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 THE GENERAL 
 
 PRINCIPLES OF ANALYSIS; 
 
 BEING 
 
 a k>tvit$ of (^rtgtnal Entesttgatfcms; 
 
 IN 
 
 DIFFERENT DEPARTMENTS OF PURE MATHEMATICS. 
 
 
 PART I. 
 THE ANALYSIS OF NUMERICAL EQUATIONS. 
 
 BY 
 
 J. R^OUNG^ 
 
 LATE PROTESSOB OP MATHEMATICS IN BELPAST COLLEGE. 
 
 LONDON: 
 
 C. C. SPILLER, 102, HOLBORN HILL; 
 
 SAMUEL MAYNARD, MATHEMATICAL BOOKSELLER, 
 
 EARL'S COURT, LEICESTER SQUARE, LONDON. 
 1850. 
 
 PRICE TWO SHILLINGS AND SIXPENCE. 
 
^>. M fa & „/ J/ 
 
 CAJORI i 
 
 NOTICE. 
 
 The work, of which this is the First Part, was intended to be 
 issued in monthly portions, and to embrace the following topics : 
 1 . Conjugate Functions, and their application to the Analysis 
 of Equations. 2. Researches respecting the Theory of Numbers. 
 3. A Disquisition on Diverging Series. 4. On Zeros, real and 
 imaginary, Conjugate Points, and Imaginary Logarithms. 5. On 
 certain Theorems of Leibnitz, Abel, Murphy, &c. 6. On the 
 Law of Continuity in Analysis. 7. Miscellaneous Investigations 
 in the Calculus of Operations, the Theory of Probability, the 
 Law of Population, &c. 8. Prefatory matter, in a Dedication 
 to the Earl of Clarendon, containing statements and explanations 
 of interest to the friends of Science. 
 
 The author feels it necessary here to state, that the further 
 prosecution of this design must, for the present, be suspended. 
 The subscribers to the work, although including some of the 
 most distinguished names in the kingdom, are nevertheless too 
 few in number, as yet, to justify him in incurring the expense 
 of the publication. The proceeds of this First Part of the work 
 are not likely to cover one half of the outlay : as soon as 
 this obstacle is removed, the publication will be regularly pro- 
 ceeded with. 
 
 The Statements, &c, intended for Part VIII will shortly be 
 printed in a distinct form. The pamphlet will be published 
 separately at One Shilling : the subscribers to the work originally 
 proposed will not be considered as pledged to the purchase of it, 
 nor will it be forwarded to them unless expressly ordered. 
 
 One remark more must be made, though it would be gladly 
 avoided. It was the early-formed resolution of the author that 
 duplicate copies should not be forwarded to any individual sub- 
 scriber ; this resolution was founded upon reasons cogent to his 
 own mind, but which he may, perhaps, here be excused from 
 formally stating. Some of the most eminent persons whose 
 names occur in the following list, have each ordered several 
 copies of the publisher. But the author respectfully requests 
 permission to adhere to his original determination, though at the 
 risk of incurring the charge of inconsistency. His hope and 
 aim was, and is, that his book might, in some degree, be accept- 
 able to science ; and he must be suffered to indulge the delusion 
 — if delusion it be — that the countenance extended to it is 
 justified by the expectation of at least partial success in his 
 endeavours, irrespective of all considerations personal to the 
 author. 
 
ON THE 
 
 GENERAL PRINCIPLES OF ANALYSIS. 
 
 INTRODUCTION. 
 
 The present work will consist of a series of independent 
 essays upon some points of interest in analytical mathematics. 
 And although the several subjects to be discussed will naturally 
 fall under special titles, I snail endeavour to blend with the 
 examination of them, such occasional remarks and observations 
 as may seem to me to throw some light upon the management 
 and texture of algebraic operations in general ; and to bear, more 
 or less closely, upon the entire science of symbolical reasoning. 
 It is on this latter account that I have ventured to call the work, 
 'A Treatise on the General Principles of Analysis.' The title cer- 
 tainly appears an ambitious one : — I fear it may not be justified 
 by the execution. But if those who take a pleasure in abstract 
 researches, should think that I have succeeded, now and then, in 
 illustrating a previous obscurity, or in adding a new truth, or a 
 new principle, to the existing stock, however trifling such additions 
 may be, they, at least, will not, I am persuaded, be too severely 
 critical upon so unimportant a matter. 
 
 In prosecuting the work, I have not kept any particular class 
 of readers before my mind. I have certainly assumed that some 
 general acquaintance with the present state of the topics of 
 
 1 
 
 i 
 
 a » jr. i^>. mW^. a 
 
2 INTRODUCTION. 
 
 inquiry introduced has been already acquired; but, as I have 
 aimed to be intelligible and perspicuous, I think the student of 
 general analysis, whose researches have not extended beyond the 
 limits of our ordinary elementary books, may read the volume 
 with ease and, I hope, with advantage. 
 
 It seems to me to be a libel upon mathematical science to 
 pronounce any of its departments to be inherently too difficult 
 and recondite for the comprehension of a mind thus trained and 
 prepared. Analysis is too frequently and too unjustly branded 
 with this stigma, when the reproach more properly belongs to 
 those who are the exponents of its doctrines. If superior clear- 
 ness and perspicuity can reasonably be looked for in any subject, 
 surely that subject is mathematics : and yet no department of 
 scientific research can be adduced which is enveloped in so much 
 unnatural and repulsive obscurity as the science of abstract 
 analysis. Still, however, I must acknowledge, that this fault is 
 not always without its advantages ; what is of easy attainment is 
 often undervalued ; and, in the minds of some, the only way to 
 enhance the estimate is to increase the difficulty of access. It 
 was, no doubt, a conviction of this truth, that influenced Mr. 
 Horner to invest his important discovery of the general solution 
 of numerical equations, in the dress in which it appears in the 
 ' Transactions of the Royal Society.' With admirable tact and 
 sagacity, he shrouded a very simple matter in a garb of transcen- 
 dental mystery ; but, as soon as his primary object was secured, 
 the object, namely, of the early publication of his results, under 
 the auspices of a learned body, he proceeded to strip the subject 
 of its unnatural disguise, and to exhibit it in all its native 
 simplicity. 
 
 From these introductory remarks, it will be perceived, that my 
 object will be to convey the contents of this volume to the reader 
 with as little expenditure of thought and labour, on his part, as 
 he can reasonably expect to incur in the perusal. 
 
 I do not see why a principle, universally adopted in our manual 
 operations, and in our industrial economy — the minimum of 
 exertion to the maximum of result — should not be equally appli- 
 cable to our intellectual processes ; nor why the muscles of the 
 
INTRODUCTION. 3 
 
 mind should be exercised for nothing, any more than the muscles 
 of the body. Labour — notwithstanding the old motto — is surely 
 not desirable for its own sake. The youthful powers, whether 
 of mind or body, unquestionably require to be stimulated to 
 exertion, in order to their full and healthy development, though 
 nothing else be gained by the effort. But too much discipline 
 may be as bad as too little; a time should arrive when the 
 apprenticeship ought to expire, and the novice be pronounced 
 qualified to set up for himself. 
 
 I should consider these latter observations as quite superfluous, 
 but for the opinions recently promulgated by a writer of great 
 and deserved eminence. It seems to be the conviction of Dr. 
 Whewell that, till two- or three-and-twenty, the scientific course 
 of a young man should be exclusively one of intellectual disci- 
 pline. Invigorating mental exercise, and not substantial mental 
 attainment, is to be the object kept in view ; and if two paths 
 lead equally to the same end, the one long and laborious, and 
 the other short and easy, the former is to be chosen instead of 
 the latter. It is a necessary consequence of this doctrine, that 
 analysis is to be displaced for geometry. But I cannot conceive 
 upon what grounds Dr. Whewell denies to analysis the same 
 logical rigour and conclusiveness which he justly claims for 
 geometry. I should say that, if any department of inquiry be 
 destitute of these, it should be at once expunged from the 
 category of mathematics. The only difference between analytical 
 and geometrical reasoning is in the language used — the instru- 
 ment employed. I have no conception of two kinds of reasoning, 
 essentially and inherently different. All reasoning must, in its 
 characteristic features, be the same ; the subjects may be diffe- 
 rent, and the apparatus may be different ; but there can be no 
 difference in the mental process. If the conclusiveness of any 
 step in a train of algebraic deduction be questioned, the analyst 
 would justify it exactly as the geometer would justify a geome- 
 trical conclusion : that is, by showing that it necessarily and 
 syllogistically follows from what has been already admitted. In 
 both methods the mind is exercised in precisely the same way, 
 however different the visible symbols of its operations. 
 
"4 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 In the writings of geometers, as well as in those of analysts, 
 errors and paralogisms may be pointed out ; and it must be con- 
 fessed that, in analysis, the temptations to go astray often assume 
 a very seducing aspect, and have often prevailed. But this 
 admission rather tells against, than fm% Dr. Whewell's argument; 
 it is an admission that, in analysis, caution, circumspection, 
 and sagacity are pre-eminently demanded, and require to be 
 constantly exercised. 
 
 SECTION I. 
 
 THE ANALYSIS OF NUMERICAL EQUATIONS. 
 
 (1.) From the very partial success attendant upon all the 
 methods of late years proposed for the analysis of numerical 
 equations, I am inclined to think that, if any considerable advance 
 is to be made in this important subject, it must be made by pur- 
 suing the path along which Newton took the initial step, availing 
 ourselves of the light that the researches of Descartes have 
 thrown over it. 
 
 It was in this direction that Budan prosecuted the inquiry. 
 His method is usually called " an extension of the rule of 
 Descartes." But whatever aid this valuable rule afforded, and 
 however indispensable that aid in Budan's process, the process 
 itself seems to me to be just that which, in the natural order of 
 discovery, would be expected to follow that of Newton, regarded 
 as a first step in a clearly indicated direction. 
 
 (2.) A numerical equation is said to be analysed as soon as we 
 discover the several limits, or pairs of numbers, within which all 
 its unequal real roots lie individually, and its equal roots in 
 distinct groups ; that is, as soon as these unequal roots, and 
 
ANALYSIS OF NUMERICAL EQUATIONS. 
 
 groups of equal roots, are all separated and severally enclosed, 
 each between two assignable numbers. 
 
 The preliminary step in this complete analysis of an equation 
 would seem to be, first to determine limits as close as practicable, 
 within wh fo h all the roots should be comprised, and this was 
 Newton's step. He showed how close superior and inferior 
 limits were to be found without the bounds of which no roots 
 could possibly exist. Budan cutjip_ Newton's wide interval 
 into a set of partial intervals, and showed that some of these 
 partial intervals might, in like manner, be rejected; and thai the 
 real roots were to be found only among the intervals which he 
 retained. 
 
 The theorem of Budan, by which this further advance in the ^ 
 analysis of an equation is effected, was first published in ISO/.*/ 
 The same theorem is, however, claimed by the friends of Fourier, 
 more especially by Navier, who published the posthumous work 
 on Equations, by the latter distinguished analyst, in 1831 ; and 
 in which this theorem is given without any reference to Budan' s 
 printed researches ; there is certainly every reason to suppose 
 that Fourier was led to it independently ; but as priority of 
 publication unquestionably belongs to Budan, and as no docu- 
 mentary evidence has been produced in favour of Fourier's still 
 earlier claim, I shall here, as in my former work, continue to call 
 the theorem in question the 'Theorem of Budan ;'* regarding this 
 writer as the first who clearly marked out for rejection certain 
 portions of Newton's interval, within which all search for roots 
 would be fruitless. 
 
 (3.) In the investigations which follow, I attempt to advance 
 a step further in the same direction ; and as Budan pointed out 
 rejective intervals within the bounds of the extreme limits of the 
 roots, so I seek for rejective intervals among those which he ha3 
 retained. 
 
 And here I would, at the outset, remark that, in an equation of 
 
 * Some discussion, respecting Fourier's independent title to this theorem, 
 will be found in my treatise on the ' Theory of Equations,' p. 150, and in the 
 Preface to that work. 
 
 
6 ANALYSIS OF NTTMERICAL EQUATIONS. 
 
 an advanced degree, it requires, in general, no small amount of 
 trouble to obtain all Bud an' s subdivisions of tbe interval ; and 
 often very considerable labour to ascertain the true cbaracter of 
 the roots of which these subdivisions contain the indications. 
 In writings on equations, much of this necessary work is usually 
 suppressed: tbe rows of signs, due to the limits of Budan's 
 partial intervals, are all exhibited to the eye ; but the process by 
 which they are obtained is usually omitted. Indeed, throughout 
 the whole of this business of analysing an equation, preparatory 
 to actual solution, there is frequently, as every practical analyst 
 knows, a great deal of arithmetical work of winch no trace 
 appears in the finished process. 
 
 In numerical operations, few things are more dispiriting than 
 to find, after a long course of calculation has been carefully gone 
 through, tbat tbe whole might have been spared, from it ulti- 
 mately appearing that we have been unconsciously prosecuting 
 researches in a direction in which the object sought cannot pos- 
 sibly exist. 
 
 Tbis is the sort of disappointment wbich we must often meet 
 with in discussing the intervals within which pairs of roots are 
 indicated by Bud an' s rows of signs. And I think that any 
 easily-understood and easily-applied directions, which can un- 
 erringly warn us oif any of these unproductive paths, would be, 
 in some measure, acceptable to science. 
 
 (4.) In the present preliminary observations, I have not ad- 
 verted to the theorem of Sturm, because, although this theorem 
 is theoretically perfect, yet, as ample experience has shown, it is 
 impracticable beyond very narrow limits. No one, I think, will 
 be inclined to repeat the labour which I have undertaken, in 
 analysing an equation of the sixth degree, with moderately large 
 coefficients, by this theorem ; the calculations involviug products 
 of more than a hundred and fifty figuresjn_a_row.* It will 
 always, however, be regarded as one of the most beautiful dis- 
 coveries in this department of Analysis, and must ever command 
 
 * Theory and Solution of Equations of the Higher Orders; 1843, p. 277. 
 

 ANALYSIS OF NUMERICAL EQUATIONS. 7 
 
 a prominent place in every complete exposition of the general 
 theory of numerical equations. 
 
 (5.) The method now to be explained is derived from a dis- 
 cussion of the identical equation, 
 
 X = {F + x/F^TX} x {F - \/F2 - X} (I) 
 
 in which X and F may be any functions whatever ; so that if one 
 of these be fixed, or prescribed, the other is quite arbitrary. 
 
