/^^f 
 
 IN MEMORIAM 
 FLORiAN CAJORl 
 
7^-^^oc ^ Qj>^^^^^ 
 
V 
 
// 
 
 NATURE OF THE ROOTS 
 
 01 
 
 NUMERICAL EQUATIONS. 
 
 BY 
 
 JAMES LOCKHART, F.R.A.S., 
 
 Author of ' The Resolution of Equations by Inferior and Superior Limits,' inscribed, by permission, 
 to the President and Council of the Royal Astronomical Society. 
 
 \ Longum iter est per prsecepta, breve et eflScax per Exempla. 
 
 Seneca. 
 
 LONDON: 
 
 PRINTED BY C. AND J. ADLAED, BARTHOLOMEW CLOSE. 
 
 1850. 
 

 PEEI^ACE. 
 
 About forty years since, the study of Numerical Equations, 
 in England, was confined to few persons, at the head of whom 
 were Mr. Baron Ma seres and the Rev. Mr. Frend. The 
 latter was a gentleman of great mathematical ability, and an 
 accomplished scholar in classical learning and biblical know- 
 ledge. The Church- squabbles of the present day would have 
 been odious to him. 
 
 Both Maseres and He dismissed imaginary and negative 
 roots from the doctrine of equations, following the example of 
 many learned men, their predecessors in science. They were 
 at no loss to find such real roots as satisfied the conditions of 
 a problem. Mr. Erend's solution of Col. Titus's problem is 
 generally known and admired^ " ""^ 
 
 However, as the fashion of the day requires obedience, we 
 have in the following treatise complied, in every respect, with 
 its algebraical dictates. The same was done in the year 1842, 
 when we published the * Resolution of Equations by means 
 of Inferior and Superior Limits,' in which some ideas, enter- 
 tained in the present treatise, appeared. 
 
 The reader should be acquainted with the theory and solu- 
 tion of equations, particularly with that by Prof. Young. This 
 will enable him to understand each process, and to satisfy 
 
IT PREFACE. 
 
 any doubt that he may entertain. Besides this, some of the 
 equations are taken from the Professor's admirable work. 
 
 It is no part of our plan to discover the figures appertaining 
 to real roots, except in the case of equal roots, nor to find 
 those belonging to imaginary roots, which has been happily 
 < effected by our friend Dr. RuTHERroRD, in a work recently 
 published, embracing equations of every degree, and combining 
 truth with simplicity, its chief ornament. 
 
 Following the example of Dr. Prideaux, in the Preface to 
 the * Connections,' we must beg the reader to excuse, on 
 account of our age, now verging towards eighty-seven years, 
 any errors he may discover in this treaTise^ eitEer~Srcomp^^ 
 tation or doctrine. No one has assisted in either. 
 
 The later homihes of the Archbishop of Grenada have place 
 in our hbrary. 
 
 Fasnacloich, Argyllshire ; 
 May, 1850. 
 
NATURE OF THE ROOTS 
 
 or 
 
 NUMERICAL EQUATIONS, 
 
 The Equations here proposed for examination have, exchisively, integral 
 indices and coefficients, unity being the coefficient of the leading term. 
 
 Every such equation can be transformed into another, having its 
 alternate signs beginning at the second term, minus and plus. This may 
 easily be accomplished by Newton's superior limit, where the signs are all 
 plus. Alternating the signs, the nature of the roots is not changed. 
 
 None of the preceding equations can have more positive roots between 
 and 1, or between two depressing consecutive integers, or between 
 
 2n 
 intervals of such depression, than the integers contained in — , n being the 
 
 o 
 
 highest index of the equation. Thus, an equation of the 41st degree can 
 
 have only 27 positive roots so situated ; those of the 4 2d and 43d degree 
 
 only 28 such roots ; for since a quadratic can have only one root so 
 
 situated, it can only impart its two roots to a cubic under the same 
 
 circumstance, and therefore the cubic can only have two roots so situated ; 
 
 and as all higher equations are compounded of one or both of these, or 
 
 may be supposed to be so compounded, they partake of the nature of 
 
 their origin. 
 
 Take the triads (2, 3, 4), (5, 6, 7), (8, 9, 10) (41, 42, 43) 
 
 n 
 w— 1, w, w + 1 ; the middle number contains — cubics; and since each 
 
 o 
 
 2n 
 cubic can only have two positive roots so situated, — positive roots can 
 
 o 
 
 only be situated in an equation where n is the highest index, and divisible 
 by 3. 
 
 The demonstration for the indices {n—\) and (w 4- 1) follows so 
 readily, that we shall not enlarge on it. 
 
 If w — 1 =41, it consists of 13 cubics and 1 quadratic, or 27 roots, 
 w = 42, of 14 cubics, or 28 roots, 
 w = 43, of 1 3 cubics and two quadratics, or 28 roots, 
 which are all supposed to be positive. 
 
