the terms not expressed can only in^-olve powers of h higher than Jr. Denote this by F' + Q'J^. Let h - d, where € is a real positive quantity. By Art. 28 it is in our power to take t so that f^ may be + 1 or — 1 ; thus we can make F'+Q'J^i=F+Qj^^(R + Sj'^)^'+..., so that F' = F^Re"'+ ... , and F"+Q"= F"- + Q'^ 2 {PR + QS) c" -h ... , ON THE EXISTENCE OF A ROOT. 19 where the terms not expressed can only involve powers of c higher than e**'. Now e may be taken so small that the sign of all the terms involving e in the value of P'' + Q'- will be the same as the sign of ± 2 {PR + QS) e", provided PE+ QS be not zero; see Art. 14. We will first suppose that PR + QS is not zero. Then the sign of P'' + Q''-P^--Q' is the same as the sign of ± 2 {PR + QS) i", vv^hen G is taken small enough ; and we can ensure that this sign shall be negative by supposing that ^'^ is - 1 or -f 1, according as PR+QS is positive or negative. We can therefore make P"- + Q'^ less than P'+ Q\ ISText suppose that Pit + QS is zero. Then instead of taking ^"' = =^ I5 take r = = V— 1. Proceeding as before we shall obtain F + Q' ^'^=P + Q ^^^{R + S ^^) €- ^^+ ... , so that P'=P~S{"-r. ... ^ Q'=Q^R,-+..., and P"+Q" = P' + Q'^2{QR-PS) e'" + ... , . where the terms not expressed can only involve powers of e higher than e . Now {PR+QSy-+{QR-PSy- = {P'+Q'){R'+S'); and this cannot be zero, because by supposition P' + Q' is not zero, and R^+S^ likewise is different from zero. Thus since PR+QS is zero, QR—PS is not zero. Therefore the sign of P'"+ Q'^—P' — Q^ will be the same as the sigTi of ± 2 [QR—PS) 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—PS is positive or negative. We can therefore make P''+ Q'^ less than J"' + Q'. We have thus she^vii 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 v>'hich cannot be diminished; 9 9 20 ON THE EXISTENCE OF A ROOT. hence, as we liave already stated, it must be possible that Z7 = and V=0 simultaneously. 30. It remains to be shewn that a and 6 in the expression a + hj~\, which is the value of x that makes /(ic) vanish, are finite. We have /(^c) =i?,a:" Jl + -^ + A+ ••• + Ail • Substitute a + l J'-i for x ; theri/(x) becomes Take any term of the series within the brackets, for example, that involving 2^2 ^ "^^^ have p^ia+hJ-iy^'pA^'-^hy aK + ^T i^oK + ^r = A+BJ^, say. Then it is evident that A and B diminish without limit as a and b increase ^vdthout limit. Thus denoting the value of f{x) when x==a + b J- 1 by ?7+ V J-l, we have where A' and B' diminish without limit as a and h increase with- out limit. If we put a — hj—l for x we shall obtain a result which can be deduced from that just given by changing the sign of V-l '• thus therefore U' + V =-pJ{a'+ h'T {(1 + Ay+ B% and this increases without limit when a and h increase without limit; for the factor {a^+b^f 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. ON THE EXISTENCE OF A ROOT. 21 31. It -will be observed in tbe demonstration of Article 29, that the coefficients 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 Pliilosophical 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^ + V^ may ap- proach indefinitely near to some limit greater than zero loitJiout ever reaching this limit. But this objection can be removed by the aid of Art. 30. Let z stand for U' + F^, that is, for then we know that z.is, finite for finite values of a and h, and infi- nite for infinite values of a and b. Hence the least value of z must occur when a and b have finite values; and if the least value of 2; were not zero the demonstration of Art. 29 would be contra- dicted. 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- mined bv the relation z=/{a + b J'^) xf{a-b J~^). 22 PROPERTIES OF EQUATIONS. III. PEOPERTIES OF EQUATIONS. 33. Every equation has as many roots as the numher which exjyresses its degree, and no more. Suppose the equation to be of the ?i'^ degree, and denote it by f{x) = 0, where fix) ^pof + p,£c""' +;^X~" + + 2^„-i^ +Pn' ^J Chapter ii. the equation/ (cc) = has a root either real or imaginary; let a^ denote that root. Therefore f(x) is di\isible hy x-a^,hy Art. 6; so that f(x) = (x-a^)(f)^ (x), where cf)^ (x) is some integral algebraical function of x of the (?i - 1)* degree. Again by Chapter II. the equation cfi^ (x) = has a root either real or imaginary; let tf, denote that root. Therefore 4>^(x) is divisible by x — a„j by Art. 6 ; so that ^ (x) = {x — a„) (f>^ (x), where (jt^ (x) is some rational integral algebraical function of x of the {n- 2)* degree. Therefore f{x) = (x-a^{x — a_^<^^(x). By proceeding in this way we shall obtain n factors of f{x) denoted by x — a^, ^ — a.,^ - — ^ — ^,J and the only other factor must be 2^q because the coefficient of x" in.f{x) is^j>^,. Thus Hence the equation /(a;) = has n roots, becausey (x) vanishes when we put for X any one of the n quantities a^, a^, a„. And the equation has no more than oi roots, because if we ascribe to x a value c which is not one of the n values a^, a„, a„, the value o^/{x) becomes p,(c-a^)(c-a^)(c-fg (c-a^); this is not zero because everv factor is different from zero ; and the product of factors real or imagmary will not vanish if none of the factors vanish; see Art. 26. Si. The roots in the preceding Article are all eitlier real, or of the form a + h J — I, where a and b are real. And some of the roots ttj, «2, a^ may be equal so that there are not necessarily n different roots of an equation of the n**" degree. The student may perhaps be disposed to doubt the propriety of saying that an equa- PROPERTIES OF EQUATIONS. 23 tion of tlie ?i"' degree has always n roots, wlien these roots are not necessarily all different. It is however found convenient to con- sider that an equation of the n^^ degree always has n roots, although some of the roots may be equal; just as in common algebra it is found convenient to speak of the quadratic equation ax^ + hx-{-c = ^ as having two equal roots when h^ = 4a<3, rather than as having then only one root. 35. The only preceding Ai'ticle of the book which can be at all affected by the consideration of the possibility of equal roots, which has just been introduced, is Article 22. In that Article it is shewn that an equation of a certain form cannot have two (liferent jyositive roots, but the demonstration there given does not exclude the possibility of a second root or of more roots equal to the root which necessarily exists. After we have proved Descartes's Rule of Signs however it will be obvious that the equation in question can have only one positive root without any repetition. 36. If we know a root a^ of the equationy(a;) =0 we know that f {x) = (x — ftj)^! (x) w^here ^ (x) is a function of x two de- grees lower than / (x) ; and the remaining roots of the equation /{x) = can be found if we can solve the equation (x), wKere <^ (x) is a function of x of the 7^'^ 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 cfi{x) vanish; therefore by supposition every coefficient in {x) is zero. Now by Art. 7, <}> (x) = g„x" + (q^a + g>"-^ + {q^a' + q^a + g^jx""' + ; thus ^0 = because the coefficient of cc" is zero, then q^ = ^ because the coefficient of a?""^ is also zero, then q^ = ^ because the coefficient of a:""^ is also zero, and so on. 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. If /(ic) be any function of x of the n* degree we have shewn that /(a?) may be resolved into n factors of the first degree. Each of these factors will divide/(a7) so that/(a7) will admit of n divisors of the first degree. Similarly as the product of any two of the factors of the first degree contained in/(cc) 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 f{x) will admit of as many divisors of the r"" degree as there are combinations of n things taken r at a time, that is,,f{x) •n 1 -x s. n{n-\)...(n-r+\) ,. . ^ , ,, , will admit ot — ^ divisors of the r'^ degree. But 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 \ja.f{x) are not necessarily all difierent. The proposition however shews that there cannot he more than — '-^ '- different divisors of the r*** desree. 26 PROPERTIES OF EQUATIONS. 41. In an eqiuition icith real coefficients imaginary roots occur in pairs. Let f{x) be a rational integi'al function of x in wHcli tlie coeffi- cients are all real; then ii a + ^J -I is a, root of the equation f{cc) = so also is a- 13 J -I a root. For when a + fSj^l. is put for x the function /(x) takes the form P+QjB J^^, where F and Q involve even powers of /3. This is obvious, because if such an expression as a;' be expanded, where x=a + (S^-^j the even powers of (S J - I will give rise to real terms, so that J -I will occur only in connexion with odd powers of /5. A7id as the coefficients in f (x) are siqyposed real J -I cannot occur except with some odd power of (3. If then a- (3 J -I be substituted for x in /{x) the result will be obtained by changing the sign of (3 in the result obtained by substituting a + (3j -I for X] the result is therefore P -Q13 /s/-l- Xow suppose that a-\-(3 J -I is a root of/ (a;) = 0; then F-^Q(3j^l = 0, and, as a real quantity P cannot be equal to an imaginary quan- tity -Q[3 J - 1, this requires P = 0, and^ = 0. And then a- (3 J -I 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-(3j -I- The i^roduct of the two factors x-a-^ J~^ and x- a -^ (3 J - I, is {x- af + /3\ or x--2ax+a^ + P'; 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 ^j 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-b){x- c)...{x-k)cf>{x), where PROPERTIES OF EQUATIONS. 27 a, h, Cy...h are all the real roots of f{x) = 0, and (i (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 she^yn that if the coefficients of any rational integral function /'(a;) of a; be themselves rational, and the equation f(x) = has a root of the form a + Jo where Jb is a surd, the equation has also a root a — Jb. T[i\\^f{x) has a rational quadratic factor (x — a)' — b. 45. To investigate the relations between the coefficients of the function f (x) and the roots of the equation f (x) = 0. Let /(a;) = a;" + ^ja;'*~^ +^;2a;""'+ ... +2:>,^_^x+p^^', and suppose that the roots of the equation y(x) = are o,, ff,, .••«„; then f{x) = {x — a^ (x — Cf ,^) ...{x — a J. Since these two expressions for^* (cc) are identically equal, relations exist between the coefficients |:)j , P2, •- -P,, > and the quantities a,, a„,...a ; these relations we shall now exhibit. By ordinary multiplication we obtain {x — a^) (x — a^) =x^ - ia^ + a„) ic + a^a^^, {x - a^{x — a,^(x - a^ =x^ - {a^ +a„+ a^) x^ + (a^a^ + a^a^ + a^a^) x - a^a^a.^. {x - a) {x - a,) (x - a^ {x - a J = cc^ - {a^ +a, + a^ + aj a;^ + (a^a^ + a^a^ + a^a^ + a^a^ + ap^ + a^a^ x' - {fi^a^a^ + a^ajx^ + a^a./.t^ + a^^a^a^ x + a^a^a.^a^. Kow in these results we see that the following laws hold : I. The number of terms on the right-hand side is one more than 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 tlie 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 no^ prove that these laws always hold whatever be the number of simple factors. Suppose these laws to hold when n~l factors are multiplied together; that is, suppose (x - a;) (x-a^)...{x- «,_^) = x"-' + q^of-' + q^x"-' + ... + q^^_^x + g„_, , where o = 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 x — a^; thus (x - a^){x -a^...(x- a) = a;" + {q^ - a) x"-' + (q.^ - q^aj of-' + fe - 5',^ ^""' + -9'„-i«.- Kow q^-a^^ = -a^-a^-...-a^^_^-a^ — the sum of all the terms — a, , — a„, ... —a ; 1 ' 2 ' n' = the sum of the products taken two and two of all the terms —a,, —a,.... — a ; 1 ' 2 ' n' 1z - UK = ^3 - «n («1«2 + «2«3 ^ ' ' ') = the sum of the products taken three and three of all the terms —a-, —a,,... — a : 1 ' 2' «•' q^_^a^ = the product of all the terms —a^, —a^y... — a^. PROPERTIES OF EQUATIONS. 29 Hence if tlie laws hold when n — 1 factors are multiplied together they hold when 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 A Igehra, Art. 506. Since if a^, a^, ... a„ are the roots of the equation x"" + p^x''"'^ + p^x"'^ + ... +p„_^x+p^=-Oj the left-hand member is equivalent to the product of the factors x — a^, x-a^^ ...x — a^^, we have the following results. In any 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 every two of the roots with their signs changed; the coefficient of the fourth term is equal to the sum of the products of every three of the roots with their signs changed; the last term is the product of all the roots with their signs changed. Or we may enunciate the laws thus : 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)')9^ = the sum of the products of every r of the roots. 46. 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, c; then — a — h - c=p^^ ah + hc + ca =p^ — abc=p.^. In order to eliminate b 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 a^, and the second by a, and add the results to the third. Thus — a^— a'b — a'c + arb + abc + ca^- abc = p^a" 4- p/j^ + p)^ ', that is, <^^^^Pi <^^' + i^o« + i^3 = ; we have thus the proposed equation vrith 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 «, 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- natino; 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^ are the roots of the equation x' +p,x''-' +p^x'"'+... +^?„_l.^' + p„ = 0, we liave —p, = a, + a„+a^+... + a , p., = a^a,_ + a^a^+ ...+ a/i^ + . . . ; thus Px - -P, = «i'+ <+ <+ • • • + «„% TRANSFORMATION OF EQUATIONS. 81 tliat is 2^1 — '^Po is equal to the sum. of tlie squares of tlie roots of the proposed equation. If then in any equation p^--p^ 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- amDle (-l)''~^p^_j = the sum of the products of the roots ?^-l at a time, (— l)"p„ = the product of all the roots ; therefore by division » , 1 1 1 ■L n 12 )i = the sum of the reciprocals of the roots. Also r), ■^^- =(«, + «,+ ... + «)(— + — + ... + — ) Jr n ^ \ 2 n' a, «, a, a^ a„ a^ a^ a^ therefore -^+-^+...+-^+ -^ + ... =-^-^^-^1 - ?2. 2 3 13 J- n lY. TRANSFOEMATIOX 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. 50. To transform an equation into another the roofs of ichich are those of the projyosed equation icith contrary signs. Let f(x) = denote the proposed equation; assume 7/ = — x, so that when x has any particular value, y has numerically the 82 TRANSFORMATION OF EQUATIONS. same value but with the contrary sign; thus x = — y^ and the required equation is y (— y) — 0. If f{x)=p^x''+p^x''-' +i>,:«"-'+ ... +;?„_jaJ+;;„, the equation f{—y) = is Po(-2/r + i^i(-2/)""'+i'2(-2/)'"'+.--iVi2/+i\-0, that is, py -l\y^~' +py- '" ^P„-,y ^P„ = 0; thus the transformed equation may be obtained from the pro- posed equation by changing the sign of the coefficient of every other term beginning tvith 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 a" + 3x' -4:X^-4x+7 = 0, into another in which the roots shall be numerically the same but with contrary signs. Put x = — y, and we get y^ - 37/ + 4:y^ + iy+7 = 0. We may if we please write the original equation thus, x'+ 3x' + Ox''-4:X^ + Ox''-4:X + 7 = ; then the transformed equation according to the rule in Art. 50, is / - 3y' + Oy^ + 4y'+ 0?/" + % -j- 7 = 0, that is, 2/' - Zy' + 4?/' + 4?/ + 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, tliat 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. TRANSFORMATION OF EQUATIONS, 33 52. To transform an equation into another the roots of wldch are equal to tJiose of the 'proposed equation multiplied hy a given quantity. Let f{x) ~ denote the proposed equation ; and let it be required to transform it into another the roots of which are h times as large. Assume y = hx^ so that when x has any par- ticular value, the value of 3/ is ^ times as large ; thus a; = ^ , and the required equation is yf yj = 0. 53. For example, transform the equation , oX OX Ji _ into another the roots of which are h times as large. Put a; = ^ and then multiply throughout by 1^ ; thus we obtain 3 3V 57v-?/ \Lh^ ^ 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 k = 6, the transformed equation is Generally, suppose the proposed equation to be x" + p^x''-' +p^x"-''+ ... +jo„_i.^- + l\ = 0, then if we put x = j , 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 2^My"~% every prime factor which occurs in the deno- minator of p^ shall occur to at least as high a power in k". T. E. 3 34 TRANSFORMATION OF EQUATIONS. 54. To transform an equation into another the roots of vjhich shall he less than those of the j^roposed equation hy a constant difference. Let f{x) = denote the proposed eqiiation ; and let it be required to transform tliis equation into another the roots of ■which shall be less than the roots of the proposed equation hy a constant difference k. Assume y = x~h^ so that when x has any particular value, the value of y is less by /j; thus x = k + y, and the required equation iiif(k+y) = 0. By Art. 10 the expanded form of the equation f{h + y) = is / (^■) + 2//'(^-) + T^ /" ('>■) + S/"' (/.■) 4- . . . + 2/-^ = 0. Thus if f(.r) =2j^x" + p^a:"-' + p^^x''-'+ ... +p,_,x+p„ the equation f {k + y) = when arranged according to descending powers of y is by Art. 1 2 Py^ (p. - W^) 2/"-' + [p, + (^^ - l)pfi -^ '-^^ Pj^f-' + ... r / i\ 7 nin—V)...{n-r^V) .) „_, ^\Pr^\^^-r^\)p,_,h^ + ^ '— ^ pjr^y^ ' + ...+/(Z;) = 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 A, we use the method of the preceding Article. Let the proposed equation be denoted byy(a;) = (), and suppose y = x + h', then x=^y-h, and the required equation is f(y-h) = 0. Thus we have only to put — h for k in the result of the preced- ing Article, and Vv-e obtain the required equation. But in fact this is included in the preceding Article; for that Article does not require k to be necessarily a 2)ositive quantity. 5G. The princi[)al use of the transformation in Art. 54 is to obtain from a proposed equation another which wants an as- TRANSFOKMATION OF EQUATIONS. 25 signed term. Thus if we wish the transformed equation in y to be without its second term, we take k such that p^ + npjc = 0, that is, ^ = — ^ . If we wish the transformed equation in y to be without its third term, we must find k from the quadratic equation ?, + {n -l)pjc + \ „ > ,F = 0. And generally, if we wish the transformed equation in y to be without its (r + iy^ term, we must find k from an equation of the r"" degree, namely r r(r-l) \r\n-r pjf + - pfi-' + ^ -/, p,Jf-' +... + '--==- p^ - 0. ^ ° n ^ n {71-1) ^ \n ^ "^ We shall see hereafter that the solution of an equation is some- times facilitated by first remo\ing some assigned term. 57. For example, transform the equation x^ - 6x^ + 4x + 5 = into another Avithout its second term. Here 2\==^y 2^i^—^ ■> ^^'^^ k = 2, and the required equation is (y + 2)^-6(y + 2r+4(2/ + 2) + 5 = 0, that is, 2/'-8y-3 = 0. Again, transform the equation x^—2x'—4:X + 9 = into another without its third term. Put y + k for x ; the transformed equa- tion is (y + hY-2(y + Jcy-4:{y + k) + 9^0, that is, y'+ y- (3/j -2)-vy {?>¥- ik-i)+ k'- 2k'- 4Z: + 9 = 0. If the third term is to disappear k must be found from the equation ok'-4k-4: = Q; this gives k=2 or -|. With the value k = 2 the transformed equation is 2/'+ 4/+ 1 = 0. With the value ^ = - f the transformed equation is 3 , 283 „ y'-iy-+-—=0. 3—2 36 TRANSFORMATION OF EQUATIONS. 58. To transform an equation into another the roots of which are the reciprocals of the roots of the proposed equation. Let y(.7:) = denote the proposed equation. Assume 2/ = -, 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 Thus if /(a^) =i?o^''+i'i^"~^'^i'2^''~"'^ ••• +i'n-i^+/'n *^^ equa- y y y 2/ that is, pjj''+i\-,y^''-^p,,-,y''~'+ "• +P:y + Po = ^' 59. To transform an equation into another the roots of which are the squares of the roots of the proposed equ/ition. Let f{D:) = denote the proposed equation. Assume y = x^, so that when x has any particular value the value of y is the square of that value : thus x = Jy and the required equation ^^f{Jy) = 0. Thus if f{x)^pjc"+p^x''~'+p.^x'"-+...+p„_^x+p^ the equa- tion fiJy) = is n Tt-1 n-2 , Poy'+p^y' ^p^' +-'-+Pn-,y'+Pn=^- By transposing and squaring we have (n n-2 n^ V / n~l n^ V py+p^y' +p.y'^ +•••/ =\Piy' +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+ i^PoP.-pn y''"+(^-p,p.+p:'^-p.p:) y""+ - = ^' 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. 45. (1) If the roots of the equation ^x^-vpx^ + 5'a3 + r = be a, 5, c, form the equation of which the roots are a h c b + c^ c + a^ a + b' Denote the required equation by y'^ + Ff + Qy + E^O. Then we have, by Art. 45, ^ a b c b + c c + a a + b ab be ca ^^ {b + c){c + a) '^ {G + a){a + b)^ {a + b){b + c)' abc (6 + c) (c + a) (a + 6) ' and a+b + c = ~Pi ab + bc + ca = q, abc = -r. Thus we may now proceed to express the values of P, Q, and R in terms of p, q, and r. For example {b + c){c + a) {a + b)' now by actual multiplication we find {b + c) (c + a) {a + b) = {a + b + c) {ah + bc + ca) - abc T therefore R = . r-pq Similarly we can express F and Q. 38 TRANSFOEMATION OF EQUATIONS. But we may evade the trouble of this process by an algebraical artifice. We have a a cb b + c a + b + c — a —2) — 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 b + c' -^ y are respectively and c + a a + b' Thus the required equation will be obtained by elimiuating x X between the proposed equation and y = . Hence x = - , : and by substitutiuf; this value in the pro- 1 + 2/ posed equation we obtain pY ^ pV pqy , ^ _ n {\+yf {l+y'f 1 + 2/ or r (1 + yf +/?/' (1 + y) -pqy (1 + yY-pY = 0» that is (r -pq) y" + (3r + p^ - 1pq)y' + i?T-pq) y + r=^0. Hence by this method we arrive indirectly at the values of P, Q, and R : we see that a h c _ 3r + p^ - 2pq b + c c + a a + b r -pq ' ab be ca 3r-pq + -. r-, 7T + (6 + c){c + a) (c + a) {a + b) {a + b) (b + c) r- j)q abc T (b + c) {c + a) {a + b) r -pq ' The last result has already been obtained by direct investiga- tion. TRANSFORMATION OF EQUATIONS. 39 (2) Kequired to transform tlie 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, hy c denote the roots of the proposed equation; then, by Art. 45, a + 5 + c = 0, a'b + hc + ca = q, ahc = — r; therefore a" + 1^ + c^ = — 2q. The roots of the transformed equation are to be (a — hf, (b — cy , and {a — c)^; now {a-iy = a^ -2ah + h' = a' + 1' ^c' -lah-c' = or + h'- + 0^ c' 2r c 2r thus a y = — 2q -\ cc", when x takes the value c the value of y is {a — bY; and similarly when x takes the values a and h, the values of y are respectively (b — cy and (c — a)'. Thus the transformed equation will be obtained by eliminating x between the proposed 2r ^equation and y = — 2q-^ x^. Thus x^ + qx + r^O, and x^ + {2q + y)x-2r = 0; therefore (q + y) x-'3r= 0. 3r Hence x = : substitutinor this value in the proposed equation q + y ^ ^ ^ "■ and reducing, we have finally y^ + Gqy^ + 9q^y + STr^ + 4q^ = 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 27?^ + 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 EULE OF SIGNS. (3) Required to transform the equation o^ + ipo:? + 5^0; + r = into another the roots of which are the squares of the differences of the roots of the proposed equation. Put x = x' — -) thus the proposed equation becomes o that is, x'^ + q'x +r ^Oj where n =q — i~ r = - — ^ + r. ^ -^ 3 2/3 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 ,f + ^rl,f^^ri->j + 27/-'^+ 4/^ = 0; that is, Hence if a, h, c are the roots of x^ + px^ + qx + r^O, we see that {a - by +{b- cf + {c-af = -2 {3q -/), ^a-hy(h-cy + {b-cy{c-ay + {c-ay{a-hy = {3q-py, {a-by(b-cy{c-ay=-^f^{22f-0pq-^27ry + 4.{3q-prY Y. DESCARTES'S RULE OF SIGNS. 61. "We have already in Arts. 21... 24 given instances of the connexion wliich exists between the signs of the coefficients in /"(x) 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. "When eacli term of a set of terms lias one of the signs + and — before it, then in considering tlie terms in order, a continua- 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^ — 2>x^ — ix^ + 1x^ + 2>x^ + 2£c^ — x' — x + l, there are four con- tinuations and four changes ] the first continuation occurs at — ix^, the second at + 3a:*, the third at -\- 2x^, the fourth at —x; the first change occurs at — 3x^, 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; 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 y (a:) = 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(a;), although it may happen that the number of changes in /{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 any equation, complete or incomjylete, the numher of positive roots cannot exceed the numher of changes in the signs of the coefficients, and in any complete equation the numher of negative roots cannot exceed the numher 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 +H + — h h. \Ye have to multiply the polynomial by a binomial in which the signs of the terms are + — . 42 DESCARTES'S RULE OF SIGNS. Then wj-iting down only the signs which occur in the process and in the result we have + + + - + + + - + + h- + — + h + + - + - + + - + ±-^=FH h — =F + - 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 coi-respondiug 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 sig-n is introduced at the end. Xow in the new polynomial take the most unfavourable case and suppose all the ambiguities to be replaced by continuations ; by the second la,w 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- DESCAKTES'S RULE OF SIGNS. 43 nuations of sign become changes of sign. And tlie 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 f{— x) = 0, that is, the negative roots oif{x) = are the positive roots oi /(—x) = 0; and whether the equation be complete or not the number of the positive roots oi/(—x)=0 cannot exceed the number of changes of sign iny*(— x). Thus the whole rule of signs may ])e enunciated in the following manner : an equation f{x) = Q cannot have more positive roots than y (a-) has changes of sign, and cannot have more negative roots thany(— cc) has changes of sign. 65. For example, take the equation x*" + 3x' + 5a:; — 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* + 3x" — Sa; — 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 ■—(), 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. — X for X, "we obtain an equation with one change of sign, so that the original equation cannot have more than one negative root. In tliis 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 'if - Qqy^ + ^q^y + 27r^ - 4^^ = ; and by Descartes's rule of signs, or by Art. 24, if 277^ -i(f is negative, the last equation has no negative root, and therefore the original equation no imaginary roots; also if 27r^— 43-^ is positive, the last equation has a negative root by Art. 20, and therefore the original equation must have two imagiriary roots. Q)Q. The student should observe that the results given in Axt. 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 tliat 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 chanf^ces of sicjn 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) = are real^ the number of positive roots is equal to the number of changes of sign in i (x), and the number of negative roots is equal to the number of changes of sign in f (— x). Let n denote the degree of the equation, m the number of positive roots, and m' the number of negative roots, /x the number of changes of sign in f{x), and /u.' the number of changes of sign in.f[—x). Since all the roots of the equation are real m + m' = n. Also m cannot be greater than /x, and m cannot be greater than /m', by Art. 63. Therefore /a + /x' = 9z, for the sum of jw, and /i,' cannot exceed n. Thus 7?i. + m' = /a + /a'. And m cannot be greater than fx] nor can m be less than /x, for then m' would be greater than ix,\ ^Yhich is impossible. Thus m - ju, and m' = jx'. In this proposition we assume that f{x) has a term in- dependent of X, so that the equation f(x) = is not satisfied by x = 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 havo m positive roots, m negative roots, and the root zero repeated r times. Then we have m + m' + r = n) so that ni + m = n — r. And we can shew that /A + /x' can be neither less nor greater than n — r; so that fx + fx' = n — r. Then as before m = fx and m = /x'. 69. Suppose fx the number of changes of sign in f{x), and /u.' the number of changes of sign in /(— x). Then the equation f(x) = 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 qi is greater than fx + fx' the equation f{x) = must have at least 7i — jx - jx 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 any equation there are at least as many imaginary roots of tlie equation. Suppose the 2r terms which might occur in f (x) between x^ and x^~^~^ to be deficient ; then the equation f{oc) = will have at least 2r imaginary roots. Let A and B denote the co- efficients of a;"* and x"^~^''~^ respectively in f{x), and suppose the deficient terms introduced with coefficients g^ q^, q^,.-.; and de- 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 + 1. But now let the hypothetical terms be re- moved; then if A and B 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 B are of the same sign there will be one change of sign for f{-x) 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 F (— x). 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 y* (a:) = as the sum of the num- bers of terms in such deficient groups. 71. If any group consisting of an odd number of terms is deficient in any equation^ the equation has at least 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 tJoan that number of imaginary roots if the deficient group is between tico terms of contrary signs. Suppose the 2r-\- 1 terms which might occur iny(a;) between a;"* and a;"*""''"^ to be deficient. Let A and B denote the coefficients of x"" 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 £ are of contrary signs the equation / (x) = has at least 2r imaginary roots. Suppose the deficient terms introduced with coefficients q^, g,, q,,...; and denote the new function by F(x). Then in the ex- pression Ax^ + q^x""-' + q.x'"-' + . . . + q,,^,x—''-' + Bx^'-' the number of changes of sign together with the number of continuations of sign is 2/* -1-2; 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 fl^c) and one change of sign for /(- x) if A and B have contrary signs. Therefore the loss of 2/- 4- 1 terms from F {:■:) ensures the loss of 2r-7- 2, or of 2r, from the sum of the number of changes of sigTi in F {x) and in i^ (- .t), 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 /{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 /{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 /(a:) = 0, the difference is always an even number. This appears from the examination of the two possible cases iii Arts. 70 and 71. That is, with the notation of Art. G9, the number ?z-/>t — /x'is always an even number. This might have been anticipated from Art. 41. 48 ON EQUAL ROOTS. Yl. ON EQUAL EOOTS. 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. "VVe 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 thej 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 function of x and f'(x) the first derived function ; then will ^ ' x — a x — o x — c x — K where a, b, c,...k, are the roots real or imaginary of the equation f(x) = 0. For let p^ be the coefficient of the highest power of ic infix), then we have identically by Art. 33, f{x)=p^{x — a) (x — b) (x-c) ...(x-Jc). (1) Put y + z for X] thus f(:y + z)=p^{y-¥z-a){y + z-h){y+z~c)...{y+z-h)] expand each side in a series proceeding according to ascending powers oi z] then the left-hand side becomes by Art. 10, /(2/)+/(2/)«+/"(y)l^+- Thus the coefficient of z is f'{y), and therefore f'{y) must be equal to the coefficient of z on the right-hand side, that is, to that is, to f(y) Jill) ^/iy) ^ ^f{y) -r J -r 1- . . . -t — - . y-a y-b y-c y~k ON EQUAL ROOTS. 40 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 /'(^)=/(fl^/M+/M + ... + /M. (2) ^' x—a x—o x—c x—k ' The result here obtained is true if among the quantities a, 6, c, . . .h, 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 atid (2) may be written X — a X -0 X — G 75. The equation f (x) =0 has or has not equal roots according as f (x) and f ' (x) Jiave or have not a common measure which in- volves X. Suppose a, 6, c,...h the roots real or imaginary of the equation fix) = 0, so thaty (ic) =Pq{x- a) [x - b) (x— c) ... (x-k); then /' (x) ^2J^ {x - h) {x — c) . . . (x - k) + p^ix - a) {x — c) . . . {x — k) + . . . If a, 6, c,...k are all unequal, none of the factors x — a, x—b, X - c,... X - k will divide y'(a;), for (x - a) for example divides every term iny'' (.x), except the first; and no product of any number of them will divide /'(ic). Thus if /(^c) has no equal factors f{x) and y'(ic) have no common measure. Hence ii f{x) and f'{x) have a common measure the factors oi f{x) cannot be all unequal. Next suppose that the equation /(x) = has equal roots; f;nppose that a occurs r times, that b occurs s times, that c occurs t times, and so on. Then /'(x) =ft(,r- a)'(x- 6)'(a,- c)'... {^ + -^ + -i- + ...J . In this case the factor (x- aY~^ {x— by~^ {x-cy~\.. occurs in every term oif'{x). Thus if/(aj) has equal factors, f{x) and/'(a3) have a common measure. Hence if f{x) and f'{od) have no com- mon measure y (it) has no equal factors. T. E. 4 50 ON EQUAL ROOTS, 76. For example, consider the equation f{x) ^x^-Ux^ + Uaf - 76x + 48 = 0. Here f'{x) = 4x'-333f+SSx-7Q. It "will be found that f{x) and f'{x) have the common measure cc — 2 ; this shews that (x — If is a factor ot/(x). It will be found that f{x) = {x-'2y{x'-7x + 12) = {x- 2y{x -S){x- 4); thus the roots of the equation / (a?) = are 2, 2, 3, 4. Again, consider the equation f{x) = 2x'-Ux'+l9x'-6x + 9 = 0. Here /(a;) aiidy'(a;) will be found to have the common measure x -3; and/(x) = (x - 3)^(2cc^ + ^)' Thus the roots of the equation /(.) = 0are3,3,+ y(-l), -^{-^^. 