 This identity, as will be at once seen, is nothing but an imme- 
 diate inference from the simple algebraic truth, that the sum of 
 any two quantities, multiplied by their difference, furnishes the 
 difference of the squares of those quantities. But the only 
 instances I am acquainted with, in which even any particular case 
 of the general principle (I) has been employed in analysis, are 
 two in number; one such particular application will be found in 
 Mr. Gompertz's 'First Tract on Imaginary Quantities,' and the 
 other in a Paper by Mr. Cockle, in the 'Philosophical Magazine' 
 for February, 1849. Besides these, I know of no other instances 
 of the introduction of an arbitrary function into a pair of conjugate 
 factors ; and I believe the general form (I), notwithstanding its 
 axiomatic simplicity, had escaped notice till the publication of it 
 by myself in the Magazine for April, 1849. 
 
 (6.) Since F in the formula (I) is entirely arbitrary, though X 
 be fixed, we are at liberty, in any particular case of (I), to write 
 afterwards F +/ for F, where /is equally arbitrary ; that is, we 
 may always change (I) into 
 
 X={F+/+\/F 2 +2F/+/2-X} x {F+/-VF~ 2 +2F/+/ 2 -X}. .(11) 
 
 from which we see, that having decomposed any function X into 
 a pair of conjugate factors, we may always afterwards add any 
 quantity /to the rational part of each factor, provided we, at the 
 same time, introduce the expression 2F/ + / 2 under the radical 
 sign ; and it was the recognition of this privilege, that suggested 
 the following course of inquiry. 
 
8 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 (7.) Before proceeding to more general forms, let us first con- 
 sider the quadratic equation 
 
 x*+px + q = (1) 
 
 the first member of which, when written thus, 
 
 x 2 — ( — px — q) 
 immediately suggests its simplest pair of conjugate factors, namely, 
 
 X + y/( — px — q) 
 
 x — \J{— px — q) 
 
 which, x being put for the arbitrary quantity F, and x 2 +px + q 
 for X, agree with the form (I). 
 
 If, now, we introduce /as in (II), and determine it so that 2/ 
 
 may be equal to p, that is, make f equal to - p, x, under the 
 
 radical, will disappear ; and another pair of conjugate factors, of 
 the proposed quadratic expression, will be obtained, namely, 
 
 *+\p + s/^-*) ( 2 ) 
 
 *+ -zP — V(tP 2 -?) ( 3 )- 
 
 These, therefore, are the two factors of the first degree, of 
 which the product is the proposed expression of the second 
 degree ; so that each of these equated to zero must furnish the 
 two roots of the quadratic equation ; that is, the roots are 
 
 x 
 
 = -^±V(^-2> 
 
 And this seems to me to be as easy and as elementary a process, 
 for arriving at the formula of solution for a quadratic equation, as 
 the method in common use. 
 
 If x 2 have a coefficient other than unity, the steps are much 
 
 the same ; thus, if 
 
 ax 2 + bx + c = 0; 
 
 .'. Aa 2 x 2 + Aabx + 4ac = 0, 
 
 or 4a 2 x 2 — ( — 4abx — 4ac) = 0, 
 
ANALYSIS OF NUMERICAL EQUATIONS. 9 
 
 of which the unmodified factors of the first member are 
 
 2ax ± \/( — 4abx — 4ac) ; 
 and these, by the principle above, may be replaced by 
 
 2ax + b ±\/(b 2 — 4ac), 
 the factors of solution, as before. 
 
 (8.) I have introduced these simple applications of the general 
 formulae (I), (II), chiefly for the purpose of preparing the way 
 for a remark of some analytical interest. If the reader turn to 
 the first factor in either of these formulae, I think he will be dis- 
 posed to affirm, from an examination of it, that it is impossible 
 for any value of x, which causes X to vanish, to cause also the 
 entire factor to vanish ; inasmuch as he will perceive that, for 
 X = 0, the factor referred to becomes 
 
 F + VF 2 , orF+/+V(F+/) 2 > 
 that is, 
 
 2F, or2(F+/), 
 
 which is zero only when F or F +/ is zero, values of F and F +/, 
 which we here suppose to be excluded. 
 
 But, from what is done above, we see that this apparent im- 
 possibility has been actually effected ; for the same value of x, 
 namely, 
 
 * = - 2 V ~ V(^ P 2 ~ 2 ) ( 4 )> 
 
 that causes 
 
 X = x 2 -+- px + q 
 
 to vanish, causes also the entire factor (2) to vanish. 
 
 For the purpose of clearer illustration, suppose we write this 
 factor (2) in exact accordance with the general form in (II), 
 without any contraction; thus: 
 
 no + -p+ <\l x 2 +px + -p 2 —(x 2 +px + q) (5), 
 
 2 4 
 
 or, which is, of course, the same thing, 
 
 x + -p+ V (* + 2^) ~( x ' 2 + P X + 9) ( 6 )* 
 
10 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 Now, it is plain, that the value of x, in (4), substituted in 
 x 2 + px + q, reduces that expression to zero ; and that, suppressing 
 
 this zero, the entire factor (6) can become zero only for x+ -p=0, 
 
 or x = — - p ; and therefore cannot become zero, like the factor 
 
 (2) for the value of x, for which x 2 + px + q has vanished, namely, 
 the value (4). And yet (6) and (2), or which is the same thing, 
 (5) and (2), are virtually identical. 
 
 The unavoidable conclusion is this, namely : If we substitute 
 in (6) the value of x, exhibited in (4), before the subtraction 
 indicated by the minus is executed, we get a certain result, 
 
 _,Y= - 2 a/' . _ 
 
 ] y 4*- v v v v 4 
 
 but, if we make the substitution after the subtraction is per- 
 formed, we get a different result, namely, zero. 
 
 This is a truth little likely to have been anticipated ; but it is 
 only one among the many apparent paradoxes of analysis which 
 seem to force themselves upon the attention of the investigator, 
 to teach him the wholesome lesson of caution and discrimination. 
 The expression under the radical sign in (6) is not a perfect 
 square ; and therefore, in taking the square root, the plus before 
 the radical is not overruled by any law of generation controlling 
 that expression. Before anything is substituted for x, we per- 
 ceive that this is a general truth respecting it ; and that, therefore, 
 from the prefixed sign, the root, when extracted, is to be taken 
 positively. If, by any management or contrivance, we force, in 
 a particular case, a violation of a general law, I need scarcely say 
 that our result will be inadmissible. 
 
 (9.) Trifling as the matter here dwelt upon may appear to 
 some, it is nevertheless one among the many delicate points of 
 analysis in which the cultivators of pure science take a peculiar 
 interest. The examination of such points generally calls forth an 
 increased degree of that careful scrutiny and wary circumspection, 
 which analytical investigations so often demand. He who desires 
 to cultivate his reasoning powers, his penetration, his sagacity, 
 
ANALYSIS OF NUMERICAL EQUATIONS. 11 
 
 should study geometry ; but if he wish to bring all these into 
 still more vigorous exercise, he must apply himself to analysis.* 
 
 (10.) Returning from this digression, I shall now proceed with 
 the general investigation : 
 
 Let 
 .r 2a + ax 2a - x + to 2 "" 2 + cx- a -' A -i- (&?»-*+ .... + A = 0....(1) 
 
 be any equation of an even degree, into which every equation of 
 an odd degree may be converted by multiplying all its terms by 
 x ; the following expressions obviously form a pair of conjugate 
 factors of the left-hand member of the equation, namely, 
 
 # n ± V{— aaP^-baP-t-capo-Z-da?*-*— .... - A-} ... . (2), 
 
 and, in virtue of the principle (II), these expressions may be 
 successively changed into the following forms : 
 
 x n + - ax n ~ l 
 T 2 
 
 ± ^Hl a 2 _ ^ x*-* - cx 2u - s -da?*-*— .... -k\ .... (3), 
 
 x n +]-a a*" 1 -3 (-a 2-6)^-2 
 2 2 \4 / 
 
 * This is in direct opposition to the opinions of the distinguished writer, 
 already alluded to ; and I think many, besides myself, will regret that so 
 eminent a person should have published such views as those contained in his 
 recent work, ' On a Liberal Education.' The statements pervading that book, 
 and the arguments by which they are defended, are, I think, anything but 
 satisfactory and conclusive. Dr. Whewell says that " analytical operations 
 in mathematics do not discipline the reason." Few persons are likely to 
 concur with him in this assertion. But if the reason only is to be disciplined, 
 let the disciple read Euclid ; if this prove insufficient, let him read it again. 
 But to dictate that the Principia should be studied as Newton wrote it, is to 
 put the dial of science " fifteen degrees backward." The results of Newton 
 are imperishable truths ; the methods by which he reached them are, for the 
 most part, obsolete. 
 
12 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 and so on ; the principle of these successive changes being, that 
 we are to take care always to introduce into the rational part, 
 common to both conjugate factors, a quantity, such that twice the 
 product of it, by the leading term, may be equal to the leading 
 term under the radical, and of opposite sign. 
 
 It is plain that, by continuing this process, we shall at length 
 arrive at a pair of conjugate factors, such that the expression 
 under the radical will be of the degree n — 1; so that when the 
 proposed equation is of the fourth degree, the expression under 
 the radical, in the corresponding conjugate factors, may be re- 
 duced to the first degree ; when the proposed equation is of the 
 sixth degree, the expression under the radical may be reduced to 
 the second degree, and so on. 
 
 (11.) Let now the first member of the proposed equation be 
 represented by X, and the rational part, common to any pair of 
 its conjugate factors indifferently, by Xj ; put also X 1 for the 
 expression under the radical sign. Then we shall have 
 
 x = (X! + vx 1 ) (X, — vx 1 ) = o 
 
 and since each root of the equation X = must satisfy one or 
 other of the conditions 
 
 x, + vx'=o 
 x, - vx 1 = o, 
 
 we deduce the following obvious but important principle, namely, 
 that, in seeking the places of the real roots of X = 0, all those 
 values of x, which cause X 1 to become negative, must be rejected ; 
 and which principle may be expressed by the following 
 
 THEOBEJI. 
 
 (12.) In seeking the places of the real roots of X= 0, all 
 those intervals from x = a to x = b, throughout which X 1 con- 
 tinues negative, must be rejected ; and those only retained for 
 further examination, throughout which X 1 continues positive. 
 
 We thus see the importance of examining the auxiliary and 
 comparatively simple function X', before proceeding to discuss 
 
ANALYSIS OF NUMERICAL EQUATIONS. 13 
 
 the doubtful intervals which, in equations of a high degree, 
 usually occur in the partial analysis furnished by Budan's 
 functions; that is, by the several limiting or derived palynomials 
 deduced from X. The method I am explaining is not wholly 
 independent of all reference to these polynomials, or rather to 
 this preliminary process of Budan ; but, in the use hitherto made 
 of Budan's theorem, whether for the purpose of preparing the 
 way for a final and complete analysis by the method of Foukier, 
 or by that which Budan himself devised for the same purpose, 
 a great deal of tedious labour, must in general, be expended upon 
 the successive contraction of intervals within which we, at length, 
 find no roots can possibly exist. Those who have had much 
 experience of this laborious waste of calculation, will readily 
 appreciate the value of any external guidance in a procedure of 
 so much uncertainty ; and in search after real roots, within the 
 limits of their existence, as assigned by Budan's theorem, will 
 be glad to avail themselves of any indications which may mark 
 with certainty unproductive intervals. 
 
 (13.) Let us now examine into the general theory of the auxiliary 
 function X 1 , it being however understood that, in this examination, 
 the object is simply to ascertain its character and peculiarities 
 under different circumstances, and not to describe the steps neces- 
 sary to be taken in completing the analysis of an equation. 
 
 Let us imagine, then, that, for any equation X = that may be 
 proposed, we have Budan's partial analysis before us. Those 
 intervals in this analysis, which comprehend only single roots, 
 furnish definite information of the fact, and require no further 
 examination ; those which supply indications of pairs of roots, 
 are the doubtful intervals ; and it is the character of these alone 
 that remains to be ascertained. If, for the values of x through- 
 out any one of these intervals, X 1 continue negative, the doubt 
 is removed ; and we may at once conclude that the roots indicated 
 are imaginary. It may happen that, throughout all the intervals 
 within which the doubtful indications lie, X 1 continues negative : 
 we may then safely infer that all the pairs of roots indicated are 
 imaginary. 
 
14 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 Should we find that X 1 is such that X 1 = has all its roots 
 imaginary, then, provided only that the leading sign in X 1 be 
 negative, we may conclude, without any reference at all to 
 Budan's intervals, that the equation X = has no real roots. 
 
 Suppose we were actually to solve the equation X 1 = ; we 
 should then ascertain precisely the limits which separate the 
 positive and negative intervals in the function X 1 ; and should 
 pronounce those pairs of roots of X = 0, indicated by Budan's 
 analysis, in any of the latter intervals, to be imaginary. 
 
 It may happen that the roots of the equation X 1 = 0, or some 
 of them, may be equal. In this case, the equation X = will 
 offer some interesting particulars. First, suppose the equation 
 to be of the sixth degree, for which, as we have seen (p. 12), 
 X 1 = may be reduced to a quadratic. If the roots of this 
 quadratic be equal, then X 1 being a complete square, it follows, 
 that if it be preceded by the minus sign under the radical, the 
 equation X = can have none but imaginary roots ; except, 
 indeed, the roots of X 1 = equally belong to X = ; that is, 
 except X be divisible by X 1 ; or, which is the same thing, X,, by 
 vX 1 . In this case the remaining roots, all of course imaginary, 
 wdl be given by the biquadratic equation 
 
 X (X,) 2 
 
 _ = 0or 1^+1=0 
 
 If, however, in the case of equal roots of X 1 = 0, the sign of X 1 
 be plus, so as to furnish no rejective intervals, then the proposed 
 equation becomes decomposable into the two cubic equations 
 
 X, + ^xi = 
 
 x l - vx 1 = o. 
 
 It is scarcely necessary to observe that, in all cases, the product 
 of every pair of conjugate factors, into which a function X is 
 decomposed, always reproduces that function in one or other of 
 the forms 
 
 X = (X,) 2 + Xi 
 
 X = (X 1 )2-Xi. 
 
 If the equation of the sixth degree, which we have been here 
 
ANALYSIS OF NUMERICAL EQUATIONS. 15 
 
 discussing, were originally one of the fifth degree, converted into 
 the sixth by multiplying all its terms by x, then X 1 , being in this 
 case necessarily of the form 
 
 X 1 = ax 2 +bx + /3 2 , 
 
 in which the last term is essentially positive, could not be a 
 square preceded by the minus sign, whatever signs a and b may 
 have : the sign of the square would always be plus, and y'X 1 
 would be of the form px -f- /3, or px — (3 ; and since /3 enters 
 into Xj, as well as here, we should have, after suppressing the x 
 in that conjugate factor in which /3 becomes cancellea, a de- 
 composition of the equation of the fifth degree into two others of 
 the following forms, namely : 
 
 x 2 + gx + k = 
 
 X s + ffx 2 + mx + 2/3 = 0, 
 
 in which all the coefficients are known. And it is easy to see in 
 what way these deductions become modified, when the equations 
 under discussion are of the third or fourth degree, instead of 
 those of the fifth or sixth.* But, in reference to equations of 
 the fourth and sixth degrees, there are still some circumstances, 
 suggested by the preceding examination, which seems to deserve 
 special notice. The consideration of these, however, is not to 
 be understood as much advancing the object more immediately 
 aimed at in the present inquiry, but rather as supplying collateral 
 information, which we may as well stop a moment to pick up by 
 the way. 
 