77. In the enunciation of Art. 75, the words " ivhich 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 certaiii sense be considered as a common measure of/{x) and/' (cc). As we are here for the first time making an important use of cominon 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 /(jc) and <^{x) denote two rational integral functions of x; then /(a:) and <^ (x) may be resolved into factors, so that f{x)=p^{x-a,){x-aX^-(^^)-'-^ {x) = q,{x- h,) {x - 6J (a; - 63)... ; and each of the functions can be thus resolved in only one way. Hence the function of x of the hisrhest decree which will di%adr ON EQUAL ROOTS. 51 both fix) and {x) is the product of all the common factors of the first degree in x'y and this we may call the greatest common measure of /(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 ?/, we may find the greatest common measure of these functions of y. 78. Suppose / (x) =p^ (x - aY(x - 5)*(cc - c)^ . . ; then we have found in Art. 75 that/(a;) and /^{x) have the common measure {x - ay~^{x — h)'~^{x — c)'~"\ ... Thus the common measure involves all the equal factors which occur in/(a3), but the exponent in each case is less than the corresponding exponent in f{x) by unity. If we divide f{x) by the common measure oi/{x) and/' (x), the quotient involves all the factors which occur in /{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 /(x) = has. 79. "We see that if the factor {x - ay occurs inf(x) the factor (x - ay~^ occurs m/'{x) ; so that the equation /'(a;)=0 has r- 1 roots each equal to a, l^ow/"{x) is the first derived function of f'{x); thus if r - 1 be greater than unity /' (a-) and f" {x) will have a common measure, and the equation /"{^) = will have r - 2 roots equal to a. Thus in this way we can shew that if (a; - a)" is ?. factor of f{x) then the derived functions /'(a;), /''(a;),... /""^x), ail 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 — a; put x = a + z; thus z''(l>{a-i-z)=/{a + z) =f{a) +/'{a) z+...+r W t+"' +/" W ^" 4—2 52 ON EQUAL ROOTS. As the left-hand member of this identity is divisible by ^ the ri'^ht-hand member must be so too. Therefore we must have /w=o, /'(a)=o, r-\o)=^. And as the left-hand member is not divisible by a power of z hi'^'-her than z^ the rif?ht-hand member cannot be, and therefore f'ia) is not zero. Thus the number of terms in the series /(a?), / (■^)j /"Wj • • -^liich vanish when x-a^ is the same as the exponent of a;- a iny(a;). For example, suppose f{x) = cc' + 2x* + 3a;' + 7x" + 8a: + 3 j here it will be found that /'" {x) is the first of the series /(a;),/' (a:), . . . ^^•llich does not vanish when a; = - 1 ; thus the factor {x + 1)^ occurs i}i/(a:). It wiU be found that/(,r) = (a: + l)'(a:= - a; + 3). For another example we will investigate the conditions which must hold in order that the equation cc* + ^-jc' + ra; + s = may have three equal roots. Here f{x) = x* + qx^ -^rx^s, f [x] = ix^ + 2qx + r, f"{x) = l2x' + 2q. Hence from/" {x) = we obtain .2 ? »='=-! <')• Substitute this value in/(x') = and /'(a:) = 0: thus bq From (3) we obtain -^^^+ra: 4-5 = «, 00 ^(_^ + 2^)-fr=0 (3). 3r n^ Aq ON EQUAL ROOTS. o3 and substituting tliis in (2) we have ^-4^-36=^ (^)- And from (1) and (4) r=' = -^- (6). Hence (5) becomes s = — -^ (7). Thus (6) and (7) express the required conditions. Conversely if (6) and (7) be satisfied, it will be found that 3r f(x), f'{x) and f"{x) all vanish when ^ = — 2~ • 80. We will briefly indicate another way in which the test for equal roots may be investigated. If the equation f{x) = ha.s more than one root equal to a, then it follows that if f{x) be divided by a; - a the quotient will vanish when x = a. Hence bv taking the form of the quotient given in Art. 7, we must have Wi'o^^""' + (^ - l)i?xa"-^+ ... + 2a;?„_3 + ^„_^ = 0; that is, f (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 /(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 f{x) = 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. 82. Suppose f{x) = to be an equation which has equal roots ; and let /(*)=X,X/X3'X.'...X„", where the product of all the factors which occur singly in f {x) is denoted by X^, the product of all the factors which occur just twice is denoted by X^, 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 Xj, X^, Xg,... will be unity, if there ij 54 ON EQUAL ROOTS. no factor in f{x) which is repeated just the corresponding number of times. Now form the first derived function /' (a;) of /(a;), and then obtain the greatest common measure oi f{x) and f'{x). We will denote this greatest common measure by f^(x), so that f^{x) = X^XiX:...X-\ Next obtain the greatest common measure of /^{x) and its first derived function f^{x), and denote it by f^{x), so that Proceed in this way and form in succession /^{x) = X^X^\..X-% Kow form a new series of functions by dividing each term of the series f(x),f^{x),/^{x),.,./Jx) down tof„_^{x) by the immediately succeeding term. Thus we get = X,X^...X„, = <^,(rr)say, ^j^^^=X^_,X„, = <^_,{a:)say, Then finally iW Y ^=^(^)-Z ^-^ (^) = X , (x) = X . Thus the factors X^,X,,...X^ are now separated, and by solving tbe equations X, = 0, X, = 0,...X„ = 0, we obtain all the roots of the proposed equation /(a;) = ; and any root found from X^ = occurs r times in the equation /(x) = 0. ON EQUAL ROOTS. 55 83. For an example of the process of tlie preceding Article suppose that f{x) =^x^'¥x'- ^x^ - 6x' + 21a;' + 9x^ - 22a^ - 4a; + 8. Then retaining the notation of the preceding Article we shall find that /^(x) = x*+ x''- 3x'-x + 2, /,{x)=x-l, ^{x)=x*-5x''-\- 4:, + 1 is a superior limit of the positive roots of the equation f (x) = 0. 88. In the equation f(x) = put - y for x, and if w is an odd number change the sign of every term so that the coefficient of y" may be + 1. Let q be the numerically greatest negative coefficient of the equation in this form ; then q+1 is a limit of the positive values of y, and therefore —(q + 1) is a limit of the negative values of x. e Hence all the roots of the equation f[x) = must lie between p + l 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 fix) = be the proposed equation ; since all the terms which precede cc""'' have positive coefficients f{oc) will certainly be positive for a positive value of x if ic"- jo (a;"-'+ 0;'-''-'+ ... +a:' + a; + 1) be positive, that is, if a;"— jt? = — be positive. Hence, sup- posing X greater than unity, f{x) will be positive a fortiori if £c" — jt? Y is positive, that is if cc" {x — 1) — joa?" ""^^ is positive, that is if oj*""^ {x — \)—p is positive, that is a fortiori if {x — Vf is equal to or greater than p. Hence if a; = 1 + [Jp or any greater value, f(x) is positive, that is 1 + Ijp is a superior limit of the positive i-oots 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''-*+ ... -px''-'+ ... +i?,.. Now we have x'"= {x - 1) (0;'"-'+ u;"'--+ ... + a + 1) + Ij 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-\)x''-' + p^{x-l)x--'-^p^{x-\)x''-'+...+p^{x-l)+p^ ■¥ p^{x- \)x"*-{-p^(x- l)aj"~^+ ... +p^{pc- 1) +pj ■¥p^{x-l)x''-^+... ■^p^{x-\)+p^ + ... Consider now the successive vertical columns of this expression. Where there is no negative coefficient the value of the column is positive if cc is greater than unity. To ensure a positive value of the columns in which a negative coefficient occurs we must have (Po +Pi -^Po) («^ - 1) greater than p^, (Po+Pi-^Ps-^ ■■■ +Pr-i) (^ - 1) greater than p^, Therefore x must be oreater than ^ + 1,... and ereater Po^Px+P, than ^ + 1, ... Therefore if a be taken equal Po+Pi+P2+'-' -^Pr-i ^ to the greatest of the expressions thus obtained, that value of a;, or any greater value, will make/(a:) positive; that is, the greatest of the expressions is a superior limit of the positive roots of the equation /(x) = 0. 91. We will now illustrate the rules by two examples. First, take the equation x' + 8x'-14:x'-53x'+b6x - 18 = 0. By Art. 87 we have 53 + 1, that is 54, as a superior limit of the positive roots. By Art. 89, since n = 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- 14 53 18 pressions ; - — ^ + 1, — — ^ + 1, - — - — -- + 1, that is, we must take i + O l+O J+O + UO 53 -g- + 1 ; so that 7 is a limit. 60 LIMITS OF THE ROOTS OF AN EQUATION. Again, take the equation Here Arts. 87 and 89 give 70 + 1 as a limit; and Art. 90 gives 70 -— - + 1, so that 19 is a limit. 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 coej65cients 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 o£ 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 superior limit than the general rules supply. Consider the first example of the preceding Article. HeT'e we have to find a superior limit of the positive roots oi f{x) = 0, where f (x) may be written thus, x\x^- 53) + ^x^ (x-~\ + 56 (^ -4) ; 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 f{x) = 0. Again, consider the second example of the preceding Ai'ticle. Here we may write f{x) thus, x\x^-bx-U) + 2x- + x-10', now by the aid of Art. 87 we see that ic' — 5ic — 13 is positive ific=13 + l or any greater number, and obviously 2x^ + x — 70 is positive when u; = 1 4 or any greater number. Thus 1 4 is a superior limit of the positive roots of the equation f{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 ic, and multiply by 2/" and divide by p^, so that the transformed equation is ^ Pn P. Pn Pn 1 ,et a superior limit of the positive roots of this equation be found by one of the preceding Articles, and denote it by L] then - is an inferior limit of the positive roots of the proposed equa- tion. Suppose that we use Art. 87 j 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 — -= — 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 ^„. For example, in the first equation of Art. 91 we have jt?^=- 18 . that IS =— -18-56' '74 — 18 18 and 2?^= 56; thus — r-^ — ^, that is =j , is an inferior limit of the positive roots. C2 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 f [x) = denote the equation which is to be considered ; put A + 2/ for £C and expand f {h + y) by Art. 10. Thus the equa- tion becomes f(h)^yf{h) + ^/"{h) + ... + ^/»(A) = 0. Now suppose h positive and of such a value that /(A), f'{h), f"{h), -/"(A) are all positive; then no positive value of y can satisfy the above equation. But y = x—hy and as y cannot be positive, X cannot be greater than h] thus 7i 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 /"(A) is neces- sarily positive, being equal to ]n. 95. For example, take the equation x' + x"- ix^ - 6x' - rOOic + 500 = 0. Here / (A) - A'+ A'- W- 6h'- 700A + 500, /'(A) = 5A^+ 4A'-12A-'-12A - 700, lf'{h) = lO/f+6h'-l'2h-6, i/"'(A) = 10A'+4A-4, -1/""(A) = 5A + 1. It is convenient to begin with the last function of A and ascend regularly. Any positive value of A makes /""(A) positive; A=l makes f"'{h) positive; A = 2 makes f"{h) positive; A=4 makes /'(A) positive; A= 5 makes /(A) positive. Then it will be found that A= 5 makes all the functions of A positive; and there- fore 5 is a superior limit of the positive roots of the propose 1 equation. It must be observed, that when according to the method hero given we begin with the last function and increase the valine LIMITS OF THE ROOTS OF AN EQUATION. 63 of h suitably as we ascend to the otlier functions, we shall not require ever to re-examine the sign of those functions of h which we have passed. For suppose, for example, we have ascertained that a certain value a when put for h renders all the functions of A positive up to /"(A). TJien put a greater value for h, say a+h] and since f"{a -f h) =f'\a) + hf"{a) + j^r'X^) + • - and all the terms on the right-hand side are positive by sup- position, f"{a + b) is positive also. Hence in the preceding ex- ample, when it was found that h=0 rendered / (A) positive, it was unnecessaiy to try whether this value of h rendered the other functions of A 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 - y 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*- I5x'+ 3x'+ 4:X+iS = 0; put — y for X and we obtain y'+ ry- 15/- 3/+ 4?/- 48 = 0. 48 By Art. 90 we have r — - — j + 1, that is 5, as a superior •^ 1+7 + 4 48 limit of the positive roots, and by Art. 93 we have j^ — = as 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 must lie, we proceed to give some theorems with respect to the 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 TJieorem. 98. If ice suhstitute successively for x in £(x) two quantities which include between them an odd number of roots of the equation f(x) = 0, we shall obtain results with contrary sigiis ; if we suh- stitute successively two quantities which include between them no root or ail even number of roots we shall obtain results with the same sign. Suppose \ and /x two quantities of which X is the greater; let a, 6, c, .. ., h, be all the real roots of the equation /(cc) =■ which lie between A and jx] by Art. 43 we have fix) = {x — a) {x — h) {x - c) . . .{x - Jcjxj/ (x), where i}/{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 X and /x. Substitute successively X and (x for x; thus f{X) = {X-a){X-b){X-c)...{X-k).p{X), Ah) = {H'-a){tJ.-b){fx-c)...{fx.- lc)i{/{fx). Now all the factors X-a, X-b, X- c,... X- k, are positive, and hU the factors fx - a, fx-b, fx- c,... fx-k, are negative; and \j/ (A.) and ij/ (/x) have the same sign. Therefore f(X) and /(/x) have the s'fme sign or contrary signs, according as the number of the roots a, b, €,..., k, 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 ( ) uantities ; 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 cz, 6, c,...,k, to be aU unequal; only LIMITS OF THE ROOTS OF AX EQUATION. 65 it must be remembered that a root repeated 7n times is to be counted as m roots. We see that if /(A.) and f{ix) be of the same sign, either no root of the equation / (re) = lies between X 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 }x or any greater value of x makes /(:c) positivej therefore jx is a superior limit of the positive roots of the equation /(cr) = 0. It must be observed that by the Avords makes f{x) ijositive, we mean makes f{x) a jyositive quantity and not zero. For example, iif{x) = {x - 4)' (a; - 1), then if x is gi-eater than unity f{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 f{x) cannot become negative for any value of x gi^eater than /x, we cannot infer that there is no root greater than fx; 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 /(a:)=0 and the roots of the equation /' (.t) = 0, where /'(^*) is the first derived function of f{x). The theorem is sometimes called by the name of Rolle, who first used it. 102. A real root of the equation f'(x) = lies heticeen every adjacent 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, b, c,...k. Let {x) cannot change its sign. Then by Art. 43 f{x) = {x-a){x-h){x-c)..,{x-?c)<^{x). In this identity put y + z ior x; thus f{y-^z)^{y + z-a){y+z-h){y + z-c)...{y + z- k) (t/ + z). T. E. ^ 66 LIMITS OF THE ROOTS OF AN EQUATION. Suppose each member of this identity expanded in a series pro- ceeding- according to ascending powers of z. The coefficient of z on the left-hand side will be f'{y); see Art. 10. The coefficient of z on the right-hand side will be By equating these coefficients of z, and changing y into x in the resulting identity, we have f'(x)=hx-h){x-c)...{x-Jc) + {x-a){x-c)...{x-Jc)+...\cj,{x) + (x-a){x-h){x-c)...{x-h) cji'{x). Now put successively «, 6, c,..., h, for x; the last term on the right- hand side of the identity vanishes in every case, and therefore the sign of f'{a) is the same as the sign of {a-h)(a- c) ...{a- k), the sign of/'(6) is the same as the sign of (h-a){b-c) ...{b-k), the sign ■of/'(c) is the same as the sign of (c -a){c -b) ...{c- k), and so on; and these signs are alternately positive and negative, for the first expression has no negative factor, the second expression has one negative factor, the third expression has two negative factors, and so on. Hence by Art. 99 an odd number of the roots of the equation /' (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 f{x) = {x- ay{x - by{x - cj. . . cj>{x), /'{x) = '{x). LIMITS OF THE ROOTS OF AN EQUATION. 67 Let fi{x) denote the greatest common measure oi f{x) and fix), that is, let /, {x) = {x - ay-'{x - by-^x - cy-\ . . Then fix) ( ) 'LAJ^ = (fi(x) 'r{x- h){x-c) ... +s{x-a)(x-c)... + ...r + {x- a) {x-h){x-c) ...(})' {xy Call this exjDression i^(a;); 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)F{x), whenever F (x) vanishes so also does /'(a?). 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 f{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 f'{x) = 0; 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 /(.^■) = include ?' - 1 intervals, in each of which a root a occurs of the equation /' (x) = ; 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 f{x) = can lie between any adjacent two of the roots of the equation /'(x) = 0. For if there could be more than one there would be a root or roots of the equation /' (a;) = 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 f{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 /'{x) = 0. If the equation /(x) = has all its roots real, so also has the equation /'(a:) = 0; for the latter equation is of a degree lower 5-2 68 LTMITS OF THE ROOTS OF AN EQUATION. than tlie 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 /'(x) = has certainly in- 1 real roots, and may have more. 105. Since /"(cc) is the first derived function oif'{x), the equation f"{p:) = has an odd number of roots between every two adjacent roots of the equation f {x) =0. Thus if the equa- tion fix) = has m real roots, the equation f'(x) = has at least 7?i — 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 /(cc) = has m real roots, the equation /'' (a?) = has at least m - r real roots^ Hence if the equation /''(od) = has jx imaginary roots, the equation f(x) = has at least /x imaginary roots. Per if the equa- tion f{x) = had less than /x imaginary roots it would have more than n- jx real roots, supposing n the degree of the equation ; thus the equation f''{x) = would have more than n- jx-r real roots, and as this equation is of the degree n - r it could not have so many as fx imaginary roots, which is contrary to the supposition. Por example, let f(x) = cc" (1 - a:)"v The equation f(x) = has all its roots real, namely, oi equal to zero, and n equal to unity. Herice the equation /" (x) = will have all its oi roots real and all lying between and 1; this equation is . nn+1 n{7i-l)(n-hl){7i + 2) „ 106. P'l^om Art. 105 we may deduce the following simple test, which will ofteii indicate the existence of imaginary roots in an equation. ^^^ Pr-iJ Pr) ^^(^^ Pf+i ^^ ^^^^ coefficients of three consecutive terms in f(x), then if p/ {s less than p,_iP,+i there must he a ixdr of imaginary roots in the equation f (x) = 0. LIMITS OF THE ROOTS OF AN EQUATION. G9 Take tlie {n-r-iy^ derived function of f{x) and equate it to zero; thus — = 1- .. . H ^ ^ i r 1 -t r + l r + 1 i . -5 Put- for X, and multiply by f^\ and divide by ^j^^, |7Z-r- 1 j y thus If the roots of this equation are all real the sum of their squares is positive; and therefore, by Art. 47, {n - rfp ^ _ {n-r + l)(n-r)p^_^ is positive. Therefore 1 n-r+1 Ij; IS greater than —— — - l\-d\^,^ / 6 / and a fortiori 20^' is greater than 2^r-iPr+\' 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. 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, /?, y, ..., k, arranged in descending order of algebraical magnitude. Substi- tute for x in f{x) successively a, (3, 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 AX EQUATION. Tlie equation /{oii) — has one root algebraically greater tlian tt, or none, according as ./"(a) is negative or positive ; and it lias one root algebraically less than k if the equation be of an even degree and/ (k) be negative, or if the equation be of an odd degree and /(fc) be positive, otherwise not. See Arts. 98 and 101. 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 + co , a, /B, y,... k, — oo , for x in /(x) suc- cessively. If however y (a:;) vanishes when any of the substitutions are made, it indicates that the equation /(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 f'{x) = 2)X'—q, so that the roots of the equation f'{x) = are ^ / ( - j ; let a = + / \o) THen /W - (|)* - . (|)^ — ^ (|) r. :3J ' ' V6. ^O, First suppose ( - j greater than [i]', tlien if ?• be positive /(a) andy(/5) are both positive, and the equation /(if) =^ has only one real root, which, is algebraically less than /3; if r be negative /{a) and /{I3) are both negative, and tlie equation /{x) = has only one real root, which is greater than a. Next suppose l^j less than ( j ; then /(a) is negative and y(y8) is positive, and the equation /(a?) = has three real roots, namely one greater than a, one between a and (3, and one algebrai- cally less than f3. LIMITS OF THE ROOTS OF AX EQUATION. 71 109. A metliod of discovering the situation of tlie real roots of an equation %Yas indicated by Waring, and reproduced by Lagrange, which we shall now explain ; it is called Waring^s Method of sejycwating the Boots. Let us suppose that the 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—Jc, 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 S. Then ^8 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, Wariug'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 T^E ROOTS OF AN EQUATION. 110. As an example of Waring's method take tlie equation By Art. 60 the equation which has for its roots the squares of the differences of the roots of the 2)roposed equation is Put y = l; thus 49s'- 441;s=+ 42;s - 1 = 0, that is, i9^{z -d) + i2f^-~)==0; thus 9 is a superior limit to the values of ^, and therefore — is an inferior limit to the values of y, Hence . / q j that is, ^ , is less than the diflference of any two roots of the proposed equa- tion. JSTow 4+1, that is 5, is a superior lin^it of the positive roots of the proposed equation, by Art. 87. And — (1 -:- \/l3) 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 ancl — 5. By substituting in succession for x the values 5, 5 — i, 5 — -|, ... it will be found that one root lies between 3 and 2^, Qri.e root between 21- and 2^, and one root between — 2 and — 2i. 111. We will conclude this Chapter with a proposition which may serve as an example of some of the principles already est£i- blished. In the equation /(^c) — 0, where /{^) = P^ + Vi^'~^ + . . . + a; — r, if <^ is the numerical value of the numerically greatest coefficient, and T is positive and less than ^r — — , there is a real positive root less than 2r. When X is zero f{x) is negative. !N"ow a positive value of x will make fix) positive, a fortiori^ if it make x-r- q{p^+ x"~^+ ... -v o?+x^) 1 -x"^ positive, that is, if it make x — r- qx^ — positive. 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 kst ex- pression and it becomes r{\-1r-'^qr}, and this is j^ositive be- cause by supposition r (2 -i- iq) is less than unity. Thus fix) is positive when x = 2r; and f{x) is iiegatiye whe» 03= j therefore a root of the equatioii 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 ij— r tUe equation /{x) = has a root betweei}. and t- 2r. VIII. COMMENSURABLE ROOTS. 112, By a commensurable root is meant a root which can be expressed exactly in a finite forn;, whole or fractional ; so that it involves no irrational quantities. "We shall now shew that when the coefficients of an equation are j-ational numbers, whole or frac- tional, the commensurable roots gf the equation can easily be found. We have seen in Art. 53 that if the coefficients of an equation are rational but not all iategers, 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, If the coe^cients of an ec^uatiqii are ivhole numbers^ and the coefficient of its first term unity, the equation cannot Jiave a rational fractional rOQt Let the equation be x''-i-2^^^''~'+P,x"-'+ .., +2\-2^'+P„-,^+2\ = ^, and if possible suppose it to have a rational fractional root which in its lowest terms is expressed by - . Substitute this value for x, and multiply aU through by 6""' ; thus 7^ COM^IEXSUEABLE 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 j caiuiot be a root of the proposed equation. 114. Thus we are only concerned with the investigation of integrcd commensui'able roots, and we shall now explain the method by which they may be found. The method is sometimes called the Meiliod of divisors, and sometimes Newton's 21ethod. Let the equation be a;" + 2\x''-' -f jJ^jxT" + . . . +p,^_^^ + p,_^c +2\ = 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 a Hence — must be an integer; denote it by q^ and divide a again by a ; thus Hence — — ^^' must be an integer; denote it by q and divide a af^ain by «, and we shall find that ^- ""° must be an integer. Proceedin:: in this wav after dividin;:^ n times bv a we shall arrive at a result denoted by ^^^^^ — — ^1=0. COMMENSUEABLE ROOTS. 75 Hence the following conditions are necessary in order tliat tlie integer a may be a root of the equation y(x) = 0. The last term of the equation must be divisible bv a. Add to the quotient thus obtained the coefficient of x 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 — 1 divisions have been effected, add to the quotient the coefficient of a;"~^ ; 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/ (a;) = ; 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:'-' or 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 Ai^t. 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 Avithin these limits. 116. For an example take the equation aj'-3a3"-8a;-10-0. Here 1 + 10 is a superior limit of the positive roots, by Art. 87; and by writing — y for x we obtain the equation 2/' + 3^(^^-?UlO = 0. 76 COMMEXSUKABLE ROOTS. for which 3 is a superior limit of the positive roots. Hence all the roots of the proposed equation lie betw^een 11 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 4- 5 + 2 + 1 - 1 - 1 - 2 - 5 -10 10 - 9 -10 -13 -18 2 - 2 -18 - 2 - 5 -21 - 5 - I ^21 + 5 o — O In the first line all the divisors of the last term are written which it is necessary to try, g.nd 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 qu.otiei;t — 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 fuljtilled, so tliat 5 is a root. With respect to + 2 and — 2 we arrive at points where exact di\dsion is not possible, so that these numbers are not roots. With respect to + 1 and —1 the filial condition is i7.ot satisfied, so that these numbers are not roots. Thus the only commensurable root is 5 ; and denoting tlie equation byy(a:) = 0, we know that gc- 5 is a factor of/{x). The other factor will be found to be a;' + 2x -i- 2. For another example take the equatioii. x' + 5x* + x'- IQx' - 20^ - 16 = 0. It will be found that the comraensurable 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. COMMENSUEABLE 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 numbers a, 6, c,..., are the only commensurable roots of an equation /(a;) = 0, it still remains to determine whether any of these roots are repeated. We may divide / {x) by the product {x-a){x-h){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 f{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 ujiUi/ 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 difference in the resulting conditions is that the last quotient must be -^j^ and not -- 1. Suppose for example 2a;'-12a;-+13a;-15 = 0. Here -^ + 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 5 3 - 3 -5 10 8 2 -10 - 2 Thus 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 EOOTS. It must be remembered tliat 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{:c) - ; for x put 7)1 + y, then a — m is a value of y which satisfies the equation firii + y) — 0. The term independent of y in. this equation is/'(wi), and all the coefficients of y are integers, if the coefficients in y(ic) are integers and m also an integer; see Art. 10. Thus if a be an intesrer a — m is an intesjer and must therefore divide y(m) by Art. 114. Thus any integer a which divides the last term of f{x) is to be rejected if a — m does not divide /{/n). 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 -f 1 does not divide f{- 1) which is — 9; thus 4 cannot be a root of the proposed equation. Again, take the example re'- 20x- + lG4:c - 400 = 0. This equation has no negative root by Art. 24; and by writing it in the form x^{x- 20) + I6i Ix- —--) , ^^'e 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 4a;''- 1 la:" + 7a; - 6 = 0, that is, x'--^x +-.T-- = 0. OF THE DEPRESSION OF EQUATIONS. 79 Fii'st, put X = I , in order to transform tlie equation into one with, integral coefficients; see Art. 53. Thus that is, 2/'^0/- 11/+ U?/- 24 = 0. By Arts. 89 and 96 all the roots of this equation must lie between 1 + ^24 and - (1 + ^24); and we see bj 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 ; 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 -IG -3-5 4 -3 4 -1 - 1 Q Thus 3 and - 4 are roots; and since x = '-, we have ^ and - 2 as roots of the original equation. IX. OF THE DEPRESSIOX OF EQUATIONS. 121. In the present Chapter we sball 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 tivo equations have a root or roots in common, it is required to determine the root or roots. 80 OF THE DEPRESSION OF EQUATIONS. Suppose the equations y (.2;) = and F{x) = to have a com- mon root a; then y (a:) and F [x) have the common factor ic — 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 of /{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 /(x)^0 andF{x) = 0. If any factor is repeated in /{x) and F (x) it will also be rejDeated in their greatest common measure* 123. Suppose, for exampile, we have the two equations x^+Ssc'-bx^'-Gx-S^O and x''+ x^- 9^.-+ lOrc - 8 = 0. The greatest common measure of the expressions which form the left-hand members of these equations is x' + 2x — S', and if this be put equal to zero we obtain cc = — 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 y (a;) = 0, the relation pa + qh = r; it is required to determine these roots. Since a and h are roots of the eqtiation f{x) = 0, we have /(«) = 0, and f{h) = 0; but h = "Jll^ . therefore / (^Jll^ = q. — ^ ) = 0. Hence a may be found by the preceding Article. Thus a is known and then h from the relation 2)a + qb = r. Hence /(x) 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 /{x) = 0. OF THE DEPRESSION OF EQUATIONS. 81 125. Suppose, for example, tliat we have the equation a;*-7£c' + lla;--7a: + 10 = (1), and that it is kno^^n that two of its roots a and h are connected by the relation h = 2a+l. Substitute 2a; + 1 for a: in (1) ; thus (2x+iy-7{2x+iy+n{2x+iy-7{2x+i) + io = o, that is 16a;* - 24a;' - 16a;^ - 4a; + 8 = 0, or 4a;*-6a;'-4a;--a; + 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 & = 5 ; that is, 2 and 5 are two of the roots of the proposed equa- tion. Then it will be found that a;* - 7a;' + 1 la;' - 7a; + 10 = (a; - 2) (a; - 5) (a;'+ 1), so that the other roots are ± ,J(—1). 126. It may happen that another pair of roots a and /8 is subject to the same condition 2')a + q^ = r. In this case the ex- /T* — 1^x\ 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 h are both repeated in the equation /(a;) = 0, the factor x — a will be repeated in the greatest common measure o£/(x) and /(^) . 127. Generally suppose that two roots a and h of the equa- tion y (a;) =0 are connected by the relation b = 4>(a). Then the equations /(a;) = and f{(fi(x)} = 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 DEPRESSION OF EQUATIONS. for example, we liave an equation f{x) — 0, and it is known tliat the roots of this equation occur in pairs, and tliat each pair of roots a and h satisfies the relation a + h= It. Then accordinix to Art. 12-i 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 f{a) = 0, that is, f(2r -b) = 0, and f{b) = 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. 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. AYe may proceed thus. Assume a — b=2z, so that we have simultaneously f{a)=0, a + b^2r, a b = 2z. From the second and third of these equations a = z-\-r. Substitute in the first equation, so that /(2; + r) = 0. From this equation values of z must be found, and then corresponding values of a and b. 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 b be one pair of roots of the proposed equation, a and /5 another pair, and so on j then f{x)^(x-a){x-b)(x-o){x-[i) ... f(z + r) = {z + r - a) {z + r - b){z + r - a) (z + r - (3) . .. (■ a + b "" ' a + B ^ .