 (14.) We have hitherto been reasoning upon the hypothesis of 
 certain definite and distinct conditions as to the composition of 
 the function X 1 . When, in actual practice, these are not accu- 
 rately fulfilled, there may still be an approach to them ; and we 
 
 * In equations of the fourth degree, the final form of X 1 , that is, the form 
 which terminates the process, is only of the first degree ; the examination, 
 therefore, of this final form, as to its change of sign, will he very easy. But, 
 for reasons to be hereafter given, it will in general be more satisfactory to stop 
 the operation at the quadratic form of X'. 
 
16 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 may anticipate consequences more or less approximative to those 
 which here follow with unerring precision. 
 
 Suppose that, in an equation of the sixth degree — and the 
 observations founded upon this supposition apply equally to an 
 equation of the fourth degree — suppose that the quadratic 
 function X 1 , under the radical, and preceded by the minus sign, 
 as above, be not a complete square ; but that the equation 
 X 1 = has unequal roots, x = a, x = b, of the same sign.* 
 Then, by the general theorem already given (p. 12), the equation 
 X = can have no real roots, except such as may exist between 
 the limits a, b ; this is evident, because, from x = oo, down to 
 that one of these limits which is nearest to cc, X 1 , omitting the 
 prefixed minus must continue plus ; and from x = — cc, up to 
 the other limit, it must also continue plus ; so that, taking the 
 prefixed minus into account, the function under the radical, 
 throughout these ranges, remains minus. 
 
 If, upon transforming the equation X = for the above- 
 mentioned limits, there be no loss of variation, we, of course, 
 conclude that the roots of that equation are all imaginary ; but 
 if a loss of variations takes place, we may then apply the following 
 considerations, bearing in mind that the lost variations must be 
 even in number, because all the real roots must lie within the 
 limits a, b. 
 
 The final sign of X 1 , here supposed to be included in a vinculum, 
 with the minus sign before it, is necessarily plus, because the roots 
 of X 1 = have, by hypothesis, the same sign ; also the final 
 sign in each of the transformations (a) and (b), of the original 
 equation X = 0, must be plus, because no real roots of X = lie 
 without the limits of a and b. 
 
 Now it is plain that, by modifying the final term of the 
 proposed equation X = — which final term, of course, enters, 
 with contrary sign, that of X 1 in each of the conjugate factors — 
 we may render the roots of X 1 = 0, thus changed, equal ; and 
 the foregoing inferences will then apply accurately to the modified 
 
 * We need not examine into the case in which the signs of a and b are 
 different, because, in the analysis of an equation, we always discuss the posi- 
 tive and negative intervals separately. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 17 
 
 equation X=0 : — all the roots of it, on each side of the limit x=p, 
 p representing one of the equal roots thus brought about, must 
 be imaginary; and although, as we have seen (p. 14), p may be 
 a root of the changed equation, it is impossible that it can be 
 a root of the original equation, because of this change. 
 
 Whether or not p be a root of this modified equation, may 
 be ascertained with tolerable ease : since, as shown above (p. 14), 
 it cannot be a root of it without also being a root of the corre- 
 sponding equation Xj = ; which, when the proposed is of 
 the sixth degree, is only a cubic ; and when the proposed is of 
 the fourth degree, only a quadratic, reducible to a simple equation 
 by dividing by x. 
 
 Suppose it turns out that p is a root of Xj = ; then, since in 
 the original equation X = 0, the unmodified final term is positive, 
 if this final term have been increased to bring about the above- 
 mentioned equality of roots in X 1 = 0, we may infer that a pair 
 of roots in the doubtful interval [a, b~\ are real ; for if they were 
 all imaginary, no increase of the final term, already positive, 
 could ever render any of them real, much less equal. (See 
 'Theory of Equations,' page 161.) 
 
 The equal roots p would thus be those into which two unequal 
 roots of the original equation X = have changed, in conse- 
 quence of the change which has been made in the final term of 
 that equation. 
 
 If, however, the final term of X require to be diminished to 
 render the roots of X 1 = equal, then, of course, all the roots of 
 X = must be imaginary. 
 
 (15.) It would not be difficult to generalize this inquiry, and 
 extend it to the case of any equation of which a pair of roots is 
 indicated between two values a and b that are roots of X 1 = ; 
 but, as already observed, the prosecution of the theory in this 
 direction would not much contribute to advance the object of 
 the present discussion. It may not be amiss, however, here to 
 remind the reader that two distinct classes of imaginary roots 
 may exist in the same equation, — the one class having immediate 
 reference to, and depending on, the final term of that equation, 
 
 2 
 
]8 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 so that a change in this final term alone, without disturbing the 
 other coefficients, will convert two such imaginary roots into a 
 real pair. The other class is independent of this final coefficient, 
 and the imaginaries belonging to that class — though under a 
 modified form— would still enter the equation, however the final 
 term be changed, and even though it be removed altogether. 
 
 (16.) Fourier has dwelt upon these peculiarities at sufficient 
 length to render any further reference to them here superfluous ; 
 and I thus briefly allude to them for the purpose of observing 
 that the imaginary roots, spoken of above as indicated between 
 a and b, are only those of the first class here mentioned. Second 
 class imaginaries, in the primitive equation, become first class 
 imaginaries in some of the derived or limiting equations, to which 
 equations the preceding examination may be applied in suc- 
 cession.* And it is pretty obvious, from what has here been 
 said, that although no first class imaginary roots may be indicated 
 between the limits a and b, yet the indications of secondary pairs 
 may lie concealed in that interval, — a fact which it is well to 
 bear in recollection, seeing that, in the present mode of dis- 
 covering rejective intervals, the final term enters as an essential 
 element of the process. Yet it must not be inferred that the 
 practical advantages of this process disappear, even when the 
 above distinctions are neglected; but only that to obtain, in 
 certain cases, the largest amount of information respecting the 
 texture of an equation, and the character of its roots, they must, 
 in general, be attended to. 
 
 It may be noticed further, that in speaking as above of the 
 
 * The eminent analyst alluded to above, though fully recognising the 
 marked difference between the two classes of imaginary roots, mentioned in 
 the text, does not distinguish them, as I have thought it convenient to do. It 
 seems desirable that some such distinctive appellations should be adopted ; it 
 might, perhaps, be allowable to call the roots of the first class, primitive 
 imaginary pairs ; and those of the second class, secondary imaginary pairs. 
 We should thus sufficiently indicate the fact, that the imaginaries of the first 
 kind discover themselves, in a direct manner, from an examination of the 
 primitive equation ; and those of the second kind, only in an indirect manner, 
 from examining, in like manner, the derived or secondary equations. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 19 
 
 substitution of the limits a, b, the roots of the quadratic equation, 
 X 1 = 0, I do not necessarily mean the employment accurately of 
 these numbers : any other pair of numbers which embrace these 
 will do just as well, provided they do not leave between them an 
 interval so great as to inclose indications of other roots. But, 
 as stated at the outset, I am not now undertaking to point out 
 the most desirable method of proceeding when, in actual practice, 
 we meet with a critical case requiring a more than ordinarily 
 minute examination. It is not my object here to give directions 
 for performing, in a new manner, what may be as readily, or, 
 perhaps, even more readily, effected by existing methods. My 
 purpose is to do away, as much as possible, with the intolerable 
 labour of pursuing roots through regions where they cannot 
 possibly be found, by furnishing decisive indications of barren 
 intervals. The rejection of these is the saving of so much profit- 
 less work ; the intervals that remain being all that require any 
 examination; the roots in these, as just remarked, may, in pecu- 
 liar circumstances, be slow in unfolding their character ; but I 
 have elsewhere given the results of much careful and continuous 
 consideration of these delicate cases ; and my settled conviction 
 is that, in general, the best way of proceeding is to treat the 
 doubtful roots, in such cases, as if they were known to be real ; 
 that is, to enter upon their actual development : then, whether 
 they are imaginary or not, is a circumstance that will unam- 
 biguously declare itself by indications, occurring sooner or later, 
 which I have sufficiently explained and exemplified.* 
 
 I have but one observation more to make : — The limits a, b, 
 above, have been considered to have the same sign, because in 
 Bud an' s partial analysis — and, indeed, in every analogous method 
 — the several intervals lie each of them wholly in one region ; 
 that is, wholly in the positive region, or wholly in the negative 
 region. It appears, from what has here been shown, that when 
 these limits a, b, are roots of the quadratic equation X 1 = 0, and 
 that the minus sign is prefixed to X 1 , all the roots of the equation 
 X = 0, of which the indications lie without those limits, are 
 
 * See the ' Theory and Solution of Equations of the Higher Orders,' p. 298 ; 
 and the ' Researches respecting the Imaginary Roots of Numerical Equations.' 
 
20 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 imaginary ; if two only are indicated ivithin the same limits, we 
 have seen how their character may be ascertained ; should they 
 prove to he imaginary, we may conclude, not only that they are 
 imaginary, but that they belong to the first class of imaginaries. 
 If a very trifling modification of the final term of X would 
 convert these into equal real roots, such equal roots might be 
 received as approximate solutions (Theory of Equations, p. 310). 
 
 (10.) The foregoing details respecting the minute analysis of 
 a doubtful interval, as well as the more general considerations 
 which precede them, I have entered upon, mainly on account of 
 their theoretical interest, as throwing some light on the consti- 
 tution and peculiarities of what I have called the conjugate 
 factors of an equation, and as illustrative of the nature of the 
 information, respecting the composition of an equation, which 
 those factors are capable of supplying. 
 
 But there are some deductions, arising out of these general 
 considerations, which claim more than a mere passing notice, as 
 they are practically useful as well as theoretically interesting. 
 I have deferred the formal enunciation of them in order, first, 
 to establish another method of obtaining the functions Xj, X 1 ; 
 as this other method will suggest what some may consider a 
 more convenient form of expression for the theorems I have in 
 view. 
 
 If we suppose the general function X 1 , under the radical, to 
 become spontaneously zero, then, as each conjugate factor reduces 
 in that case simply to Xj ; we infer that X, is the complete 
 square root of X ; in other words, that X is a complete square. 
 It follows, therefore, in all cases, that X 1 is the quantity to be 
 added to X to make this latter polynomial a complete square ; so 
 that — X 1 is nothing else than what remains when the operation 
 of extracting the square root of X is stopped ; that is, — X 1 is 
 the correction which must be applied subtractively to X, in order 
 to render that function a complete square. And thus the 
 improvements in the analysis of an equation, which will presently 
 be exemplified, would have been thought of before, if it had only 
 occurred to any one to extract the square root of the first member 
 
ANALYSIS OF NUMERICAL EQUATIONS. 21 
 
 of that equation, and then to discuss the character of the re- 
 mainder; and, as a collateral circumstance, a simple and compact 
 method of extracting the square root of a polynomial — much 
 more convenient than the straggling process in common use — 
 would probably, at the same time, have suggested itself. 
 
 (17.) It will, perhaps, be an excusable digression if I offer 
 here an illustration or two of this improved arrangement; ob* 
 serving that, as the remainder due to each step of the operation 
 is to be exhibited with the proper signs, as in the common method, 
 the terms which contribute to the formation of X 1 , as already 
 explained, are now to be subtracted, not added, as in the fore- 
 going pages. 
 
 Let the polynomial proposed for extraction be 
 
 x 
 
 6 + $ x 5 _ 4^4 + 2*3 — 6x 2 — 20* + 17, 
 
 then the successive approximations to the root, accompanied with 
 the corresponding remainders, or portions of the polynomial still 
 unexhausted, may be arranged in the following manner, upon the 
 principle explained at the commencement of the present discus- 
 sion (p. 7), the observation above, respecting the formation of 
 the remainder, being attended to. 
 
 Root. Remainders. 
 
 a ,3 +4a; 2 -20* 4 + 2* 3 — 6* 2 — 20*+17 
 
 #3 + 4* 2 — 10* 82* 3 — 106* 2 — 20*+ 17 
 
 #3 + 4* 2 -10* + 41 — 434z 2 + 800tf— 41 2 + 17 
 
 And, if requisite, this approximation may be continued on for 
 negative powers of *. 
 
 Should the leading term of the proposed polynomial require 
 a factor to make it a square, we may introduce this factor into 
 each of the terms ; and if, after the extraction, continued as far 
 as may be thought necessary, we divide the root by the root of 
 the factor, and the remainder by the factor itself, the results will 
 be correct. 
 
22 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 As another example, let it be required to extract the square 
 root of 
 
 X 2 + X + 1 . 
 
 Root. Remainder. 
 
 1 3 
 
 X+ 2 4 
 
 so that, taking the remainder, with a changed sign, the conjugate 
 factors of the proposed expression are 
 
 3 
 4' 
 
 Lastly, let the polynomial be 
 
 # 4 + ax 3 + bx 2 + ex + d. 
 
 (* + -) + V — j aQ <* (^ + 2)""^' 
 
 Root. Remainder. 
 
 x 2 + - ax lb a 2 ) x 2 + ex + d, 
 
 zz+laz+l (4-I4 {_la(»-i«») + i}*-i(«-i#)*+4 
 
 from which the conjugate factors of the polynomial may be 
 obtained as before. 
 