+ -2- -a Z + a + /3 '2 -,y^ 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 —^r- for a root would also have — ^r— as a root ; and such' is the case. II. We may also proceed thus. Assume z = oh. Then ■ (a; - a) (a; -6) = aj^ - (a + 6) £c + a6 = cc^ - ^rx + z. Hence if 2; be suitably determined, x^ - 2rx + z ^^dll be a factor of f{x). Perform the process of dividing f{x) by x^-1rx+z until the remainder takes the form Px + Q, where P and Q are functions of z, but do not contain x. Hence the necessary and sufficient conditions for x^ -Irx + z being a factor of f{x) are P=0 and ^ = 0. Find by Art. 122 a value of z which will satisfy both these equations ; then find a and h from a + h = 2r and z = ah. 130. Suppose we know that between three roots a, h, c of the equation f{x) = 0, the relation pa + qb + qx = s exists ; it is required to determine these roots. Since a, h, and c are roots of the equation /(x) = 0, we liave /(a)=0, f{h) = 0, /(c) = 0. Thus /W = 0,/(6)=0,/(i^^^^)^0. Suppose h 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 be the greatest common measure of m and n, and let 7n=pix^ and n=pv. Let a denote a value of ^'\, and ft a value of ^/l ; then ^ 1 x i^l may be written thus ^/a x y/?, or ^a^. Now a^ has /a x v values, and as each jo**" root of a^ has p values we have in all ^:>/xv values, that is r values. And these values are all different. For let a denote another of the /x, values, and (^' another of the v values, and suppose if j^ossible that ^aft' = Ja(3; raise both sides to the^" power, then a'/3'=a^; ' R therefore — = 7^ . The left-hand member is a root of the equation a (5 of- — 1 = 0, and the right-hand member is a root of the equation ic" — 1 = ; and these equations can have no common root except unity by Art. 147. Thus there are /xi/ different values of a/?, and r different values of ^1 x ^1. 1G4. The essential part of the preceding Article is sometimes treated thus. "VVe have r/1 x "/I = 1 ""* , and if be reduced ^ tnn CUBIC EQUATIONS. 99 to its lowest terms, the numerator will be an integer and the m + n 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 m+n relation 1" x 1"= 1 ""* , in the sense in which this relation is here required. XII. CUBIC EQUATIONS. ■ 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 teiTQ. 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 + %, 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^ + {^z + g) (?/ + ^) + 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—2 100 CUBIC EQUATIONS. to make another assumption j suppose then that St/z + q = 0. Thus we have 2/^ + «' + r = 0. Substitute for z m terms of y; thus '''-^(-i)--^' that is f + ri/'-^ = 0. Hence , ^^-i^yt+iz)' and — ^=-iV(M-;)- 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 ;s^; for distinctness suppose the upper sign taken. Therefore X {-lV(M;)}'*{-5-y(M?))'- Thus the expression for x is the sum of two cube roots, and as ever J 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--+ / \ -r + ~] , then 2 V V4 27/' the other cube roots are ma and ma?, let n denote one of the cube roots of--- /(t + ^)j ^^^^ ^^^ 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 g in fact the process of solution requires that yz = — -x, 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 = — \', thus we can have y = 7)i and z = n as ad- missible values. Then we can also have y = a.m and = a^n; and we can also have y^a^rti and z = an; 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 = — ^ or — — ~ instead of — |- . o o 167. For example, suppose o;^ + 6u; — 20 = 0. Here g' = 6 and r = -20j thus a; = (10 + 7io8)3 + (10 - J"m% By numerical work it may be ascertained that (10 + 7108)^ = 2-732 ..., and (10- Vl08)* = --732..., so that we may presume that ic = 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^ + Qx—20 is divisible by cc — 2, and we find that jc' + 6a; - 20 = (^ - 2) {x^ + 2a; + 1 0) ; therefore the other two roots of the proposed equation may be found by solving the equation a'4-2a;+10 = 0; thus these roots are -Idfe^rg^ thatis-liSy^. 102 CUBIC EQUATIONS. Ill the preceding e^s^m^le we may verify by trial that (10 + 7108)3= 1 + V3 and (10-7l08)* = l-V3, 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 + sjb)^ in an infinite series ; in this case in order to obtain a convergent series, we must expand in ascending powers oi sjb or of a, according SiS s/b is less or greater than a; see Algebra, Chapters xxxvi. and XL. 168. "We have seen in Art. 166, that although apparently nine values are furnished for x only three are really admissible. "We may see a reason for the occurrence of the nine values. For the relation yz = —~ was assumed, but this was transformed into o y^z^ = — ~ 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 x^ + qx-hr = 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 --^{m + n)+^{m-n)J-'i, and --^{in + 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 2 3 r Q m = Uf that is, y^ = z^, is that j" + 97 = ^• 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; 2 3 TO. that is, suppose that — + ^ is a negative quantity. We know from Art. 142 that y^ and z^ will each have cube roots of a cer- tain form. We may therefore suppose that m = fji + vj—l, and as z^ only differs from y^ in the sign of the radical, we can take ')i = ^ — y J —\, In this case the roots of the proposed cubic equation are all real, namely, fx + vJ-1+iJ.-vJ-l, that is 2fx, ({JL + V J - I) a + {[JL - V J - I) a, that is - ju, - v ^3, and {fjL + vJ-l)a"+{fjL-vJ-l)a, that is - /x + 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 this case exhibited in a form which is alge- 104 CUBIC EQUATIONS. braically correct, but arithmetically of little value. For example, take the equation Here r = — 4 and q = — 15. Hence we obtain that is, aj = (2 + 11 J^)^ + (2 - 11 J^)i Now here we have no obvious mode of extracting the cube roots. It may be verified by trial that and (2 -11 7^)3 = 2 -V^. Thus x=2 + J^ + 2 -J^ = 4. Hence 4 is a root. The other roots can then be found by the method of Art. 169; or we may proceed thus, X*-15X- i-(x-4:){x^ +4:X+1). We have therefore to solve the equation x^ + ix+l =0', the roots are — 2 ± ^3. Again, consider the equation £C^ — 3 ^2a3 - 2 = 0. Here r = - 2 and g = - 3^2. Thus x={l+J^)^^ + {l-j:^)i. It may be verified by trial that n 4- /~T\4 - ^^ ■^'i 4- v^^~^ /"T K^+sf ^) - 2 112 24/2 ^ ' ^^^"24/2 2 ^2 ^ • Thus _ V3 + 1 ^3-1 ,— ■ V3 + 1 J3-1 .— ^73+J ^~"2iy2 "^ 2^2 ^ '"^ :iV2 2 ^2 "^ ' ^2 * The other roots may then be found; they are — ^^~ and - ^2 ""^ ^2* CUBIC EQUATIONS. 105 171. The case in wliich 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 /azYs 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 inconyenient 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 p + qj— 1. For if g' be numerically less than p we can expand (jj + 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^)^ wiU have an approximate value P-Q J — 1; and the sum of the two cube roots will be 2P. But if q be numerically greater than p we may proceed thus; P + qJ^l = J~l{q-2^J~l); hence {p + qj~lf = (J^YH^ -pj^)^^- Now -^/- 1 is a cube root of ^- 1 as we find by trial, so that we have {p + qs/-'^)^ = - J^(q-pJ^)i And we can expand {q-pj-iy 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 oi p + qj—l 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— 1 in a form involving Trigonometrical functions. 172. It appears from the preceding Articles that the cubic equation x^+qx + r=^0 may always be solved by Cardan's process 106 CUBIC EQUATIONS. without any difficulty when 5' is a positive quantity, and also when q is Q: negative quantity provided ^^ that — f- -i_ = 0, the proposed cubic equation has two of its roots equal V r by Art. 60. We have by Art. 166 in this case m = n= / --: ; and the three roots are 2m, — m, and - m. In every 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 ad(? + Zbx" + 2)cx + d = 0'y assume x = z — , then we obtain a z^ + qz + r = 0, ^c W d Zhc 2b^ where q=o 5- , t = ^ + — - . Hence by Cardan's method "=V-2-'\/4'"27J '■V-2-V 4'-27J ' The condition which must hold if there are equal roots is — h -^ = : 4 27 ' CUBIC EQUATIONS. 107 that is (26' - ZaU + aHY + 4 (ac - 6^ = 0. It will be found by common Algebraical work that this can be put in the form {ad - hcf - 4 (6' - ac) {c" - hd) = 0. 174. Some cubic equations in wliich the coefficients have special values may be solved without using Cardan's method. For example, suppose This may be written x^+3x = (a — J +3 I a — ), that is, x^-Uc — j+3\x-(a — )[ = 0j and now we see that one root is given by x = a — . a Again, suppose we have the complete cubic equation x^ + ax' + hx + c = 0, and that the relation Sac = 6' holds among the coefficients. The proposed equation may be written — x^ = ax^ + hx + Cf therefore - 3abx' = Sba^x^ + Sb^ax + ¥, therefore (a' - Zab) x* = aV + 35a V+ Wax + ¥= {ax + b)% therefore x^/a^- 3ab ^ax + bj h therefore aj = J a^— 3ab — a 175. A process is given in the Trigonometri/, 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 vre will sliew how the Trigonometrical Tables may also be used for examples ^Yhicll do not belong to the irreducible case. Suppose x^+qx-\-r = 0; then If q is positive, assume ^ = -7 tan^^; then we get If g is negative, and ^L(f' numerically less than 27r^, assume K^ = — -j sin^ ^; then we get / r r As / ^ '/' n\^ x= { -- +- cos 6 j +f---;^cos6'j (-2 + 2 = (->•)* {(cos |)^+ (sin ^)'}. 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 /(x) = are all real, where y(aj) denotes {x -a){x- h){x - c) - a'^ (x-a) - V' {x-h)- c' {x-c)- 2a'h'c. The equation may be written thus, {x - a) Ux - h){x -c)- a"\ - ib"(x - b) + c'\x -c) + 2a'b'c'\ = 0. CUBIC EQUATIONS. 109 Let h and h denote the roots of the quadratic equation {x-h){x-c)-a'^ = ^, and suppose h not less than h. Then by solving the quadratic equation it will be seen that h is greater than 6 or c, and that k is less than h or c. Substitute successively + go , A, ^, — oo for X iny(£c); the results will be respectively + 00 , - 16 V(/i - 6) + c' ^{h - c) y , |5' ^(5 - h) - c'J{c - h)X , - 00 . Thus the equation f(x) = has three real roots, one greater than A, one between h and k, 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 = ky (2) that in which h or k 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 b = c and a'=^ 0. Hence it will be found that c is a root of the cubic equation; and on dividing /{x) by cc — c and equating the quotient to zero we obtain a quadratic equation which has real roots. (2) Suppose that A 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 k be a root of the cubic equation, it has a real root greater than h; and 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 A is a root of the quadratic equation and also of the cubic equation. Since A. is a root of the quadratic equation, we have (\-h)(\-c)-a" = (1); 110 CUBIC EQUATIONS. and since X is also supposed to be a root of the cubic equation, we obtain b"{\-h)+c"{\-c) + 2a'h'c'^0 (2). From (1) and (2) we deduce b" {\-h) + c" (X - c) + 2h'c'J(X-b)(k-c) = 0, that is, |6V(A -b) + c'J{\ - c)J = ; therefore b''{\-b) = c"{X-c) (3). From (2) and (3) we obtain ^-^=—^^ ^-""^--^ W^ and therefore b — -rr = c (5). 6 c ^ ^ 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. li 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 ^ 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. Ill Hence the 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, Vc' ^ da! ah' a - = b--7T = c — - . a G Conversely, if these conditions hold the cubic equation has equal roots. For denote these equal quantities by r, so that h'c' , c'a! ah' substitute for «, 6, c in the cubic equation, and it becomes so that the root r occurs twice, and the other root is 'i.t h'c' c'a' a'h a c This assumes that a', h'j and c' are all different from zero. Suppose now that one of these quantities vanishes, say a'. Then from the quadratic equation {x — h){x—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 h' = Q and (c-a)(c-5)-c'-' = 0. If a J h' and c all vanish then in order that there may be equal roots, two of the three a, h, c must be equal ; if they are all equal the cubic equation reduces to (x — af = 0. 112 BIQUADRATIC EQUATIONS. XIII. BIQUADRATIC 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 de23rived 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* + qx^ + raj + 5 = 0. Assume aj* + qoif + rx + s={x^ + ex +/) {pc? — 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-f) = ry g/=s; that is, g +/= q + e^, g -/- - , gf^ s. Find g and / in terms of e from the first two of these equa- tions, and substitute in the third; thus (j + e'+r)(y + e^_r) = 4.. From tliis 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. W'hen e^ is known we can find e, and then g and/ become known. Thus the expression x'^ + qx^ + 7'x + 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. BIQUADRATIC EQUATIONS. 113 182. It will be observed that in one of the 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 x^ 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 4 3 in ways, that is, in 6 ways j and thus we see the reason why t 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 g, and this has no influence on the results which are obtained by solving the biquadratic equation. 183. Suppose, for example, that a^— lOa;^— 20cc— 16 = 0. Here q = -\0, r = — 20, 5 = — 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 j therefore x"- lOaj"- 20aj- 16 = (of + 2a;+ 2)(a;^ - 2^-8). The four roots of the proposed biquadratic equation will be found tobe 4, -2, -1 + y^, and -1-7^. 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 the following pro- position. The auxiliary cubic equation will not fall under the irreducible case when the biquadratic equation has two real roots and two imaginary roots. For suppose the imaginary roots of the biquadratic equation to be denoted hj a + ^J-l and a-^J-l; 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, =t (y + ISj-l), and ± (y -(Bj-1). Thus the three values of e' will be {laf, {y + jSj-lY, and (y_^yri)2. •£ ^ -g ^^^ 2ero 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^ will not fall 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^ and -a^yJ-1. By taking the sum of every pair of these roots we obtain the expressions ±2a, =i=(/5 + y)>/-l> and ±(/5-y)^-l; thus the values of e" are 4a-, -(^ + y)^ and — (;S — yY, 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 forms 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 BIQUADRATIC EQUATIONS. 115 negative, there must be one positive root ; and as the 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. (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?, ^^ y^ denote the three values of e^ ob- tained from the cubic equation . Then by Art. 45 we have r" = aft'y^, and -2q = 0^ + fi^ + y". Thus we may put r = a/?y, and take a as a value of e ; therefore x^ + ex -f/= x^ -\- ax + -^Iq + (I - -\ = X' + ax + j{a'- ^--y^-2py). By solving the equation x^ + ex +f= we shall therefore obtain Similarly, by putting x^ — ex + g = we shall obtain x = -^{o--^ + y), or x = ^{a + (3-y). Thus the four roots of the biquadratic equation are l(-a-/5-y), i(-a-^^+y), ^ (a - ^ + y), ^(a + ^-y). 8—2 116 BIQUADRATIC EQUATIONS. In order that the biquadi'atic equation may have equal roots the auxiliary cubic equation must have equal roots. For suppose, for example, that _(_a-^-y) = -(-a + ^ + 7), then (S + y = 0, therefore i^^ = y^ ; and a similar result will follow in any other case. Hence we can express the condition which must hold in order that the proposed biquadratic equation may have equal roots; for by Art. 173 the condition in order that the auxiliary cubic equa- tion may have equal roots is (27r - 72qs + 2qy = 4:{q' + l 2s)^ It will be seen, by Art. 79, that the conditions which must hold in order that the proposed biquadratic equation may have three equal roots may be expressed thus : 27r'-72qs + 2q^=0, and g'+ 125 = 0. It will be useful to note the forms of these conditions for a complete biquadratic equation. Let the equation be ax* + ^hx^ + 6cx^ + idx + 6 = 0] assume x^z — , then we obtain a z* + q^ + r;2; + s = 0, where o = -„ , _U_ 126c W ~ a d? a^ ^ _e ihd 6b'c W a a a a BIQUADRATIC EQUATIONS. 117 Hence we sliall find that a and 27?-^ - 72qs + 2q' = ^^^-f^ {ad' + eh' + c' - ace - '2hcd). Thus the condition for equal roots is (ae - 45cZ + 3c-)' = 27 {ad' + eb' + c' - ace - 2hcdf; and the conditions for three equal roots are ■ ae-ihd+2>c' = 0, and ad' + eh' + c' - ace - 2hcd = 0. 188. Another mode of solving a biquadratic equation has been given under slightly different forms by various mathema- ticians; and thus it is sometimes called Ferrari^ s method, some- times Waring's method, and sometimes Simpson's method. We ■will now explain it. Let the biquadratic equation be X* + px^ + qx' + 7'x + s = 0; add to both sides aaf + hx + c, and then let a, h, c be so determined as to render each side a perfect square. We have then x^ + px^ +{q + a)x'+ {r + b)x + S + c = ax' + hx + c. The right-hand member will be a perfect square if 6^ = 4aG. Sup- pose the left-hand member to be equal to + m by comparing the coefficients we obtain 2m + '^ = q + a, pm = r + h, m' — s + c. These three relations express a, b, c in terms of mj substituting the values of a, h, and c in the equation h' = 4:aG we obtain {pm—r)' = 4:(2m+---qj{m'-s). 118 BIQUADRATIC EQUATIONS. From this cubic equation m must be found, and then a, b, and c. And since we now have ( '2 cc^ +'^ + ??i j = ay? + 6aj + c = ax^ + 6ic + -j— , , , . o px 2aa3 + h we obtain x +-Tr- + m = =fc ^ . — . Thus we have two quadratic equations to solve, namely, „ px 2ax + b ^ ^ „ px 2iax +b „ ocr+-^ + m+-^r—, — = 0, and x^+-^-r- + m — -r-— — = 0. 2 2 s/a 2 2 Ja 189. It may be shewn that the auxiliary cubic equation which this method requires us to solve will in general fall under the irreducible case, unless the proposed biquadratic equation has two real roots and two imaginary roots. For let a, /?, y, 8, denote the four roots of the proposed biquadratic equation; then from considering the two quadratic equations obtained in Art. 188, it follows that m + must be equal to the product of two of the four quantities a, (3, y, 8, and m — . must be equal to the pro- duct of the remaining two. Suppose then thus ^ = o (^^ "^ y^)' Hence we infer by symmetry that the other two values of m will be ^ (ay + ^8) and - (aS + /?y). It is obvious that if a, /?, y, 8, are all real, these three values of m are all real; and it may be shewn that such will be the case if a, /?, y, 8, are all imaginary. If however two of the four quantities are real and two imaginary, it will be found that two of the values of m are imaginaiy and one real, or else they are all real and two of them equal. BIQUADRATIC EQUATIONS- 119 190. We will now give Euler's method of solving a biquad- ratic equation. Suppose the equation to be x^ + qx^ + rx->r 8=0. Assume x = y + z->ru; thus x^ = y^ + z^ + u^ + 2{'i/z + zu + uy), that is, x--y~-z^- ir = 2(yz + zu + uy). Square both sides; thus x' - 2x'(y' + z' + u') + (y' + z' + u^ =i{yz + zu + uy)' = 4 (yh^ + z^u^ + u^y-) + ^yzu (y + z + u). Put xiov y ■\-z + u, and transpose ; thus a;"- 2x'{y'+ z'-¥ u') - Sxyzu +{i/ + z'' + u'f - 4 {y'z^ + z\i^ + u^) = 0. In order that this equation may coincide with the proposed biquadratic equation, we must have q^-2{y^+^ + u% r = - Syzu, s = {y' + z' + uy - 4 (2/V + zho' + uY). Thus y' + ^+u'=-^, 2/V + .W-fuy = l(|-.)=^^^i^ ,2~2 , ^2„,2 , „.2^,2 2 2 2 Therefore it follows fi'om Art. 45, that y', z^, and u^ are the values of t furnished by the following cubic equation, ^^2^"^ 16 ^ 64"^- Let the roots of this equation be denoted by t^, t^, and t^', then If we substitute these values in the expression for x, namely, 2/ + Z + U, we obtain eight different results on account of the am- 120 BIQUADRATIC EQUATION'S. biguities in sign. But tliese results are not all admissible; for we T must have yzu — - -k-, so that the sign of the product of y, z, and u^ must be the contrary to the sign of r. If we suppose r 'positive^ we have the following admissible values of cc, If we suppose r ^legative, we have the following admissible values of x, 191. The reason why eight values of x present themselves in r the preceding Ai'ticle is because the relation yzu = — ^ was squared and used in the process in the form y~:^ii? = ^ ; for since the relation in the latter form is not changed by changing the sign of r, the process really determines the roots of the biquadratic equation x^ + qx' — rx + s-O, as well as the roots of the biquad- ratic equation x*+ qx" + rx + s = 0. The auxiliary cubic equation of Art. 181 will be found to coincide with that of Art. 190 by supposing e' = 4:t; thus the re- marks made in Arts. 184... 186, respecting the connexion between the roots of the auxiliary cubic equation and the biquadratic equation, and the circumstances under which the cubic equation falls under the irreducible case, apply to Euler's method of solu- tion as well as to Descartes's. 192. It may happen that special forms of biquadratic equa- tions admit of simpler solution than the general equation. The following is an example. The biquadratic equation x^ + px^ + qor -h rx + s = can be solved as a quadratic equation ii j/— ipq + Sr — 0. For the equation x* + px^ + qx' + 7^x-{- s = may be written x' STURM'S THEOREM. 121 and this may be solved as a quadratic equation, if —5 = ^ , that is, if 'p^— 4:pq + 8r = 0. Some valuable remarks on Biquadratic Equations by Professor R. S. Ball will be found in the Quarterly Journal of Mathematics, Vol. VII. 1866. XIY. STURM'S THEOREM. 193. In the preceding Chapters of the present work we have demonstrated various theorems respecting the roots of equations, and have given the algebraical solution of equations of the third and fourth degrees. We are now about to enter upon a different part of the subject, namely, the methods of finding approximately the numerical values of the roots of equations ; the present Chapter commences this part of the subject by proving Sturm's theorem, the object of which is to determine the situation and the number of the real roots of any equation. We shall enunciate and prove the theorem in the next Article; we shall then give some remarks connected with the theorem, and finally apply it to some examples. 194. Sturm's Theorem. Let f{x) — ^ be an equation cleared of equal roots, and let f^ (x) be the first derived function oif{x); let the operation of finding the greatest common measure of /(x) and /j {x) be performed with this modification, that the sign of every remainder is changed before it is used as a divisor, and let the operation be continued until the remainder is obtained which is independent of a?, and change the sign of that remainder also. Let X(cc),/^(cc),...y^(£c), be the series of modified remainders thus obtained. Let a be any quantity, and /3 another which is 122 Sturm's theorem. algebraically greater, then the number of real roots of the equa- tion f{x) = between a and (i is the excess of the number of changes of sign in the series f{x), /^{x), f^{x), ...f^{x), when x = a, over the number of changes of sign when x = ^. We shall call the whole series f{x),fi{x), fsi^), '••/^{^), Sturm' s functions, and we shall call the series /j (re), /2(^)) •••fm{^)i the auxiliary /unctions, so that the auxiliary functions consist of Sturm's functions omitting /(x). Let q^, q.,, •" q„i-i} denote the successive quotients which arise in performing the operations indicated; then we have the following relations, /W = 51/1^-/2(^)5 /lW = 52/W-/3W' /2W = 5'3/3W-/4W' /^_2 W =?,n-l/„-l W -A(4 From these relations we can draw three inferences. (1) The last of the functions/^ (a;) is not zero; for by supposi- tion it is independent of x and if it were zero /(a?) and/^(aj) 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 auxiliary functions would vanish including f„X^); and this is impossible by (1). (3) When any auxiliary function vanishes the two adjacent functions have contrary signs. Suppose for example that /g (a;) = ; then from the thii^d of the above system of relations we have /H = -/W- 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 STURM'S THEOREM. 123 of sign is lost by Sturm's functions, and tliat no cliange of sign is lost or gained in consequence of x passing through a value ^Yhich makes one of the auxiliary functions vanish. I. Suppose c a root of the equation /(x) = 0, so thaty(c) = 0. Let h be a positive quantity. iN'ow 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, by Art. 14; that is, the sign otf{c - h) will be the same as the sign of- hf^{c) since y(c) = 0. The sign of f\ {^ ~ ^ ^^^ ^® *^® same as the sign ofy^ (c) when h is taken small enough. Thus if x = c — k and h is taken small enough, /{x) and y (cc) have contrary signs. Similarly, it may be shewn that if x = c + h and h is taken small enough, y (a?) and/j(x) have the same sign. Thus as X increases through a root of the equation f (x) — 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, /^{x), so that /]. (c) = 0. Then f_^{c) and f^.^{c) have contrary signs, and thus just before x — c and also just after x = c, the three terms y_j(a3), /^{x), f+^{x) will present one permanence of sign and one change of sign ; for if f^_^{x) andy(ie) have the same sign, /^(x) andy^^(£c) 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 f{x) be one of them, it follows by I. that a change of sign is lost as x increases through that value, and iif{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 124 STURM'S THEOREM. number of clianges 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 ^. 195. "We have shewn that no alteration occurs in the nuniber 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 y (x) = and that a is less than h; then y(.^•) and /j(.^') have contrary ?,igns, just hef ore x=a and have the same ^ign. just after x = a. 'Now just before x=h the signs of /(x) and f^{x) are again contrary. In fact the equa- tion /^{x) = has one root between x= a and x = h, and so f (x) must pass from positive to negative or vice versa between x= a and x = h. This transition of f (x) from positive to negative or vice versa between a and 6, 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 h. 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 inditferent 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 f{x) = 0, we may first put — go for x and then 4- oo for x in Sturm's functions ; the excess of the number of changes of sign in stukm'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 — CO the sign of any one of the functions will be the same as the sign of the highest power of £c in that function. 198. Let Qi denote the degree oi/(x); then the number of the auxiliary iu.JiGtionsf^(x),f2{x),...wi\\ 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 oi f(x), and we will suppose that the highest power of ic iny(ic) has a positive coefficient. (1) If the first terms in all the auxiliary functions have posi- tive coefficients all the roots of the equation y*(a;) = are real. For all Sturm's functions will then be positive when a; = -i- oo , and they will be alternately positive and negative when x= — oo ; thus qi 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 sign 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 7?z changes of sign and n — m continuations of sign in Sturm's functions. Now change x from +00 to — oo ; then the changes of sign are replaced by con- tinuations of sign and the continuations of sign by changes of sign, so that for x = — co there are n — m changes of sign. The excess of the number of changes of sign when x = - co 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 X(ic), 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 f{x), f^ (x), f^i^)) ...f^{x). For in the original 126 stuem's theorem. demonstration of Sturm's theorem tiie necessary property of the last auxiliary function is that it should not vanish, and as fX^) cannot vanish, the demonstration will hold for the series This remark is of practical importance, because the labour attendinof the formation of Sturm's functions is considerable in examples of equations of high degrees, and thus it is useful 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. f(x), and suppose that 4*{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 y^ (x) and deduce the remaining auxiliary functions from f{x) and ^(a;) instead of iromfix) andyj(a;). 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 wheny(cc) 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 ]) times and the root h occurs q times in the equation /{x) = 0. Let y(^) = {^ — cf'Y {x — i>y{x — c)(x — d)... then y {x) = {x - ay~' {x - by lp{x- b){x -c){x-d)... + q{x — a){x — c){x — d) . . . * ] stukm's theorem. 127 Thus {x-a)^^{x— by ~^ is the greatest common measm^e oi /(x) and f^{x), and this expression will divide all the auxiliary {unc- tions f^(x),f^(x),...f^(x) which are formed as in Art. 194. Now let if/ (x) ^(x — a) {x — h) {x — g){x — d) . . . and <^{x)=p{x-h){x-c)(x-d)... + q{x — a)(x-c){x — d)... + (x - a)(x — b)(x — d)... + ... Then cfi(x) is not the first derived function of xl/{x), for that would be what [x) would become if ^j) = 1 and q=l; but (x) has the same sign as the first derived function of {{/(x), when we make x = a, or bj or c,... Hence, by Art. 200, we may determine the situation of the real roots of the equation i/^(a;) = by taking il/{x) and c}>{x) as the first two of Sturm's functions and forming the rest from them. But the series of Sturm's functions formed from /(x) and f (x) only difiers from the series formed from if/^x) and (fi(x) by reason of the additional factor {x - af~^ (x-bj'^ 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{y^ and f^ (x) we can ascertain how many of the roots of the equation \l/(x) = lie between assigned limits, that is, how many distinct and separate roots of the equation f{x)=^0 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. 128 stuem's theorem. 202. We may observe that 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 f{x) = x^— 3x^— 4:a; + 13 = 0. Here /j (x) = 3x^- 6x - 4, f^{x) = 2x-5, /3(^) = + l. 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. f(^) /.W /.W Ai^) + - - + 1 + - - + 2 + - - + 3 + + + + Here there are two changes of sign when cc = 2, and none when £c = 3 ; thus there are two positive roots between 2 and 3, and no other positive roots. Sturm's theorem. 129 It Tivill be found tliat ^rlien x = -3, the succession of signs is - H — +, 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=2h; then the succession of signs is + , and thus we have only one change of siorn, whether we consider the to carry the si^n + or — . Thus a change of sign is lost in proceeding from 2 to 2h, and therefore one root lies between 2 and 2-V; hence the other root lies between 2h and 3. Again, suppose /(rr) =a;*— 6x^+ 5x^+ 14x - 4 = 0. Here jT (x) = 2x^- 9x^+ 5x + 7, omitting a factor 2, /^[x) — 17a;"— o7x — 5, f^{x) = 152a; -457, In this example it will be found that the calculation oi f^{x) is somewhat complicated; it is sufficient for our purpose however to know the sign, and thus when we ascertain that it is positive we need not calculate it exactly, but merely put down /^ (a;) = +. 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. /w /,w fl^) U^) fP) -2 + - + - + -1 _ _ + - + _ + - - 4- 1 + + - - + 2 + _ _ _ + 3 + _ _ _ + 4 + + + + + There is one change of sign lost between — 2 and - 1, one be- tween and 1, and two between 3 and 4. T. E. 9 ISO Fourier's theorem. If we put 3^ for x the succession of signs is - -f + 4-, and thus there is only one change of sign, so that one root of the equation lies between 3 and 3^; therefore another root lies between 3^ and 4. Again, suppose/ (:<:) = 2x^—Voj:'+ 10^; - -49 = 0. Here f^ (x) = ix^— 13ic -f 5, omitting a factor 2, f^{x)^-13x'-lDx+9S. It is easy to see that the roots of the equation ^^(a:) = are imaginary, that is, /Jx) 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 o: = — co the succession of signs is H h, and when a; = + go the succession of signs is + + + ; thus the equation has two real roots and two imaginary roots. One of the real roots is positive and the other negative by Art. 21. XV. FOrEIER-S THEORE.U 204. Sturm's theorem constitutes the complete solution of a problem which has engaged tlie attention of many of the most eminent mathematicians during the last two hundi'ed years; this theorem was published in the volume oi Memoir es presentes 2)ar 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 French waiters 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 Fomier had given the theorem in a course of lectures delivered before the publication of Budan's work. We ^\"ill now enunciate and prove the theorem. FOURIER S THEOREM. 131 205. Fourier'' s Theorem. Let fix) be an algebraical function of the n* degree; \^\j f^ix), f{x)^...fjx) be the successive derived functions of f(x). Let a be any quantity and y8 another which is algebraically greater; then the number of the real roots of the equation f{x) = between a and )3, cannot be greater than the excess of the number of the changes of sign in the series /{x).,/^{x),f,(x),...f(x), when x = a, over the number of the changes of sign when x = (3. "We shall call the whole series/(x),/^(x),/^(x),...f^(x), Fourier'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{G — h) is the same as that of — hf(c), and the sign of f (c — h) the same as that oi f{c) ; see Art. 14. Thus if x^c — h and h is taken small enough, y (a;) and/j (x) have contrary signs. Similarly it may be shewn that if x — c + h and h is taken small enough, f{x) and fjix) 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 jign. II. Suf)pose when x = c that y (a;) vanishes and also the de- rived hinctioTi.^f{x),f^{x),... up tof_^{x), and thaty(a;) does not vanish. Put c — h for x where h is a positive quantity; then h may be taken so small that the signs of the series of terms f{c-h), f^(c-h), flc-h), ./,.,(c-A), fXo-h) shall be respectively the same as the signs of the series of terms (- mx'), (- 'O'-'/w- (- ^'■rvx'^i ■■■- ¥Xo), /w ; 9—2 132 Fourier's theorem. see Arts. 10 and 14. Thus if x = c-h and 7i is taken small enough, the first 7' + 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 lety].(a:) vanish when x-c but neither y]._j (a:-) nor J\j^^{x). Then if h is taken small enough, when x^c -h the signs of the three terms fr-\{^)i /ri''-^)) fr+ii^)) ^^'® rcspcctlvely the same as the signs of /r-]{^)^ — ^fr+ii^)^J'r+i(.^)' ^^^'^ whcn X — G + Ih thc signs are the same as the signs of /..^(c), hf^^,{c\ f^,{c). Thus if f_^{c) and f^^{c) have the same sign, Fourier's functions lose two changes of sign as X increases through c, and if y]._j(c) and f+J^c) have contrary signs Fourier's functions neither gain nor lose a change of sign. lY. Suppose when x = c that several successive derived func- tions vanish; for example, suppose when x = c that the m i\mc- tions fjx), f^^(x),..f^^^^{x) vanish, and that /._^ (a;) and /^,„(a3) do not vanish. By proceeding as before, and supposing h taken small enough and positive, we shall obtain the following results with respect to the m + 2 terms, f_,{x), /X^)v/.+„.-iW, /r+mW- (1) Let m be even. If /^_j(c) and/^^^^(c) have the same sign, the terms present m changes of sign when x = c — h, and no change of sign when x = c + h. If /^_^(c) and f^^J{i/) negative, and 2/= 11 makes ^ (y) positive; therefore the required value of y must lie between 10 and 11. Assume v = 10 + - : then c/,(10) = 10'- 10 . ICf- 6 . 10 - 1 = - Gl, (/)'(10)= 3. 10- -20. 10-6 -94, i<^"(10)= 3.10-10 =20, and the equation in 2; is — 61;:'+ 945;"+ 20;: +1 = 0, that ls, 61^^-94^^-20^-1=0, saj »//(;:) = 0. Here z = 2 makes 1/^(5) positive, so that the required value of z must lie between 1 and 2, Assuuie z = l +- ; then If, i/:(l) = 61 . r- 94 . r- 20 . 1 ~ 1 = - 54, i/.'(l)= 183.1-^-188.1-20 =-25, Jf'(l)= 183.1-94 =89, and the equation in it is — 54?^'— 25u^+ 89u +61 = 0, that is, 54i<^'+25i*'-89i^-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 + 10.-^ 1. ' 1 +(fcc. Lagrange's method of approximation. 139 The convergents corresponding to tliis continued fraction are 2 21 23 44 V, TTc, v-r, 7TV, See Alqebra, Chapter xliv. The difference 1 ' 10' 11 ' 21' ^ ' ^ 44 . 1 between — r and the real value of the root is less than ^^ aix^ .xxo.ca. v...ua u. .x.. xwu. xo x... .^c.^ 21(21 + 11) ' that is, less than ^r^^- . D/2 Tor another example take the equation x^ —1x + 1 = 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 1^^, and that another root lies between IJ and 2. Therefore if we put x = -^ 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 x' is ('^) - 7 'J + 7 = 0, that is, x"- 28 x' + 56 = 0. The root which lies between 2 and 3 will be found to be 2.— 1 1. 1 2 + &c. The root which lies between 3 and 4 w^ill be found to be 1 3 + 2. ' 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 x = l + - ; thus fl + iY- 7 ("l + i") + 7 = 0. This will give y'-4:f+3y+l = 0, say <^(2/) = 0. Here <^(1) is positive, ^(2) is negative, and makes f{x) positive, so that a root of the equation /(a;) = lies between 2 and 3. Again, x=-2h makes /(.r) positive, so that the root lies between 2 and 2^; also x = 1"2 makes /(ic) positive; thus the root cannot differ from 2*1 by so much as '1. Suppose then c = 2 -1 ; then /(c) c'-2c-5 _- -061 „- .... , c = c - -^ -c = 2-1 - r — ;-:r = 2'1 - '0004 uearlv ; ""^ / /'(c) 3c^-2 11-23 thus Cj = 2-0946 nearly. Then for a new approximation we have c, - A^ = c - -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 /(c) is an approximate value of the root, and that c^ = c — jrj-^ > ^^ ^^^ not sure without further investigation that Cj 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 positire, and this shews that the root lies between 2-0946 and 2; thus 2-094G 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 inile guarantees the success of New^- tou's method when certain conditions are satisfied. Fourier's supplement to Newton's method depends upon a property of the fii'st 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 f(h)-f{a) = {h-a)f'{X). For let F{x) denote f{x) ~/{a) - 1^ |/(6) -/(a)\ ; then F{x) vanishes when x = a and also when x=^b. Therefore bv Art. 102 the equation F'(x) = has a root between a and b. And, by Art. 11, F'{x) =f'{x) ^ — ; hence some quantity X intermediate be- — \th^tf{\)-l^zl^ =.()■, there- -^ ^ ^ b-a ' toTe/{b)-/(a) = {b-a)f'{X). 224. Suppose that b is greater than a: then /(b) is algebrai- cally greater or less than y (a) according as /'(A.) is positive or negative. If /'(ic) is positive between x = a and x = b, then /'(A) is necessarily positive, and if/'(x) is negative between x—a and x^b, then /'{X) is necessarily negative. Hence we have the following result; ify'(a:) is constantly posi- tive through any interval, /{x) increases with x through that interval, and ii/'(x) is constantly negative, /(a?) decreases as x increases through that interval. By the increase or decrease of /(x) we mean algebraical increase or decrease. We may however state our result thus; if /'(x) retain the same sign through any interval, then as x increases through that interval /(x) increases numerically/ when it has the same sign as f'{x), and decreases nuniericalbj when it has the contrary sign. tween a and b must exist, such that f'(X) -^-^-M — ^^^-^ = 0: thei -^ ^ ^ b-a ' WITH Fourier's additions. 145 225. "We sliall now enunciate and prove Fourier's rule. Let f{x) = be an equation which has one root and only one between a and ^; and suppose that the equation f'{x) = has no root between a and /?, and also that the equation /"(a;) = 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(ic) andy"(x) have the same sign. It follows from our suppositions that f{x) changes sign once and only once between a and ^, and that /' (x) and /" {x) do not change sign between a and /?. We mil suppose ^-a to be positive. (1) Suppose that f{x) and f"{x) have the same sign when x=a. Take a for the first approximation; then Newton's second approximation is a - ~^ . Let a-¥h denote the true value of the root ; then /(a + h) = 0. Now by Art. 223, we have /{a. + h)-f{a) = hf'{\)j where \ lies between a and a + h; thus h = - ^77^, and the true value of the root is a - ^7^ . We have J (a) / (A) then to shew that a - V^ is nearer than a to the true value J iv of the root. Since h is necessarily a positive quantity, /(a) and f'QC) are of contrary signs, and /(a) is of the same sign as /"(a), and therefore /'(A.) and /"(a) are of contrary signs. Hence/' (a;) decreases numerically as x increases between a and /?, by Art. 224, so that /'(A.) is numerically less than /'(a); therefore - jq-r 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. fe Let a = a - ^tVt ; then /(a') and /''(aj have the same sign, f (v and the approximation can be continued from a^. T. E. 10 146 Newton's method of approximation (2) Suppose tliat /(a?) andy"(a;) have the same sign when X — (3. Take ;8 for the first approximation, then ^N'ewton's second approximation is yS — ^/ . Let P -h h denote the true value of the root; then f(p + h) = 0. Now, by Art. 223, Ave have f(j3 + h)—f(^) = h/'{\), where A. lies between f3 and (3 + h; thus h = — ^^~. • We have then to shew that /? — ^./ is nearer than yS to the true value of the root. Since h is necessarily a negative quantity, f{(3) and /'(A) are of the same sign, and/(/3) is of the same sign as/"{P), and therefore /'(A) andy*"(/3) are of the same sign. Hence y (a;) increases numerically as x increases between a and ^, by Art. 224:, so that f {X) is numerically less than/'' (y8). Therefore .^^ is a positive quantity which is numerically less than the positive quantity -~r^^ • This shews that Newton's second approximation is nearer to the true value of the root than the first approximation. Let p, = /3- jTf^. ', then f{(3^ and f"{/S^) have the same sign, and the approximation can be continued from /5^. 226. The preceding Article shews that the conditions given by Fourier are sufficient to ensure the success of Kewton'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 — >/\ v , while the true value of the root would be obtained by addin^; — - ,,,/, . Hence the WITH Fourier's additions. 147 permanence of sign of f'{x) is necessary in order that we may be sure that f (c) and f'{^ have the same sign ; if these quantities do not have the same sign the Kewtonian correction has the wrons: 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 y"'(A) is numerically less than f'{c). If this is not the case the Newtonian correction is numerically greater than the true correction, and thus, suj^posing the correctiou to be of the right sign^ the true value of the root lies hetiueen 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 /(:/;) =0 has ouly one root between 2 and 2-1, and that the equations f {x) = and f"{x) = have no roots between these limits; also f (o:) and f"{x) are both positive w^hen x~2'l. Hence the Newtonian approximation will certainly succeed if it be besun and continued from the limit 2"1. For another example take the equation o;^ - 7a; + 7 = 0, say f{x) = 0. It may be shewn by trial that the equation has one root between 1*3 and 14; the equations /"(a;) = 0, andy"(x) = 0, have no roots between these limits; also y (a;) emd J"^' (x) are both positive when x = l''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 f(c) the true value of the root is c — \., , , so that the numerical value / {v f(c) of the error at this stasre is .,;/, , which we will denote by r. f(c) The next approximate value will be c — ,., , . , and now the nu- 10—2 148 XEWTON's method of ArPROXDLVTlON merical value of the error is .,^/. - ^t^t j ^^^^ is, r .. . . ^ ^ , / (^) / (c) / (c) And by Art. 223, we have /'(c) -/'W = (c- A)/"(/x), where /A lies between c and A.; thus the error is — ^ — jrr\ — -^. Xow A. lies between c and the real value of the root, so that c — A. is less 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^» Por example, in Art. 220, the root lies between 2 and 2-1. Thus to find q we divide the value of 6:c when x = 2'1 by the value of 3a:"— 2 when x = 2] therefore q^\-2Q', 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 Kewton^s approximation. Suppose PQR to be a part of the curve determined by the equation y =f{x). Then we may be supposed to know 02f and ONy 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 jP, since the curve at P is convex to the axis of x. Draw the tan;?entPr: let OJ/=a, then JIT ^ - '^^ , as is kno^^^l by the / (a) ^ Differential Calculus; so that, starting from M the Xewtonian approximation proceeds to T. And as 5' falls between J\I and Q it is obvious that the method succeeds in this case, and that the approximation can be continued from 2'. WITH FOURIER S ADDITIONS. 149 At the point R we have /(a;) positive and /"(a:) negative. Draw the tangent RS ; then, starting from N the Newtonian ap- proximation proceeds to aS', and *S' and N are on opposite sides of Q. Moreover there is no security that QS is less than QJV, 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 JVS is less than 'N3f we may start from ^, as the point >S' will then fall between Q and 3f, and the approximation can be continued from >S'. Let ON' =(3; then we may start from lY if {An!, is less than P — a. Messenger of Mathematics, Y ol. iii. page 40. 150 hoener'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. For 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. Let /(a:) = be any equation ; then/(« + a?) = is an equation the roots of which are less by a than the roots of the first equation. The equation f{cc + x)=-0 becomes when developed J \ / J \ / Y '2, I 3 \n 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^ + nx' + Ex + F; then f{a) =Aa' + £a^ + Ca^ -i- Da' + Fa + F, f (a) = 5Aa' + 4:Ba' + 2>Ca' + IDa + E, ^ /"(«) ^ 1 {)Aa^ + Q£a' + ZCa + Z>, J: f'"(a) =lOAa'+iBa + C, 1 /""(«) =5Aa + JJ, ^ f""\a)=A. (1) We may calculate f{ct) in the manner explained in Art. 5, thus ; hokner's method. 151 A =A, Aa + B =P say, Pa + C=Aa^ + £a + G = ^ say, Qa +D=Aa^ + J3a^ + Ca + D = E say, Ha +E = Ad^ + Ba^ + Ccr + Da + B = S say, Sa + F= Aa' + Ba" + CaJ" +Dd' + Ea + F =f(a). Here each line is obtained by multiplying the preceding line by a, and adding on in succession the tenns B^ C, D, B, F. (2) We may now calculate^ (a) in the same way asy(Aa + ^ = TF say, Wa+U= QAa- + 2>Ba ^C = X say, ' Xa + F= 10.1a' + (SBa' + 2>Ca + D = hf"{a). (4) We may now calculate -^/'"{a) in the same way, using A, W, X; A =A, Aa+ W= iAa -\- B=Y say, Ta+X= \OAa' + iBa + (7= ^/'"(«)- 152 Horner's method. (5) We may now calculate rif""{ci) in the same way, using A and Y\ A =A, Aa-¥Y=5Aa + B='^f""{a). (6) Lastly, A=^-f""'{a), The above process may be conveniently arranged thus ; A B G D E F Aa Pa Qa Ra Sa F ~Q 'f: 'F fj^) Aa Ta Ua Va T U ~y f\a) Aa Wa Xa 2' Aa Ya T ~ 3/'» Aa \f""{<^) The quantity under any horizontal line is obtained by adding the two quantities immediately over the line. We have thus shewn Homer's process of forming the coeffi- cients of the equation f{a + a?) = when the equation is of the Jifth degi-ee ; 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) = 3x* - a' + 4a;' + 5x-S. Horner's method. 153 -8 114 106 -1 + 4 + 5 6 6 12 11 22 26 52 57 6 ' 24 70 192 12 35 96 249 6 18 36 71 142 238 6 24 48 119 6 30 Thus /(2 + x)^ Zx'+ 2>0x'^ 119a:'+ 238aj'+ 249aj + 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. ISText 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 HORXEBS METHOD. has a root lying between and '01. Tlius tlie first equation lias a root whicli 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 ap^^roximate 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 /(x) = be the proposed equation, and supjoose that by one or more 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 /[c + x) = 0, and suppose that this last equation has a small root. Then c is an approximate value of a root of f(c) the original equation; hence by the preceding Chapter c — ,, . . will / (w f/A be in general a nearer approximation to that root. Thus — ^^tt-- is an approximate value of the number which we want in order to continue the operation, 234. Example. Let f(x) = 2x''-4:73x' - 234:(: - 711. It \vill be found by trial that/(200) is negative and/(300) positive, so that the equation f(x) = has a root between 200 and 300, We proceed to diminish the roots by 200. 2 -473 -234 -711 (^200 400 -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 2a;'+ 727a;'+ 50566a; - 2967511 = ; so that /(200) = - 2967511 and /'(200) = 50566. Horner's method. • 155 Hence — ^,\^^ { is more than 50. We then proceed to dimi- /(200) nish the roots of the equation just given bj 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 y*(250) = 1628289 a positive quantity, while y (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 liable 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 80 32280 3313840 "807 82846 346329 Thus 40 is also too large, for/(240) is positive. We then try 30. 727 50566 -2967511 (^30 60 23610 2225280 787 74176 -742231 60 25410 847 99586 60 - - 907 Thus /(230)= - 742231 a negative quantity, so that 30 is the right number. Hence the equation which has its roots less than those of f{x) = by 230 is '2x^+ 907^'+ 99586a; - 742231 = 0. Here/'(230) = 99586 so that - jk^-. = 7 approximately. 156 ■ Horner's method. . We proceed then to diminish the roots of the equation just given by 7. 2 907 99586 -742231(^7 14 6447 742231 921 106033 ' This shews that y(237) = 0; so that 237 is a root of the original equation. The whole operation is usually exhibited thus; -473 -234 -7in237 400 -14600 -2966800 -73 -14834 -2967511* 400 65400 2225280 327 50566* -7422311 400 23610 742231 727* 74176 60 25410 787 995861 60 6447 847 106033 60 OOTt 14 921 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 the 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. horxer's method. 157 -3 3 3 3~ 3 •1 "ei *i ?2 •1 •02 6-32 •02 6-34 '02 6-36+ •008 6-368 * -008 6-376 •008 -2 -2 9 5 (^3-128 -6 -1* •761 7* •61 - •239t •167128 7-61 •62 - -071872+ •068273152 8-23T •1264 -•003598848 8-3564 •1268 8-4832+ •050944 8-534144 •051008 8-585152 6-384 Here to find tlie second figuf e of the root we Lave — — , so that '1 is the nearest number to be tried ; to find the third figure — -239 of the root we have — , so that "02 is the nearest number to be tried ; to find the fourth figure of the root we have 158 horxer's method. — '071^7'^ < o o o ' ^^ *^^* '^^^ ^^ ^^® nearest number to be tried. In all these cases tlie 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 ; -3 -2 5 (,1-2016 1 -2 • - -4 -2 -4 1000-"- 1 -1 -992 -1 - 500'-^ 8000000- 1 4 -4879399 00* -496 3120601000;': 2 8 -2927060904 2 -48800001 103540096 2 601 4 -4879399 2 602 600t -487879700J 1 36216 601 -487843484 1 36252 602 -487807232 1 G030X 6 6036 6 6042 6 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 mil be suflScient. 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 workino- column, and so on if there are more than three working 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 fi^^'ure considered as an integer would be approximately given by 8000 , T . . . — __4^QOAA J ^^<^ *^s IS less than unity ; so a cipher is put in 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 do^vn 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^ + ?>x' -2x-6 = 0. We will first perform the operation at full until five decimal places of the root have been determined. 160 1 HORNER S METHOD. 3 1 -2 4 2 5 700 189 889 198 -5 (1-33005 2 4 1 -3000 2667 5 1 -333000 332337 60 3 - 663000000000 564352475125 63 3 108700 2079 -98647521875 66 3 110779 2088 \ 690 3 112867000000 3495025 693* 3 112870495025 3495050 696 3 112873990075 699000 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 -446624425T 338625624 1128751574 489 - 107998801 11287520631 2 112875208 2 112875210 At the point wliere tlie 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 two. 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. The mark * indi- cates where the first stage of the contracted operation finishes. Now 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 1G2 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. 11287521,0 -107998801 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 differing 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^— Zx^— '2x + 6- 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 HORNER'S METHOD. 163 positive roots to more places of decimals than ^ye 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 one which no person will very often perfoiTa, 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." Penny C ycloiJcedia, 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"-' + p^x''-'+ ... +p^^_^x+p^^, for X put y + a, and suppose thaty(a;) then becomes we have to prove that q^^, q„_i, •••q^, q^y ^^'^ found correctly by Horner's process. It is obvious that 5-^=^^. Since y = x — a the following expressions are identically equal, p^x" + p^x"-' + 2l^''~'+ ... +P„_iaJ+i\, and q^ {:c - a)"+ q^ {x - «)"-'+ q^ {x - «)"--'+ . . . + q„_, {x - a) + q,^. Therefore q^^ is the remainder that would be found on dividing f(x) hj x-a; also the quotient arising from this division must be identically equal to q^{x-aY-'+q^(x-ay-'+q^{x-ay-'+... + q,_,. 11—2 16-i horner's method. Then again q , is the remainder tliat would be found on di- O J. n — \ viding the last expression hj x — a; also the quotient arising from this division must be identically equal to Then again a „ is the remainder that would be found on di- viding the last expression \)j x-a; also the quotient arising from this division must be identically equal to q,{x-aY-'+q^{x-ay-'+q,{x-ay-'+...+q„_,', and so on. Thus 5'^, 5'„_j, q^^_^, ^n-a^ ••• ^^® *^® successive remainders which occur in dividing, first /(x) hj x — a, then the quotient by x-a, then the new quotient hj x-a; and so on. And we see by Arts. 5, 7, and 9 that Homer'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 + bJ-1 is an imaginary root of an equation /(a;) = 0; then since the real and imaginary parts of/ (a + bj^) must separately vanish, we obtain two results, which we may denote by P = and (? = 0, as in Art. 41. Here F and Q will be functions of a and b, and if we eliminate a or & from the equations F^O 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 liquations. SYMMETRICAL FUNCTIONS OF THE ROOTS. 165 XIX. SYMMETEICAL 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" + 6^ + c^ is a symmetrical function of the three quantities a, b, c; so also is ab + be + ca ; for each of these functions is unaltered when we interchange a and b, or a and c, or b and c. 243. The coejicients of an equation are symmetrical func- tions of the roots of the equation. For by Art. 45, if the equation be £c" +^i£c""' ■¥'p^~^ + ... = 0, we have — />j = the sum of the roots, IJ^ = 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 /(a;) denote x" +p^x"~^ +212^^"'^ + "• + P„, and let a, b, c, d,... denote the roots of the equation f{x) = 0. Let Si = a + b + c + d + ..., S^ = a^+b-+c'^ + d^ + ..., iS^ = 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 FUXCTIOXS OF THE ROOTS. By Art. 7-4 we liare *^ ^ ' X— a X- x — c Tlie divisions indicated on the right-hand side of this identity can all be exactly performed by Art. 7 } and we have n-3 =^ ic" ^ + (« + «,) a}" - + (a- + pjJb + « J di x — a ^ /(^) fi^) c ' 1-3 , -T- . . . „n — m — \ ■ • • and similar expressions hold for " j , " By addition then we obtain fix) = iix^-^ + (^1 + np,) x"-' + (*S'^ +^i*S'i + np^ x"- Also /' {x) = nx""-' + (/I - l)p,x''-' + (n - 2) ^v^""' + . . . Equate the coefficients of the same powers of x in the identity ; thus /S'l + np^ = (/I - 1 )p^ or >S\ + p^ = 0, and generally In this general result m is su^^posed to be less than n. By means of the general result we can express the sum of the w*^ 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. 1G7 Next suppose that m is not restricted to be less tlian n. Multiply the given equation f{x) = by a;'"""; thus x'^~"f{x) = 0, that is, a;"* + pj£c"'~' + p^x^'^ + ... +px'^~'' = 0. Substitute for x successively a, b, c, ... and add the results ; thus m ^ I m — 1 1 2 m— 2 ^ n rn—n 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, ^Ve have therefore /.,, X /{^) fi^) fi-^) ^' X — a x — o X — c xf'ix) XXX /{x) x — a X — b x — c ( xj \ xj \ xj S S S Thus, if we actually divide xf'{x) hj /(m), the coefficients of the terms in order will he 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 f{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 y. 168 SYMMETRICAL FUXCTIOXS OF THE ROOTS. Or we may make m successively equal to ?i - 1, ?^- 2, ti- 3,... in the result of the preceding Article; and thus obtain suc- cessively S_^, s_,, S_^,.... 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 a, b, c, d,... denote the roots of a given equation. By Art. 244 we can express in terms of the coefficients the value of a"' + b"' + c'" + dr + ... This function may be said to be of the Jirst 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'^h" ■¥ard'-¥irc^+ ... 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 2a'"6^, as it is the sum of all the terms which can be formed like oTlf, A. function may be said to be of the third order when each term involves three of the roots, as a'"b'c'+ ared'' + oTlfd^ + . . . Here every permutation is to be formed of the roots taken three at a time, and the exponent m placed over the first root, j) over the second, and q over the third. We shall denote this function by '^a'"b^c\ as it is the sum of all the terms which can be formed like a^'b'c'. 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 iSa^^b^. We have aS^^ a" + ^>'" + c"* + ... , By multiplication we obtain that is, >S^^*S; = S^^^ + 2ci"'5^, and therefore ^^"6^= .S' S, - S,„,,. This supposes that m and p are unequal. If we suppose p equal to m the terms in "Xa^^ become equal two and two, so that this sum may be expressed thus, 22 (ah)"'', and therefore 170 SYMMETRICAL FUNCTIONS OF THE ROOTS. 249. To find the value of the symmetrical function of the third order loJ^^d^, We have 2a'"6^ = a'^h'' + V^c^ + a'"c'+ . . . , S^^a'+h' + c'+ ... By multiplication we obtain S^^a'^h' = fr^^6^ + 6"+ V + c'-'W + . . . + a'"6V+ ... The terms on the right-hand side form three sets, which in our notation are denoted by ^a"'^?6^ Sa^^^^"*, ^a'^Uc''; thus S^^a^^h^ = ^r^-^h^ + %a'^''ir + ^a^lfc\ Substitute for ^a^^h^, Xc^^'b', and ^a^^^"" their values from Art. 2-i8, and we obtain ^a'^'b^c^ = S^^SpS^ - S,^^pSg - S^^.gSp - S^^^S^ + 2S„^^p^g. We have suj)posed 771, p, q all unequal. Suppose, however, that m=2^} then, as in Art. 248, we have 2S (aJ)"o' = -S:^, - S.„S, - iS^^S^ + 26'„,,. I£ m=i) = q, the sum 2a"4'c' reduces to 2 . 32 {abc)"; thus 6S(«6or = 6:-35,„^„ + 2^3,, The method of this and the preceding Article may be con- tinued to any extent, and thus a function of any order like ^a""}/ and %a'"¥c'' may be expressed in terms of the coefficients. Hence by Art. 246, the object proposed in the present Chapter can be attained. 250. We have shewn how the function denoted by S^ 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 ^(rc) denote any rational integral function of cc, it is required to express in terms of the coefficients the sum <^ {a) + eft (b) + (ji (c) + ... SY:iDIETRICAL FUNCTIONS OF THE ROOTS. 171 w 7 f'{^) 11 1 We have "^-^ = + . + + . . • / [x) X — a X — x — c therefore *M^ = *ff:) . ^ . M ^ . . . j\x) X — a X — x — c 4>(x)-<}>(a) <}>(x)-4>(b) ^(x)-4>(c) , + — - — ■ — , 1- x—a x—b x—c Ma) 4>(h) ^cj>(c) T i r H 1- ... X — a x — o x — c In this identity the integral parts and the fractional paiis will 1 4. 1 TIT, • <^(^)-<^W be separately equal; also such expressions as — —^ ^-^-^ are in- tegral by Art. 7. Let cf>(x)f'(x) be divided hy /(x), the process being carried on until the remainder is an integral function of x of lower degree than /(a;); let li be this remainder. Then by considering the fractional parts of the identity we have E _4>{a) , <^(6) Mcl^ J^yx) x — a x — b x — c Multiply up; then B = x''-'U{a) + {b) + cji{c)+ ...\ + terms involving lower powers of a: than cc"~\ Thus {b) + -P^Ss-PoA-Ps^,-4^P,= 19 + 105- 1-24:= 99, ^s = -pA-pA-pA-pA= 99 + 133-15- 6 = 211, Se = -P^^5-pA-Ps^s-2\'S,= 2n + 693-19-90 = 795, and so on. 172 SYMMETRICAL FUNCTIONS OF THE ROOTS. Put - for X in tlie given equation; then . 1 3 7 , 1 1 . Thus for the sums of negative powers of the roots of the original equation we have o_,- ^o_, .^^ 6A 6 36~36' 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. Again, suppose we requii-e the values of S^, S,^, S^ and S^ in the biquadratic equation x* + px^+ qx^+ rx + s = 0. '^i '^P - ^j therefore S^ = -p, ^2 +P^i + 2? = 0, therefore ^S'^ =/ - 2q, *^3 + P'^2 + ^^1 + 3r = 0, therefore S^ — -p{p~- 2q) -k-pq - Zr = — ^^+ '^pq — 3r, aS', +^^3 + g^^ + r.S'i + 4s = 0, therefore S^~—p{^'p^-\- Zpq — 3?') — q {p"^ — 2q) + rp — 4:S = p^ — 4:p^q + 4:rp + 2q^ — is. As another example, let a, (3, y, 8 denote the four roots of the biquadratic equation x'^ + px^ + qoe? + ric + s = 0; let^ = i(a;8 + y8), J5 = i(ay + /38), C=^(aS + ^y); and let it be required to find the value of the following sym- metrical functions of the roots of the biquadratic equation, (1) A+B + C, (2) AB + BG+GA, (3) ABG. SYISOIETRICAL FUNCTIONS OF THE ROOTS. 173 (1) A+B + C=^(a(3 + ay+ad+(3y + /3S+yS)=^ , (2) AB + £C + CA=^ (a^ySy + a^S +...)= ^ ^a'i^y = I ('S'.'.S', - S,' - 2S^S^ + 2SJ ; by the method of Art. 249. Tlien the values of S^^ S^, S^ and S^ may be substituted which have already been obtained, and the value of - iSa'^y will be known. Or we may proceed thus, a pT And a/SyB = s, and ^k=- 4, by Art. 48 ; therefore AB + BC + CA = - {pr - is). (3) ABC = ^ (a^/3yS + . . . + a^/3 V + ...)=! ^^'iSyS + | Xa'/^Y- 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, Sa^ySyS = aPyBSa' = S {p' - 2q), fcr 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 ABC = ^{r' + p's-iqs). The values of the functions of 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 SYMMETKICAL FUNCTIONS. XX. APPLICATIONS OF SYMMETKICAL PUNCTIONS. 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 difierences 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 ivhich has for its roots the squares of the differences of the roots of a given equation. Suppose the given equation to be of the ^^ degree, and denote its roots by a, b, c,... . Then the roots of the required equation will be (a — b)', {a — c)", ... (6 — c)", . . . ; the number of these is the same as the number of combinations of 7i thino-s taken 2 at a time, that is, -n{n — l); and this number will therefore denote the degree of the required equation. Put m for -n(n — l), and suppose that the required equation is denoted by a:- + q^x-' + g^^-^+,„+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,,...5^, and then the coefficients of the required equation will be found in succession by the formulae of Art. 244, namely, s^ + q^ — 0, s„ + q^s^ + 2^, ==■ 0, and so on. Let (a) + (ji{b) + 4>{c)+ .., Now let *S'j, S^, S^,... denote the sums of the powers of the roots of the given equation ; thus {x) = nx''-- 2rS^x''-' + \ ., ' S^x"-'-,.. + S^^. APPLICATIONS OF SYMMETRICAL FUNCTIONS. 175 Put for X in succession a, 5, c,... and add; thus A. t ^ 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^, - 2rS^S,,, , + — rrj^ ^^^-'- -■■■ 2 ^ ^ \r 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 by/(a;) — 0, so that /(x) = (x — a){x — b)(x — c)... Then /'(x)= (x — h)(x — c) ... + (x — a)(x-c) ... + ... ; thus f'(a) = (a — h){a — c)...f fih) = {h-a){h-c)... Hence g^=/'(a)/'(6) /'(c).... ISTow let a, p, y,... be the roots of the equation/'(a;) = 0; then f'{x)-^n{x-a.){x- (S){x-y)... y therefore f'{a)f{h)f'{c)... = 'nr{a-a){a-/3)(a-y) ...{b-a){b-(3)...{c-a).... But (a-a)(5-a)(c-a)... = (-l)"/(a)..., (a-/5)(5-A)(c-^)... = (-l)y(/3)..., and so on ; 17G APPLICATIONS OF SYMMETRICAL FUNCTIONS. tlms f'{a)f'{h)f\c)... = n" (-ir<^-V(a)/(iS)/(7) ... ,,,,,,,, =^^yw/(«/(y)..., for (-l)""'-^)^!. Now y(a)/(/3)/(y)... is a symmetrical function of the roots of the derived equation/' (cc) = 0, and maj 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 differences 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 imaginary 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 ^ixj-i; in this case the new equation will only have real negative roots. It ^vill 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 4- c = 0. rri 1 ^ • 4:(b^-ac) The product is 2 . a (2) x^ +px" + qx + r = 0. By Art. 60 the product is APPLICATIONS OF SYMMETRICAL FUNCTIONS. 1/7 If tlie equation be ax^ + 2>hx^ + ^cx + c? - 0, this becomes - — \ (26^ - 3a6c + a-df +4:{ac- Vf \ , or more symmetrically .'^Uad-Uy- 4:{b'-ac){c'-bd) j . (3) x'^ 4- qx^ -{- 7'X-i-S= 0. By Art. 187 the product is (a^'-^T(/3-'- /)=(/- ay, where a^, /3', y^ are the roots of a certain cubic. Hence the product is If the equation be ax* + ibx^ + 6cx' -r4:dx + e = 0, this becomes by Art. 187, ^ («e- 46c; + 3cy- 27(acZ' + e6^ +c' -ace- 25aZ)n . 256. "We shall now shew how to eliminate one of the unknown quantities from two equations containing two unknown quantities, by the theory of symmetrical functions. Let the equations be p^x- + p.x-' +p^c-' +>-+P„ = 0, and q^x" + q^x"~^ + g^cc""' + . .. + ^„ = 0. The coefficients Po, P,, Ps^---, g^t q^y S'a'--- ^^^ supposed rational integral functions of a quantity y, and £c is to be eliminated. T. E. 12 178 APPLICATIONS OF SYiDIETEICAL FUNCTIONS. Suppose that from the first of these equations the values of x could be found in terms of 7/ ; let these values be denoted by a, b, c,.... Substitute them in the second equation, and we obtain 7n equations for determining y, namely qy + q^a-"-' + q^a"-' + ...+q^ = 0, qJ>'^+qfi-' + q.^b"-^+... + 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 /5 is put for y ID. a, that the value is a; then a; = a, y = ^ will satisfy the two original equations. For these values obviously satisfy the second equation j and the first equation is satisfied by o: = a, vjhatever y may he, and is therefore satisfied when we take x = a and give to y in a the value /?. 