 It thus appears that the process for the extraction of the 
 square root of a polynomial, whether conducted as above, or by 
 the common method, not only furnishes us with the exact root, 
 when the proposed expression is a complete square, but, by 
 analysing that expression, when not a complete square, into con- 
 jugate factors, it enables us to ascertain whether or not it is 
 composed of the sum or difference of two squares ; a fact well 
 worthy of notice, although I believe hitherto unobserved. The 
 polynomial, in the last example, is composed of the sum or 
 difference of two squares if 
 
 4 ( b — l - a 2 ) d = c 2 , 
 
 the sum, if the two factors in the first member of this equation 
 are positive, and the difference if they are negative. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 23 
 
 (18.) It must not, however, be supposed that when the poly- 
 nomial is thus constituted, the decomposition here adverted to 
 will invariably be effected by the particular conjugate factors to 
 which our attention has been exclusively confined in the present 
 inquiry. As noticed at the outset, the pairs hitherto dwelt upon 
 — and which are those we shall almost entirely restrict ourselves 
 to throughout — are not the only conjugate factors that may be 
 employed ; we may, indeed, decompose a polynomial into as 
 many distinct pairs of conjugate factors as we please ; as is 
 evident from the arbitrary character of F and f in the general 
 form at page 7. Wben, therefore, a polynomial is suspected to 
 be composed of the sum or difference of two squares, or when 
 our object is to put the matter to the test, we may find room, in 
 certain cases, for the exercise of some ingenuity in the discovery 
 of the proper component factors. An easy example will, perhaps, 
 sufficiently illustrate this. Suppose the polynomial were 
 
 x i — 4# 3 + 7x 2 — 4x + 1, 
 
 then, proceeding as in the foregoing instances, we have the 
 following results, namely, 
 
 Root. Remainder. 
 
 x 2 — 2x 3x 2 — 4x + 1 
 
 3 1 
 
 x 2 — 2x + - 2x—\- 
 
 2 4 
 
 The last remainder shows that the expression cannot be a com- 
 plete square ; but we cannot affirm that it is neither the sum nor 
 the difference of two squares from the form of the preceding 
 remainder. To come to a decisive conclusion on this point, we 
 must first ascertain whether any quantity, introduced into the 
 partial root a; 2 — 2x, can render the remainder a square. And 
 it is easy to see that 1 is such a quantity ; so that, varying the 
 second step, the results may be written thus : 
 
 Root. Remainder. 
 
 x 2 — 2x 3* 2 — Ax + 1 
 
 x 2 _ 2x + 1 x 2 
 
24 ANALYSIS OF NUMERICAL EQUATIONS 
 
 from which we infer that the proposed polynomial is the same as 
 
 (x 2 — 2x + l) 2 + x 2 . 
 
 If the polynomial, here discussed, were the first member of a 
 biquadratic equation, that equation might, therefore, be replaced 
 by the two quadratics 
 
 x 2 — (2 — V— 1)«+ 1 =0, 
 and 
 
 x 2 — (2 + x/—l)x+ 1 =0. 
 
 And similarly in other like instances. 
 
 (19.) The theorems to which I have prospectively ' alluded 
 above, in addition to what has already been shown, will require 
 the proof of the following proposition, which, in itself, is not 
 without independent interest. 
 
 Every equation of the form 
 
 (Fx) 2 — fx = (A) 
 
 must necessarily have two real roots, provided Fx be of an odd 
 degree, that (Fa?) 2 be of higher degree than fx, and that the roots 
 of fx = are all imaginary. 
 
 For, since (Fx) 2 is of higher degree than fx, a real value so 
 great may be given to x, such as to render the left hand member 
 of (A) positive. But, as Fx is of an odd degree, a real value may 
 be given to x, such as to render (Fx) 2 zero ; in which case, seeing 
 that fx is invariably positive, the left-hand member will become 
 negative Hence the equation must have one real root ; and, 
 since it is of an even degree, it must necessarily have another ; 
 and it is plain that we may amplify the proposition by the fol- 
 lowing addition to it, namely : 
 
 And if Fx be of an even degree, the other conditions remaining, 
 the equation (A) will still have a pair of real roots, provided the 
 roots of Yx = be not all imaginary. 
 
 (20.) This proposition, taken in conjunction with what has 
 previously been proved, warrants the inference of the following 
 theorems, namely: 
 
ANALYSIS OF NUMERICAL EQUATIONS. 25 
 
 THEOREM 1. 
 
 If, in an equation of the sixth degree, we arrange the terms all 
 on one side, and extract the square root of the polynomial, then, 
 if the quadratic remainder have its leading sign positive, and its 
 roots imaginary, the proposed equation cannot have a real root. 
 
 But if the leading sign be negative, the roots of the quadratic 
 being still imaginary, then the proposed equation must have one 
 pair of real roots at least. And, generally, if the equation be of 
 any even degree, of which the square root is of an odd degree, 
 and that any remainder at which we stop have all its roots ima- 
 ginary, then, if the leading sign of this remainder be positive, 
 the equation cannot have any real roots ; but if the leading sign 
 be negative, it must have one pair of real roots at least. 
 
 THEOREM IT. 
 
 If, in any equation of an even degree, of which the square root 
 of the polynomial forming the first member is also of an even 
 degree, any remainder at which we stop have only imaginary 
 roots ; then, if the leading sign of this remainder be positive, the 
 proposed equation cannot have any real roots. But if the leading 
 sign be negative, then the equation must have one pair of real 
 roots at least, provided the roots of the polynomial, which is the 
 result of the extraction, be not all imaginary. 
 
 It is further obvious that the two real roots, shown in these 
 theorems to exist under the prescribed conditions, must be un- 
 equal roots ; for the left-hand member of (A) could not furnish 
 a change of sign for two substitutions for x, of numbers between 
 which equal roots only were comprehended ; and, as before 
 noticed, the presence of one root in an equation of an even 
 degree, as indicated by such change of sign, necessarily implies 
 the existence of another root of different value. 
 
 It must be noted, too, that in these theorems, the remainders 
 
26 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 spoken of are those that present themselves in the operation for 
 extracting the square root, "whether that operation be conducted 
 in the usual way, or as above exhibited. The function X 1 , in 
 the preceding pages, as also in those which are to follow, is this 
 remainder with changed signs ; and it is of importance to keep 
 this in remembrance when any reference to these theorems may 
 be made hereafter. 
 
 (21.) It is the consideration of this function X 1 , as we have 
 already seen, that guides us to the rejective intervals between the 
 extreme limits of the roots. When such rejective intervals exist 
 in conjunction with others not rejective, then at each boundary, 
 between the two sorts of intervals, X 1 , of course, vanishes ; and 
 the corresponding value of x makes X plus, unless, indeed, this 
 same value of x cause Xj to vanish also, in which case that value 
 will be one of a pair of equal roots of the equation X=0. Hence, 
 whether this happen or not, if Xj= only have a root in a non- 
 rejective interval, for such root, as Xj vanishes, X must be minus. 
 And thus we obtain a third theorem, namely : 
 
 THEOREM III. 
 
 If X, = have a root in any non-rejective interval, the proposed 
 equation, X=0, must have real roots (a, pair of real roots at least, 
 X being of even degree); one will lie between the root of Xj= 
 and + oo, and another between the same root and — go. If there 
 be any rejective intervals at all, a real root will be situated between 
 the root of Xj = 0, here spoken of, and the boundary of that re- 
 jective interval which is nearest to it, on either side ; so that if a 
 boundary exist on each side, the situation of two real roots within 
 defined limits will be ascertained, X being, of course, understood 
 to be of even degree. 
 
 I venture to regard the theorems here announced as of some 
 importance in the analysis of equations. They are, moreover, a 
 little remarkable ; since, although we have several criteria for 
 
 \ 
 
ANALYSIS OF NUMERICAL EQUATIONS. 2/ 
 
 detecting the presence of imaginary roots, yet, beyond the qua- 
 dratic, if we set aside the theorem of Sturm, I believe there exists 
 no rule for enabling us to affirm the presence of real roots, in 
 any equation of an even degree, with the last term positive, 
 unless, indeed, the utterly impracticable tests first proposed by 
 Waring, and afterwards re- discovered by Lagrange, can be 
 regarded as an exception. 
 
 (22.) There is frequently a strong temptation, in the inves- 
 tigation of such criteria, to assume a converse proposition from a 
 direct one. I think Maclaurin must have yielded to some such 
 temptation when he undertook the demonstration of Newton's 
 rules for the discovery of imaginary roots ; for he commences the 
 inquiry evidently under the impression that these rules apply 
 both directly and conversely; an impression, however, which 
 meets with no confirmation as he goes on, and which he closes 
 his paper without undertaking to justify. 
 
 Most persons engaged in these reseaches, I dare say, have had 
 sedulously to guard against similar temptations. I acknowledge 
 that, for myself, even in the course of the present discussion, 
 views and considerations have presented themselves with such 
 plausible allurement, and supported by so many examples devised 
 to test their validity, that I have found some resolution to be 
 necessary to escape the snare, and to keep myself honestly within 
 the pale of rigorous demonstration. And I hope it will be found, 
 accordingly, that I have here advanced nothing respecting the 
 real roots of an equation, merely from the non-detection of ima- 
 ginary roots. All the positive information, which I have felt 
 justified in deducing, about the real roots of an equation, is com- 
 prised in the theorems here given, and in a few other general 
 inferences that will be found towards the close of the essay. But 
 I entertain the expectation that additional theorems of interest 
 may be hereafter derived from a further examination of the con- 
 jugate factors into which it is here proposed to divide the first 
 member of an equation. 
 
 (23.) I shall now proceed to practical applications of the theory 
 
q 
 
 28 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 developed in what has preceded ; and, in order to secure the more 
 ready means of instituting a comparison between existing methods 
 and that which this theory suggests, I shall, in general, take 
 examples which have already been discussed by Budan, Fourier, 
 Sturm, or other writers, who may have employed their processes. 
 As I shall quote the authors from whom these examples are ex- 
 tracted, I do not conceive that it will be at all necessary that I 
 should encumber these pages by copying from those authors the 
 initial steps of the analysis of an equation by the theorem of 
 Budan. It will, indeed, be seen, in the subsequent operations, 
 that we may frequently dispense with the theorem altogether ; 
 for, in the present method, it will usually be best to ascertain the 
 rejective intervals, by an examination of our function X 1 before 
 any portion of Sudan's analysis is undertaken, as it is of no use 
 to search out the particular intervals furnished by that analysis, 
 within which roots, already known to be imaginary, are indicated. 
 Neither shall I, in the course of the analyses to be given, pay 
 much attention to the criteria already in existence for testing the 
 character of the roots of an equation by an examination of the 
 coefficients, as I wish the method here proposed to be judged of, 
 independently of such extraneous aid. Of the criteria adverted to, 
 those given, without demonstration by Newton, in his Arithrnetica 
 Universalis', and which have been glanced at above, are, perhaps, 
 the most important ; but, no proof of their truth having been 
 given till very recently, they have been altogether neglected by 
 modern analysts. Their demonstration, as already noticed, was 
 
 <Qnclertaken by Maclaurin ; and Montucla seems to think with 
 success ;* though a careful examination of their reasoning will 
 show that all that Maclaurin and Campbell did in this way is 
 insufficient to enable us to detect, with certainty, more than two 
 imaginary roots of an equation, even under the most favorable 
 circumstances. A complete demonstration of Newton's rule, 
 was, I believe, for the first time, given in my ' Researches re- 
 specting the Imaginary roots of Numerical Equations,' published 
 \ in 1844. Other rules, analogous to this of Newton, will be given 
 towards the close of the present essay. 
 
 * Histoire des Mathematiques ; tome hi, p. 31. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 29 
 
 EXAMPLES. 
 
 1. Fourier, 'Analyse des Equations,' p. 137, 
 
 x± — 4x s - 3x + 23 = 0. 
 
 The proper conjugate factors of the first member of this equa- 
 tion are 
 
 x 2 - 2x ± s/ (4x2 + 3 X - 23) 
 
 = #2 _ 2x — 2 ± sf{7x — 19), 
 
 .-. X 1 = 7x— 19. 
 
 As, for all negative values of x, X 1 is negative, we infer that 
 the equation cannot have any negative roots. And as X 1 con- 
 tinues negative for all positive values of x, up to 2 inclusive, it 
 further follows that, in substituting numbers below 10, the 
 superior limit of the roots in Budan's (or Fourier's) functions, 
 we may commence with 2. The substitution of this number 
 furnishes but two variations of signs. We conclude, therefore, 
 that the equation has two imaginary roots. The substitution of 
 3 causes one of these variations to disappear, so tbat the equation 
 has two real roots, of which one lies between 2 and 3, and the 
 other between 3 and 10. (See Fourier, pp. 107 and 138.) 
 
 Fourier arrives at these results only after a laborious process, 
 involving an operation for the common measure of X, and its first 
 derived function. 
 
 It will be observed that the preceding expression for X 1 shows 
 that the root, afterwards found to lie in the interval [2, 3], lies 
 in the still narrower interval [2-5-, 3] ; a circumstance not un- 
 worthy of notice when actual development is the object. 
 
 2. Fourier, page 143, 
 
 aA _ a? + 4a; 2 + x — 4 = 
 
 x 2 
 
 1 ,/ 15 , A 
 
 --x± s /[--x 2 -x+4y 
 
 .-. X 1 =-(^r 2 + x-4). 
 
30 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 It is obvious, without proceeding further, that since X 1 is 
 negative for se = I, and for all higher values, the positive roots 
 of the equation must all lie between and 1 . 
 
 Applying these limits in Budan's functions, one variation is 
 lost, and two variations still remain ; and since, as just shown, 
 these cannot be lost by the passage of real roots, we infer that 
 there is a pair of imaginary roots in the positive region ; and, 
 consequently, that there is a single real root in the negative 
 region. 
 
 (3.) Lockhart. « Resolution of Equations,' p. 26, 
 
 12a? 3 - 120.r 2 4- 326* - 127 = 0, 
 
 or, 
 36a? 4 — 360* 3 + 978* 2 — 381* = 
 
 6* 2 - 30* ± V(- 78* 2 + 381*) 
 6* 2 _ 30* + Z? ± V- (9* - H- 2 ), 
 
 .-. X i = - 9* + —• 
 4 
 
 13 2 
 Hence X 1 is negative from * = — --r-9, onwards ; that is, from 
 
 4 
 
 4'7 toco. But Budan's functions give two variations for 4 - 7; 
 hence the roots in the positive region are imaginary. 
 
 This example, although only of the third degree, is one of 
 considerable difficulty. According to Mr. Lockhart, the com- 
 plete analysis of it, by Sudan's method, is all but impracticable;* 
 and it is also exceedingly troublesome by the process of Fourier, 
 as I have elsewhere shown. j- 
 
 (4.) Let it be proposed to determine the criteria of imaginary 
 roots in a general equation of the fourth degree, with its last 
 term positive, 
 
 x 4 4- ax 3 -f- bx 2 4- ex + d = 0, 
 
 * Lockhart' s ' Resolution of Equations ;' Oxford, 1837, p. 28. 
 s^ f Theory and Solution of Equations of the Higher Orders ; p. 244. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 31 
 
 x 2 + - ax ± \f — ( b — - a 2 J x 2 — ex — d, 
 
 ... X != -Ub — ^a 2 )x 2 + cx + d\ 
 
 Consequently, if 
 
 4 (b-^-a 2 }d>c\ 
 
 the roots are all imaginary. 
 
 So simple a criterion as this has not, I believe, been given 
 before. 
 