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. ISow in this product no alteration is*made by interchanging any two of the quantities a, h, c, ..., so that the product is a syimnetrical function of these quantities, and the value of it can therefore be expressed in terms of the coefiicients 2^01 Pi^ Pzf-- ^^ 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 p^x^+p,x^ + p^x-¥p^^O, and S'o^ + g'lO; + S', = 0, where the coefiicients are supposed functions of y. litre with the notation of the preceding Article we have APPLICATIONS OF SYMMETRICAL FUNCTIONS. 179 {q^a? + q^a + q,) {qjf + qfi + q^ (q/ + q,c + q^) = 0, that is, g/ + q^'abc + q,'a%'c' + q^q.^a'h' + q^q.^a'K'c + g,Xo2«& + 9i^/^^ + S'o^'s'^^" + %qi'^CL'bc + q,q,q2^a'b - 0. Also ahc = - - , p and qp may be of the degree p in y, but not higher. 12—2 180 APPLICATIONS OF SYMMETRICAL FUNCTIONS. l^ow suppose that x is eliminated by the method of Art. 257; the first member of the final equation in ?/ then consists of a series of terms, each of which is the product of }n factors, and is of the form q a""*" x qj)"'' x qfi"~' x ... And as we know that the series of terms forms a symmetrical function of a, h, c,..., the aggregate of the terms with the exponents just indicated will be Now the degree of q^q^qf" is not higher than r + s + t+ ..., so that we have only to shew that the degree of ^a"~''6"~ V"^ . . is not higher than n-r + n — s + n-t+ ..., and then it will follow that the degree of the product is not higher than mn. The re- quired result follows from two observations. (1) From the formula 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^hf^c^ ... will involve powers and products of S^, S.^, S^, ... S\j^.f^+j,+^,^; and in each term the sum of the subscript letters attached to the symbol S is X + [X + 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 cVAlgebre Superieure. SUMS OF THE POWERS OF THE EOOTS. 181 XXI. SUMS OF THE POWERS OF THE ROOTS. 260. By Newton's method, wMcli is explained in Art. 244, the sums of the powers of the roots of an equation may be found successively; M^e 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, h, c,... denote the roots of an equation f(x) = 0, so that we have / (x) = {x — a) (x-b)(x- c)...; and suppose the equation of the of" degree. Then Take the logarithm of both sides, and then expand the loga- rithms on the right-hand side; thus log — 7r = -- (a + b + c + ...) X x^ ' 2x' 3x' ^l-^a^^b^ + c^-,...) 1 S Thus on the right-hand side the coefficient of — is "^ : x"" m S 1 hence we have -^ = the coefficient of — in the expansion of f/x) . - log -—' in descending powers of x. X This supposes m positive; if the sum for any negative integral power is requii-ed we can change x into - and find the sum for the corresponding positive power of the roots of the equatior in 2j. 182 SUMS OF THE POWERS OF THE ROOTS. 261. For example, find the sum of tlie m^^ po^ve^s of the roots of the equation x^ —px + q = 0. Here /(?) = l_f^-lV X' \X X' J -log/fe) = -log{l-(?-4)j X \ \x X J ) =^_i,.i(^_iy lpi_:n%...,l(^^_^)\... X X 2\x x'J 3 \x X J ill \x X J The complete coefficient of --. may be obtained by selection X fix) . from the various terms in the value of -log' — 5^ in which this ° x" power of X can occur; these terms written in the reverse order are ■m \x x'J m— L \x x" J mi — A\x x J The coefficient of — - is therefore X m^ m - 1 \ ^ ^ m-'l 1.2 :hus s^^r- mf-'-q, + ^il!!i^ jr-y- m— 4^2 _ • • • + (_!)' m(»»-r-l)...^..(m-2-,--t-l) ^^.,,^^, ^ ^_ Suppose ^=1, then the quadratic equation is a reciprocal equation, and its roots are of the form a and-; see Art. 133. a Thus we have « + - = », and also a SUMS OF THE POWERS OF THE ROOTS. 183 ^_2 , m(m-3) „_, (Jb X * jU m + (-i; r^ p +... We have thus obtained a general expression for clT + —^^^ in terms of powers of a + -; see Art. 138. Again, suppose q = -\; then the roots of the quadratic are of 1 . / IN" . the form a and — : thus we get an expression for a"* + ( — j in terms of powers of a — . a 262. Again, let it be required to find the sum of the m^ powers of the roots of the equation a;" - 1 = 0, -rr /W 1 1 ^ /W 1111 Here the coefficient of -^ is zero unless m is a multiple of w, and then the coefficient is — : so that *S'„= unless m is a multiple m of n, and then *S'„= n. This result is often useful, and we will give three applications of it in the following three Articles. 263. AYe will shew how to find the sum of selected terms of a given series. Suppose that the sum of a^ + a^x 4- a„o:^ + . . . ad infimtuni is known, and denote it by ^(x) : and let it be required to fbid the sum of the series a x"" -i-a ^ x"'^" + a ^ x""^'" + ad infinitum. Let a, (3, y, . . . denote the n^ roots of unity, that is, the ?^ roots of the equation x" -1 = 0. Multiply both sides of the given identity by a"""*, and then change x into ace; thus 184 SUMS OF THE POWERS OF THE E00T3. SimLlarly, 7 9(y^)='^n7 + ^*i7 ^+^27 ^ + •••, and so on. j^dd 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 a »"• + a ^ »;'"+" + « , , x""^-" + . . . OT m+n 7n+2ii = l(a"-Xaa;) +/3"-"'{x) = e. Thus the required sum ^OW a=l, ^ = 2 » V = 2 •2 2 Hence cl>{^x) = e '^ = e 2/cos^-4-^/-lsin-^j; ^{yx) = e -fcos-2--V-lsin-2— ^• And finally the required sum is 1 , 1 -?/ xJZ .- . a:V'3\ SUMS OF THE POWERS OF THE EOOTS. 185 264. Again, by means of Art. 262 vre can prove tlie following 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 ti be a positive integer of the form 6m + 1. Let 1, a, ^, be the three cube roots of iinity, that is, the three roots of the equation ic^ — 1 = 0. Then the product of these roots is 1, that is, a(3 = l, by Art. 45; and 1 + a" + /J"* = 0, provided m be not a multiple of 3, by Art. 262. Thus x^ + xy + y^ = {x- ay) {x — ^y). Hence {x + y)" — cc" — t/" is divisible by x^ ■¥ xy + y^ provided it vanishes when x = ay, and when x = (3y ; and it is divisible by (x^ + xy + y^Y, provided its derived function also vanishes when x = ay, and when x = fiy : this derived function, by Arts. 11, 13, is n{x + yY~'^ -nx''~'^. "When x = ay we have (^ + ^)»_^«_2;»=2/«|(l + a)" - a"-l| = 2/" |(-^)"-a"-l|, and this vanishes when n is an odd integer which is not divisible by 3. Also, when x = ay, n {x + yy-'-7ix''-'=7iy''-' |(1+ a)''-'-a"-^| = w/-^ |(- /3)"-' - a"-'\ ; this vanishes if n — 1 is an even integer and a multiple of 3, because a^=l, and j8^=l. And if n~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)" - cc" — v/" also vanishes. The same results would be obtained by putting f3y for x. Comjjtes Rendus Vol. ix. p. 360. 186 SUMS OF THE POWERS OF THE ROOTS. 265. The last application we siiall make of Art. 262 is to pi.'ove the following theorem. Let S denote the sum of the series n-3 (7^-4)(?^-5) (n- 5)(n - 6)(?z-7) ^ 2 "^ |3 j4 ■^■•* \r 3 . . Then S=- ii n is an odd positive integer divisible by 3 , >S' = if n is an odd positive integer not divisible by 3, S = — if 72, is an even positive integer divisible by 3, 2 . . S^ - if n is an even positive integer not divisible by 3. In Art. 261 put xi/ for q and x-i-y forp, so that 6\ = x" + 7/" ; thus, if 7^ is a positive integer, (x + yf- x"- 2/"= nxy {x + y) Ux + ?/)"-' - -y- xy {x + y)"-' Let 1, a, 13, denote the three cube roots of unity; put x = ay, then the right-hand member of (1) becomes ™a(lH-a)2,"{(Uar-=-'-lZ^a(Uar-=+(!^=l|fi=i)a=(Uar'-...}. But a/3=l, and therefore I3' = a^^=a; also a + ^+l = 0, so that — /? = a + 1 ; thus a = (a + 1)^ Hence the right-hand member of (1) reduces to /I xn „ fi n-o (?i-4)(n-5) ) ^^(l+^)T|l — ^+- j9 ---j, that is n{-l3yy"S. SUMS OF THE POWERS OF THE ROOTS. 187 Also wlien x = ay the left-hand member of (1) becomes ,f |(1 + a)"- a"- 1 j , that is, f |(-^)'' - a" - l| . Therefore (-^)"-a"-l =7i(-/5)"^..; (2). If n is an odd positive integer divisible by 3, the left-hand member of (2) is equal to - 3 by Art. 263; therefore - 3 = - TiyS"^ = - 72;S' ; 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 ; therefore >S'= 0. 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 S== . ^ n If n is an even positive integer not divisible by 3, the left-hand member of (2) is ;S"-a"-l, that is 2/3", since a" -i- /S" -f 1 - ; 2 thus 2B'' = 7iB"S, and therefore aS'=- . n It is to be observed that the series denoted by S consists of a Jlniie number of terms 3 in fact if n = 2m or 2m 4- 1 there are ??i terms in the series. Crelle's Mathematical Journal, 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 Of -n(2 cos Oy^ + ^^f~^^ (2 cos Oy-" - ...H.(_i)-"("-'-^)('^-^-2)-("-2'-+i\2coser"+... Put ^ = -q ; hence, transposing and dividing by ?z, "we obtain .4(l-2cosf). 188 SUMS OF THE POWERS OF THE ROOTS. 266. As another example of the theorem of Art. 260 we will shew how to express x" + y" + {^-x- y)" in terms of x^ + xy + y^, and xy (ic + y). Let a = x^ + xy + y^, h = xy(x + y), and put z for —x — y. Then x + y + z-=Oj xy + yz + zx = xy—{x + yY^ — a, xyz = — h; thus X, y, and z are the roots of the cubic equation t^-at + h=0 ; and therefore - (cc" + y" + z") is equal to the coefficient of — in the a b' expansion of — log ( 1 — ti + ts ) • Now -\og(l-j, + js 1 / ^\ 1 / hV 1 =?l"-7r2?r-7;-^3?i"-7j + "We can then expand (a — j , (a — ),..• and collect the coeffi- cient of any assigned power of — . If 71 be an even number we thus obtain a formula for (x + T/Y + x^' + y"; and if n be an odd number for (x + yY-x^'-y" The following are special cases : {x + yY -x'' -y"^ = 7a^b = 7 (x^ + xy + y^Y^V {^ + 2/)» {x + yY + x^ + y^ = 2a'' + Sab' = 2 (x- + xy + y") {(x^ + xy + y-Y + ^xry' {x + yY). SUMS OF THE POWERS OF THE ROOTS. 189 The general formulae maybe easily obtained by putting 2m and 2m -r 1 for n. Thus it will be found that {x + yy-'" + x'-+y"'' _a- , m - 2 (m - 3)(m- 4)(m- 5) 2^ m^l.2'' ^"*" 14 "" ^ (m-r -l)(m-r-2)...(m-3r + l) ^^_3,^., , , + ...+- r^^^ a +..., and that 2m + 1 [3_ (m - r - 1 ) (m - r - 2)^^^(m-3r) ,_ + , 2r+l 267. 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, h, c,... denote the roots of an equation; suppose them all real and a numerically the greatest. We have S ,, a"'-'' + h"'-'' + c"'^'+ ... o„ a +0 +c + ... -©"*©"- Thus if m 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 hftw far the result of the pre- ceding Article is modified by the presence of imaginary roots. Let fi + y J — 1 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 /5=.y7ri=/,g=tZ7H). Assume - = cos 6, and - = sin $, so tlmt tan ^ = ^ and /x" = /3" -f y' ; thus fji is tlie modulus. Then the conjugate roots may be put in the form [x (cos ^ ^^ sin 0) ; and bj De Moivre's theorem the sum of the 7n}^ powers of the two roots is 2/i,"' cos m9. Thus if the numerical value of the greatest real root be greater S 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' limitinc: value of — ^- . Example, c/ - 2a^ - 5 = 0. Here the series S^, 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 roots 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 1-5; thus the modulus is not very S small compared with the real root, and so the expression -~^ fit 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^ = a- + ir + c- + ,.,, m+l '^m + 2 Therefore SJ„^,, - S\,, = aV {a - hf + aV {a - c)' + 6'"c"'(6-c)'+... AVe will denote this by u^y so that ---iria-ir {i^ricTTi) *A^h) ^-j- ^„. Hence by proceeding as in Arts. 267 and 268 we may obtain the folio win o^ results. (1) If all the roots are real -^^ can be brought as near as m we i^lease to the product of the two numerically greatest roots by increasing m sufficiently. (2) If there are real roots nuuierically greater than the modulus of any imaginary root, there is a limiting value of '"^ , namely, the product of the two greatest of these roots. u u (3) If there be one or more moduli greater than the numeri- fit cally greatest real root there is a limiting value of -^^^ , namely, m the square of the greatest of these moduli, that is, the product of the corresponding imaginary roots. (4) Thus the only case in which -f^ can fail to have a Limit 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 EOOTS. From the values o£ S„, S .,. S .„, and S^_^^. we shall obtain + &'"c'"(6 + c)(6-c)^+...; we will denote this by t\^. Then u^ having the meaning assigned in the j)receding Article, we shall find that there is a limit of — ^ in the cases named in the preceding Article, and that this limit m 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. 269 we can get the product of two roots by Art. 269 and their sum by Art. 270; and in cases (1) and (2) we can get the sum of two roots by Art. 270 and the greater of them by Art. 267. 272. Example, x* + x^ + ix^ - Ax+l^^ 0. Here we obtain the following values : £orS^,S^,...-l, -7, 23,-3, -116, 227, 202,-1571,...; for u^, u^,...- 72, - 508, - 2677, - 14137, -74961, - 397421,...; for v^, v,,...164., 881, 4873, 25726, 136382,... Here no definite limit is obtained by dividing each term in the series S\, 'S'g,... by its predecessor; we are therefore sure of the existence of imaginary roots. By dividing each term of the series u , u_^,... 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 t\, v^,... by the corresponding term of the series w^, u^,... we 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 'o\J L . . . imaginary. ELBnXATION. 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 ic^ = ci"b'" {a - hf and neglecting the other terms, the error is about -^ of the whole quantity; and hence we can judge of the accuracy of our result. For example ; we have given above the values of u^ as far as u^ and it^, so that we can depend upon having found the product of the roots with an error /l\th of only about ( —5 ) part of the whole. XXII. ELIMIXATIOK. 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 nj denote the unknown quantities ; then if one of the equations involves a;"* and no other power of x, we can immediately find a;"' 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 by J. = 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 jDroposed equations are obtained by solving the simultaneous equations 6^=0 and F= 0, ^"=Oand F' = 0, U'=0 and F= 0, U' = and V = 0, U" = and F= 0, U" = and V = 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 -4 is identical with one of the factors of B ; for example, suppose that U and F are iden- T. E. 13 194 ELIMINATION. tical. Then any values of x and y which satisfy the equation U=-0 will satisfy the simultaneous equations A=0 and B = 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 1/= 0, 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 ^^(x^ 2/) = Siud fg{x, y) = 0. Suppose that x^a and y = (3 are values which satisfy these equations ; then the equationsy^ (x, ^) = and / (x, /5) = are satisfied by the value x = a. Hence f^{x, (3) and f„{x, (S) 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 2/ = /5 the proposed equations are satisfied. Suppose then that we- arrange f^{x, y) and/^(a?, 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 ^ (y). Then no values of y will be admissible except such as mako (f)(y) = 0; for unless ^ (.y) vanishes /^ (.t, y) and /^(^j 2/) have no common measure and therefore do not vanish simultaneously. It is not however true conversely that every value of y which makes (?/) 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 cp{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 bv any finite value of y, and anv value of y which makes (f> (y) vanish, combined with the corresponding value of x deduced from the equation /^(x, y) = 0, will msikef^{x, y) vanish. But if q is a fraction, involving y in its denominator, q may be infinite when <^ {y) vanishes, and/,(.r, y) will not necessarily vanish when c^(?/) = ATidifJ^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 coeflicients. Suppose, for example, that we multiply /(.r, y) by a quantity c in order to avoid the fractional coeflicients which are functions of y \ and suppose we now have If we fiud y from the equation cfi{y) = 0, and then x from the equation /^(ic, y)=0, the values so obtained must necessarily make c/\[x, y) vanish ; but it does not follow that /^ (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 Ave shall now proceed. AVe shall suppose that in finding the greatest common measure the usual precautions are taken to avoid fractional coefficients. We may assume that in the equations which we shall denote by ^ = and ^ = 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 the Algebra of MM. Mayer and Choquet. 276. Let the two simultaneous equations be denoted by ^ = and B^O; we will suppose that neither A nor B has a 13—2 196 ELIMIXATIOX. factor Avliich is a function of y only, and that B is not of a liiglier decree 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 G denote the factor by which B must be multiplied in order that it may be divisible by E ; let q^ be the quotient and tJa^ 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 73. Thus we shall have the following identities : c A ^ qB + rR , c^B =q^R -^r^R,, (1) Let d be the greatest common measure of c and r, let d^ be the greatest common measure of -,^- and o\, let d.^ be the greatest cc.c^ common measure of -^ and 9' , let d^ be the greatest common measure of ^^^^ and r . We shall now prove tliat the solutions of the equations ^ = and ^ = will be obtained by solving the 1 followincf svstems : - = and^ =0, d ^-f = and 7? = 0, ^ = and 7?, = 0, ^=0 andi? =0; (?) that is, we shall shew in the first place that all the solutions obtained from (2) do satisfy the equations ^ = and ^=0, and iu ELIMINATION. 197 the second place that all the values of x and y which satisfy the equations A = and ^ = are included among the solutions obtained from the system (2). Divide both members of the first identity (1) hy d ; thus 5^ = 1^-^^ (3)- Now, by hypothesis, - and - are both integral functions of y; qB . thus — 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 the equations - = and B = 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 - = d and ^ = satisfy the equations ^ = and B — 0. Again, multiply both members of the identity (3) by c , and substitute for c^B its equivalent obtained from the second of the identities (1); thus a d d ^ The expression ' ^^ ^ is integral, for r and q are divisible by d; moreover this expression is divisible by d., for d divides^ ^ d and rj and does not divide R. Di\ide by d^ ; then, for shortness, putting M for -. and J/, for ^i^^J ^q liave d ^ dd. ^A = M^R.r^niE^ (4). 198 ELIMINATION. (» Multiply both members of the second of the identities (1) by ~- ; thus a a a ^ ^ CC CO Since d will divide -^} and r, , it will divide -J' R ; but it is CO not divisible by d^ and therefore -y must be. Divide by d^ ; then, C CO for shortness, putting N for -^ and X^ for - ^^ , we have d d """^ ^' ' ^^' dd ^^B^N^B.^^NR (0). The identities (-4) and (5) shew that all the values of x and y which make -p and R vanish, make -^ A and -yy- jS vanish : but -y4- and -7- have no common factor, and therefore all the solu- dd^ «! T tions of the equations y = and i?=0 satisfy the equations "1 ^ = and i? = 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 CC^C ~dd f^ A = (qjf^ + 1^ J/) R, + r, J/,i?, . By hypothesis d^ divides the first member of this identity, and also divides r^; it must therefore divide ( g^ J/j + -y-^ il/ j i?, , but c 7* R^ is not divisible by d^; therefore q.2^^i + -'j- ^^ is diWsible by cfg. »! Denote the quotient by M^; thus cCjC, , ,, _ r ^ = J/,7?, + -^ ^^A^2 (6). dd.d^ ^ d^ ELIMINATION. 199 Multiply both members of the identity (5) by c^, and substitute for c^R its equivalent from the third of the identities (1); thus We may prove as before that the coefficient of R is divisible by cZ^, and denoting the quotient by iVg "^^'^ have The identities (G) and (7) shew that all the val\ies of x and y T which make -j- and R^ vanish, make the first members of these CC C 7* identities vanish: but , / ^ and -^ have no common factor, and dd,d^ d^ ' T therefore all the solutions of the equations ;^ = and R^ = Q satisfy the equations ^ — and £ = 0. In the same way as before if we multiply both members of the identities (G) and (7) by c^, and substitute for Cgi?^ its equiva- lent from the fourth of the identities (1), we obtain to/=^'»^'*^^' ('')■ where 3f^ and iV^ are integral functions of x and ?/. The identi- ties (8) and (9) shew that all the solutions of the equations -7 = and -^2= satisfy the equations ^ = and £ = 0. We have thus proved the first part of the proposition, namely, that all the solutions obtained from the system of equations (2) do satisfy the equations A = and £ = 0; we have now to shew that all the values of x and y which satisfy the equations -4 = and £ ^ are included among the solutions obtained from the system (2). 200 ELIMINATION. The identity (3) may be written KA-MB = '^^li (10). Multiply (4) by B and (5) by A and subtract; thus {M^B - F^A)R + {2fB ~KA)-fR^ = 0, and therefore by (10) and therefore ^^.^-'^v=5''- ^"^- Multiply (6) by B and (7) by A and subtract; thus d 2 and therefore by (11) and therefore (KB-K^)R.-^^P^A = o, ^'^"^-^=^ = -ig^" ^''^- Similarly from (8) and (9) we deduce Tf T T The identity (13) shews that all the values of x and y which T T T V make A and B vanish make -■,-^-~ ^ vanish; so that one of the d d^ d^ d^ T T T T factors -. . -f . -^ , and -^ must vanish. Hence the equations d d^ d^ G?3 T T T T d ^' d, ^' d, ~ ^' ^^^^ d~^' 12 3 supply all the admissible values of y. ELIMINATION 201 Suppose then that x==a and y^fS are values which satisfy the equations ^ = and ^ — 0. First suppose that /8 is a root of the equation -, = 0; then it is manifest that the values x-=a and l/ = ^ satisfy the equations - = andi?-0. a Next suppose that ^ is not a root of the equation - -= 0, but is a root of the equation ;/ = 0; since -^ does not vanish when 2/ = A it follows from (10) that the values x = a and 2/ = /5 make R vanish, T and so they satisfy the equations y = and ^ = 0. If Next suppose that /? is not a root of the equation y = 0, nor of the equation -^^=0, but is a root of the equation y = 0; since i does not vanish when 2/= A it follows from (11) that the d d^ values x = a and y = ^ make R^ vanish, and so they satisfy the equa- tions ^^0 and^ =0. Next suppose that ^ is not a root of any of the equations !1:=0, ^ = 0, ^=0, but is a root of the equation -7 = 0; since d d^ d^ d^ !!. !j _2 (Joes not vanish when y = P, it follows from (12) that the dd^d^ values x = a and y = ^ make R^ vanish, and so they satisfy the T equations y = and i?^= 0. This proves the second part of the proposition. A« A« A« rt« The equation -: -^ -~^-r = which gives all the admissible ^ d c/, d^d^ values of 2/ naay be called the^^ia^ equation in y. 202 ELIMINATION. 277. Examples. (1) x'-r'6yx'^+{?.y'-y+l)x + y'-y'+2y = 0, ar+ 2yx + y-— y = 0. Here we have x+2y for the first remainder, so that r = 1, and y^—y for the second remainder, which is independent of a;. The only solutions are those furnished by -7^ = and B = 0, that is, by y'—y = 9indx + 2y = 0. (2) x'+2yx'+2y{y-2)x + y^--4. = 0, af+ 2yx + 2-1/- 5?/ + 2 -= 0. The first remainder here is (?/ — 2) (x + ?/ + 2) ; so that r = y — 2 and B.=x + y-\-2', the second remainder is y^— 5y + Q, which is T independent of x. The solutions are those furnished by - = and ^ = 0, that is, by y - 2 = and x^-t2yx + 2y^-Dy + 2 = 0; and those furnished by y = and E^O, that is, by y'—5y+ 6 == and x + y + 2 - 0. The, final equation in y is (?/ - 2) (3/^— 5?/ + 6) = 0. (3) x^+ 3?/x'- 3x"+ 2>y^x — Qyx - x ■\- y^— 2>y' — y +3 = 0, ^— dyx'+ 3x'+ Zy'x — ^yx — x-y^+ 3^^+ 7/ — 3 = 0. The first remainder is 2(?/ - 1) (3a;' + v/-— 2y- 3); the second remainder is 8(y^— 2y)x; the third remainder is y^— 2y — 3. The solutions are those furnished by y-\ =0, and o^- 3yx^+ 3x^+ 'dy'x - 6yx -x-y^+ 3y^+ y - 3 = 0, by 2/'- 22/ = 0, and 3x'+ y'-2y -3^0, and by y^—2y— 3^0, and x = 0. The final eqication in y is (y - 1) (2/'- 2?/) {y'- 2y-3)=- 0. ELIMINATION. 203 yx^—bx+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; + 'if+ Qy. In order to carry on the division we now multiply yx~— 5x + iy by 3?/ — 1 0, and perform the following operation : {3y-10)x + y' + 6y\ {3y - 10) yx' - (3?/ -I0)5x + {3y - 10) iyfyx (3y - I0)yx'+ (/+ ey)yx -(y' + 6y'+15y - 50) x + 12y' - 4:0y 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 3y — l0, which will give rise to the same remainder as if we had originally multi- plied by (dy — 10)". Thus we continue the operation as follows: -(2/^-f6/-fl52/-50)(3y-10):«+(12/-40y)(3^-10)|-(/+6/+152/-50) -y+6y+15y-50)(3y-10)a:-(/4-6/+157/-50)y-f6y) 2/'+12?/*+87^'-200/+100y "We have here a remainder independent of x, which is the value of r^; and cZ^ here =y; so that the solutions are those furnished by y'+ I2y'+ 87/- 200y + 100 = 0, and {3y - 10)a^ + y'+ 6y = 0. 278. The following remarks may be made on the process of Ai-t. 276. I. We may always take c such that c and r have no common factor. Eor if cl be the greatest common measure of c and r the division of -, ^ by ^ can be efiected 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 d=l. Similarly we may take c,, 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, c^, c^, Cg,... so that d = l, that d is the greatest common measure of c and r^ , that d^ is the cc greatest common measure of -— and r„ , that d, is the greatest com- mon measure of -r^-^ and r , and so on. II. Suppose that the remainder independent of x which has been denoted by r^ is zero; then li^ is a common measure of A and B. Hence the solutions of the equations ^4 = and ^ = consist, (1) of an infinite number of values of x and ij which may be deduced from the single equation li^ = 0, (2) of the finite number of values of x and y which may be obtained by solving A B the equations -r- = and ^c- = 0. But since r. = it follows from the identities (1) of Art. 276 that R^ divides R and 7^,. Divide the identities (3), (4), (5), (6), (7), (10), (11), (12) of Art. 276 by It ; we thus obtain new identities in which A, B, R, R^ and R^ are replaced by -r- , -— , -r- , -^.^ and — ^ . By means of these identi- ^ ^ R^' K^' R./ i?2 R^ ^ ties we can prove, as in Art. 276, that all the solutions of the A B equations ^- = and -vr = ^ ^''^^^ ^® obtained by solving the following systems: r B -=0 and 77- = 0, d U^ 4^ = and 4^ = 0, ^1 ^2 -3-^ = and^' = 0. ^2 ^2 ELIMINATION. 205 Eor example, suppose x^+yx'-(f+\)x + y-y^=0, and x^- yx'- {y'+ Gy + 9)x + y^+ 6y'+ 9y - 0. Here the first division gives 2 < yx^+ {3y -)- 4) a; - (y^+ 3y^+ iy) \ for the remainder, so that we may take R = yx-+ {2>y + i)x-(y'+ 2>y-+ Ay). 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 d> {y^ -r 2>y + 2) {x — y) for the remainder r^R^. 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 x-y = 0, (2) of the finite number of values of X and y which may be obtained by solving the equations 2/^+ 3y + 2 = and yx + y"^ + 2>y -\- i = 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 examj)le that {y —l)x' —2x^y'' =0 ; then so long as y is not equal to unity the two values of x furnished by this quadi-atic equation are finite. If y approaches indefinitely near to unity one value of x increases indefinitely; see Algebra, Chapter xxii. Thus when 2/ = l 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 x'= ; 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 FUNCTION IN SERIES. 279. Suppose we have an equation connecting two unknowTi quantities x and y. If we could solve the equation so as to obtain the values of y in terms of a:, 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 Lagrange; it was suggested by a process given by Newton which is called Newton's 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. P. M. Talbot, published in 1861. 280. Let the equation be denoted by Ay"- + B/ ...+ Ky''+... + Sy" ^Q), where A, B, ...K, ...S, are all functions of x. We suppose a, ^, ...K, ...o- to be arranged in descending order of algebraical magnitude; and throughout the investigation such words sls greater 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 ^ be of the degree b, , K of the degreed, , 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 + jSt, k + Kt, s + at. EXPANSION OF A FUXCTIOX IX SERIES. 207 Begin by supposing that ^ is + oo ; the first term is then greater than any of the others. As t diminishes eacli 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 gi-eatest value of t which can be obtained from the equations a + at-h + ^t, a + at — c + yt, ...a + at--k + Kt, ...a+ at = s + at, that is, the greatest value of f must be found from the set b — a c — a k — a s — a i ) J ~ -■ ~ .. a— a n 1 J a — p a — y a — k h — a Let be the gi^eatest 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 r. a— K a — K Let t continue to diminish from the value t until Tz + kI 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 !:,-{■ Kt = I ■\-\t, h + Kt = m + jxt, h + Kt = s + at, that is, the greatest value must be taken from the set l — k m — k s — k K-\' - ■' K— fJL 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 + 'ult, 7i + vt = s + x) + y^ (x^ + 2x^) - y {Ax' + 3) + 3x" = 0. _, -, b — a c ~a .... , The set of terms p, , , ... is, in the present case, a — fd a — y - — ^ , - — 3- , - — ;r-. The second and third of these are equal to 4-24-14-0 1, which is greater than ^ , which is the value of the first term. Thus T=l. Hence we put y = x{u+U), and substitute in the proposed equation. The highest power of x is then x^j and the term involving it is x' Uu+ Uy - 4: {u+ U) + 3] , The coefficient must vanish when x is infinite ; this gives U'- 4:21+3= 0. It is obvious that u=l is a solution, and as the derived func- tion 4:U^-4: also vanishes when w=l, the root 1 is repeated. T. E. 1^ 210 EXPANSION OF A FUNCTION IN SERIES. Divide w^- 4w 4- 3 by (u-lY; the quotient is u^ + 2u-^3. Thus the other values of u are furnished by the equation w"+ 2w + 3 = 0, and they are - 1 ^ J-2. We infer then that the proposed equa- tion will only furnish two real values of y in terms of x, and that X is the first term in each of these values when they are expanded in series according to descending powers of x. "We may now put x{l + U') for ?/ in the proposed equation, and proceed to find the values of U; we will resume this example presently. 284. The following inferences may be drawn from Arts. 281 and 282. (1) If a 4- a, h + (3, ... , Jc + K,..., s + a- are all equal, the quantities t, t, t", . . . are all equal to unity. (2) If of the quantities a + a, b + (3, ... , k + k, ... , s + a; two or more are equal and greater than all the rest, then unity occurs among the set t, t', t", ... For it is obvious that ^ = 1 is a suitable value in the investigation of Art. 281, since this value makes two or more of the terms there given equal, and greater than all the rest. These two inferences involve the theory of the rectilinear asymptotes of algebraical curves. In the remainder of this Article we suppose that a, /?, y, ... are all integers, and that cr is zero. (3) The first equation for u in Art. 282 will have a- k roots, the second will have k — v roots, and so on ; thus on the whole we get a values for the first term of y, as should be the case, since the proposed equation is of the degree a in y. (4) Suppose that the degrees of all the functions of x from K to N inclusive are equal and higher than any of the others. Then out of the values of y there will be a - k which begin with a positive power of ic, and k — v which begin with the zero power of X, and v which begin with a negative power of x. For the K — v values of y which begin with the zero power of x arise EXPANSION OF A FUNCTION IN SERIES. 211 from the fact that by hypothesis the value ^ = makes all the following terms equal and greater than any which follow them, k + Kt, 1+ Xt, ...n + vt. The a — k values of y which begin with a positive power of x arise from positive values of f, and the corre- sponding values of u obtained relative to the exponents a, (3, ... k. The V values of y w^hich begin with a negative power of x arise from negative values of t, and the corresponding values of u ob- tained relative to the exponents v, ...a; where a =0. (5) If ^, jB, ... S, are all of the same degree except i¥, and J/ is of a higher degree than the rest, there are a- fx values of y . , ni — a in which the highest power of x has the positive index — — - , and /A values of y in which the highest power of x has the nega- . m-a tive index - 285. A remark should be made respecting the equation in U which is obtained \vhen the second terms in the values of y are required; see Ai't. 282. Suppose we assume y = x*iii+ U), where u and t are known, and substitute this value of y in the proposed equation. We thus obtain an equation in U of the same degree as the original equation in y. However in general only some of the values of U will be admissible. For, by supposition, U vanishes when x is infinite, and so we must reject any value of U w*hich commences with a positive power of x or with the zero power of x. These rejected values of U must belong to the other values of y with which we are not at the moment concerned, since by supposition we are seeking only that particular value of y which commences with icx% or those particular values which so commence if there are more than one, where u and t have known values. 286. Let us now resume the example in Art. 283. We have to substitute x {u + U) for y, and make u = \. We shall thus ob- tain the following result after dividing by cc, U'(x\..)+U\4:x\..)-¥U\^x\..)-U(\0x*...)-2x\.. = Q. 14—2 212 EXPANSION OF A FUNCTION IN SERIES. Here in tlie coefficients of the powers of U we liave only ex- pressed the highest powers of x. Form the fractions according to Art. 282; thus we obtain 5-5 5-5 4-5 4-5 4-3' 4-2' 4~1' 4^0' Here the first two terms are zero, and are algebraically greater than the others ; but a zero value is to be rejected as explained in the preceding Article. We therefore proceed in the manner of Art. 2ol, supposing that t = 0, and that we have to find t'. Thus ^ e form the fractions 4-5 4-5 2"^' 2"^' Of these the second, which is — -, is algebraically the greatei*. Accordingly we put U=ux~'^, and to find %(, we obtain the equa- tion Qic^ - 2 = 0, so that w= TTT. Thus the first term of U is ±v/3 1 1 .^, or -,-:zr=. Therefore, as far as we have gone, we have si'dx \f3x 287. The nature of the values of U may be seen by examin- ing the formation of the general equation in U. Let us first put x'u for y and then change u into u + U. AVhen we put x^u for y the left-hand member of the proposed equation will take the form where n^, n^, n^^ ... are supposed in descending order of magnitude. Denote this expression by (f>{u); then the equation in U will be ^ ; hence one of the values of U begins with a negative power of x, namely, with £c~("i~"2). And this is the value of U which we are seeking, because Xi(^0' ^ ^^ the equation from which u is to be' found according to our process. If however the equation Xi (^) = ^ has equal roots, we obtain more than one suitable value of U. Suppose, for example, that the particular root which we have selected occurs four times ; then ^^ will be of the degree n^ in cc, while cj)^, (jj^, ^^ , will only be of the degree ?i2. Hence, by Art. 284, there will be four suitable values of U, each commencing with x raised to the negative power --(n^-n^). "We have here supposed that Xo(^^) ^^^^ i^^ derived functions do not vanish for the value of u which is considered. 288. In what we have hitherto given we ha^•e investigated values of ?/ 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 ij proceeding according to ascending powers of x, so that the first term in a value of ^ 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. \ye must suppose a, ^, ... cr arranged in as- cending order of algebraical magnitude; and A^ must vanish when x vanishes and not when x is infinite, so that of must be the lov-est power of 03 in ^ and not, as before, the highest power; a similar 214 MISCELLANEOUS THEOREMS. change of meaning must be made in B^ and 5, and in the remain- ing similar quantities. Then when ^ is + oo the following quantities are in ascending order of magnitude, a + at, b + /3t, ... k + kI, ... s + at. As before, the gi-eatest value of t is to be found from the equations a + at=^b + pt, a-{-at= c + yf, ... a + at = k + Kf, . .. a -^ at= s + crt. XXIY. 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' Cf- . A'^ . _ + r + + . . . + - A = 0. x-a x-h x-c '" x-k If possible suppose that p + qj-l is a root; thenp-^^-^ is also a root. Substitute successively these values for x and sub- tract one result from the other; thus ( A' B' C^ K' ) ^ {{p-ciY-^q' ^ {p-hy+f "" {^cy'-^'r ^'"^ii^ ^f'^f) ^ ^' and this is impossible unless q = 0. Or we may prove the theorem thus. Denote the left-hand member of the proposed equation by cf)(x), and suppose a, b, c,...k, in ascending order of algebraical magnitude. When a; is a little greater than a the first term of cfi(x) is very large and posi- tive, and by taking x sufficiently near to a we may ensure a positive value for {x) changes sign for some value of x between a and b. Similarly, (j>{x) changes sign for some value of x between b and c; and so on. In this way we may shew that the roots of the equation (f>(x) = are all real and unequal. MISCELLANEOUS THEOREMS. 215 The form in which the equation (2c) = is presented, enables us to recognise more easily the property we had to prove. But our result will not be affected if we clear the equation of fractions, so as to bring it to the standard form ; that is, in fact, if instead of (f) (x) = we consider the equation "-^,.) vanish; then x^ {a;-' + c^a^-' + c^a;-' +... + c^,_,a^ + c„_^) = h. From the assumption with respect to c^_j, c^_^,...c,^, c^, it follows that CTg, «3, ...a, are the roots of the equation z"'' + c^z''-^ + c.f-^ + ... + C^_^Z + C„_j = 0. Therefore the left-hand side of this equation is identically equal to Hence substituting a, for z the equation which determines x may be put in the form x,{a,-a^){a,-a.y..{a^-a^) = h. Thus x^ is known; and the values 0^ x^, x^,...x^, can be deduced by symmetry. 216 MISCELLANEOUS THEOREMS. 291. Required the values of the n quantities x, y, z,... from the following n equations, X y z k^ — a k^ — b k^ — c X y z A*2 — a A*2 - h k„ — c X y z - k — a k —h k n n n We may regard the n quantities k^ , k^,... k^ as the roots of the single equation X 7/ z - k — a k — b k — c which is of the ?i*^ degree with respect to h. Assume k = a — t', it will follow that a — k^^ a — k^, a — k^,... are the values of the roots of the following equation in t, X y z ^ 1 + - + — V- — + + . . . = 0. tt+b — at + c — a Multiply by the product of the denominators so as to put this equation in the usual form ; thus r + Af-' + A^f-' + . .. + ^„ =-- 0, where the tenii independent of t, that is A^^ is xib — a)(c — a)... Therefore, by Art. 45, (a - kj {a - y {a -k^)...= {- iyx{b -a){c-a)..., (a-k){a-k)(a-k^)... that IS, x = ,' , , , ^ — i^a— b){a — c)... From this expression the values of y, z,... may be deduced by s}Tnmetrical changes in the letters a, b, c... Grunert's Archiv der Mathematik luid Physik, Yol. xxiii. p. 235. MISCELLANEOUS THEOEEMS. 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) ^" (^l)(c^-l)...(c"'-l) "" Assume (rr+c)(a^ + r)...(x4-c'')=a;"+7;,T''-^+...+jt?„_^,r + 79„ (1). Then bj Ai't. 45 we have to find the value of p,„. In (1) change X into - and multiply by c"j thus c From (1) and (2) we obtain Equate the coefficients of ic""'"'^' in the two members of this iden- tity; thus c"'(c"-'"+^-l) ,., therefore Vm=Vm-x ^m^^ ^ ^' And ^j = c + c" + . . . + c" = — ^ Y^ ; then by means of (3) we can determine successively 'P^i Vz") V^^-") ^^^ ^y^s, we shall arrive at the required value for p^. 293. Let there be n quantities a, h, c,...; let s^ denote their sum, s^_j the sum of any n—\ of them, and so on; and let S denote («,.)'- 2(^„-,)' + HK-i'- - + (- ir-'2(«,)'. Here 2(sJ'' denotes the sum of such terms as {sj'" formed by taking all possible selections of m quantities out of the n quanti- ties a, h, Cy... Then we shall shew that aS*=0 if r is less than n, 218 MISCELLANEOUS THEOREMS. and that S is divisible by ahc... if r is equal to oi or greater than ?i; and in particular that S=\nabc...y whenr = w, \n + l and S = -=^= (a + h + c + ...)ahc..., when ?' = ?i4-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, and the latter part thus, where 1^ indicates certain of the terms formerly included under 2, and ^2 indicates the remainder. Now supj^ose a - 0, then S vanishes; for we have in this case Similarly, we may prove that S vanishes when 5 = 0, and when ' c= 0, and so on. Thus we conclude that S is in general divisible by each of the quantities «, 5, c,... and therefore by their product. But the product will be of ?^ dimensions, and therefore if aS' be of less than n dimensions it must be identically zero. And as S is of r dimensions it follows that *S' vanishes when r is less than n, and is divisible by abc... when r is not less than 7i. When r = ?i we have therefore S=Xahc..., where X is some numerical quantity which is to be determined. To determine A suppose that a, h, c,... are all equal to unity; then >S' becomes „._„(„_l)"+!L^L:il)(„_2)--..., that is 'n, by Algebra, Chapter xxxix. MISCELLANEOUS THEOREMS. 219 Next, suppose r = n + 1. Then aS' is divisible by abc...; and as S is of w + 1 dimensions, it must have a factor which is of one dimension and symmetrical with resj^ect to a, h, c...; this factor must therefore be a + 6 + c + . . . Hence S = ij.abc...{a + 6 + c + ...), where /x is a numerical quan- tity which is to be determined. To determine /x suppose that a. b, c,... are all equal to unity ; then S becomes and this must equal [xn. Hence by Algebra, Chapter xxxix. we \n+ 1 have fj. = -— — . 294. Let [c]^ denote c(c - l)(c — 2) ... (c — r + 1), whatever c may be; then will [» + *!= W. + « W.-.i + -t"^ M.-M + • • • + U^h For suppose that a is a positive integer; then we know that this theorem is true for an?/ positive integral value of b, for it follows by equating the coefficients of a;" in (1 +0?)""^* and in (1 + x)" x (1 + a;)*. Hence since this is true for more than 7i 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 is 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 {x) . 1 /i ^ ^""'^ log -^-^— = log C 4- log I 1 + , h x"-' b /, x''\ where z= = — ( 1 - -- ) . cxccx\o/, "We have now to pick out the terms involving - ; we shall obtain such a term from z, from z"'^\ from z^'''^\ and so on. Hence we shall find for the root the series b_b^ 2nb'"-' 2,n{Zn-l)b^" 222 MISCELLANEOUS THEOREMS. 298. Let <^(a:) = be an equation of which a^, a^,...a^^^, are roots, so that we may suppose <}>{x) = {x-a;){x-a^) ...(x-aj^(x); Take the logarithms of both sides; then, as in Art. 295, we infer that — (a^ + a^+ ... +ctj) is equal to the coefficient of - in the expansion of log --,7^ . See Murphy's Theory of Equations, pages 82 and 83. As in Art. 296 we may conclude that the process will give the sum of the numerically least m roots. 299. We shall now give some theorems relating to the decom- position of a rational fraction into other fractions, which relatively to the original fraction are called partial fractioiis. Suppose that <^ (x) is a function of x of the n^^ degree ; let the roots of the equation (x) = be all unequal and let them be denoted b}'" a, 6, c,.,.k. Let \l/(x) be a function of x which is of the {n — If^ degree or of a lower degree. Then the following relation will be identically true, it;(x) A B C K ^)-{ = + r + + + r (fi [x) x — a X — X — c X — k provided proper constant values be assigned to A, B, C,...K. For in order that this relation may be identically true it is neces- sary and sufficient that the following should be identically true; ^(^)=j*M+5l(^) + ci^ + +A-^>. ^ ^ ' x — a X- x — c X — k MISCELLANEOUS THEOREMS. 223 Tlie members of this equation are not of a higher degree than that expressed hy n-1, hence the relation will be identically true if n values of x can be found for which it is true; see Art. 39. And by properly choosing A, B, C,...K the relation can be made true for the n values a, b, c,...k, of x. For suppose x = a, then all the terms on the right-hand side vanish, except that which involves -4; and we obtain \^x — aj,^=(i that is, by Art. 74, il/(a) = A(fi\a). This determines A ; and similar values will be found for x> , C , . . . A . 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 x, and /(x) is of a lower degree than <^ (x). We may then decompose —)-{ into partial fractions in the manner shewn in the preceding Article. Since we have x{;{x)=^i>{x)F{x)+/(x); it follows that ij/ix) andy*(ic) have the same value when <^(a:) vanishes. Hence the partial fractions corresponding to ^-(, when determin- {x) 22-i 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 l": 1 9^ 1 {x+\){x + 2)...{x + n+l) x + \ lie +2 1.2 x+K> x + n-\-\' For we may assume that the left-hand member can be put in the form ^, ^. -^, ^ ^1 x+1 x + 2 x+3 a; + 71 + 1' and then we may determine A^, A_^, ...A^_^_^: this is effected by multiplying both sides by (a; + l){a: + 2) ... {x + n+l\ and then substituting for x in succession the values - 1, — 2, ... Again, if n be any positive integer 1 n n(n — '[) x+l {x+l)(x + 2) {x + l){x + 2){x + 3) ' (-l)"k ^ 1 (x + l){x + 2) ... (x+n+ I) ~ x + n+l* For we may assume that the left-hand member can be put in the form ^1 -^. -^, ^ 4., ^- + •;. + ^ + + ^' • X+l x+2 X+3 £C+?i+l' multiply both sides by (x+l){x+2) ...{x + n+l) and then sub- MISCELLANEOUS THEOREilS. 225 stitute for x in succession the values — 1, — 2, .... Thus we shall obtaiQ .d,=(i-ir=o, ^,=«(i-ir'=o, A=^^i-ir'=o, and by proceeding thus we find that A^^ A^,...A^ are aU zero, and that ^ ^, = 1. n + l Again, if m be any positive integer ^ ^^ (x+1 {x + l){x+'2) \-y m{m-\) f y \\ y {x+l){x+-l){x+'i) \l-y^ ^ '— {x-\'\){x + 2) ...{x + m + \) \1 - ?// 1 m7j m(m-l) ,f (- 1) V + :; : 7^ + + x-\-l £c + 2 I .'1 a; +3 x + m+l' Tills may be demonstrated in the way already exemplified by assuming that the left-hand member can be put in the form A, A„ A„ A, -1 + 2_ ^ 3_ + ^ -^5±i- £c + l a; + 2 x+'6 a; + m+l* then we deduce A=(i-2/+2/r=i, and so on. Or we may establish this result by the aid of the second ex- ample. Eor 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 factors, 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 which we now proceed. Suppose that (f){x) is a function of x which involves repeated factors ; for example, let {x) ^ (1) Any factor x-k which is not repeated will give rise to a single term - , . ° x-k (2) The factor {x - ay will give rise to the series of terms A A^ ^ A^ ^ ^A^ (x-af {x-ay {x-ay ^ x-a A similar series of terms will arise from each of the other repeated factors. (3) There will also be an integral expression if il/{x) be not of a lower degree than {x) Now let A be determined by the equation ij/ (a) -Ax{a)^0; then xp{x)-Ax{i^) vanishes when x=a, and is therefore divisible by a; - a. Therefore with this value of A we may put il^{x)-Ax{x) = (x-a)x{;^{xy and therefore ij^jx) _ A xf^,{x) which is more than 70 ; and the rule of Art. 89 would give 1 + / — = — , which is more than 11. The following is a brief statement of the theorem. Divide the Avhole expression into successive positive and integer lots, j[ -B +(7^-i>, + ... ; 2h 5'j '% *v representing the last expo- nent of X in each lot. Divide A^^-B^ by x\ and ascertain a value of X, say X, which makes the quotient positive ; let I be this quotient. Divide Ix^ + C^-D, by x\ and ascertain a value of x, say /x, which is perhaps not greater than X but must not be less than X, which makes the quotient positive ; let m be this quotient. Continue the process with mx' -irE^ — F^, and so on to the end. The last value of x used is greater than any root of the equation ; and the first value of x, namely \, is very often the last also. XXV. CAUCHY'S THEOREM. 307. We shall devote the present Chapter to the demonstra- tion of a remarkable theorem given by Cauchy, the object of which is to ascertain how many roots real or imaginary lie within as- signed limits ; in fact, the theorem proposes to effect with respect to the roots in general what Sturm's theorem effects with respect to the real roots. CAUCHY S THEOREM. 308. Take any rectangular axes, and let cc, y be the co-ordi- nates of any point. Let <^ iz) be any rational function of % ; then (^ {x + y J — 1) can be expressed in the form p + q J — \. 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 the number of radical points which lie within the contour is - (k - I). M O X 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 ^ (s) = ig 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 2)ositive 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 moving point passes round the contour. The theorem is proved by fi!rst considering tlie 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 If) 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, 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, 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 nof; 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 p 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 j9 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. 2.34 cal'chy's theore:m. Let a, h be the co-ordinates of this radical point ; and put x = a + r cos ^, and y=^h^ r sin 6 ; thus x + y J~^ -=a + bj-l+r (cos 6 ■\- J -I sin 6\ = a -r-h J -\ -^ V. say. Suppose now that the equation 6{z)=^0 has the root a + h J-l repeated m times ; then cfi {a ^ b J - I + r) takes the form cv^' + c v"''^'^ + c„v"'^-+ ..., where c, c^, c^,... are certain imaginary expressions of the standard form; so that we may suppose c = 7i(cos a + 7 - 1 sin a), c^ = A, (cos a^ + ^ - 1 sin a^), ... Hence, by De Moi\T.'e's theorem we shall obtain p h cos{7n0 + g) + A,r cos{(7?i + 1) ^ +a.} +/y'cos {(//i + 2)6 + a^}+. . . q ^ h sui{mU -f a) + A^r sin{(wi + 1)0 ^a^\ + hysin[{ni^2) 6^ + aj+ ... We may suppose r so small that the number of changes of sign shall be unaffected by r ; that is, we may proceed as if - = cot (??i^ + a). And as 7nO 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 -f to -. Thus we have k = 2???, and l^O-j so that ^ (k -l) = m, 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 ABCD 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 ;^ 2 {h - I). 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 tivo contours, and is therefore cauchy's theorem. 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 -%ik- 1) is unaffected. Hence the interior lines of division may be omitted, and the moving point constrained to describe the contour ABCD 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. SujDpose the contour ABCD to be a cii'cle with the origin as centre and an indefinitely large radius. The value of - will now depend only on the term in- volving the highest power of z in ^(;s); and if we suppose that term to be h (cos a + ^/ — 1 sin a);s", we shall have - = cot (nO + a). Thus we shall obtain k = 2?^, and ^ = 0; 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 27r is gained in the circuit. The demonstration will not be affected by the admission of the kind of figure here contemplated; for the injinitesimal contours 236 Newton's rule and Sylvester's theorem. may still be supposed, if vre please, ovals wliicL. 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 maybe 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 we have only tt, so that m occurs instead of 2»i as the number of changes of sign. 314. Cauchy's Theorem is given in the Penny Cydojjcedia, Article Theory of Equations, in Mr De Morgan's Trigonometry and Double Algebra, 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. XXYI. NEWTON'S RULE AND SYLVESTER'S THEOREM. 315. Ne%vton 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 the Philosophical Transactions for 18G4, the publications of the London Mathema- tical Society, No. II., and the Philosoj^hical Magazine for March, 1866; from these sources the exposition which we shall now give has been essentially derived. 31G. We begin by enunciating in substance Newton's rule. Newton's rule and Sylvester's THEORE^r. 237 "Let fix) = be an algebraical equation of the n*^ degree; and suppose /./ \ „ n-i n(n — \) J {x) = ajx + oia.jxr + — ^ — ~ a^x ' + ... + na^_ ^x + a^; then «Q, 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^, A^, A^,...A^ may be termed the quadratic elements oif{x). "We shall call /"-'(«)./"-'(<)> /'(<),/(«), as t increases. No 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 Q^^{t +h) to be expanded in powers of h, and take the coefl5cient of h. -rJ/"'-'W+¥"'(<) + •••}{/""(«) + ¥■"*'(«) + •••}• The coefficient of h is Now it is easily seen that fm+l thus ©'„(«) = —/"(«)/"*'(0-7./'""W/"'"W 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 ^ = c, sa7/''(c)=:0. Let h be an indefinitely small quantity ; this will be the mean- ing of h throughout the investigation : then f^ (c + h) has the sign of hf^^c). Thus the associated terms r''{o+h), r{c+h), r-\c+h), have the same signs as f'*'(c), hf'*\o), /'-'(c), {f'"{c)]\ -f'*'{c)/'-'{c), {/'-(o)}'. li/'"*'^(c) aiid/''~\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 the series of quadratic elements intennediate between the first and last vanishes when t^Cf say G^{c) = 0. Since G^{c) = it follows that/''~^(c) and/'^^c) have the same sign. Thus the associated terms r^^c+h), r{c+h), r-\c+h\- G'r^^io + h), G^o + h), G^_,{c + h), have the same signs as /-'(c), f'{c), f'-\c), Gfr.A'=), ''^^VW. <^,-W, and by Art. 322 the sign of G\{c) is the same as that of Newton's rule and Sylvester's theorem. 243 If/'"'^^(c) and /'■(c) have contrary signs, the terms here con- sidered have no double permanence. If/'"'"^(c) and /'(c) have the same sign, and G^^_^_^(c) and G^_^{c) have contrary signs, there is one double permanence whether we suppose h negative or positive. If/'""^^(c) has the same sign as/'"(c), and ^,+i(c) the same sign as G^_ (c), there is no double permanence when h is negative, and there are two when h is positive : thus in this case two double permanences are gained when t increases through the value c. 325. Suppose that several consecutive terms of the series of simple elements vanish when t = c, say /'•+-^(c) = 0, /'■^-Xc) = 0,.../'(c) = 0. Thus we suppose s consecutive terms to vanish, and as /" (c) is a constant which cannot vanish, r + s cannot be gTeater than n : we suppose that r is not zero. We have to consider the changes in the signs of produced when t increases through the value c. Let ^ (c) stand for /'"*■' (c) j then when t = G + h, the signs of the simple elements here considered are the same as the signs of ^(c), Ac^(c), h^{c),...h'4>{c), r-\G). We proceed to investigate the signs of the quadratic elements : ^r+.{c) = { /*"*"* (<^) [ J which is positive, G^_^(c) = \ /""^{c) [ , which is positive, 16—2 244 NEWTON'S KULE AND SYLVESTER'S THEOREM. 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 We shall now shew that the other quadratic elements which we have to consider are positive. For let IX stand for r + 5 - m ; then by expanding and picking out the term which involves the lowest power of h we obtain { '^W^'X A. I \ ^^''"' A/^ ^^""^ the sign of this is the sam.e as the sign of /x+ 1 that is as the sign of r + s — 77i + 1 71 — m + 1 that is as the sign of T ^r s — m n — m 1 r + s — m n — m Now r + s is not greater than n so that the sign is never negative ; the case in which •/• + s = n will require further examination. In this case and as/''(^) is of ?^-r dimensions in t it follows that all the roots oif{t) = Q) are equal to c. Thus /'(^) is of the form C(t-cy-'' where C is a constant. I Newton's rule and Sylvester's theorem. 245 Then f-' (t) = C(n- r) (t - cy-^-\ ' f^^ (t) = C{n- r) (n -r-l){t- c)"-"--^ ; and thus it will be found that G^+.^{t) is identically zero. And in like manner it will be found that G^+^{t), Gr+z{i)y ••. ^„-iW are all identically zero. We will adopt the convention that these quadratic elements which are identically zero shall be supposed to have ih.Q positive sisn : and thus the case in which r + s = n will lead to the same results as that in which r + 5 is less than n. Thus finally the signs of the terms of the associated series which we have to consider are the same as the signs of 4>{c\ lMc\ h'cf^ic), h'cf>{c), /^-(c), "We can now ascertain the number of double permanences; the following results will be easily obtained : Suppose s even, and <^(c) and f'^ic) of the same sign; when h is negative there is one double pennanence, 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 4>{c) and f'^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 - 1 : thus s — 1 are gained. Suppose s odd, and <^(c) and f^~\c) of contrary signs; when A is negative there is one double permanence, and when h is positive there are s : thus s - 1 are gained. Hence an even number of double permanences is gained when t increases through the value c. 246 NEWTON S RULE AND SYLVESTERS THEOREM. 326. Suppose tliat several consecutive terms of the series of quadratic elements vanish wlien t = c, saj 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 alements, we have the following conditions holding among the simple elements comprised between /^+,{c) and /^_^{c), both in- clusive : /r+,W» fr+s-2{(^), fr+sJp)^-'- ^ro aU of thc same sign; /r+*-i(c),/.+.-3(c), fr+^{c),... are all of the same sign. If the teiTQs 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 Q^^{t) be any two consecutive quadratic elements comprised between G^^^it) and G^(t), both inclusive: then ^^+1 (c + A) and G^ (c + h) shall have contrary signs when h is negative and the same sign when h is positive. For by Art. 322, Put c + h for f, and expand in powers of h. Suppose that in G^ (c + h) the term which involves the lowest T) power of A is — h'', so that H is the value of the p*^ differential P coefficient of G^{c) with respect to c. Then the term which Tj) involves the lowest power of A in G'^{c + h) will be zr^'K NEWTON S RULE AND SYLVESTER'S THEOREM. 247 Hence from the above equation the term which involves the lowest power of A in 6^„,+j (c + h) will be /'"Wlp-1 Rh' Hence finally G„.{g + h) has the sign of Rh^ and G^^^ (c + h) has the sign of Rh^~^ ; so that G^^^(c + h) and G^^c + h) have con- trary 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 G^^^(c + h) and G^[c + h), both 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 Gj._^{g) 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^_^(g) 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 G^^^{c) and G^_^{c) of the same sign; when h is negative there is no double permanence, and when h is positive there 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. f"{t) and G^if) are constants, and can never vanish; and G{t) is essentially positive. Suppose however that /'"'(c) = 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->rh, as f\c), h/^c), hrf'ic), h-T{c)> hT{c), 3_ _s_ 2 '^'-^ s-1 + 5 2-7,.i, ^-y,_2 7~T~y^'> ^^ Here when h is negative there are no double permanences, and when h is positive there are 5: thus s double permanences are gained when t increases through the value c. o"^ 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 ju, cannot exceed 'Uj{jj) — ■st (X). If -we know that some of the double permanences gained arise from the vanishing of any of the elements except fit) we can of course make a corresponding reduction in the extreme number 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 tzr (/x) - ot (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, 1 7r+l = 2-y. NEWTON'S EULE 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. thus ^r+l ^r _'^r^2 ^r+, ^r+1 therefore u^^^ ^1u^^^ + u^ = ^. The solution of this equation is where A and B are constants. A-¥B{t-\) Hence r.= ^^Bt (7 + r-l (7 + where G stands for -^ , The student who is not acquainted with the Calculus of Einite Differences may easily verify that this value of y^ satisfies the relation 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 : 250 newton's rule and Sylvester's theorem. Form the simple elements corresponding toy(a3 + /\,) and to f{x + fx). Let p (X) and p (/x) denote the corresponding numbers of permanences of sign ; and suppose /x greater than X. Then pin) ~P{^') ^^ either equal to the number of roots of the equation /(x)=^0 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 proposition so as to convert the resemblance into a coinci- dence. For take the equation there obtained, ,,, ln-r)p , {n-r-¥l){n-r)p^, ,_i „ ^ Pr^X ^ 1-2 P>r^, this equation has at least as many imaginary roots as any of its derived equations. Take the (r- 1)^ derived equation, which is (r+l)r „ r(n-r)p^ (n-r+l){n-r)p^_^ r-rl This equation has imaginary roots if (n - r) rp^ - (t?. - r + 1) (r + V)Pr_^ 2^r+i is negative ; and hence in this case the original equation p^x" +p^x"~'^ +P2^"~' + • • • + 1\ = *^ 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 which the reason and extent of that rule remained REMOVAL OF TEEMS FKOM AN EQUATION. 251 undiscovered hj mathematiciaiis, among whom Maclaurin, Waring, and Euler are explicitly included, ^ve 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 Tschimhausen. 334. Suppose we have the equation x"+2\x''-''+p^x''-'+ +Pn-i^+P„ = ^ (!)• Assume y = a^ + a^x^ a^} + + 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 cc 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, ...?ij and by means of (1) de- press the exponents of cc, so that none of them shall exceed n-\, in the following way. substitute for cc" its value from the preceding line, and we have aj"+^ expressed in terms of a^""^ and lower powers of x ; then mul- 252 •REMOVAL OF TERMS FROM AN EQUATION. tiplj by X and substitute as before, and we have o(^^^ so expressed ; and so on. Thus we shall obtain results of the following form : 2/' = &o + ^1-^ + S^a^' + . . . + &„_,a;" \ 1 n — 1 ,1-1 (3). Here h^, h^, ... 5„_j are integral homogeneous functions of the second degree of the undetermined quantities a^, a^, •••^mi ^^^^ c^, c,,...c , are inteori'al homogeneous functions of the third 0' 17 n— 1 o o degree of o.^ , a^, ... a^; and so on. Let Sj, §2, Sg, ... denote the sums of the first, second, third, ... powers of the roots of (1) ; and let S^, S^, S^, ... 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 ^, = ^A + ^^1 + ^2 + ••• + ^n-l^n-l' ^„='^^K+K^ + K^+ '"+K-A-1' (4). Thus, as the sums of th3 powers of the roots of the equation in y are known we can construct the equation; see Art. 244. Or we may proceed thus : from equations (3) we can obtain the values of x, x^, ... a""^ 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 cc 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^, ...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 REMOVAL OF TERMS FROM AN EQUATION. 253 But from equations (4) we see that S^ is of the first degree with respect to ao, a^, ... «^, that S^ is of the second degree, S^ of the third degree, and so on. Hence by Art. 259 the determi- nation of these quantities a^, a^, ... a^^, of which one may be assumed arbitrarily, will depend on the solution of an equation of the degree \m. 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 function of the second degree of n variables can he expressed as the sum of the squares of v linear functions, the number v not being greater than n. Let V be an integral homogeneous function of the second decree of the n variables x,, x„,...x . If 72, = 1, the function contains only one variable, so that it is of the form jSxj^, that is, (x^ >Jf^y. 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 Xi we obtain where yS is a constant, () is a linear function of the n-1 varia- bles, x^, iKg, ... cc^, and B is an integral homogeneous function of the second degree of these n—1 variables. Put X. = a., + |, F. = i?-|; thus r=(xji3y+r„ and Fi is an integral homogeneous function of the second degree of 7^ - 1 variables at most. Next suppose that V does not contain the square of any of the variables ; then, arranging V with respect to two variables x^ and ojg, we have 254 EEMOVAL OF TERMS FEOM AN EQUATION. Put X, =^(^1+372+ ^2-2(^1-^2--^) thus V=p {X; - X/) + F, = (X, JPf + (X, V^)' + F^. Here X^ and X2 are linear functions which may involve the n variables x^, x^, •■- x^; and V^ is an integral homogeneous function of the second degree which involves at most ti — 2 variables. Thus in the first case the function V which involves n vari- ables is made the sum of a certain square and of F^, where Fj involves only n-1 variables at most ; and in the second case F is made the sum of two squares and of F^, where V^ involves only 01-2 variables at most. Then by continuing the process on F or F we can finally express F as the sum of not more than n squares. 338. Let there be an equation a;" -f ^^cc""' + ;?X"^ + . . . + ^„ = 0. Assume y=^% + ^^ + (■^■^ + ^^ + ^^^^' Let the equation in y obtained by eliminating x be de- noted by Now from Art. 334 it will follow that q^^ q^, ^'3, ...are re- spectively of the first, second, third, . . . degrees with respect to the quantities a^, w^, a^, a^, a^. Suppose then that we wish to make the second, third, and fourth terms of the equation in y disappear. We put ^.=--0, ^2=0, ^3=0. EEMOVAL OF TERMS FROM AN EQUATION. 255 The first of these equations is of the first degree. Suppose ■we obtain a^ from it in terms of a^, a^, a^, a^, and substitute in the second and third equations; and then denote these equations by Here q' is an integral homogeneous function of the second degree with respect to ay, a^, a^^ a^; and q\ is an integral homo- geneous function of the third degree. By Art. 337 the equation q\^ = may be put in the form where ^ 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^ and a. and substitute in the equation g'g = ; and then denote this equation by 2"a=0. 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 + q = (), x^+px^ + q = Oj x^ +px^ ^q-=0, cc^ +^:>ajV 2' = 0. 256 REMOVAL OF TERMS FROM AN EQUATION. 341. The foregoing Articles of tlie present Chapter have been derived from Serret's Cours d'Algebre Super ieure. The reduction of the equation of the fifth degree to the form of the preceding Article was given bj Mr Jerrard; it appears from a paper by Mr Harley in the Quarterly Journal of Mathe- matics, "Vol. VI., that the result had been previously obtained by E. S. Bring, a Swedish mathematician. Mr Jerrard considered that the algebraical solution of equa- tions of the fifth degree could be efiected ; his proposed method formed the subject of an enquiry by Sir W. K. Hamilton in the Reports of the British Association, Yol. vi. Most mathema- ticians admit that Abel has demonstrated the impossibility of the algebraical solution of equations of a higher degree than the fourth. .An abstract of Sir W. K, Hamilton's exposition of Abel's argument will be found in the Quarterly Journal of Mathematics, Yol. v. A simpler demonstration due to "VYantzel will be found in Serret'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. XXYIII. 