 If the second term ax*, be absent from the equation, then the 
 criterion becomes 
 
 4bd > c 2 , 
 
 in which case, we have a confirmation of Newton's rule; for, 
 since d is necessarily positive, b must be positive likewise ; so 
 that, as we know, from the criterion of De Gua, that two imagi- 
 nary roots are indicated by the absence of the second term, tbose 
 on each side having like signs, and as two are also indicated by 
 the three terminating coefficients, and none by the middle ones, 
 Newton's conditions are fulfilled ; and, as here shown, the roots 
 are, in consequence, all imaginary. 
 
 There never existed any doubt as to Newton's rule indicating 
 a pair of imaginary roots, when these conditions are fulfilled; 
 but, till recently, as before noticed, it was matter of question 
 whether we could ever safely infer the existence of more than 
 one pair. 
 
 The roots of the preceding equation are still imaginary, though 
 the condition should be 
 
 4(b — -« 2 )rf=c2, 
 
 unless the roots of X 1 = also satisfy the equation 
 
 x 2 A — ax = 0, 
 
 2 
 
32 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 in which case, the proposed equation would have a pair of roots, 
 
 each equal to — - a ; the other two roots must be imaginary 
 
 (page 14). 
 
 It may not be superfluous to remark here, in reference to 
 equations of the fourth degree, that all the roots in the positive 
 region will be imaginary, if the roots of x 2 + ax + b = are 
 imaginary, and if, moreover, c and d be positive ; and those in 
 the negative region will be imaginary if, the other condition 
 remaining, c be negative, and d positive ; for, under these 
 restrictions, all positive substitutions in the one case, and all 
 negative substitutions in the other, give positive results. 
 
 (5.) Midy. 'Du Theoreme, de M. Sturm,' p. 35, 
 
 x 6 + x 5 — x 4, — x 3 + x 2 — x + 1=0, 
 
 x & 
 
 2 8 
 
 3 l 2 5 3 _u 
 
 2 8 16 
 
 + x* 2 ± VV + x 3 — x 2 + x — l 
 2 4 
 
 /3 /B 2 \ 
 
 V( 
 
 or, 
 — (204a; 2 - 3 16a? + 247). 
 
 As this is preceded by the minus sign, and the roots of it are 
 imaginary, we conclude (page 14) that all the roots of the pro- 
 posed equation are imaginary. 
 
 It is in equations such as this, where all, or a considerable 
 proportion, of the roots turn out to be imaginary, that the method 
 here explained discloses, in the most obvious manner, its peculiar 
 
ANALYSIS OF NUMERICAL EQUATIONS. 33 
 
 advantages ; and it is precisely in 6uch equations that extra assist- 
 ance was more especially needed. I have, elsewhere, analysed 
 this equation, both by the theorem of Sturm and that of Fourier, 
 but not without a large amount of very tedious labour, notwith- 
 standing that every expedient to abridge the processes was 
 resorted to.* The work above did not occupy more minutes 
 than these methods required hours ; for, in such an extent of 
 calculation as is exhibited in tbe aualysis alluded to, frequent 
 and careful revision of the steps of the work become necessary 
 to ensure accuracy. Budan's method, also applied to this 
 equation, in the same place, happens to be shorter, but is much 
 inferior to that above. 
 
 (25.) As it is not my wish that the present mode of treating 
 an equation submitted to analysis should be overrated, I have not 
 resorted to the stratagem of framing examples peculiarly fitted 
 to its powers, but have taken them at random from other works; 
 and I take this occasion again to state, that the value of the 
 method mainly consists in its readily making known to us the 
 rejective intervals in which it would be in vain to seek for real 
 roots. This is always an advantage ; but when there happens to 
 be no imaginary roots in the equation proposed, the advantage 
 may then be considered as reduced to its minimum, although it 
 does not wholly disappear. And nearly the same may be said, 
 when, as will sometimes happen, imaginary roots exist in non- 
 rejective intervals. But in no case whatever can the work be 
 entirely fruitless, as the theorems at page 25, and what precedes 
 them, sufficiently show ; and as will be still further seen, in a 
 scholium appended to these examples. 
 
 As already observed, the pair of roots which are indicated in a 
 non-rejective interval, if submitted to the process of actual de- 
 velopment, unfold their character in a perfectly unambiguous 
 manner. 
 
 (26.) The following example is selected for the purpose of 
 exhibiting the method in its least favorable aspect. 
 
 * See the ' Theory and Solution of the Higher Equations,' p. 234. 
 
 3 
 
34 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 (6.) Sturm. 'Third Example,'* 
 
 # 3 + 11a; 2 — 102a; + 181 = 0, 
 
 or, 
 a* + 1 la* — 102a; 2 + 181* = 0, 
 
 .-. X 1 = 529a; 2 - 724x. 
 
 Here we have but a very small rejective interval in the positive 
 
 r 724-1 
 region, namely, the interval 0, — — ; and no rejective interval 
 
 in the negative region. 
 
 In Budan's functions, therefore, we commence our substi- 
 tutions, in the positive region, with x = 2 ; and we find two 
 variations to disappear between 3 and 4. We may, therefore, 
 apply the process of development to the roots thus indicated in 
 a non-rejective interval, and shall find them to turn out real, and 
 to be 
 
 3-213 and 3229 
 
 as Sturm has determined, but not without much more trouble. 
 As no other root can exist in the positive region, we conclude 
 
 that the remaining root is negative ; and since x = — — is a 
 
 root of _ _ 1 1 
 
 X, = a? + - w <= 0, 
 
 we know, from Theorem in, page 26, that this negative root 
 
 11 
 
 and — 20. 
 
 must be greater than — . It is found to lie between — 10 
 
 (27.) And here it may be proper to remind the reader that, in 
 dealing with equations of an odd degree, we always, as above, 
 
 * See Mr. Spiller's translation : ' The Solution of Numerical Equations, 
 by C. Sturm ;' p. 30. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 35 
 
 introduce an extraneous root, namely, the root x = ; and that 
 the theorem, just referred to, applies to the equation, in its full 
 extent, only after this modification; so that when a root is shown 
 by that theorem to exist between certain assignable limits, we 
 must take notice whether this be only the extraneous root x = 0, 
 temporarily engaged, or a root really belonging to the original 
 equation. If this point be not first settled, we cannot deduce 
 any conclusion as to the roots of the equation proposed. 
 
 (7.) Foukiek, page 145, * 
 
 x 5 + x i _j_ x 3 _ 2x 2 + 2x — 1 = 0, 
 
 or, 
 x a + x » _|_ #* _ 2a? + 2x 2 — x = 
 
 gS + I x 2 ± ^/ ^_ 3 x 4 + 2x? , _ 2x 2 + x J 
 
 a? + -x 2 + -x ± a/ — X s — - x- + x); 
 
 28 V \ 8 64 / 
 
 .-. X 1 = (152*- — 119* + 64)*. 
 
 As the roots of the quadratic in the vinculum, are imaginary, 
 every substitution for x, made in it, must give a positive sign. 
 Consequently, on account of the other factor x, the substitution 
 of negative numbers must always give a negative sign ; hence 
 there are no real roots in the negative region ; so that the roots 
 must all be sought in the positive region. 
 
 In Sudan's functions, three variations only are given for A'=0; 
 or, which is the same thing, without referring to Budan's func- 
 tions at all, three variations only occur in the original equation. 
 Consequently, the equation must have two imaginary roots at 
 least; and these are indicated in the negative region. Indica- 
 tions of the remaining three roots occur in the non-rejective 
 interval [0, oc]. (See page 46, in reference to this.) 
 
 As stated at page 18, it will be desirable to reject secondary 
 imaginary roots if there be any ; so that, taking the derived 
 equation, for the purpose of ascertaining this, we have 
 
36 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 5# 4 + 4a? 8 + 3a; 2 — 4x + 2 = 
 or, 
 25a 4 + 20a.- 3 + 15a 2 — 20* + 10 = 
 
 5x 2 + 2« ± y/(— 1 l* 2 + 20x — 10 ) 
 .-. X l = -(ll 2 -20* + 10), 
 
 the roots of which are imaginary ; so that the proposed equation 
 has but one real root, and that is positive ; as, indeed, we other- 
 wise know from the sign of the final term. Had the interval 
 here, instead of being wholly rejective from — oo to -f- cc, been 
 rejective only from to 1, the same conclusion would have fol- 
 lowed ; since, by Budan's functions, a pair of roots of the derived 
 polynomial, here examined, are indicated in that interval. 
 
 (28.) As already noticed, the preceding examples are all dis- 
 cussed, without any aid from criteria, respecting imaginary 
 roots, as furnished by the coefficients of the original equation. 
 Assistance, spontaneously offered, has thus been disregarded. 
 But the rule of Newton, and that of De Gua, which is com- 
 prehended in it, as they often supply valuable information from 
 a mere inspection of the coefficients of an equation, are of con- 
 siderable use in particular cases, whatever general method of 
 analysis be employed. If Newton's rule had been appealed to 
 in the discussion of the preceding equation, we should have dis- 
 covered the existence of the four imaginary roots at once. 1 
 shall avail myself of the information supplied by this rule in the 
 next example. 
 
 (8.) Fourier, page 111, 
 
 x 7 — 2x b — 3a 3 + Ax 2 — bx + 6 = 
 
 or 
 a& _ 2x 6 — 3a l + 4a? 3 - 5x 2 + 6x = 
 ay* — a? ± ^(4x± — 4a* + 5x 2 — 6x) 
 aA _ gfi _ 2 ± N /(_ 4rf + g x 2 _ 6 X + 4) ; 
 .-. X 1 = — (4x* - 9x 2 4- 6x — 4). 
 
ANALYSIS OF NUMERICAL EQUATIONS. 37 
 
 By Newton's rule, the original equation has at least four 
 imaginary roots ; although we cannot infer from that rule the 
 particular intervals in which indications of the imaginary pairs 
 lie. The cubic equation, 
 
 4» 3 — 9x 2 + 6x — 4 = 0, 
 
 by the same rule, has also a pair of imaginary roots ; and we see 
 at once, that its real root lies between 1 and 2; so that [1, 2 J 
 is an interval partly rejective and partly non-rejective ; and, 
 throughout the entire region, beyond 2, X 1 is rejective. It is 
 further obvious that \J2 is a root of 
 
 X, = x 4 - x* - 2 = ; 
 
 and, with a view to the application of Theorem in, it remains to 
 be ascertained whether this root, y/2, is in the rejective or in the 
 non-rejective part of [1, 2]. 
 
 For x = y/2 } X 1 is 22 — 14 >/2, which is obviously plus; 
 therefore the root \/2, of Xj = 0, lies in a non-rejective interval, 
 the non-rejective part of [1, 2]; and, consequently, by Theorem m, 
 one real root, at least, must lie between 1 and 2. 
 
 By substituting these limits in the proposed equation, we find 
 no change of sign in the results ; hence there must be two real 
 roots in the interval [1, 2]. We therefore conclude that the 
 equation has two positive roots between 1 and 2, one negative 
 root and four imaginary roots ; and this without any reference 
 to Bud an' s functions. 
 
 In this example, Newton's criteria have afforded considerable 
 assistance ; but, valuable as they are, in certain circumstances, 
 their capabilities must not be overrated. The unfavorable feature 
 in them is, that, although in such circumstances they make 
 known the existence of a pair of imaginary roots, yet they do 
 not indicate the interval, but only the region, to which they 
 belong ; so that, when pairs of roots occur in different intervals 
 of that region, Newton's rule does not enable us to distinguish 
 the interval of imaginary from the interval of real roots. It is 
 remarkable, however, that no mention of this rule is ever made 
 — even in its simplest application — by any of the modern conti- 
 
38 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 nental writers on equations ; and even English authors, when 
 they do mention it, content themselves with a mere passing 
 notice of it, with the single exception of the venerable Mr. 
 Lockhart, a gentleman who, at the age of eighty-seven, is still 
 prosecuting his scientific researches with the same ardour that 
 animated his early years.* 
 
 The following example is from the "Appendix" to that gentle- 
 man's work, on the ' Resolution of Equations by means of Inferior 
 and Superior Limits,' published in 1843. 
 
 (9.) Lockhart. 'Appendix,' &c, page 9. 
 
 x 4 * — 7'2* 3 + 13-44^ 2 — 1728* + '0576 = 0, 
 
 x 2 — 3-fw ±V(— ' 48 * 2 + 1728* — -0576), 
 
 x 2 — 3'6x + -24+V0- 
 
 The first member of the proposed equation is, therefore, a 
 complete square ; so that it has two pair of equal roots, one of 
 each pair being furnished by the quadratic 
 
 x 2 — 3-6x + -24=0. 
 
 The above equation has been introduced for the purpose of 
 showing the decided advantage of this method, whenever the 
 polynomial is a complete square ; it always makes known to us 
 whether such is the case or not ; — a feature peculiar to the form 
 of the conjugate factors more especially dwelt upon in the present 
 essay. 
 
 The reader can readily analyse, in the same way, the equation 
 
 #4_ \0x s + 3\x 2 - 30* + 9 = 0, 
 
 taken from Mr. Lockhart's tract on 'The Nature of the Roots 
 of Numerical Equations,' alluded to in the foot-note. The com- 
 ponent quadratics will be found to be 
 
 x 2 — 5x + 3 = 
 x 2 - bx + 3=0 
 
 * I may here mention, as a fact, not without a certain kind of interest, 
 that Mr. Lockhart is at present (May, 1850) engaged in carrying through 
 the press some of his recent investigations on the Resolution of Equations. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 3 l J 
 
 Before terminating these examples, I shall apply the method 
 to the following general problem the discussion of which is nut 
 without interest : 
 
 (29.) To determine the conditions for which the roots of the 
 general equation of the sixth degree must all be imaginary ; the 
 last term of the equation being, of course, positive. 
 
 (10.) x G + ax s + bx* + ex 3 + dx 2 + ex + f= 0, 
 
 x 3 + - ax 2 ± V[ (- «" — *)«* — ex 3 — dx 2 — ex — A 
 
 or putting b — \ a 2 = p, and dividing by p, the expression within 
 the brackets will become 
 
 X* + - x + -a? + -x + -, 
 P P P P 
 
 and this, equated to zero, will have all its roots imaginary, provided 
 the condition established in example 4 have place ; that is, pro- 
 vided 
 
 or, removing fractions, and restoring p, provided 
 
 If a = 0, then it is plain that X 1 remains the same, whether 
 # 6 be absent from the original equation or not. In fact, if we 
 merely expunge a, the conjugate factors become 
 
 x 3 ±\/ { — bx l — cx% — ■ dx 2 — ex — f } , 
 and if x G be also suppressed, they may be written 
 
 ± </ { — bx^ — cx z — dx 2 — ex —/}, 
 the X 1 being the same in both cases. 
 