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 demonstrate the principal general theorems of the subject, and in the following Chapter we shall give some applications to the theory of equations. IXTRODUCTIOX TO DETERMIXANTS. 257 Consider the simultaneous equations from these equations we obtain X = y "1-2 2^1 12 2 1 The common denominator ap^ — afi^ is called the determinant of the four quantities a^, 6^, a^) ^2' ^^^ ^^ denoted by the following symbol, ^ij ^1 «2' ^2 The numerators of the values of x and 7/ are also determinants ; and we may exhibit the values of x and y thus, «2> ^2 a? :J/ «.' ^1 «2' ^2 343. 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. The quantities a^, b,, «„, 5„ which occur in the determinant cifir,-aj)i are called tlie 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. "VYe shall now indicate some properties of determinants of the second order. Since we have I «25 ^o ^ad)„ — ah. 12 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,, ^1 '__ K «i ^2' ^2 K^ «2 «2' K \ K «.J «1» K ^' «1 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 p «15 ^ -P «1' \ Thus if each constituent in one row or in one column is multiplied by a given quantity, the determinant ls multiplied by that quantity. 347. We Lave = 0, «2, «2 = 0. Thus if two rows or two columns are identical the determinant vanishes. 348. It may be proved by developing the determinants that «2 5 ^- «2 + «8') ^2 + V 4- a/, h^ «i, V I «2'» ^8 ' I «2 5 ^/ + Tbus 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 a^ = h^ and a^'^h^; then the second of the above four determinants vanishes by Art. 347, and we have ds + hj, b^ + b^ «2J h + a,, b,' «3, b^ b,, b,' INTRODUCTION TO DETERMINANTS. 259 349. By Art. 348 we have a^a^ + 5, A, a,a^ + bA '^2«l + ^2/^1 J (^2^2 + ^2^2 ajttj, a^a^ + ^•A, h(^2 4. «i«i> &./52 4. bA. a,a. ^2*^1 ) ^*^2^2 bA. hA 1^ ^2^^!, bA i^ Mx, (^2(^2 a,a^ «2> ^2 + PA b,, b, &2, 62 ^"A a,, b, «2> ^2 ^A«2 b,, a 62, a 1 2 by Art. 346. By Art. 347 the first two of the four determinants just written vanish. And by Art. 345 &x, «X «x, ^x ^2, «2 »2> ^2 Thus we have left . (a^-fta^) «1, &i that is ttxj A ^2? A X «!, b^ a,, b. Therefore ax, A V^ «x, ^ ! _ a,a, + b,(3„ a,a, + b,(3. «2J A X ^3, h a, .ttl + 62^ I, « ,a2 + 62/3 2 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^, P^, a^, A *o be respectively equal to the constituents a^, b^j a^, b^; then we find that the square of the determinant £^2 5 ^2 is equal to the determinant < + &i', «1«2 + ^^ »l«2 + ^^2» «2' + V 17—2 260 INTRODUCTION TO DETERMINANTS. X- 350. We will now proceed to determinants of the third order. Consider the simultaneous equations from these equations we obtain and similar expressions for the values of y and «. The denominator of the value of x is called a determinant of the third order, involving the nine constituents a^, h^, c^, a^, b^, c^, a , b , c ; the determinant consists of six elements, each element beinc' the product of three constituents. This determinant is de- noted by the following symbol, a„ b,, c, «3, K^ ^3 Since the value of this determinant is «i (^/3 - h^) + «2 (^.^i - ^^3) + «3 (^^2 - K<^l\ we may express it in terms of determinants of the second order thus + «„ Kj ^3 61, Ci + a„ ^2' ^2 The numerator of the value of x is also a determinant of the third order; we have only to change a^, a^, a^ into d^, c?^, d^ 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 properties as determinants of the second order. INTRODUCTION TO DETERMINANTS. 261 351. Suppose «! = 1, ^2 = 0, and ^3 = 0; then we liave 1, 6i, Cj, 0, ^2) ^2) 0, 63, c. 3) ^2, C2 ^3) <^3 Thus the determinant of the third order reduces in this case to a determinant of the second order. The values of h^ and Cj 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 su23X30sing certain constituents to vanish. 352. It may be shewn by developing the determinants that «1 h. c. + ^2 h. Cs + «3 &., c. h, ^3 h, Ci 62, c. =«1 K h + *, Co, ^3 + ^1 CI. 2 , Ctg C2, Cz a„ «3 h, h that is, a„ ^, Ci a,, h, C2 = «3» K ^3 «!, «o, «3 K h, h ^1, Ci, fg Thus the determinant is not altered by changing the rows into columns. 353. The following identities may be easily verified, by expressing the determinants of the tliird order in terms of deter- minants of the second order and developing : «n &1J ^1 «2J ^2 5 ^2 — — «35 ^3) ^3 Oj J cv^ , Cj = - 62, ^2) ^2 ^3 J ^3 ) ^3 ^2 5 ^2 J ^2 ^3) ^3> ^3 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 detei-minant 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 it follows that if two columns are interchanged and also two rows the determinant is unaltered ; so that a„ h, Ci a„ K ^2 r= «3J K Cz K a,, <^2 h, a„ Cx K «3J Cz 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 CTo ^3, h 3) ^3 = and ^2 J ^2 > ^2 = 0. Thus if two rows or two columns are identical the determinant vanishes. 356. It is easy to see that the determinant «! + «/ + «/', 6i, Ci is equivalent to the sum of the three determinants «1> ^1) Ci a/, K Ci «1 , K «. a^, 62, C2 ^s'j K Co <, h, Cj «3> h) ^3 J a,', hz, C3 ) <, K <^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 b^, b^, b^ 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 the follov>ang determinant may be seen to be equiva- lent to the sum of 27 determinants : «! + a-l + «/', h^ 4- 6/ + 6/', Ci + c/ + Cj a2 + (^^l + «2"j ^o + h.l + 62"? ^2 + c,' + c ^3 + tto' + o^,", ^3 + h^ + 63", Cg + €3'+ c. 2 // 3 The 27 determinants are to be formed by taking instead of each column one of the partial columns ; thus for example three of these determinants will be the three which are given above. 357. As a particular case of Art. 356 we will take the follow- in 2; determinant : ^3°^3 + ^3i^i+V/l' ^3«2+^A+C372' «3«3+^3/^3 + ^37 It will be found that of the 27 determinants of which this may be considered the sum, all except 6 vanish by Arts. 35-i and 355. For example, we have for one of the 27 determinants, ^2^5 «2"2) ^2/^: «^3«1) «3«2> ^3^: «l«2/^3 that is, '') 2 «2> ^3) ^3) ^3 by Art. 354; and this determinant vanishes by Art. 355. One of the six determinants which remain is «!«!> ^1/^2 > <^i7 that is, ttg) ^2 J ^2 ^3 ) ^3 > ^3 Another of the six determinants which remain is ^^l^lJ ^l72' ^lA "172^3 a,a^, c^y^, 6^/33 that is, «3 -372 J ^3A\ - «l72^. that is, ^2 ) ^2 J ^2 ^^3 J ^3 > ^3 «1, ^IJ ^1 '^2> ^2J ^^2 ^3 > ^3 J ^3 by Art. 353. 2CA JXTRODUCTIOX TO DETERMINANTS. The result is that the six determinants which do remain constitute {■ 537, - I^Js) ^«3(ft7.-fto7,)} »2 ^2 J ^2 «3> ^3; ^3 ^' /^i' 7i X «n ^ij ^1 «2 5 ft' 72 «2' ^2' ^2 S> ft' 73 «3, 63, C3 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^, h^, ... respectively equal to a^, fS^,... we obtain a determinant of the third order which is equivalent to the square of a determinant of the third order. 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. 2G5 359. The value of the determinant «., ^o, Co The first element here is afi,^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, 6, c in all the difi'erent 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 afi.^c^; 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 aj)^c^ ; this can be derived from the second element by interchang- incr 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. 360. The following examples are particular cases of determi- nants of the third order, which the student may verify : (1) (2) (3) a, h, g h, h, f =ahc-af--hg'-cli'+1fgh. 9, S\ c \ ^v Vx 1' ^'- 2/2 o J 1. ^z^ 2/3 = ^xV^ - ^^Vx + ^22/3 - ^32/2 + ^Wi - ^12/3 ■ 1, «, + «„ a^a^ 1, c, + c,, c^c^ \ 266 PR( W 1, -y> A 7j 1, -/5 a 1 PROPERTIES OF DETERMINANTS. = l+a^ + /5^ + 7^ XXIX. PROPERTIES OF DETERMINANTS. 361. Let there be n symbols 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 c^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 23ermutation a^a^a^a^ there are four disarrange- ments, namely a_,a^, cc/i^, a^a^, and a^a.^. 362. The permutations of the symbols a^, a^, ...ct^ may be di\dded 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 symhols interchange their places ivhile the others remain unchanged the number of disarrange- ments is increased or diminished hy an odd number. Let g and h denote two symbols of which Ic is the higher. Let A denote the group of symbols before g and h, let B denote the group between g and k, and let C denote the group after ^ and k', so that the permutations which we have to compare may be denoted by AgBkC 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 fS^ of them are higher than g and (Br. of them higher than k. Then in the group gBk; besides the disarrange- ments in B itself, there are jS - j3^+ fS^ disarrangements ; for g is higher than /3 - /?, of the symbols in B, and there are /S^ symbols PROPERTIES OF DETERMINANTS. 267 in B higher than h. In the group TcBg, besides the disarrange- ments in B itself, there are y3 - /?2 + ^i + 1 disarrangements ; for h is higher than /S -(3^ symbols in Bj and there are (3^ 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 (y3i - jSg) + 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 ^vith 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 square, thus <^l,l> ^1,25 ^1,3) ^l,n ^^2, 1 ) ^2, 2 > ^2, 3 > ^*2, n ^«,1> <^«,2J ^*n,35 ^n.n Here for any quantity a^^ ^ the first suffix, r, indicates the row, and the second suffix, Jc, indicates the column in which the quantity «r.* appears. The above symbol is used to denote the determinant of the 7i^ 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 aj^j, a.^^^i a^^,...a„,^, which lie in the 268 PROPERTIES OF DETERMINANTS. diagonal drawn from the upper left-hand corner of the square to the opposite corner ; we shall call this diagonal tlte 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 '^*^:,i'^2,2'''3,3---^«,n hy permutations of the second suffixes, the first suffixes being left unchanged. The sign + or — is to be prefixed to each element of the determinant accordinof 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 determinant is said to be of the ?z*^ 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 determinant. 367. Instead of the above symbol for the determinant, it is sometimes denoted by ^ =*= c^i,i ^0.2 <^3. 3 ••• '^n.n; ^^^^ i^j "^^^ fi^'^^ ^"-^'" ment is written and the symbol % ± put before it to indicate the aorcrre2:ate 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; thus sometimes ({, k) is used instead of a^,., and in this case we must remember that (^, k) and (k, i) in general denote difi'erent 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 suffix:es ; 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. 269 first suflBxes remain unchanged ; these elements may however be derived in a different way, namely, by permutations of the first sufiixes while the second suffixes remain unchanged. Tor suppose that a, ^, y,-'-v represents a certain permutation of the n numbers 1, 2, 3, . . .72 ; then a^ a cts, /s ^3, v • • • ^n, v ^^ ^^ element of the determi- nant which arises from the first element by changing the second suffixes 1, 2,... 92, into a, /?, y, ...v, respectively. This element may however also be derived from the first element 0^1, 1 «2,2"-''^i, n i^ the second suffixes are left unchanged and the first suffixes are suitably changed, namely, a to 1, ^to 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 rows are changed into successive columns ; that is '^l, 1 J ^1, 2 > ^l,n a,2^ 1 ) ^2 2 5 ^2, n ^n, 1) ^n, 2J ^n, n ^1,2) ^2, 2 > ^n, 2 Tor 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 tivo rows or two columns are interchanged, the sign of the determinant is changed. 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 PEOPERTIES OF DETERMINANTS. in R' which arises from the first element in R' 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! Avith contrary signs prefixed. 371. If two TOIL'S or two columns are identical, tJie determinant vanishes. For by interchanging two rows or two columns, a determinant is changed from ^ 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 J? = — i?; and therefore R = 0. p- 372. W7ien all the constituents except one of a rovj or of a column vanish, the determinant reduces to the irroduct of that con- stituent and of a determinant of the next inferior order. Consider, for example, the determinant a^ ^i> Ci, d^ a 2 J ^2, ^2 J d„ ^*3J ^3 5 *^3» ^3 0, 0, c,, 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, a 1 J a 2J 5i, Ci, cZi ^2) 2> 2 «3J ^3J 0,0, d^ =(-irx c„ 0, 0, Cj, ttj, Oj, ttj ^•2^ ^2) 2) 2 ^3 5 ^3) ^3) ^3 The first element of the determinant on the right-hand side is c cihd^. and the other elements are to be derived from this by permutations 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 wliich does not vanisli, 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 (-ir^. tto, 5^, d^ This mode of demonstration applies, whatever may be the order of the proposed determinant. The negative siga which arises in this example from (— 1)^ may if we please bo removed by interchanging two rows or tvro columns in the determinant of the third order. 373. The top row of a determinant of the Ji*** 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 by 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 : {-ly (-1)- <^.2) ^25 ^2, d. (-ir «3, h, C3, ^3 (-ir 0, 0,c^, a3» ^3J C3, d. 0, 0, c„ a,, 61, Cj, d, 0, 0, ^4' «1, &n c,, d. «2J K <^2> ^2 a„ b„ Cl, d. > «2, &2, C2, d^ i 0^3, 63, C3, d^ 0, c^, 0, (-1)" ^4' 0, 0, (-1)"^. d„ «i, I, K Ci, d„ a. Cl, dn «!, 5j d., ^2 J ^2 h, <^2, d^, »2 ^2, d^y «2J ^2 d,, «3, ^S h, C3, d^, «3 > ^3> ^3, «3» ^3 > 272 PROPERTIES OF DETERMINANTS. «1, K Cl a 2, K C2 = «3> K, C3 1, 0, 0, i A «n ^i, Cx == y, 0^2 5 ^2, c. 8, a^, ^3, C3 374. A determinant can ahvays he expressed in the form of a determinant of any higher order. For example, by Art. 373, 1, 0, 0, 0, /., 1, 0, 0, V, A a^, h,, c, P, 7j «2 5 ^2» ^2 cr, 3, a^, &3, C3 where /3, y, S, /a, v, p, a, 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, ...n; let R denote the determinant S =•= «i i^o 2 • • • <^n. n ^ and let A^^j. denote the coefficient of a^j^ in R. Then each of the expressions and «i. i^i, ;(. + % i^2. A. + • • • + ««. i^n. u is equal to R or to 0, according as i and Z; are equal or unequal. For every element of R contains as a factor one out of the constituents a^-i, a^^i ^i. 3> ••• <^;, «? which form the i^^ row. And since -4,.^ ^ denotes the coefficient of a^^ * in R, we have Similarlv we have In the first of these expressions for R, put «;, 1 = %, d ^i, 2 = %. 2 3 ••• ^^^ ^*^ *^^ ^ ^^^s we obtain the value of a determi- nant with two rows identical, which is zero by Art. 371. Similarly, in the second expression for it put a^ ^ = a^ ^ , a^^i^cto J., ... and so on; thus we obtain the value of a determi- nant with two columns identical, which is zero by Art. 371. PROPERTIES OF DETERMINANTS. 273 376. If every constituent in one row or one column is multi- 2)lied hy a given quantity^ the detei^minant is 7nulti2olied by that quayitity, Tor Ii = cli^y_A^^■¥a^„A^^^_■\^ ... +ai^A.^^; and if every term in the i**" row is multiplied by 2^ ^^ must put 2^cc^i for a^^, pa^ ^ for a. 2, and so on; tlius we obtain 2^ times tlie 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 the constituents in one row or one column is the sum of m terms^ the determinant can he considered as tJie sum of m. determinants, ^ - Suppose, for example, that each constituent of the i^^ row is the sum of m terms ; and suppose that «i, 2 = 2^2 + 5'2 + ^'2 + • • • Then i? = a,_iJ,, + a,_.^.„4- +«i,„^i.« ^ PAi,l+ 2^2^,2 + + PuAn + ^l^i.l+ 5'2^i.2+ + QnAi,„ + o\A.,+ r,A.2+ + r„^,„ + Thus ^ may be considered as the sum of m determinants which have for their i'^ rows respectively 2\y P2^ Pn^ 5'l, 5'2) 9n^ ^1. ^' ''„' T. E. 18 274 PROPERTIES OF DETERMINANTS. 378. We shall now sliew 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 «< j, and those alone, we may suppose all the consti- tuents in the i'^ row to be zero, except a^y, then putting 1 for a.^ we shall obtain the required coefficient. In this way we get ^»-l,15 ••• ^i-X.k-XI ^t-l,*> ^j_l,A+l> Cb i—\ n 0, ... 0, 1, 0, ^i+l.lJ ••• ^i+l.i-1' ^i + 1,*' ^i + l,4 + l» ••• ^i+l,n ^n. U ••• ^7i, i-15 a 71, h 1 Ch. '7!,*+l> ... ct. n, n Thus ^. . is here exliibited as a determinant of the riH^ order. We may, without influencing the value of A..^^ put for each constituent in the U^ column except that which is 1. By Art. 372, or by Art. 373, we may exhibit A^j^ as a determinant of the {n-Vf order. Thus, adopting the method of Art. 373, we make i-\ cyclical changes in the rows and ^-1 cyclical changes in the columns. Therefore ^.^j_€X «i+i,4^i, ^*i + l, nJ ^^i + 1, 1' a. t+i,ft-i Oj.. 4+1) ••• "h, n) ^^n, 1> ••• ^n,i--l a 1,4 + 1' *^h, n J ^h, 1 5 a a t-l, 4 + lJ ^h-hnJ ^*i-l.l> 1,4-1 «i-1.4-l where c= (_ iyi-i+*-^)('.-i) = (_ iyi+*)("-i). 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 (71 -ly^ order; the determinants of the {n-lf^ order may themselves be similarly treated ; and the process continued to any extent. For example, PROPERTIES OF DETERMIXAXTS. 275 , ^i» Cj, cZj , ^2' ^2' ^h ' ^3' <^3' ^3 ' ^4' ^4' < CO. h,, c„ f/^ -^ ^3' ^3' < ^.' ^4' < < Co, (?2, ^2 + ^1 ^3' ^^3' «3 c„ <, a. , ^2» ^2' h - ^1 ^3' «3» ^3 «2, ^2. ^2 «3' ^3. ^3 «v K, C4 = «, 1^2 ^3' ^3 I I C4, d^ + ^2 I ^3» ^. d^, h. -^1^. ^3' «3 + ^2 «3' ^3 ( ^4 5 C4 + cZ„ + «.„ ^3» ^3 ^3, d^ C4, c/4 ■■{ cZ„ «3' ^3 +«2l^3» ^3 1+^21^^3. ^3! 1 a^, *4 64, 6/4 di, «4 , ^ ^ ^3» ^3 +^2 1^3' «3 +^2 ' «3' ^3 h^, C^ 1 C4, 0^4 I ^4, ^4 380. "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, a 1,15 a 1, p' a and 71,1) ^1,1) a n, p) *-.i. n,p From these let a third set of symbols be formed, '1. 1) '1, n J ■n, 1) ' n, n ) these symbols being determined by the general relation Let i? denote the determinant S± c, iC „ ... c„ „. "\Ye shall now prove the following results : 18—2 276 PROPERTIES OF DETERMINANTS. (1) Suppose p less than n; then R = 0. (2) Suppose 2^ = n; then R is equal to the product of the two determinants ^vhich consist of the two given sets of symbols in the order they occupy. (3) Suppose p greater than n ; then R 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 svmbols for one determinant, and the corresponding n columns out of the other given set of symbols for the other determinant. The first element of R is q,iCo,o ... c„ „, and the value of this is where in the first factor % denotes a summation with respect to r, in the second factor % denotes a summation with respect to 5, in the third factor 2 denotes a summation with respect to t, and so on ; and all these summations extend from 1 to jh both inclusive. Thus the product may be obtained by taking the sum of the values of the expression where r, s, f, ... take all integral values from 1 to "We may denote this sum by The other elements of R are derived from the first element by permutations of the second suffixes and prefixing the proper sign. Kow 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 cr, but the first suffixes of the symbol b are changed and these alone. Hence we obtain a result which we may denote thus, PROPERTIES OF DETERMINANTS. 277 Here S=^&i ,^2 ,&3,t ••• constitutes a determinant o£ tlie 7^* order, which is formed from the second given set of symbols by taking certain cokimns, and the S refers to changes of the first suffixes ; see Art. 368. We shall denote this determinant by Q. Now, in the first place, suppose ^:> less than n. The suffixes r, s, t, ... are n in number, and none of them can exceed 2^ ', 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 R vanishes. Secondly, suppose ;j = n. Then the system of suffixes r, s, t,... can be a permutation of the n symbols 1, 2,...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 2±«i,iO'2,2 ••• ^«,«» where ^ refers to changes of the second suffixes. Therefore when 23 = 71, JR ^1,1 J ••• ^],»i «„,i; •••«,.,,. ^H, 15 ••• ^n, n Lastly, suppose |; greater than n. Then the system of suffixes T, s, t,... can be any combination of 7i numbers that can be formed out of the p numbers 1, 2,... 2); and the number of such combina- tions is I — = . Let P denote what Q becomes by changing in p — 7l h into a. Hence, as in the second case, we shall obtain FQ for one term in ^, which arises from the selection of a definite combina- \P tion out of the .-^-j possible combinations. Therefore when \n\p — 7i p is greater than oi we have i? = !ISP(2, where % refers to the sum- \P . • mation of ; — , terms arisino; from all the possible combina- ^n\p — 7i tions. 278 PROPERTIES OF DETEPtMINAXTS. 381. By the second case of tlie 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. Suppose we wish to form the product of the two deter- minants ^1.13 ••• ^^,n and 5i_„...5i,„ By Art. 3G9 we may change the successive row^ into successive columns in either or both of these determinants. Thus, if we denote the product by ^1, 15 •• • ^1, n ". 1 > • • • ^n. 1 PROPERTIES OF DETERMINANTS. 279 "we may form the new constituents in four ways, for we may adopt either of the following laws throughout, or C^t = <^l,iKh + ^2,ih,k+ "•+^n,iKk' 383. Let ^ , J; denote the coefficient of a. ^ in a determinant E. The system of symbols A A 4 A A ■^2, 1 J "^2, 2V 2, n 4 -^ ri -^71,15 ''^fi, 2J" •''^n, rt is called the reciprocal of the system of symbols ^2, 1) ^2, 2J • • •^'^2, n ^n, 1> ^'^n, 2i"'^n,n ' 384. ^Ae determinant of a system ivJiich is tJie reciprocal of a 2)roposed system of n^ symbols is the {n-iy^ power of iJie determi- nant of the proposed system. If we multiply the determinants A A and «i,i5 •••«!.„ we obtain for the product 2S0 PPvOPERTIES OF DETERMIXAXTS. where c- ^ = J . ^ a^_ ^ + J , „ f^r., o + . . . + ^ ,-, „ «a, n • Hence, by Art. 375, the coustituents of the last determinant have the value it or according as i and k are equal or unequal. Thus this determinant reduces to its first element Ci^^Co ^...c^ „, that is, to E". Therefore therefore ^E n-l 385. Suppose we have a determinant of the n"" order, and in the square symbol denoting it suppose 7n columns and m rows destroyed; the remaining symbols may then be supposed moved close np 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 m 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 complementary to each other. 386. Let R 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 ^"'~^ into the complementary of the corre- sponding partial determinant of the original system. Let f, g,...r, s,... denote one permutation of the n numbers 1, 2,...n; and let i, h,...u, v,... denote another permutation. And suppose/, g,... and i, k,... to be groups of 7n numbers each, while r s,... and 21, r, ... are groups oi n — m numbers each. Thus A /.«•' A^., A '■g.i') V.*' • is a partial determinant of the reciprocal system of the order m ; we shall denote it by aS'. PROPERTIES OF DETERMINANTS. 281 Now ^^s, i J ^*», * J • • • ^''3, u1 ^^s, V J = €R wliere c is + 1 or — 1 ax^cording as the permutations f, g,...r, s,... and i, k, ... ^i, 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 ii by inserting additional constituents ; see Art. 374. Thus we may put for iS the following determinant, A J A A ^..,' ^,. *» • -A. u> -K. v5 • •' s,.>, -8. 45 • ■■£r u) ^r. »?••• ^'..., ^„ * ? • -K fl • • • • • • »)• •• 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 eR, 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 S and iR] we shall select the first of these, according to which c^^^ is obtained by multiplying respec- tively the constituents in the ^j*^ row of S by those in the q^^ row of ei?. Thus 282 PROPERTIES OF DETERMINANTS. Therefore by Art. 375, we have c, j, c,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 (.R are zero. For the first term in the (rti + 1 f^ 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 (rii + 1)^ row we have siuiilarly Proceeding in this way, we find that the (m+1)^'" row in the product of S and iR is the same as the (m+ 1)"* column in c7t. Similarly, the {rii + 2)'^ row in the product is the same as the {in + 2)*^ column in eR. The determinant then which is equivalent to SeR reduces by Art. 372 to the product of R"" and the following determinant of the (?z — my^ order, ^ r, « > ^r, V J • • • (I , a , ... Thus S=€Pr-' a^„, 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. 0) 0, a, /?, y I A y., 0, a, [ = aV+/3^/3/+/y;-2aa^/?/?^-2aa,yy,-2/?/3,yy, PROPERTIES OF DETERMINANTS. 2S3 — a (2) i 0, a, ft y 0, 7p ft A -7., 0, a. = («", - /5^, + 77,)' (3) W 0, a. — a A 7 ^» 7i. ft ft -7iJ ^j «^i y, -ft,-a^, e 6' + 6^ (a^ + /3-^ + y' + a^-' + (^; + y;) 1 <^, 9 9^ « 9^ « //^ = a a, h, g 1 9^ « «, /i K ^ / y;/^ h,h 9^f^ c (5) ?, a /,c A, 6 K 9 h, h 9 J = h a, h, g K b, f 9^/, G (6) h,f f,c h, b /,c K 9 9^f f,c c,9 9,^ ^9 9^ « f,h h,b 9^ « a, h 9 J f.h h,b = tlie square of a, h, g h,b,f 9,fy C (7) a, b c, d + b, c «, d + c, a b, d \ 0. 2S4 APPLICATIONS OF DETER:MIXANTS. XXX. APPLICATIONS OF DETERMIXANTS. 388. Suppose we have to find the values of oi unknown quan- tities x^, x„,...x^^ from the following n simple equations Let it denote the detemiinant 2=^ai,iCf2,2---^Vn; ^ii<^^ ^^^ A* denote the coefficient of «. ^ in B. Then the values of the unknown quantities will be given by the formula where h may have any value between 1 and n both inclusive. For let the given equations be multiplied respectively by A,,. .4.,,... J ;.; and add the 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. AYe may write the formula which gives x^ thus Ex^ = S, where S is also a determinant, namely the determinant which is obtained from E by removing the k^^ column of E and substituting for it the column formed oi ic^, u_^,...io^. 389. Suppose that the determinant E vanishes; then the values of the unknown quantities become infinite. This indicates that the given equations are inconsistent; see Algelra, Chapter xv. APPLICATIONS OF DETEPMIXAXTS. 285 390. Suppose tliat u^, u.2^...u^ Yanish, and that R 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 oi n-l of the unknown quantities to the remaining unknown quantity. These ratios can however be at once assigned : we shall have where i is any integer not greater than n. For since R=0, we have by Art. 375, for all integral values of i and h between 1 and n^ «*. i^c 1 + «t. 2^i, 2 + «*. 3^t. 3 + = ; and thus when x^ , o:.^ , x^, . . . are taken in the ratios assigned above, we have k 1 1 /* " ■' k 3 3 * — By taking n-l of the given equations, and supposing «tj, u^^...iir^ all zero, we shall obtain in general a single definite value for the ratio of each of n — 1 of the unknown quantities to the remaining unknown quantity. Hence it follows that when R=0 the ratios 4 'A 'A are independent of i. 391. If u,, u,,...u all vanish, and R does not vanish, the system of equations in Art. 388 has no solutions, except we suppose 5Ci, 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 examj^le, in order that the equations a^x + h^y + c^z— 0, a^x + hjj + c^z = 0, a.j):^-h^y-vc.^z = 0, 2SG APPLICATIONS OF DETERMINANTS, may admit of solutions which are not zero we must have If this condition is satisfied the equations may be satisfied by X : y : K C2 • • ^0, «2 • i«2> ^2 K Cz C3, «3 1 «3, ^3 or X '• y K C3 : C3, ^3 • i ^'3? ^3 h, ^1 1 ^15 «1 «1, 5. or X ' y \ z w K c ' • Cl, «1 • • «n h h, C2 ^2, «2 1 «2» h These three forms of sohition coincide by Art. 390. 393. From the given equations in Art. 388 we have deduced u,A,^ 1 + ^^^2. 1 + ^*3^3, 1 + ... + w.^4„ J = Iix„ ^h-^\, 2 + "^2-^2, 2 + ^^a^'^s. 2 + ' • • + ^^H-^*. 2 — Ii^2^ u.A, ^ + u^A^ ^ + u,A^ ,+ ... +u„A, „ = ^x" . Let p denote the determinant 2±-4j j^g g...^^^^; and let a. ^ denote the coefiicient of A^^^ in p. We may from the above equa- tions find the values of ti^, w,,...w^; and by proceeding as m Art. 388 we shall obtain the general result pu, - Fl ^x,a, 1 + x.^a,^ 2 + • • • + ^nH r, c • By comparing this result with the given equation in Art. 388, we have, since the values of u^ must be identical, -k,X -~ ^k, <• APPLICATIONS OF DETERMINAXTS. 287 But p=-B''-' bj Ai-t. 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 n quantities be denoted hj 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 ^ = (S - «i) (S - «i) • • • («» - «i) («3 - «2) k - S) • • • K - a„-i)- Then F may be exhibited as a determinant of the order 7i. For consider the determinant 1, aj, a{^, ...a^ 1 2 i, tto, tto ,... tto n-1 n-1 n-1 1 2 1 i, a , a .... a This determinant is a rational integi'al function of the quantities a , a^,... a^; and it varnishes when any two of these quantities are equal, by Art. 371. It is therefore divisible by the product which we have denoted bv P. Also both the determinant and ft/ the product P are of the degree ^^ \^^ in powers and products of a , a,,...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 rt}^ order consists of \n, terms. The product P prior to simplification and cancelling would n a2» '^2^ a, n-l a.. n-l 1 2 1, a , a ,, a n-l X 1, ttj, Oj-, 1, «2 5 «2"j a, n-l a„ n-l 1, a , a ",, a n-l 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 P'^ 0' 'l' ^1' ^2' n-l 5„_i) S^,5 ^n + l5 ••• "^2tt-2 where S^ = 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 x^ , cc, , ... x^ from the equations + re = 1 , X^+X^ + X^ + ^l«i + ^2^+^3«3+---+^n«n t, X^a; + X^a; + X^a^^ + 2 j2 X a = t , n n ' x,ar' + X2<" + ^'^s<~' + • • • + '^.<~' = ^" '• The values of the unknown quantities will be determined by the formula ^' (a^ - a^ (a^ - a.) . . . (a._ - a j (a.^ - a,) . . . (a . - a.) ' APPLICATIONS OF DETERMINANTS. 2S9 For by Art. 388 we have BXi = Sf where E= 1, 1, 1, a a If 2 1 J a a 2' 2 3 > a a 3» 2 3 » ...1 , .. a , .. a„ and*S' = „ n— 1 „ n — 1 »i — 1 n— 1 X* •••X* X« X* cavX 2 2 j2 2 2 a n— 1 /,"— 1 •*'«~1 1 » „n— 1 an n-l ,.. a «-! Xow let the ^^ column in jS be placed first, and the ^^ 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 vre obtain the value of a?, 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 deteiminants may also be used to obtain the resulting equation when certain quantities are eliminated from given equations. Suppose we have to eliminate x from the equations /(cc) = and (^ (x) = 0, where f{x) = % + a^x + a,x' + CTgCC^, {x) = 1^ + h^x + h^x'. We may proceed thus f{x) =a^ + a^x + a^x^ + a^x^ + 0, xf{x) = + a^x + a^x^ + a^ + a^cc*, ^ {x) = &o + ^1^ + ^-^^ + -t- 0, X(^ (oj) = + h^x + h^x- + h^^ +0, a;-(^( cc) = + 4- h^x- + h^x^ + h^x^. T. E. 19 290 APPLICATIONS OF DETERMINANTS. Let^= CT^J ^iJ ^2' ^3' ^ 0, a^, a^, a,, «3 K, K, K, 0, 0, K, h„ h,, 0, 0, 6o, ^, ^», then since by su2:)positiony(a:;) = and ^ (x) = 0, and therefore also xf{x), xcfi (x), and x^cf3 (x) are all zero, it follows by Art. 391 that jK = is the necessary relation which must hold among the coeffi- cients of f{x) and ^ (x). 400. We have given a particular example in the preceding Article, as the general investigation to which we now proceed will thus be more intelligible. Let f{x) = a^ + a^x + ajxi^ + . . . + cb„x"'= 0, (^ {x) = 6o + h^x + h^c(? 4- ... + 6,03" = 0; and suppose we have to eliminate x between these equations. AVe have f(x) = a^ + ^1^ + ^2^"" + • • • + ^„,^"*j xf{x) = a^x + a^x^ + . . . + a^_^x'^ + cijc'^^\ a^x"" ^ + a^cc" + . . . <^ (aj) = 6^ + 6jCC + h^x^ + . . . + 6„cc", a<^ {x) = Jo«^ + S^a;' + . . . + 6„_ja!'' + 6„a;'""S Let R denote the determinant of the order m + n which has for it;; first n rows «oJ ^u ^2» •••^m' ^j ^> ^' ••• 0, a„, a^, ...«„_p a^, 0, 0, ... 0, 0, «„,...«„.,, a„._^, «„., 0, ... APPLICATIOXS OF DETER^NANTS. 291 and for its next m rows then ^ = is the necessary relation among the coefficients in order that /{x) and <^ {x) may simultaneously vanish. The relation E = has been called the resultcmt or the elimi- nant of the proposed equations /(ic) = and ^ (a?) = 0. 401. The terms in the quotient obtained by dividing one algebraical expression by another may be exhibited as deter- minants. Let and let the quotient of <^ (x) divided by ij/ {x) be denoted by g^cc"*"" + ^^cc"'"""' + . . . + qjxT'""' + ... Multiply by the denominator, and equate the coefficients of a'""'' on both sides. Thus Similarly, «r-l = qr-iK + 9r-2^l+--+9oK-i> «^r-2 = ^r-2^0+---+^0^r-3» »1 = 9A + 9oKy »0 = %K We may regard these as r + 1 equations for finding q^^ q^_^, ...^o. 19—2 292 APPLICATIONS OF DETERMINANTS. We have 0, 0,... \ Therefore by evaluating the determiiiant on the left-hand side, and rearranging that on the right-hand side, we obtain a^ a. ^2 &,, 5,_,, ^_2J ^-3' a 402. AYe 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 ?^ 'independent variables X X , ...X ; and let these variables be expressed in terms of n new independent variables y^y^y-yn'^J means of the following 71 linear equations : ^2 = ^2, l2/l + «2, 22/2 + • • ♦ + ^2, n2/„ .