 * It is, of course, to be understood, as in the criteria of Newton, that the 
 second member of each inequality, and, consequently, tbe first, must be 
 positive. 
 
40 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 Hence, for the biquadratic equation 
 
 bx*> + cot? + dx 2 + ex +/= 0, 
 the condition (A) becomes 
 
 4 {«- 1 «■}/>*■ (B), 
 
 which, if 6 = 1, becomes, 
 
 as already determined. 
 
 In like manner, making b = 0, c = 0, we have for the qua- 
 dratic equation, 
 
 dx 2 + ex + / = 0, 
 the condition 
 
 4df> e* 
 
 the well known criterion of ordinary algebra. 
 
 And this is the only criterion of the absence of all real roots, 
 hitherto discovered in the general theory of equations, which 
 holds both directly and conversely ; whenever this has place, the 
 quadratic equation, whose coefficients furnish it, has its roots 
 necessardy imaginary ; and when its roots are imaginary, this 
 condition necessardy has place ; a circumstance which is con- 
 nected with the fact that, for a quadratic equation, we have a 
 general formula of solution, not in the least controlled by any 
 special condition ; but no such formula for the higher orders of 
 the equations. It is on account of this peculiarity that, even in 
 equations of the fourth degree, examined in the foregoing exam- 
 ples, I have usually preferred to stop the operation when X 1 has 
 reached the second degree, rather than to extend it a step further. 
 It is often prudent, however, in equations of the fourth degree, 
 to glance, at least mentally, at the leading term of X 1 in the 
 final step ; for if the last term of X happen to be a complete 
 square, X 1 may wholly disappear in that final step ; showing 
 that X itself is a complete square, as in example 9 ; or, when the 
 last term of X is not a square, still X 1 may vanish in part, and 
 
ANALYSIS OF NUMERICAL EQUATIONS. 41 
 
 become reduced to px 2 or to a single number. In either case, the 
 proposed equation becomes decomposed into a pair of rational 
 quadratics. But, if fractional coefficients enter Xj, when the 
 coefficients of X are integral, neither of these things can happen. 
 
 The following equation will illustrate what is here said ; and 
 will also furnish an additional exemplification of the principle 
 noticed at page 23. 
 
 The equation was first proposed in 1658, by Hudde ; and is 
 quoted by Mr. Lockhart. 
 
 (11.) Lockhart. 'Nature of the Roots,' &c, p. 14. 
 a? 4 + 4«3 — 3a? 2 — 8x + 4 = 
 x 2 + 2x ±s/(7x 2 + 8x — 4) 
 x 2 +2x — 2±\/3x 2 . 
 Hence the component quadratics are 
 
 * 2 + (2 + y/3)x — 2 = 
 x 2 + (2 —y/S) x — 2 = 0. 
 
 (30.) Since the criterion (B) above is the same, whether x 6 be 
 absent or present, and as it is obvious that the same would hold 
 generally, for the decomposition here considered, we infer that 
 when the second term is absent in any equation of an even 
 degree, we need examine the criterion only in reference to the 
 terms beyond the absent one to the right, taking care, however, 
 in this case, to divide all the terms by the coefficient of that 
 immediately succeeding the absent one, or, at least, not to omit 
 it in the statement of the condition. 
 
 Instead of the second term from the beginning, if the second 
 term from the end, namely ex, should be absent, the condition 
 under which all the roots will be imaginary, is equally simple ; 
 for, if e = in (A), we must have 
 
 4 (b — - a 2 ) d > c 2 , 
 
42 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 f, of course, being positive ; and which is the same condition as 
 we should get if ex and/ were both expunged, and the remaining 
 polynomial reduced to one of the fourth degree. 
 
 (31.) Many other criteria, enabling us to affirm, with confi- 
 dence, the entire absence of real roots in an equation, might be 
 investigated. In the above general equation of the sixth degree, 
 we might have proceeded on to the quadratic, instead of stopping 
 at the biquadratic form of X 1 ; and we might also consider diffe- 
 rent pairs of the innumerable systems of conjugate factors into 
 which a polynomial may be decomposed. But I do not think 
 that the resulting general forms would be of much practical 
 importance ; as the particular forms, in individual cases, may 
 very easily be obtained, as shown in the special examples already 
 given ; from which it will be seen how we may reach conditions 
 under which real roots must be absent from a particular region, 
 either the positive or the negative, as well as those under which 
 they must be absent from both regions. 
 
 In equations of an odd degree, we may apply such criteria to 
 the limiting equations, after the manner in which we have treated 
 example 7. And, by thus applying the proper limiting or derived 
 equations, we may determine conditions which, when fulfilled, 
 will enable us to affirm that an equation, of the degree 2n or 
 2n + 1, must have 2, 4, 6, &c, or 2n imaginary roots. 
 
 Those who are familiar with the writings of Maclaubin, 
 know that the limiting equations, here alluded to, may be made 
 to assume a variety of forms ; so that these criteria will be sus- 
 ceptible of different changes. Into the consideration of these, 
 however, as just remarked, I do not conceive it to be of any 
 importance to enter ; and with thus briefly alluding to the general 
 deductions, which any one can readily make for himself, after 
 what has here been done, I shall terminate this part of the 
 inquiry ; simply noticing another form for the conjugate factors, 
 when the second term of an equation is absent, and which may 
 sometimes be useful ; and adding, in conclusion, a statement of 
 the more interesting inferences which the principles now estab- 
 lished suggest. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 43 
 
 (32.) If the second term be absent, the general equation of an 
 even degree (1), at page 11, will be 
 
 a&a + bx 2n ~ 2 + cx 2n ~ 3 + dx 2 *-± + + k = 0, 
 
 which, by the method here employed, may be decomposed into 
 the following conjugate factors, namely, 
 
 x n + 1 bx n ~ 2 ± x /{ — CX 2n ~ 5 + (- b 2 — d\ X 2n -"* — ....—/.} * 
 
 If the equation be of the fourth degree, 
 ar 1 + bx 2 + ex + d — 0, 
 then the expression X 1 , within the brackets, will be 
 
 — ex + - b 2 — d= - b 2 — (ex + d), 
 
 4 4 v ; 
 
 so that, within those limits of x, for which this is negative, no 
 real root of the equation can exist. 
 
 The following final example will be illustration sufficient of the 
 above decomposition ; and as I have all along spoken of Bud an' s 
 functions, and Bud an' s rows of signs, I sball exhibit, in this 
 last example, the method of obtaining those rows winch I have 
 hitherto tacitly supposed to have been adopted. 
 
 (12.) 'Memoires de Turin,' tome vi, p. 171; andLocKHART's 
 'Nature of the Roots,' &c, p. 11. 
 
 x'° — 5x 2 + 1G = 0, 
 
 or 
 
 x G — 5x :] + 1 6x = 0, 
 
 5 ,/ .„ 25\ 
 
 & 
 
 _£ ±N /(-^ + a, 
 
 * It will, of course, be noticed, that this form is preferable to that which 
 would arise from proceeding as at page 11, only inasmuch as we here obtain 
 it at one step, instead of two. If a be made zero in (4), at page 11, the form 
 here given will residt, and it is plain that, however many terms after the first 
 may be absent, we may proceed in the same way. 
 
44 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 Now the equation proposed may be written 
 
 x 5 + Oa* + Oar 3 — 5 x 2 + Ox + 16 = 0, 
 
 furnishing only two variations of sign ; hence there are not more 
 than two real positive roots. 
 
 From X 1 it is plain that [1, oo] is rejective ; so that the two 
 positive roots, if they exist, must lie in the interval [0, 1], and, 
 indeed, in the narrower interval [0, *4]. Transforming them by 
 1 , the operation is as follows : 
 
 1+0 + 0- 
 
 -5+0+16 
 
 l 
 
 1 
 
 1—4—4 
 
 1 
 
 1 - 
 
 -4 — 4 + 
 
 1 
 
 2 
 
 3 — 1 
 
 2 
 
 3 - 
 
 - 1 — 
 
 1 
 
 3 
 
 6 
 
 3 
 
 6 
 
 + 
 
 1 
 
 4 
 
 
 4 
 
 + 
 
 
 1 
 
 
 + 
 
 As no variations are lost by this transformation, it follows that 
 no roots can lie in the interval [0, 1] ; and since beyond this, 
 the intervals are wholly rejective, we infer that there are no real 
 positive roots. Hence the equation has but one real root, and 
 that negative. 
 
 Inferences from the 'preceding Investigations. 
 
 (33.) (1.) When the signs of X are plus, at the limits of an 
 interval within which a pair of roots is indicated, then, if a root 
 of X, = lie in a non-rejective part of that interval, the pair of 
 roots indicated are real. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 45 
 
 (2.) When the signs of X are minus at those limits, then, if 
 that interval be not wholly non-rejective, the indicated roots are 
 real. 
 
 (3.) If the leading coefficient of X 1 under the radical be minus, 
 and of an even degree, and if, preserving this minus sign, the root 
 of X 1 = 0, which is nearest to + go be a, and the root nearest 
 to — cc, b, then no real roots of X = can lie between a and 
 -f go, nor between b and — go; for, between these limits, X 1 
 must be negative. Hence the real roots of X = are all com- 
 prised between those of X 1 = ; so that when these latter are 
 imaginary, all the roots of the proposed equation are imaginary. 
 And, moreover, there cannot be any real root of X = between 
 the second root of X 1 = 0, reckoning from either extreme root, 
 and the third root ; nor between the fourth and fifth ; and so on ; 
 it being observed, that, in here speaking- of the second, third, &c. 
 roots of X 1 = 0, we always mean the real roots taken in order 
 from either of the extreme roots. 
 
 (4.) The reverse happens, should the leading sign of X 1 be 
 plus, that is, the real roots of X = can lie only between [«, go] 
 and [b, — cc] ; and between the second and third ; the fourth 
 and fifth roots of X 1 = 0; and so on. 
 
 (5.) If X 1 be of an odd degree, and the leading coefficient plus, 
 the real roots of X = can lie only between [«, co], and between 
 the second and third, the fourth and fifth, &c, of the real roots 
 f X 1 = 0, reckoning from a. None can lie between b and — cc. 
 
 (6.) The reverse of these last circumstances happens if the 
 leading sign of X 1 be minus ; no real roots can he in the interval 
 [a, cc] ; they can lie only in the interval \b, — co], and between 
 the first and second, the third and fourth, &c, of the real roots 
 of X 1 = 0, reckoning from a. 
 
 (34.) From the foregoing deductions, we see that, stop at 
 whatever step of the process we may, information respecting 
 
46 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 the structure of the equation under examination is always de- 
 rivable ; and although we continue the steps till the operation 
 spontaneously terminates, yet we may take our choice of the X 1 
 to be finally discussed, that is, we may take it of an odd or of an 
 even degree, as may seem most eligible. 
 
 The discussion of X 1 , when it does not exceed the fourth 
 degree, may be completely and readily accomplished by Sturm's 
 theorem, as I have elsewhere shown ;* and X 1 can reach the 
 fourth degree, in the final step of the work, only for equations 
 of the ninth and tenth degree. 
 
 If the proposed equation be of the seventh or eighth degree, 
 X 1 in the final step will be a cubic : should this have but one 
 real root, a, we see, from inferences 5 and 6 above, that all the 
 real roots must lie either in the interval [a, oo] or [c — oc], 
 according as the leading sign of X 1 is plus or minus. 
 
 It is worthy of notice, that whatever X 1 we select for discussion, 
 the other forms of X 1 may often be consulted with advantage ; for 
 instance, the X 1 chosen in the analysis of example 7, page 35, 
 shows us that three roots only can lie in the positive region 
 [0, oc]. Knowing this, and finding from the preceding value 
 of X 1 , namely, 
 
 3 
 
 x 4, + 2x* — 2x 2 + x, 
 
 4 
 
 that the interval [4, oo] is rejective, as Maclaurin's limit at 
 once shows, it follows that the three roots adverted to must be 
 confined to the interval [0, 4] . A reference to the X 1 , furnished 
 at the first step, namely, 
 
 — x 5 — x* + 2X 3 — 2x 2 + x, 
 
 would supply information still more acceptable ; a glance at it is 
 sufficient to show that the interval [1, oo] is rejective; so that 
 the three roots inquired about are all indicated in the interval 
 [0, 1], We thus see how one form of X 1 may give greater 
 explicitness to the information supplied by another form. 
 
 * Analysis and Solution of Cubic and Biquadratic Equations. 
 
ANALYSIS OF NUMERICAL EdUATIONS. 47 
 
 I shall add but one observation more : — When the conditions 
 (4), above, have place, then if a, or any number nearer to + <» 
 than a, be substituted in Bldan's functions, the variations of 
 sign will show the exact number of imaginary roots indicated 
 between the number substituted and + oo. And if b, or any 
 number nearer to — oo than b, be substituted, the permanencies 
 of sign will show the exact number of imaginary roots indicated 
 between the number substituted and — oc. No real roots can 
 lie within the limits here mentioned. 
 
 SCHOLIA. 
 
 (35.) Whatever method be employed for the analysis of a 
 numerical equation, it is always assumed, as an established truth, 
 that every such equation has as many roots as there are units in 
 the exponent of the highest power of the unknown quantity en- 
 tering it, and which exponent marks the degree of the equation. 
 The equation being reduced to a rational form, there is no ques- 
 tion as to the legitimacy of this assumption, in reference to the 
 form thus obtained. But it is now pretty well known that there 
 are irrational equations which have no root ; in other words, that 
 irrational functions of x, involving, besides this symbol, only 
 given numbers, may be devised such that no substitution for x, 
 whether real or imaginary, can ever reduce that function to zero. 
 A quadratic function of x may, in general, be split into two con- 
 jugate factors, such that one of them shall never vanish, whatever 
 substitution be made for x, while the other shall become zero for 
 each of the two roots of the quadratic of which they are factors. 
 Or the conjugate factors may be such that one root of the re- 
 sulting quadratic shall satisfy one of them, and the other root 
 the other; while every different substitution shall fail. If, there- 
 fore, it be allowable to call an equation, which has only one root, 
 a simple equation, we might affirm that certain advanced equations 
 mieht be decomposed into different sets of simple equations, such 
 that any one in one set shall agree with the corresponding one in 
 
48 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 another set only in this, namely, that hoth of them vanish for a 
 particular root of the equation formed from their product. 
 
 Into whatever factors a polynomial be split, it is plain that 
 whatever reduces either of the factors to zero, the others re- 
 maining finite must reduce the polynomial itself to zero. When 
 the factors are rational, we know, not only that the different 
 roots of these belong to the equation composed of those factors, 
 but that equal roots among the former, imply the same equal 
 roots in the latter. The case is different when the factors are 
 conjugate and irrational, as an easy example or two will sufficiently 
 show. 
 
 The factors 
 
 x — a -\- \/x — a 
 
 x — a — s/x — a 
 produce the quadratic expression 
 
 x- — (2a + 1) x + a (a + 1). 
 
 The first factor becomes zero for x = a ; the second becomes 
 zero for x = a, and x = a + 1; and yet the quadratic equated to 
 zero has not the three roots, o, a, a + 1 . 
 
 Again, the factors 
 
 2#_10 + V{(4 +x) (5-x)} 
 
 2x — 10 — V{(4 + a?) (5 - x)} 
 furnish the quadratic equations 
 
 5a? 2 — 41a; + 80 = 0. 
 
 The first factor becomes zero for x = 5, so also does the second 
 factor. The first, again, becomes zero for x = 3£, and yet the 
 quadratic equation, which arises from the product of both, cannot 
 have the three roots, 5, 5, and 3£. 
 
 (36.) It thus appears that two conjugate factors, such as those 
 here considered, when equated to zero, may seem to furnish three 
 
ANALYSIS OF NUMERICAL EQUATIONS. 49 
 
 roots ; but it is evidently impossible that they can supply three 
 different roots ; since, as already remarked, whatever renders a 
 factor zero, must make the product into which it enters, zero ; 
 the two roots of the quadratic equation above are 5 and 31 ; 
 and no values, besides these, can reduce either of the conjugate 
 factors of its first member to zero. There is room for some ob- 
 servations on the circumstance of one of the roots of the quadratic 
 satisfying both the conjugate equations into which it is decom- 
 posed ; but this is not the place to introduce them ; they more 
 properly belong to a subsequent section of the work, the section 
 on zero-symbols. 
 
 (37.) At the commencement of these supplementary remarks, 
 it was noticed, that every numerical equation, into which x 
 enters throughout in a rational form, has as many roots, including 
 both real and imaginary, as there are units in the highest ex- 
 ponent of x. This truth has, I believe, hitherto been considered 
 as equally applicable to equations in general ; or, at least, it is 
 customary to admit that every equation, numerical or literal, has 
 a root, whether we can actually determine it or not ; for it seems 
 to be regarded as a settled point, that Cauchy has demonstrated 
 this important proposition. But this is a mistake. Whoever 
 carefully examines Cauchy's proof, will perceive that the main 
 points of the. reasoning necessarily involve the assumption that 
 the coefficients of the equation are real or imaginary numbers : 
 exclude this condition, and the argument is wholly nugatory. 
 
 I do not know whether any one has noticed the dilemma in 
 which the subject would be placed if the demonstrations by Abel 
 and Hamilton, that the solution of the general equation of the 
 fifth degree is impossible, be received, and also the demonstration 
 by Cauchy, that every equation has a root, be received. The 
 two conclusions are contradictory. When it is proved that the 
 solution of an equation is impossible, it is proved not merely that 
 we have not the algebraic skill to discover the root, but that the 
 supposed root has_ really; no algebraic existence. This is no more 
 than might De anticipatedironT~«"/9no>7 considerations; every 
 symbolical expression for a root of an equation, with general 
 
 4 
 
50 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 symbols for the coefficients, must of necessity be such that, 
 when it is substituted for x, in the proposed equation, the two 
 members of that equation will become absolutely identical, what- 
 ever interpretation be given to the symbolical coefficients. The 
 identity spoken of is an identity purely symbolical, irrespective 
 of all arbitrary interpretation, and holds for every interpretation, 
 however novel or extravagant. If the general expression for x, 
 in the equation x 2 -f ax = b, be substituted for x, the identity of 
 the two members remains undisturbed, whatever meanings, how- 
 ever fanciful, we give to a and b. And, in like manner, when 
 we seek for the general solution of an equation of the fifth degree, 
 we presuppose the existence of an expression, involving arbitrary 
 symbols of so very general a character, that when substituted for 
 x, in that equation the two members will become identical for all 
 values or forms that can be imagined of those symbols. It cer- 
 tainly seems more remarkable that such a supposition should be 
 confirmed, in the inferior equations, than that it should prove 
 erroneous in those of more advanced degree. 
 
 (38.) What, in one sense, is the perfection of symbolic inves- 
 tigations, is, in another sense, a disadvantage ; for it is certainly 
 unfortunate, in the matter in question, that the unavoidable 
 generality of interpretation to which our symbols have claim, 
 really prevents their accomplishing what we exclusively aim at, 
 without, at the same time, accomplishing a great deal more ; so 
 that if these superfluous offices prove impossible, those that are 
 actually required, are involved in the condemnation too. A 
 formula for the roots of an equation that should be necessarily 
 restricted to numerical values for the symbols entering it, would 
 not be algebraical. Analysis could not furnish such a formula ; 
 and this sufficiently accounts for how it happens that the real 
 roots of a numerical equation may be discovered by common 
 arithmetic, when their determination is impossible by algebra. 
 In this point of view, arithmetic has, unquestionably, the advan- 
 tage over algebra; in the former science we cannot help our 
 symbols being specific in value ; in the latter, we cannot help 
 their being general, whenever they are really the symbols of 
 
ANALYSIS OF NUMERICAL EQUATIONS. 51 
 
 algebra, and not those of arithmetic. It is not only in the solu- 
 tion of equations that these marked peculiarities operate ; other 
 inquiries might be adverted to in which the algebraic forms are 
 impressed with the general mark of impossibility, and from 
 which, therefore, no special real cases can be inferred, though, 
 at the same time, such cases exist in numbers, without any 
 arithmetical restriction at all. The formula which proves the 
 algebraical impossibility of the extension of Euler's theorem, 
 respecting the product of squares, to the case of sixteen squares, 
 is an instance in point ; the extension is always arithmetically 
 possible, though always algebraically impossible. 
 
 Application of the Conjugate factors to the determination of Two 
 Roots of an Equation, when the others are known. 
 
 (39.) The conjugate factors employed in the preceding pages, 
 to facilitate the analysis of an equation, may be applied to many 
 other purposes. Thus, putting <p (x) for X, in order that we 
 may not confound X, in what follows, with the first member of 
 an equation, and, therefore, writing the factors thus : 
 
 <p («) = JT + V(F 2 - f («)}] x [F - V(F 2 - * (*■)}] ■ ••■(!), 
 we see that 
 
 log <p (*) = log [F+ V(F 2 -0(*)}] + log [F-V(F 2 -?> (*)}]; 
 
 or, putting (x)] n for </> (a?), and then taking the nth root, the 
 factors may be written 
 
 (?) (,-)=[F + V{F 2 -[^(^)n]"x[F^V{F 2 -[0(^] n }]" 
 so that 
 
 log (.r) = -log[F+ x /{F 2 -[0W] n }]+-log[F-V{F 2 -[?.(*)] n }] 
 
 where F and n are arbitrary. And these forms for log <p (x) 
 would lead to developments different from those usually given. 
 But it would be out of place to dwell upon such applications 
 
52 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 here ; and I shall therefore confine myself to what is more imme- 
 diately connected with the doctrine of numerical equations ; and 
 shall now show how the conjugate factors (1) may be made 
 available for the purpose of determining expressions for two 
 roots of an equation when all the others have been found. 
 
 (40.) First, suppose one root a?, of the cubic equation 
 
 a; 3 + px 2 -\- qx -\- ?' = 
 
 to be known ; then 
 
 o r 
 
 x x + P x \ + 1 = 
 
 will be the product of the two remaining roots. 
 
 Let this product be substituted for (j> (x) in the factors (1); 
 then, that those factors may represent the roots themselves, we 
 shall only have to assume F, so that the sum of the factors, 
 namely, 2F, may be the sum of the two roots, that is to say, 
 we must satisfy the condition 
 
 2Y=-( P + *,); .'.P = -^l 
 
 under which restriction the two factors become 
 
 and these exhibit the values for the two remaining roots of the 
 equation. 
 
 Again, suppose two roots, x v x 2 , of the biquadratic equation, 
 
 x 4, + px? + qx 2 + rx + s = 0, 
 
 are known, then we shall have 
 
 s 
 
 x \ + V x \ + qx, + r = 
 
 a?, 
 
 ,v 
 
 .r 2 3 + pxf + qx 2 + r = 
 
 a? 2 
 
ANALYSIS OF NUMERICAL EQUATIONS. 53 
 
 .'. (xf + a?./) + p (x* —x 2 2 ) + q (x l — x 2 ) 
 s s _s(x l — x 2 ) 
 
 W Q *** ] tC I it' f> 
 
 that is, dividing each member by x l — x 2 , 
 
 (a?j 2 + x l x 2 + x 2 2 ) + p (a?, + x 2 ) +q = 
 or, which is the same thing, 
 
 0?! + x 2 ) 2 +jp(x x + x 2 ) —x { x 2 + q = 
 
 x x x 2 
 
 X\X 2 
 
 the product of the two remaining roots. 
 
 As before, let this product be substituted for <p(x) in the 
 factors (1); then, that those factors may represent the roots 
 themselves, we must assume F, so that 2F may be equal to their 
 sum, that is, we must make 
 
 2T=-(p + x 1 + x 2 ); /. F = -*±^±£?. 
 Hence the expressions for the two remaining roots are 
 
 p + X l + # 2 
 
 2 
 
 ± Vll^^-— 2 ) 2 - (*i + *& - p to + ■*> + »a - *} 
 
 which expression, to give the irrational part of it a form more 
 convenient for actual computation, may be written 
 
 p + x l + x 2 
 
 ±V{( 
 
 ■i±^ - 2 (,, + * 2 >) «±a±a + V2 - ,} 
 
 which, we see, reduces to the preceding expression for the roots 
 of a cubic when x 2 = 0. 
 
 (41.) Expressions having the same object as those above might 
 be obtained differing from these in form. In the 'Analysis of 
 
54 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 Cubic and Biquadratic Equations,' the following expression is 
 given for two roots, x 3 , x 4 , of the biquadratic equation 
 
 x 4, + qx 2 + r + s = 0, 
 where the term in X s is absent, namely, 
 
 *i + x 2. A / f/ »i ~ x 2 \ 2 , _JL_1 
 
 --2- ± VH-T"J + ^T*J 
 
 which, although less convenient for computation than the cor- 
 responding expression derivable from that just given, may deserve 
 a place here for the sake of the following inferences from it, 
 namely : 
 
 Supposing r to be positive, which is, of course, always allow- 
 able, we see, from the formation of the quantity under the radical 
 sign, that, 
 
 1. If two real roots, x v x 2 , occur in the positive region, the 
 remaining two roots must be real also. 
 
 2. If two imaginary roots, x v x 2 , are indicated in the negative 
 region, the remaining two must be imaginary also. Therefore, 
 
 3. If two real roots are detected, without regard to their situa- 
 tion, and if the remaining two are indicated in the negative region, 
 these must be real also. And if two imaginary roots are indicated 
 in either region, and if the remaining two are indicated in the 
 positive region, these must be imaginary also. 
 
 (42.) If we were to take an equation of the fifth degree, and to 
 imitate the process above (40), we should obtain three expressions 
 of the third degree, like as the expression (2) of the second degree 
 was obtained ; and if, from the double of one of these, the sum 
 of the other two were to be taken, and an obvious division per- 
 
 o 
 
 formed, we should get an expression , the product of the 
 
 x i x 2 x z 
 remaining two roots x 4 , x 5 , in terms of the three x v x 2 , x z , sup- 
 posed to be already known ; and, determining F as before, we 
 should arrive at the values for .r 4 , x b , analogous to those above. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 55 
 
 But it is not difficult to see that the product of two roots of any 
 equation in terms of the other roots is 
 
 (# 1 +a? 2 +a? 3 +.. .) 2 +i> 0] +a? 2 +a , 3+ • • •) — ^ l x 2 +x l x. i +x. 2 x s + . ..) + q, 
 since all the terms cancel, except that product. Consequently, 
 if this be subtracted from the square of half the sum of the same 
 roots, that is, from 
 
 m + *i + x 2 + .r 3 + . . . \ 2 
 
 the remainder will be the square of half the difference of the 
 same roots ; and since the half difference added to and subtracted 
 from the half sum gives the roots themselves, we have generally 
 
 p + a?, + x 2 + x 3 .. . 
 2 
 ± ^ r+tl+H + „,+ ... y _ ^ + h + h + _ )2 
 
 — # (a? 2 + x 2 + x 3 + •••) + x \ x 2 + *1*B + % r 3 + •.• — ?! 
 
 an expression which may be so modified as to render the irrational 
 part a little more convenient for computation. 
 
 Thus in the case of an equation of the fifth degree, instead of 
 
 the forms 
 
 p + x x + #2 + x 3 
 ~~ 2 
 
 ±>v /{ f + »■ + « + «. )'- (,,-,.„ + ,,). 
 
 — £ («i + A'2 + a? 3 ) + »l» 2 + "Vfc +«g* 8 — ? j- 
 
 we may write 
 
 p + x x + x 2 + x H 
 ~ 2 
 
 + a?, (a? 2 + a? 3 ) + x 2 + x s — q>. 
 
 It thus appears that the forms previously deduced might have 
 been arrived at independently of the conjugate factors ; but it 
 
56 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 was the previous introduction of those factors that suggested the 
 general considerations here employed. 
 
 (43.) In reference to the formulse here established, it may not 
 be superfluous to observe, that they will be found more especially 
 useful in those cases in which all the roots but two are real ; as 
 they will enable us to exhibit the imaginary pair, by aid of the 
 real roots, with comparatively little expense of calculation. And 
 even when all the roots are real, a saving of figures is, in general, 
 still effected by them. But, in comparing formulae of this kind 
 with the numerical process of Horner, it must always be re- 
 membered that Horner's method supplies the roots in an explicit 
 form ; whereas, in expressions for them, such as these, there yet 
 remains an unperformed operation, indicated by the radical ; 
 which operation, however, in the case of a pair of imaginary 
 roots is, of course, impracticable, and therefore nothing further 
 can be done. But in all formulae for imaginary roots, into which 
 approximate values only of the real roots enter, it is necessary, 
 in delicate cases, — that is, in those cases in which a very slight 
 change in any of the coefficients would convert real roots into 
 imaginary, or imaginary roots into real, — it is necessary, in such 
 cases, to push these approximations to a more than usual extent, 
 in order to avoid the risk of converting imaginary roots into real, 
 and vice versa; for there is no hope of attaining the imaginary 
 forms accurately, when we employ approximations only to the 
 real quantities which enter into our expressions for them ; and, 
 in such critical cases as those just alluded to, the error in the 
 approximations may be sufficiently great to lead to a false con- 
 clusion as to the character of the roots. 
 