(1); «^n=«n.l2/i+»«.22/2+-"+«n,n2/„ then by substituting the values of cCj, a^, ... a;„, the assigned fimc- tion becomes a function of y^, y^^ •••2/„' Suppose now that we impose the condition that <+«^/ + ... + a^/=2/r+2//+-+2// (2); then certain relations will hold among the coefficients of the linear equations (1); these relations we shall now demonstrate. APPLIOATIONS OF DETERMINANTS. 293 I. For eveiy value of i and k between 1 and n inclusive (3). ^2. 2 J *•• ^2,71 '^w.l' ^».2> ••• ^n,n Denote the proposed determinant by E: then E% by Art. 380, is equal to the determinant ^2, 1 > ^2, 2 > • • • ^2, 71 where ^n,l> ^n,2} ••• ^n.n 294 APPLICATIOXS OF DETERMINANTS. Thus, by (5), we have c. ^ = when i is not equal to k, and c. .= 1. Thus the latter determinant reduces to its first element, that is, to unity : therefore E^ = l. V. Let A. k have the same meaning as in Art. 388: then we obtain from (1) Hence, comparing this result with (4), we have .(6). VI. The following partial determinants which can be formed of the constituents of E are numerically equal : » «, m+1' « n,tB+2 J a n, n and ^2, 1) ^2, 2> ••• ^2,m ^TO,1J ^m,2) a 771, m Denote the first determinant by P, and the second by Q; then by Art. 386, -^i.ij -^1,2 > ••• -^i.^ •^o 1 ) -^2 2? ••• 2, m at, «t is numerically equal to R^'^P. And by Art. 376 and equation (6), /< yl A ■^i,\i ■"2,2* ••• '"2,n A' A A = E-Q, APPLICATIONS OF DETERMIXAXTS. 295 Hence, since H' — l, "we see that F and Q are numerically equal. 403. "We "^ill finisli "witli some examples. (1) Shew that a, h, G c, a, b h c, a ! = a^ + b^ +c^- dabc. Shew that a +b + c must be a factor of this determinant. (2) Shew that \a, b, c, d =a'-b' + c'-d'-2a'c' + 2b'd' d, a, b, G —4:a'bd + 4:b'ac— 4:C'bd+ 4:d-ac. c, d, a, b b, c, d, a Shew that a + b + c + d must be a factor of tliis determinant. (3) Let there be a determinant of the order ?z + 1 in which all the constituents are equal to unity except those which form the diao'onal series, and these are 1, 1+a,, 1 +a„. ...1 +a : the value of this determinant is a,a„...a . 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^ ••• ^„. ^^.d 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 all the constituents are unity except those which form the diagonal series, and these are 1 + a, , 1 + a^.... 1 + a : the value of this determinant is aa...a„ . For 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^ • • • ^, is found by considering the first element of the determinant. Quarterly Journal of Mathematics, Yol. I. page 364. ( 296 ) XXXI. TRIGONOMETRICAL FORMULA. 404. We will give in tlie present Chapter a few propositions wliich bring the Theory of Equations into connexion with Trigo- nometry. 405. In Art. 272 of the Plane Trigonometry we have an expression for tan 7i0 in powers of tan^, supposing n to be a positive integer. Sujjpose now that tan nO is given, and we require tan 6. Clear of fractions, and thus we obtain the following equation of the w*^ degree for determining tan^; „ f, n{n-l)^ „. nin-l){n-^)in-2>)^ ,^ 1 tan nd{l ^^ — '- tan"^ + — ^^ '-^—^ — '-^ '- tan'(9 -...\ I 1^ 1 4 J = n tan 6 ^^ ^ tan' Now the value of tan 716 is not changed if we put instead of 6 any one of the following angles : 6 + -, 6 + — , 6 + — , 6 + TT. n Qi n n Hence we infer that the roots of (1) are tan^, tanC^ + -V tanC^+— ^ tan ^(9 + TT Let S denote the sum, and P the jDroduct, of the n quantities just expressed ; then, by the aid of Art. 45, we may deduce from (1) values for 8 and P ; but for this purpose we shall have to consider separately two cases. TRIGONOMETRICAL FORMULA. 297 I. Suppose n even. Then (1) becomes tan nQ U. - ' '^'""^^ tan"' ^ + + (- l)^tan" ^| = ,-, tan B - ^^(^^-^K^^-^) tan^^ +... + ,,(- 1)^ tan""^ 0. In tills ease 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 nB. Thus we obtain n-2 ^ = -!^)-l~ = -7icotn^, P=-Lir=(-lf ...(2). (-l)Mann^ (-1)' II. Suppose n odd. Then (1) becomes tan7i^|l-' '^''"^^ tan-^^+ ... + 7z(- l)"~\an"-^^| = ,, tan Q - ^(^^-|)(^^-^) tan^^ + ... + (- l)'^\an'' Q. In this case in order to put our equation in the standard form, ti-i we must divide by (- 1) ^ . Thus we obtain n-l 8 = n\^T,nQ, P = (-l)^ tan?z^ (3). 406. Again, take equation (1) of Art. 405, and multiply by cot"^j thus we have tan nB {oot'd - "<"-^> cot-g + '^C^' -l)(»-^)(»-g) eof -«-...} = ?^cot"-'^ 5^ ^ ^cot" ^e ^ n (n - l)(n - 2) (n -S){n- 4) ^^^„.,^ 15 298 TRIGONOMETRICAL FORMULA. Divide by tan n9, and we obtain an equation in the standard form for determining cot^ when tan?i^ is given. Hence, proceed- ing as in Art. 40-i, we have o^\ ^ n-l \ TT n cot ^ + cot (d + y\+Qot(0 + — \+ + cot (O = 71 cot nO (4). 407. From equations (3) and (4) we see that, if w be any odd integer, the product of itan^ + tan (^ + 7) + + tan \0 + '~ — ttjI into ]cot ^+ cot f ^ + - j + + cot (6 + '- ttjI = n\ 408. Propositions like those of Arts. 405 and 406 may be easily deduced from other formulae of Trigonometry. "We will give one more. By Art. 287 of the Plane Trigonometry we have, when n is even 9 ^ T n' . „^ n-{n -T) . .^ _, cos n& = 1 - rs sm-fc' H ——, sui d - (0). 1^ 1 4 ^ ' Let cos nO = ', then we may put for any one of the follow- ing n values ^¥ti' ^2^' ^2n' '^~2n"^' Letm = ^, and a; = cosec"^; then dividing (5) by sin"^, we get 0=x"'-j^x'" ' + —^ ^cc"^- |_2 |4 The m values of x are cosec 77- » cosec" -— , cosec —7^ — tt : hence, by Art. 45, the sum of these m quantities = th • TRIGONOMETRICAL FORMULA. 299 Thus, if n be an even integer, „ TT 2 Stt 2^-1 ?^^ cosec -^r- + cosec -^ + + cosec — r — •"■ = tt- . 409. We see by Art. 142 that any algebraical quantity has n different n^ roots. If then we have found an expression for the iiP^ 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 Eecherches sur Vanalyse des sections angulaires. A memoir by Abel also treats on the same subject : see his Oeuvres Comijletes, 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 (2 cos 6)" in terms of cosines or sines of multiples of 6. The case in which n 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. Pirst suppose cos 6 positive; and let p denote the arithmetical value of (2 cos 0)", Then we may put (2cos^)" = rp (1). Now 2 cos = e^^ + e~'^, where t is used for J- 1 ; thus (2coseY={e'^ + e-'^y 300 TRIGONOMETRICAL FORMULA. But e"'^ = cos 71$ + L sin 7i$j e(n-2)id = cos {n-2)e + L sin (n - 2) 6, and so on. Hence (2 cos 6)" = c + is, where c stands for a certain series involving cosines, and s for a corresponding series involving sines. Again, Vp = (cos 2?^/x7^ + t sin 2?z/x7r) p, where p. denotes any integer. If then we were to equate this to c + ts 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^ + 15^ = (cos 2np.7r + t sin ^np.Tt) p (2). 411. If we suppose n an integer, we have c^ and 5^^ coin- ciding with G 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 PlaTie Trigonomeiry^ 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 = p.. This point has sometimes been assumed; but Poinsot gives a reason for it in the following manner. Let us supjpose to diminish without limit. Then it will be found that „ (^ n{n-l) ) c^= cos znrmr l\-\-n + —\k — - + ••.[•, • o fi n(n-\) ) 8^ = sm Inrmr il + ?i + — !^— — '- + ...>. TRIGONOMETRICAL FORMULA. 301 Tlie series witliin the brackets may be regarded as equal to 2", bj the Binomial Theorem ; so that 0=2" cos InmiT = p cos 2wm7r, 5=2" sin 2n7mr = p sin 2n7mr, Hence by (2) we get cos 2nrmr + t sin 2?zm7r = cos 2njXTr + t sin 27i/t7r ; from which we conclude that m = fi. 413. Thus, when n is not an integer, we have from (2) p= ^- - andp=-^-— (3); ^ cos 2nm7r sm 2nm7r so that p may be expressed either in a series of cosines or in a series of sines. 414. If we put m = in the fii-st 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 is positive. Again, putm = in the second of equations (3); then, since sin 27im7r vanishes with m, it follows that s = so long as cos 6 is positive. 415. Let us now suppose that cos 6 is negative ; and let p de- note as before the arithmetical value of (2 cos 6)". Then we may put (2 cos ^)" = (-!)>. Also (-l)" = cos(2/x + l)n7r + tsin(2/A+l)n7r. Hence instead of (2) we now obtain c^ + 15^ = {cos (2/x + 1) WTT + t sin (2/x + 1) mt) p (4). 302 TRIGOXOMETPJCAL FORMULA. 416. 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 foiTQer agrees substantially -w-ith 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 = ix; then, as in Art. 413, we have c s p= TT^ — "St — and p = —. — ,-, "• .,, • (5). cos {2m + I) oiTT sm (2wi+ l)n7r ^ ^ If we put m = in the first of equations (5) we obtain c p = • cos OITT This shews that when cos is negative the numerical value of (2 cos 9)" is not equal to c, but to c divided by cos mr ; and this divisor is in general not equal to unity. Also, if we put m = in the second of equations (5) we obtain s p = - ; Sm 7177 thus s is in general not zero. 418. Eeturn 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 qi is equal to some integer together \vith 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 2?im7r=0 ; thus m = is one value of m, and if s be even, m = -: in another. TRIGOXOMETRICAL FORMULA. 803 If p can be expressed by sines only, it is obvious we must have cos 2nm7r = : therefore must be an odd multiple of a ; thus s ^ s 3* if r is odd and s a multiple of 4 we may take m = - 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 ob\dous we must have T sin(2m+l)n7r = 0; therefore sin (2w+ 1) - 7r = 0; thus, if i' is odd, we can take 27n + 1 = s. If p can be expressed by sines only, it is obvious we must have cos {2m + l)mr = 0; therefore cos {2m + 1) - tt = 0; thus, if r is odd, o Q and - an odd integer, we may take 2m + 1 = k or 2??i + 1 = -^ . 420. Abel shews that the formulae here obtained for (2 cos &f hold when n has any positive value j and also when n has any negative value numerically less than unity, except for those values of Q which make cos vanish. 421. We might investigate series for (2sin^)" in the same way as for (2 cos &f \ or we may deduce the results by putting ^-6 for B in the formulae already obtained. 2 '' 422. In the Plane Trigonometry, Chapter xxiii., the ex- pression cc^" - 2x" 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 FORMULA. lately shewn by matliematicians 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 Tol. XI. of the Transactions of the Cambridge Philosophical Society. 423. The relation between successive values of »'"+-— cor- responding to successive integral values of m is given by the formula a'" "%^-=(-5(-4)-(--^.)^ ■when m=\ this becomes a \ xj \ xj x^ + -.= An exactly similar relation holds between the successive values of 2 cos mO ; thus 2 cos (m + 1)6 = {2 cos 6) (2 cos mO) - 2 cos {-/n -1)9; when m = 1 this becomes 2 cos 2^ = (2 cos ^) (2 cos ^)- 2. !N"ow let ijj,, Vj, 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 x'" + -— and a;'" of 2 cos mO ; that is to say, let m+l 1 m m — l Also, as in those cases, let v^ = 2, but let v^ be any quantity "whatever ; thus we have and so on. TRIGONOMETRICAL FORMULAE. 305 Hence it is ob\ious (1) that v^ is a definite integral function of v^ of ?i dimensions, and that the coefficient of v" is unity; (2) that iiv=x + -, thenr =x" + — : (3) that ifv = 2 cos^, then v„ = 2 cosw^. Hence 'y„ - 2 cos na will vanish when v^ is equal to any one of the following n quantities : 2 cos a, 2 cos (a + /5), 2 cos (a + 2/5), ... 2 cos (a + ?i - 1,5)^ ■ 27r where B is put for — . Therefore v - 2 cos na — ^ n " jvj - 2 cos a| L\ - 2 cos (a + /3) j jv^ - 2 cos (a + 2/5)| ... ,1 •j t?! - 2 cos {a + n- 1/3) r Now put X + - for V, ; thus we obtain ■^ a; n 1 O X + —r — -^ cos na = ' X -\ 2 cos I X a 1 1 X + - - 2 cos (a + /3) > ■ X + - - 2 cos (a + 2/?) i ... f„ , 1 ...-a; 4- 2cos(a + ?i— l^)i. This gives the required resolution. Similarly if we put 2 cos 6 for v^ we obtain 2 cos ?i^ - 2 cos 7za = |2cos^-2cosa||2cos^-2cos(a+/5)||2cos^-2cos(a + 2^)| ... ... |2cos^-2cos(a + ?r^^)|. 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 cc=- 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, cos n6 = \-% sin^ 6 + ^(^ TL?!) sin^ 6 .2 |_4 — ^- ~ -^sm'^- (1); and if n be an odd integer, Sin n^ = n sm ttt sm o H r^ sin y - . . . i^^;. Now substitute — ( ;s - - j for sin 9, where t denotes J -I', and for brevity put p for « . Then from (1) we deduce, when n is even, 1/' 1\ , «" . «'(«'- 2") . . -'<"' ^:;f-^^> /. (3); and from (2) we deduce, when n is odd, Thus we obtain the algebraical identities (3) and (4). These may be verified by the aid of Art. 244. For snp]->ose f{x)=^x^-px-\, so that the roots of/(x) = are of the form z and , and z = /?. Then z z ^W - iil^izii = 1 + _.5i±i_ = 1 + f 1 + -iVi - A - ?>)" f{x) ji^-2JX-\ yf-fx-l \ x'j\ X- x) ='*o*i)(o-.T*(-rf*(-in:-i TRIGONOMETRICAL FORMULA. 807 First suppose n even; then by Art. 244 we have 2" + ( ) 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 (l + - j M - - j is 1 + 1, that is 2. The coefficient of — in (1 + -^) (1 5 ) "^ is 1. / 1\/ l\~^io* The coefficient of— in(l+ — Wl— ^ ^ x"'"V ' xV V xV x' '^ P* (fn-4c \fn-4: _ ^N^^n-4 _ ^\ ^n-i [4lV 2 *" ,4 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 oj' + 7x^ + 3a:' + 1 7x' + I0x-14: is divided by a: - 4. 2. Expand (a + bx)" in powers of x, and then obtain the Jirst derived function of (a + hx)". 3. Shew that the equation x^ + 3a;^ + a; - 6 = has one root and only one between 1 and 2. II. 1. Find a root of the equation x^ = + >J—l. 2. Find a root of the equation x^ = — ij—^- III. 1. Form the equation whose roots are 1, 1, 1, - 1, - 2. 2. Form the equation whose roots are l±J—'2 and 2^ J — 3. 3. Form the equation of the eighth degree one of whose roots is J2 -^ J3 + J^. 4. Solve the following six equations in each of which 07ie root IS given (1) x'-x'+3x+D = 0; l-2j^. (2) x' + 4:x'+Gx' + 4:X-\-5 = 0; J'^. (3) x* + x'-25x' + 4:lx+66 = 0; 3 + J^. (4) x*+2x'-4:x'-4:X+4: = 0- J2. (5) x'-2x'-5x'-6x+2 = 0; 2+^3. (6) x'''-x'-8x* + 2x'+2lx'-9x-b4: = 0; J2+J^. EXAMPLES. 309 5. Solve the equation ic^ — x^ + 8x^- 9a; - 15 = 0, one root being ^^3, and another 1-2^-1. 6. The equation x^-ix' + x + c^O 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'^ - Qx^ + 40a;^ + QOx^ - a; - 1 = 0. 8. The equation cc* - 21a3^ + 166a;^ - 546a; + 580 = 0, has roots of the form a, /?, a + /3 + (a - /3) J- 1 ; solve the equation. 9. Find the sum of the cubes of the roots of a given equation. 10. Form the equation the roots of which a, /3, y, S, are l(l + JS^J^dy and 1 (^1 - ^3 ± Vri^^ ; 2 /D2 2 2 and thence shew that ^— + — — ~ + . . . = 0. a/5 ay 11. If a, 6, c, ...are the roots of an equation, find the value of 2 2 7 2 7 •> 6.1 ~r* iy l~ • • • ' I ^ l" ^ T* • • • c a c' 12. Assuming that the arithmetic mean of any number of positive quantities is greater than their geometric mean, shew 2 that if p^^ — 2/>2 is less than ?^/?„", 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, b, c....are the roots of an equation in its simplest form, shew that p- - '2p^p^ + 2p^ = a^6- + a^c^ + b'^c' + . . . 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'-Sx^-x'+i^O- 1. (2) x' + x+l=^0- 3. (3) x'+ia^'-x' + ll -0; -3. 2. Transform each of the following four equations into another wanting the second term : (1) x'-3x'+4:X-4:=^0. (2) x'-6x'+l2x + 19 = 0. (3) x'- 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) a;'+5a;'+8a;-l=0. (2) x'-6x'+Qx- 10 ^0. (3) a;^-8a:'+18a;''-15a; + 14 = 0. (4) x'-18x'-Q0x'+x-2 = 0. 4. Transform the equation x^+ '2x^+-rX+ k- = into another ^ 4 9 with integral coefficients, and unity for the coefficient of the first term. 5. Remove 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) x^ - ex + 6 = 0. 7. Transfoi'm x^— 12a;"+ 12a3 — 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 ic'* + o;^- 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. exa:viples. 311 9. The roots of the equation, x^ -^ px' + qx + r — are denoted by a, b, c ; transform the equation into others which have the roots assiirned in the followinc; fourteen cases : (1) a^, b'-, c". (2) b + c, c + a, a + b. ^ ^ b-i-c' c + a* a+b' ^ ^ bc^ ca' ab' (5) 5'c", cW, a'b". (6) \^{ka), J{kb), J(kc). (7) -^(b + c-a), -(c+a-b), -(a + b-c). (8) b + c + Jca, c + a + kb, a + b + kc. (9) " * " b + c-a' c + a — b^ a + b-c' (10) bc+-, ca + -, ab + -. (11) S' + c^ c^+a', a-+b\ ^ a c ^ 6 c c a a b /iq\ ^''^^' c' + a' a^ + b' ^^^^ c'^b' «^c' b'^'a' ^ ^ "W' cV ' ~^^' (14) b — c, c — b, c - a, a-c, a — b, b — a. 10. The roots of the equation x^+qx + r = are denoted by a, b, G ; transform the equation into others which have the roots assigned in the following two cases : (1) G-.)'. C-^J c V J .a -b. (2) ba+ac, cb+ba, ac + cb. 11. If a, b, c denote the roots of x^-Qx'+ 11a;- 6 =0, form the equation whose roots are 1 1 _1^ b^ + c'' c'+a'' a' + b'' 12. If a, b, c denote the roots of a;'-2.r+2 = 0, form the equation whose roots are b^ + c^ c'+_a' a^ + b^ 13. Shew that the third term of the equation x^ 4- px- + qx -rr ^ cannot be removed if ^' be less than 3q. 812 EXAilPLES. • 14. Shew that the second and fourth terms of the equation a;* + j9jCC^ + 'p_pT 4- p^: + p^ =f can be removed by the same transformation if '^1'^^ = 'P\K^V2~'P\)' 15. Solve the following two equations: (1) a:^+4.7;^+7r/r-i-6./:-10 = 0. (2) :>:^ + 4a;^ + 3:// - 2a; - 6 = 0. 16.- Shew that the equation a;^ -r 4a;'+ 6a:4- 3 = does not admit of the second and third terms being removed 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 5i^^ degree if n X (sum of squares of root?) = square of sum of roots. Y. 1. Shew that the equation ct^— 4a;"4-3 = has at least two imaginary roots. 2. Shew that the equation s;'— 2x'*+a;^- 1 = has at least '■four imaginary roots. 3. What may be inferred respecting the roots of the follow- ing two equations ? (1) a:^°-5.r'+3r-a;-l = 0. (2) a;^" - c/r" + a;" + x + 1 = 0. YL 1. Solve the following twenty equations, each of which has equal roots : (1) a/-7.-/r+16a;-12 = a. (2) a'- 3a;'- 9a: + 27 = 0. (3) a;'-or-8a: + 12"0. (4) a;'- ox'- 8a; + 48 = 0. (7) a;'+8x-+20a;+16-0. (8) a;*-ia; + -| = 0. 2 lb EXAMPLES. 813 (9) x'-Ux'+18x-8=0. 10) x*-2x'-x'-4:x + 12 = 0. 11) x'-rx^+Ux'+^x-lS^O. 12) x^-4:x^- ()x'+36x-'27 = 0. 13) x''+ 13x'+ 33a;' + 31x +10 = 0. 14) 2x'' - 12a;' + 19a;'- Gx + 9 = 0. 15) a;'+16a;'+79a;'+ 126a; + 98 = 0. 16) 8a;"+4a;'-18a;'+lla;-2-0. 17) a;^-a;^-2a;'+2a;'+a;-l = 0. 18) a;'-2a;'-6a;' + 4a;'+13.^ + 6 = 0. 19) x'- Ux'+ 67x'- inx'+ 216a;- 108 = 0. 20) x'- 3x'+ 6a;'- 3a;' - 3a; + 2 = 0. 2. Find the condition tliat x"—px^ + r-0 may have equal 'roots. 3. Shew that x* + qx' + 5 = cannot have three equal roots. 4. If x" +p^x"~^ + ... +p^^ = have two roots equal to a, shew that jj^x"''' + 2p^x"~^ + ... +np^ = has a root equal to a. 5. If x^+qx^+rx^ + t = has two equal roots, prove that one of them will be a root of the quadratic , 2q' 5t iq_ VII. 1. Find limits to the positive and negative roots of X 5a;'+a;'+12a;^-12a;'+l = 0. 2. Write x* - 8a;' + 1 2a;' + 1 6a; - 39 = so as to shew that 6 is a superior limit of the positive roots. 3. Shew that the real roots of tJje following six equations lie between the limits respectively assigned : 814 EXAMPLES. (1) x*-x^+ix^-3x + l^0; - and 1. (2) x' + x^-lOx'-x + W^O; -4:Sind3. (3) £c'+5a;V£c'-16a;'-20a;-16 = 0; -5 and 3, by Art. 92. (4) (x^-26)(x''+5x + l) + 60x = 0; - 5 and 3. (5) (x'- 4x - 2y- 43 = ; - 2 and 6. (6) x' + x'+ x"- 2oa; - 36 = ; - 3 and 3, by Art. 92. 4. Find by Newton's method limits to the roots of the following five equations : (1) cc'-a3'-5aj-+8x'-9 = 0, (2) re' - 5a;' + 6a; - 1 - 0. (3) a;'-a;'+4a;' + a;-4 = 0. (4) a;*- 5a;' + llx»- 20 =. 0. (5) X*- 2x^- 3a;'- 15a;- 3 - 0. 0. Prove that a;*4- 5a;*- 20a;" — 19a:- 2 = has one root be- tween 2 and 3, but none gi'eater 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;"- 32a;+ 20 - 0. (3) a;' -3a; + 3 = 0. (4) 4a;' + 9a;'- 12a; + 2 = 0. (5) a;'-aV+c' = 0. (6) x^'-'px'+r^O. 7. Shew that the equation 3x* + Sx^ — 6x' — 24a; -f- ?• = 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 1 0, and no real root if r is greater than 19. YIII. 1. Obtain the commensurable roots of the following twelve equations : (1) a;'- 106a; -420 = 0. (2) a;^- 9a;^+ 22a;- 24 = 0. (3) a;'*- 2a;' -25a; + 50 = 0. (4) 2a;^- 3.r + 2a;- 3 = 0. EXAMPLES. 315 (5) 3x^-2x'-Qx + 4: = 0. (6) 3a;'-26ic' + 34a:- 12 = 0. (7) a;*-2a;"+8a.'-16 = 0. (8) x'-x'-l3x' + Ux-iS = 0, (9) a;*-£c='-a;'+19aj-42 = 0. (10) x'+8x'-7x'- 4:9x + 6Q = 0. (11) a;'-3a;'-9a;'+21cc'-10a;+24 = 0. (12) x'- 7x'+ llaj^- 7x'+ 14^^- 28a;+ 40 = 0. 2. The coefficients of tlie eqiiation /(x) = are all integers : shew that ify(O) and y*(l) 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;*+3a3'-7a;--27a;-18 = 0. (3) a;* + 3^' + 2ic'+9ic-3 = 0. (4) x^ + a^- llic^- 9aj + 18 = 0. 2. Solve the following four equations in each of which the roots are in Arithmetical Progression : (1) x^-Qx' + Ux-Q^O. (2) a;'-9a5' + 23a;-15 = 0. (3) a;'-8x-'+14ic=^+8a;-15 = 0. (4) a;'+ 4a;'- 4a;'- 16a; = 0. 3. Solve the following six equations in which certain con- ditions relative to the roots are given : (1) 3a;' - 2a;'- 27a; + 18 = ; product of two roots is 2. (2) x^ — 3a;' — 6a; — 2 = j product of two roots is — 1. (3) x^— 4a;' + 5a;'— 16a; + 4 = ; product of two roots is 1. (4) 2a;^ - 5a;' + 1 la;' — 1 la; + 6 = ; product of two roots is 1. (5) x^ - 45a;'— 40a; + 84 = ; difference of two roots is 3. (6) x^ - 7a;* + 15a;' — 15a;' + 14a; - 8 = ; 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, 3a, 6a. (2) a;*-10a;'+35a;'-50a;+24 = 0; a + l, a-1, 6 + 1, 6-1. 316 EXAMPLES. (3) 6x'-4:3x'+l07x'-10Sx+36 = 0; a,h,^,~. Cb (4) x' + 8a;'' + 5x'- 50^-- 3Ga; + 72 - ; a, 2a, h, 26, a + h. (6) a;"-12x^-2x'+37ic'+10j?-10 = 0; l^ Ja, b=^J2, ^Jc. 5. Solve the following two pairs of equations, each pair having a root in common : (1) cc'-3x^-16:c-12 = 0; a;'-7a;^+5ic+13 = 0. (2) £c'-3a:'+lla;-9 = 0; a;'-5x' + lla;-7 = 0. 6. Solve a;'-7ic'+36 = 0, and a;'-3af - 10x + 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', cc'+ 4ic'- 2a;'- 12x + 8 = 0. • 8. Pind in terms of m and a the roots of the equation X* +pax^ + (m* + TJi) arx" + qci^x 4- a"* = 0, which are in Geometrical Progression; and determine ^9 and q in terms of m and a. X. 1. Solve the following ten reciprocal equations : (1) a;^-2a;=^+3a;'-2a;+l = 0. (2) x' + ix' -bx' + ^x ^l =0, (3) 2x'-^x^+Qx'-bx+2 = 0. (4) x'+ix'-\Ox'+ ix + l =0. (5) a;'-2.'c*-19a;'-19x--2a;+l=0. (6) x'-ix\x'+x'-ix+l^O, (7) 6a;'-lla;'-33a;'+33a;= + lla;-G = 0. (8) 2x^-t)x'+ix'-Ax"+5x-2 = 0. (9) ^x'-\Qx'-2bx''-Ux'+^^.0. (10) \ + x' = a{\+x)\ EXA3IPLES. 317 2. Obtain roots of the follo^ving four equations, and depress the equations : (1) x'- 2x'+ x'+ x^-2x'^l=^ 0. (2) x'^ 2x^- ^x'- 7x'- 7x^- 8x'+ 2a; + 1 = 0. (3) x'+2x'+3x'+2x'-2x'-3x'-2x-l^0. (4) a;'°-l = 0. 3. Exhibit the roots of x'^+px-+ 1 = in the form a,b, -, ^. 4. If a, b, c,... denote the roots of the recurring equation x"" +px''~'^ + qx''~- + ... +qx^ +px + 1=0, 2 2 12 12 ^2 a a c / 2 o \2 ., — + - + ...+—+-+ ...+-^+ ... = {p -2q) -n. h c" a c a 5. In the recurring equation cc'^'-j^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) a;^-l=0. (2) a:«-l = 0. (3) a:V 1 = 0. 2. Shew that the factors of a'+ 6^+ c'- 3a6c are of the form a + hi-\- ci^, where i^— 1 = 0. 3. Shew that the factors of a'{a'-ibd - c')-b\b'- iac - d') + c' {c'- ibd-a') - d\d'- iac - b') are of the form a + bk + ck- + dk^, where ^^-1 = 0. XII. 1. Solve the following eight equations ; (1) a;^-3^-2 = 0. (2) a;'- 9a; -28 = 0. 3 (3) x'-x+Q = 0. (4) x^ + 3x = ^. (.5) 3.^^- 6a;''- 2 = 0. (6) o;^- 15a;^- 33a; + 847 = 0. (7) x'+Qax'=dQa\ (8) x'-^d {a' +b')x = 2a {a'- Zb'), 318 EXAMPLES. 2. Determine tlie relation between q and r necessary in order that the equation x^ -¥ qx + r = may be put into the form x^= {x'+ ax + hy \ and hence solve the equation 8jc^— 36x + 27 — 0. 3. If the roots of the equation rc^ + px' + ^-o:; + r = are in Geometrical Progression, rp^= q^. Hence solve the equation £c'-a;'+ 2^-8 = 0. 4. If the roots of the equation 7?+qx-\-r = are diminished by h, shew that the transformed equation will have its roots in Geometrical Progression if h be such that 21rJi^— 9q'h'- (f- 0. 5. If the roots of the equation £c^+3/):ic"+ Tyqx + r^0 are in Harmonical Progression, 2q^^r{?>pq- r). 6. If the roots of the equation x^ + 3/?a;' + 2>qx + ?• = are in Harmonical Progression, the equation rx' + 2q"x + qr = contains the greatest and least of them. 7. The impossible roots oi x^ + qx + r = being put under the form a^ jS J — 1, shew that ^' = 3a^ + q. 8. If r, a + ^//3, a - ^//3, are the three roots of the equation a;' + j9 ar +2).,^ "^P — ^5 ^^ which r is real, and if x^ + m^x" 4- m^x = is the equation resulting from the diminution of all the roots by r, 771 1 shew that a = - — ^ + r and ^ = - - (/»., + 3^9., -jy^'). 9. Keduce the equation x^ + p:f?+ qx + r^O to the form of— 2>y + 711^0, by assuming x = ay-^h; and solve this equation l)y assuming y = z + - . Hence shew that if the original equation has equal roots, 4 (/- Zqf^ (2/- 9pq + 27r)^ 10. If the roots of the equation x^^ jyx'^ qx-^r = ^ are in Harmonical Progression, so also are the roots of the equation {pq - r) if- {p^- 'Ipq + Zr) 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'-Sx-3 ^0, (3) x'- 12x''+ 4:9x'- 78a; + 40 = 0. (4) x'-2ax'+{a'-2b')x'+2ab'x-a'b'=0. (Art. 192.) 2. If r^—p^s = the equation x*+27x^+ qx~+ rx + s — may be solved as a quadratic. 3. If s and p are positive, and 27p* less than 25 6s^ the roots of the equation x'^+2^x^+ 5 =^ are all imaginary. 4. Assuming that the equation x'^+ qx^+rx + s = has roots of the form a=i=f3j-l, shew that the values of a and /3 may he found by the equations 64a' + 327a^+ (W- 16s) a^- r'= 0, j3'^a'+i + ^. 2 4a XIV. 1. AjDply 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) x'-4.x^-3x + 23=^0; f^{x) = - i91x + U7l, fi{x)==-. (2) a^-4:x^^x'+6x + 2^0; /^{x) =6x'-l0x-7, /^{x)^x-l, f,(x) = + . (3) x^+cff + x-l^O; f^(x) = 3x'-l2x + \7, Art. 199. (4) x'-2x^+x^-8x + 6 = 0; /^[x) = 16x' - 23x + 9. (5) x'-h 5x*- 20x'- 19a; - 2 = ; f^(x) = 20a;'+ QOx'+ SGx - 9, /Jx)^96x'+l87x + 67, /,{cc) -43651a; + 54571, Jl{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) x^+6x'+10x-l='-0. (2) re'- 6.^'+ 8a; 4- 40 = 0. 320 EXAMPLES. 3. Determine the situation of the positive roots of the equation a;"- 2x^+ Sx''- 5x — I =0, having given /^{x) = 6x{x-iy+l9x+6. 4. Apply Sturm's Theorem to the following four equations : (1) 0;'+ a;'- 2a; -1 = 0. (2) a;'- 4a;"- 4a: + 20 = 0. (3) a;'+2a;--4a:+10 = 0. (4) a"- a; + 1 = 0. XY. 1. Shew that the equation a;'- 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;'4-10a; + 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. XVIL 1. Apply Newton's method to calculate the root which is situated between the assigned limits in the following five equations : (1) x^- 4a; -12 = ; root between 2 and 3. (2) x^- 4a;'- 7a; + 24 = ; root between 2 and 3. (3) a;-'- 24a; + 44 = ; root between 3*2 and 3-3. (4) x^— lox -5 = 0; root between 4 and 4-1. (5) a;'- 8a;^+ 1 2a;' + 8a; - 4 = ; root between and 1, EXAMPLES. ■ - 821 2. Apply Newton's metliod to calculate a root of tlie follow- ing two equations : (1) x'+3x-5^0. (2) aj'- 3^=- 3^ + 20 = 0. XYIII. 1. Apply Horner's metliod to calculate the root wliicli is situated between the assigned limits in the following three equations : (1) a;^-i- 10x"+6aj- 120= 0; root between 2 and 3. (2) ic'- 2a;'+ 21^3 - 23 = ; root between 1 and 2. (3) a;*- 5x^+ 3x^+ 3dx - 70 = j root between 2 and 3. 2. Solve the equation cc^— 17 =0 by Horner's method. 3. Calculate the real roots of the following four equations by Horner's method : (1) x'+x-3 = 0. (2) a;' + 2£c-20=0. (3) 3:<;'+5a;-40-0. (4) cc'+ 10a,-+ 8a;- 120 = 0. XIX. 1. Find the value of the following seven symmetrical func- tions of the roots a, b, c of the equation a;^+^jaj' + ga; + r = 0: (!) (a + h + ab){b + c + hc){c + a + ca). (2) {a + b-2c){b+G- 2a) {a + c- 26). (3) % {a + by {a + c). (4) S (tt + 5- 2c)(6 +c- 2a). (5) 1 ,. (G) Sy-(1+- 1+-) . ^ ^ a + b ^ ^ bc\ aj \ aj (7) {b-cY{c-a)\a-b)\ 2. If a, b, c, d are the roots of the equation a;*+^;a;^+ 5'a;'+ raj + s = 0, find the value of 2 {a + b){c + d). T. E. 21 322 EXAMPLES. 3. In the equation a^" + /'i-'C" ' + . . . + ^^ _ ^ aj + ^9^ = 0, suppos- ing tlie roots to be a, b, c, ... I find (1) la'b. (2) :^{a + h){a + c)...{a + l). (3) ^(^^. (4) Sf 4. Form tlie equation tlie roots of which are the squares of the sums of every three roots of the equation x^ + 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"~'^ +2^jL''~^+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) x'-4:X-+2^0. (2) x'+4:X+3 = 0. (3) a;'+l = 0. 2. Eliminate x from the equations ax^+hx + c = 0, ax'+b'x+ c'^0. XXL 1. Find the sum of the assigned powers of the roots of the following five equations : (1) x''-x^- ldx'+ 49aj- 30 = ; the cubes. (2) x^— 3x^— 5x-hl ='0 ; the fourth powers. (3) x'-2x^-22x''-2Sx'+72x + U4: = ; the cubes. (4) a;*+ 2a: + 1 = 0; the inverse squares. (5) x^-x-1 =0 ; the sixth powers, EXAMPLES. 323 2. If a,h,c,... are the roots of a;"- 1 = 0, find l,a'"h^. 3. If the sum of the r^ powers of the roots of the equation «.*"+ aj + 1 = be expressed by S^^ and the sum of the r"" powers of their reciprocals by 2^, prove that S.--S,=l, and V.-2„ = ^^-2(-ir. 4. In the equation a;"- x^+l = 0, find 2^""^, Sa"~^, and 2a" ; supposing ?^ greater than 3. 5. Find the sums of the r'** and {^nf" powers of the roots of the equation x"''— ^jaj''+ q=Oj supposing n greater than r. XXII. 1. Solve the equations (?/ - 1) a;^+ 2/a; + 2/'- 2?/ = I (y-l)a; + 2/ = J* 2. Solve the equations 1 {y-\)x' + y{y + l)x + ?>if-l=0 3. Shew that the following equations have no solution : yx^-{y^-2>y-\)x + y = 0y^ a;^-2/^+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^x" + 2>yx^ — y^x + 4?/ — 2a? = 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 - xY-^ 2a;y- xY + y' - Zxif+x'Y' = 0. 224! MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 1. If there be n qucantities a, h, c..., and if 7i functions of tliem be taken of the form (x -h)(x-c) ... {a — h){a — 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 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 one which has only changes of sign. 4. If 2^ and q are positive, the equation a;^"-^9a3^'"+ ^' = has — j is gi^eater less than ( -^ ) ; and it has two pairs of equal roots if fr^oV f rq or ^n — r, n — r n J \n — r. 5. If -F„_,cc"-', -K_.cc"-^ -i\_3^""', ... are the negative terms of an equation of the n'"" degree, then the greatest root of the equation will be less than the sum of the two greatest of 1 the quantities (;;„_j', 0\_,)', (Pn-s)% ■•- 6. If k be the last term of an equation of the ?^'' degree whose roots are in geometrical progression, shew that — k'^ is a root, if n be odd. Shev*^ 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 x"^ - 6x^ + 7x^ + 7x^ - 6a; — 3. Solve the equation x^- 63;"+ 7..;^+ 7a;''- Gaj- 3 = 0. [MISCELLANEOUS EXi^IPLES. 825 8. Diminish by h the roots of the equation give such a value to h that the roots of the transformed equation Ttl Oil may be of the form a, ~ , 5, y- , and shew how this equation may be solved. Example, x^— 2x^+ 1 6^j + 1 = 0. 9. Shew by the process for extracting the square root of an algebraical expression that the equation x'^+2)x^+qx"+rx + s=-0 can be immediately deduced to quadratics if p^s - 4^s + r^ = 0, or if ^/- i-j^q + 8r = 0. 3 10. Prove that the equation x'^+- qx'+ rx + s = cannot have all its roots real if q^+r' is positive. 1 1. lif{x) be a rational integral function of x, either /(a;) = or f (r/j) = has certainly a real root. 12. Shew how to find the value of the semi-symmetrical function a^b + h'c + c'a of the roots of a cubic equation. 13. Let a, b, c,... k denote the roots of the equation (^ (x) = 0, which is of the ?^"' degree and in its simplest form, and suppose these roots all unequal : shew that the expression a' b" c" k' \c) ••• 4>\k) is equal to unity if r = ?i — 1, and is zero if r is zero or any positive integer less than n- 1. Shew also, that if r = - 1 the expression = —^ — - — ^ . aoc ... k 14. If » 328 AXSWEES. 5. Let the transformed equation be yr ( 'J') J^ ) then m = —^ — . Find the sums of the powers of the roots of the transformed equation, and then the coefficients by Art. 21-i. We shaUget q,= {n-l)2?,; q,= '- ^^ P,+ {n-2)2^,; (n-l)(n-2)(7i-3) r, , c^o , k\ gs-- — ^3 — ^^'^-^ ('' ~ -)y^p-^^ ('' - ^) ^v XXII. 1. The solutions are given by 2/^ — 22/ = and (i/—l)x + y = 0. 2. The solutions are given by 2/'-l = 0and {7/-l)x + 2y = 0. I— 2 XXIII. 1. y = x+...; y-±J'dx+...', 2/ = :t7i+--- 2. Six values of the form y = x^ {u + U), where il is to be determined from 1 —u"—ii^ + u^ = ] three values of the form y = aj-3 (lo + U), where u is to be determined from 1 - 3ic^= ; and 13 four values of the form y = x '^ (u + U), where u is to be deter- mined from 3 — w^ = 0. MISCELLANEOUS EXAMPLES. 1. Call the sum ^ (x) \ then shew that ^ {x) — 1 is identically zero by Art. 39. 2. 2/' - 12/ + 65/ -840/+ 20377/ -1428 = 0. 1 5. Form a quadratic with roots ^/3 + ^/5 and ^3 — ^5 ; then use Art. 2G1; see also Algebra^ Art. 526. CA3IBR1DGE : PRINTED AT THE UNIVERSITY PRESS. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 1354 ^B 13^394 LD 21-100m-7,'52(A2528sl6)476 RARY U.C.BERKELEY LIBRARIES CD^E3Dfi713 I Engineerin:: Library UNIVERSITY OF CALIFORNIA LIBRARY mn^ ••■^ • ■ ■.-. v.* '* ',', > > I'i. , -si..