 It appears to me, however, and I have elsewhere expressed the 
 same opinion, that we may always safely regard a pair of roots 
 to be real and equal, provided they would actually become so 
 upon making changes, in the coefficients of the proposed equation, 
 so minute as to render the difference between the two states of 
 that equation of no practical moment. Whether the roots con- 
 verted into equality by such changes be originally real or imagi- 
 nary, should not form any item of consideration. There is no 
 
ANALYSIS OF NUMERICAL EQUATIONS. 0/ 
 
 doubt that real roots and imaginary roots are distinct classes, 
 marked by distinct peculiarities ; but it may be questioned, 
 whether pairs belonging to the latter class are not sometimes too 
 hastily rejected as mere symbols of impossibility. If a pair of 
 real roots, kept together for seven or eight places of decimals, 
 and then separated — a very possible case — the error committed 
 by rejecting the non-concurring figures, and pronouncing the 
 two roots to be equal, would not be considered as of serious 
 consequence; yet it is impossible that these equal roots can 
 strictly belong to the equation, for if either of them be substi- 
 tuted in the equation, the left-hand member of it, instead of 
 becoming zero, would be a minute finite value ; that is, the equal 
 roots really belong to the equation only after a minute change 
 has been made in one or more of the coefficients ; if a similar 
 change to that which thus converts unequal roots into equal roots, 
 should convert imaginary roots into equal roots, it is plain that, 
 whatever considerations justify the change in the one case must 
 equally justify it in the other; in both cases, a pair of equal roots 
 would belong to an equation differing from that proposed by a 
 quantity too small to deserve notice. But, for further observa- 
 tions on this point, reference may be made to ' The Theory and 
 Solution of Algebraical Equations,' pp. 307-12. 
 
 (44.) As a conclusion to this article, it may be noticed, that 
 anv function <f>(x) may be decomposed into as many pairs of con- 
 jugate factors as we please ; thus it is evident that 
 
 »W=[F+V}r--^ 1 / 2 ---/ n -<>W}]x[r-- 1 /{F--^ 1 / 2 .../ n .<.M}] 
 
 x[W{'-S}]>«[>.-V{v-#] 
 
 *[W(>V-j]>[3-V{^-j.}] 
 x f._, + \/{*Vi - jj] x [*U- «/{fV.- jJ] 
 
 where F, Fj, F 2 , &c, and/,/!,/ 2 , &c. are arbitrary. 
 
 /A 
 
58 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 I should not have inserted this obvious identity but for the 
 fact, that there is no foreseeing what even the plainest truisms of 
 analysis may hereafter suggest ; the simplest case of this general 
 form, considered in the preceding pages, leads to results which, 
 I venture to think, will be deemed not unworthy of notice in any 
 future account of the progress of the subject to which the present 
 essay is chiefly devoted. 
 
 (45.) One of the most simple applications of the formulae now 
 given is to the general solution of equations of the fourth degree. 
 For since 
 
 ^)=[F+V{F-/^)},]x[F-v'{F2-/^)}]....(l) 
 
 x 0, + \/{ F . 2 -)}] * [ F > 2 - \/{ F . 2 -)}] • ' ' -< 2 > 
 
 it follows, that if we have the biquadratic equation 
 
 # 4 + px 3 + qx 2 + rx + s = (3) 
 
 and that we put « for (j>(x) in the formula, the four factors (1) 
 (2) may be regarded as the conjugate factors of the four roots of 
 the equation. The first pair (1) of these roots will obviously be 
 furnished by the quadratic 
 
 x 2 — 2Yx +fs = 
 
 and the second pair (2) by the quadratic 
 
 x 2 - 2F,a + - = 
 
 so that if we multiply these two quadratic expressions together, 
 and equate the resulting coefficients with those of the like powers 
 of x in (3), we shall get the requisite equations of condition for 
 the solution of the equation proposed ; if p be zero, the quadratic 
 factors will, of course, be 
 
 (*2 - 2F* +/*) (* 2 + 2F* + y) 
 
ANALYSIS OF NUMERICAL EQUATIONS. 59 
 
 since, in this case, F + Fj must be zero : we shall thus get the 
 solution of Descartes. 
 
 It is easy to see how the quadratic factors of any equation of 
 an even degree may, in like manner, be symbolically expressed. 
 
 Wbile thus noticing one form of investigation for the solution 
 of the general biquadratic equation, it may not be out of place 
 to advert to another form immediately suggested by the mode of 
 decomposing a function employed in the preceding pages. 
 
 Resuming the equation (3), the conjugate factors of the first 
 member admit of the introduction of an indeterminate quantity 
 u, thus 
 
 * 2 + y± V {(^--v)* 2 ™- 8 } 
 
 x* 
 
 + ^ + u ± Y I (^ - q + 2ttjatP + (pu - r) x + u 2 - s\- 
 And if u be now determined so as to make 
 
 4(— — 1 + 2«) (m 2 - s) = (pu - r) 2 
 
 the expression under the radical will be a complete square of the 
 form (mx + n) 2 , where in and n are known. Hence the biqua- 
 dratic becomes decomposed into the two quadratics 
 
 x z -\ \- u + [rnx + n) = 0, 
 
 x 2 + — + u — (mx + n) ■= 0, 
 
 and thus the solution, when p is zero, becomes that of Louis 
 Ferrari. And this mode of arriving at it is certainly more 
 direct and easy than any method hitherto proposed. The steps 
 might be imitated for an equation of the sixth degree ; but to 
 obtain, as here, an equation of condition of the third degree, or 
 even of the fourth, would necessitate a determinate relation 
 among the coefficients. 
 
GO ANALYSIS OF NUMERICAL EQUATIONS. 
 
 On some Criteria of Imaginary Roots analogous to the Criteria 
 
 of Newton. 
 
 (-16.) In a former publication,* I have discussed some useful 
 formulae for discovering imaginary roots in a numerical equation, 
 from inspecting the coefficients of its terms. The new forms 
 now proposed will, I think, prove an acceptable addition to those 
 here adverted to. They are not only of greater simplicity, but 
 they will often succeed in detecting the presence of imaginary 
 roots in cases where Newton's formulae — the basis of those 
 before given — would prove inefficient. 
 
 These new forms are deduced as follows : 
 
 Let the equation 
 
 V n + A n-i* n_1 + A n _ 2 tf n - 2 + A n _ 3 tf n - 3 + . . . .A 11 r + A =0 
 
 be multiplied by x — a ; that is, let a new undetermined real 
 root a be introduced into the equation. 
 
 Then whatever imaginary roots are indicated in the new 
 equation, the same, of course, enter into the original. 
 
 This new equation is 
 
 A Q x* + i + (A n _ 1 -aA I > n + (A^-aA^O^-H 
 
 (A n _ 3 -aA n _ 2 )^-2 + .. ..(A -aA,)a? + aA = 0, 
 
 in which a is entirely arbitrary ; and may therefore be made to 
 satisfy any condition we please. It may, for instance, be deter- 
 mined so as to render any one of these compound coefficients 
 zero. And if, in conjunction with this determination of a, the 
 original coefficients be so related as to cause the compound 
 coefficients, on each side of this zero, to be of like signs, we 
 shall at once recognise the presence of imaginary roots in the 
 proposed equation. 
 
 Equating then the several coefficients to zero, one after the 
 
 * Researches respecting the Imaginary Roots of Equations, forming an 
 Appendix to the Treatise on the Theory of Equations. 
 
ANALYSIS OF NUMERICAL EQUATIONS. 61 
 
 other, commencing with the second, and determining the adjacent 
 pair of coefficients in each case conformably to this condition, 
 we shall have the following sets of relations, the existence of any 
 one of which will imply the entrance of a pair of imaginary roots 
 into the proposed equation. 
 
 A n = + , V.-aA^O, A n A n _ 2 -A2 n _,= + 
 
 A n-l~ A n A n-2= +> A n-2~« A n _, = 0, A„_, A n _ 3 -A 2 n _ 2 = + 
 
 A n _ 2 -A n _ 1 A n _ 2 =+, A n _ 3 -aA n _ 2 =0, A n _ 2 A Q _ 4 -A 2 _ 3 = + 
 
 &c. &c. 
 
 And, therefore, a being always assumed so as to satisfy one of 
 the midde equations, those on each side will furnish the following 
 series of criteria of imaginary roots, namely, 
 
 A n =+, A n A n _ 2 >A 2 n _ 1 (1) 
 
 AS_ 1 >A B A tt -, A n _ 1 A n _ 3 >A 2 n _ 2 (2) 
 
 A 2 n _ 2 > A,-, A n _ 3 , A n _ 2 A n _ 4 > A* _ 3 (3) 
 
 &c. &c. 
 
 Now, assuming, according to custom, that A n is always plus, 
 it is plain that if any one of the right-hand conditions have place, 
 without regarding those" on the left, a pair of imaginary roots will 
 necessarily be indicated ; because, although the accompanying- 
 left-hand condition should not exist, yet, by ascending to the 
 preceding pair of conditions, and thence to that next in order, 
 and so on, we shoidd evidently, at length, arrive at a coexistent 
 pair. It is equally evident that no two consecutive pairs can 
 exist simultaneously ; since the second condition, in any one 
 pair, is opposed to the first in the pair next following. 
 
 Hence, although all the right-hand conditions marked (1), 
 (2), (3), &c, were found to have place, yet more tban one pair 
 of imaginary roots could not be safely inferred ; these conditions, 
 therefore, ought not to be regarded as anything more than so 
 many concurrent indications of the same thing. 
 
 Whenever this concurrence ceases, by a failure of one of the 
 right-hand conditions referred to, or, which is the same thing, 
 
62 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 by a fulfilment of the left-hand condition next in order, then 
 preparation is made for a new indication, totally distinct from 
 the indications that have preceded. And if such new indication 
 offer itself, a distinct pair of imaginary roots may be inferred. 
 
 Hence the series of conditions (1), (2), (3), &c, furnish us 
 with criteria of imaginary roots somewhat analogous to those of 
 Newton ; much simpler, however, in form, but to be employed 
 in exactly the same manner. I shall give two examples of this 
 application : 
 
 1st. 5a; 8 — 2a; 7 + 3a: 6 - 24a; 5 — 16** + a? 3 — 4a 2 — 2x— 60 = 0. 
 
 Applying the criteria to this equation, we discover the existence 
 of six imaginary roots, the same as by the rule of Newton. 
 ('Researches,' &c, page 47.) 
 
 2d. 9a; 5 — 5* 4 + 4a; 3 — 3a; 2 + 6a; + A = 0. 
 
 In this example, Newton's rule detects only a single pair of 
 imaginary roots ; the criteria above discover two pairs ; so that 
 the equation has but one real root. 
 
 It is scarcely necessary to add that, as in the rule of Newton, 
 though the sign > be replaced by = the criteria still hold. 
 
 (47.) It will be easy to give to these criteria a more general 
 form ; it will only be necessary to multiply the several coefficients 
 of the equation, in the manner so often employed by Newton 
 and Maclaukin, by the terms of an arithmetical progression, 
 such as 1, 2, 3, &c, or, more generally, k, k + l, k + 2, &c, and 
 to replace the original coefficients by the results ; so that, in- 
 cluding the criteria already established, we thus obtain the three 
 following sets, namely, 
 
 A n A n _, > A»_, 
 
 A a _, A B _ 3 > A*_ 2 
 
 A n - 2 A n _4 > A 2 n _ 3 
 
 A 2 A > V 
 
ANALYSIS OF NUMERICAL EQUATIONS. 63 
 
 3A n A n _ 2 > 4A 2 „_, 
 
 8A n _!A n _ R > 9A 2 U _ 2 
 
 15 A 4 _ 2 A n _ 4 > 16 A*_ 3 
 
 (n2-l)A 2 A >n2A,2 
 
 /•(H2)A n A B . 2 >(Hl) 3 A--, 
 (* + 1) (k + 3) A... A n _ 3 > (k + 2)2 A*_ 2 
 (k + 2) (A + 4) A n _ 2 A n _ 4 > (k + 3)2 A 2 n _ 3 
 
 (A + w — 2) (A + n) A 2 A > (k + ?i— 1)2 A, 2 . 
 
 And in these criteria we may, if we please, write A , A,, A 2 , &c, 
 in the place of A n , A n _,, A n _ 2 , &c, and vice versa. 
 
 It will be observed, that the two latter sets of criteria are really 
 distinct from the first set, and are not comprehended in that set ; 
 for although, whenever any one of these holds, the corresponding 
 one in the first set must hold, also ; yet one of these may fail 
 without any failure in the analogous form of the first set. 
 
 The last general forms, when expressed in words, furnish the 
 following rule, which it will not be difficult to remember : 
 
 Rule. — Commencing with the second coefficient, and pro- 
 ceeding towards the right, or with the last but one, and pro- 
 ceeding towards the left, multiply the successive coefficients by 
 any series of consecutive whole numbers, and square the results ; 
 the square of each coefficient will thus be multiplied by an in- 
 tegral square. Let the product of the adjacent coefficients on 
 each side be now multiplied by the same quantity, minus 1, and 
 compare these results with the former, as in Newton's rule. 
 
64 ANALYSIS OF NUMERICAL EQUATIONS. 
 
 I here terminate the essay on the Analysis of Equations by the 
 aid of Conjugate Factors. It will be readily seen that this ap- 
 plication of those factors is only one out of many inquiries into 
 which they may be advantageously introduced. In the theory 
 of curves and surfaces, for instance, the employment of the con- 
 jugate factors will often be found of use ; indeed, any facilities in 
 the analysis of a genei'al equation must be accompanied with like 
 facilities in the discussion of a curve or a surface. 
 
 I think, too, that in the investigation of certain trigonometrical 
 formulae, the decomposition adverted to will afford acceptable aid. 
 In these, as in nearly all the general theorems of analysis, nothing 
 more is expressed but identity of value under diversity of form. 
 In fact, the more advanced parts of analysis are almost exclusively 
 devoted to transformations of this kind. Such being the case, 
 any addition to our existing analytical forms, however simple and 
 obvious it may appear, when actually exhibited, is not without a 
 certain amount of value in the estimation of the algebraist, who 
 well knows that, to such simple, and perhaps almost axiomatic 
 operations upon the previously-existing expressions, the most 
 important steps, both in pure and applied science, are traceable. 
 I need mention here, in corroboration of this, only the trigono- 
 metrical transformations of Euleb, and the dynamical principle 
 of D'Alembert. 
 
 For some additional remarks on the subject of the present 
 Essay, see Note (A), at the end of the volume. 
 
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 Collegiate Institution. 
 
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 and late Fellow of St. John's College,Cambridge. 
 
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 matical Master, King's College, London. 
 
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