NATURE OF THE ROOTS 
 
 The foregoing property enables us to decide with certainty that roots are 
 imaginary under circumstances similar to the following : 
 
 Let a?s — 2a?2 + 3a? — 1 = 0, 
 
 depress by '7, 
 
 1 +-1 + 1-67 +'463, 
 
 three changes of sign are lost, which would have taken place if the roots 
 had been real, and a superior limit taken ; but since three real roots 
 between and I cannot exist in a cubic, the loss of the signs is occasioned 
 by two imaginary roots. 
 
 This is of frequent occurrence in equations of high degree, pervading 
 many steps of examination. (A.) 
 
 Let d: r be the absolute term of an equation, or of any equation derived 
 therefrom by depression, either of them having one or more roots between 
 and 1 . 
 
 Let q be the greatest coefficient whose sign is contrary to that of r. 
 
 Consider them as positive. Then the least positive root of the equation 
 
 18 greater than . 
 
 gr + r 
 
 See all the treatises on equations. (B.) 
 
 Let an equation, whose variations of sign permit the idea of its having 
 
 two or more real roots between Oand 1, be depressed by a quantity^ less 
 
 than unity, so tliat the resulting equation may have two or some equal 
 
 number of signs less than the former. 
 
 Change the alternate signs of the equation in ^, beginning at the second. 
 
 Let c be the greatest coefficient under this last circumstance, having its 
 
 sign contrary to that of the absolute term ± c?, where they are both to be 
 
 C T 
 
 taken positive. Then if — (1 — ^) is less or equal to as de- 
 rived in the last article, the roots passed over are imaginary, agreeing with 
 the equal number of signs lost. 
 
 q r 
 
 Thus, let a?2— la?+ 1 =0, 
 
 c d 
 
 depressed by (-6), 1 + -2 + "76 
 
 V \ c 
 
 = •— and , — (1 — '6) is negative. 
 
 q -\- r 2 c -^ d ° 
 
 This is explained in the work alluded to in the preface. (C.) 
 
 In any part of the process of examination, if a quantity equal to or 
 
 less than (which of course is changed in value at every depression) , 
 
 should, as a depressor, cause two or some equal number of signs to dis- 
 appear, two or some even number of imaginary roots are discovered. (D.) 
 
 The contradiction to article (B) confirms the present. 
 
 Two imaginary roots, at least, exist in every equation whose signs are 
 alternately — -f , beginning at the second term, if the coefficient of the 
 second terra, taken positively, is less than the highest index ; and even if 
 
OF NUMERICAL EQUATIONS. 
 
 the second coefficient is equal to the highest, there are imaginary roots, 
 unless all the roots are equal to each other. (E.) 
 
 In the equation x^—bx-{-c = 0, all the roots are imaginary if n is even, 
 
 and 
 
 b^ 
 
 Tx — =i greater than ^, 
 
 if n is odd, there is but one real root, of which x^ — 21a? + 38 = is an 
 example. (F.) 
 
 The preceding means are employed in respect of the following examples, 
 which the reader may suppose to be derived from equations with greater 
 coefficients, and brought by depression to a proper state for the exami- 
 nation of their roots. 
 
 1 r 
 For the most part — has been taken for a depressor, when was 
 
 less than — , for such substitution generally teaches whether we should 
 
 advance or retrograde. 
 
 It often occurs that a single depression is not satisfactory, for no means 
 are yet known, the method of Sturm excepted, to exempt us from trials 
 in almost every department of search after the character of roots. 
 
 Example 1. 
 
 q r 
 
 a;8 _a?7 + 4x^ — 2x^ + 6x^-'2x^ + 5a?2 — a; + 2 = 0, 
 
 by (B) i» > iTj by which depress the roots, the result is ; 
 1 + -5 + 375 -f- + + + + + ; 
 
 since no change of sign takes place in the depressed equation, all the 
 roots are imaginary. (D.) 
 
 Example 2. 
 
 q r 
 
 ^6_7^5 _|_ i^x'^ — llx^ + 21a;2 — 9a?+ 1 =0, 
 finding that no root exists so great as unity, if a positive root exists it is 
 
 greater than — . 
 28 
 
 Depressing by — ■, 
 
 q r 
 
 l_-4 4.5.25 — 4 + 1'1875 — -75 — -640625 («.) 
 
 Thus, by the change of sign in the absolute term, there is a positive root 
 between and -5 ; but as five changes of sign still remain, we are yet 
 unacquainted with the nature of four roots. 
 
8 NATURE OF THE ROOTS 
 
 Depressing the last equation by '5, making the whole depression unity, 
 there results 
 
 1 — 1—1 — 1 — 1 — 1 — 1, showing the loss of four signs. 
 
 in (a) is a positive quantity ; 
 
 b„tby(C), _i-^_(,_^)=^-i=0 = 
 
 .-. four roots of the original are imaginary, and two real. 
 
 Example 3. 
 a?5 — 7a^+ 16a^— 18a?2 + 9a;— 3 = 
 
 in (a?- -4) 1 —5 + 6-4 — 4-88 + 0-616 — 1 {a) 
 
 c d 
 in (a?— -5) l — 5-5 + 4*5 —3-25 - 0-1875 — 1-40625 (b) 
 
 in(a?-l) i_2-2~2-2 — 2 (c). 
 
 Depressing by -5 two signs are lost ; but as one of the terms is very small, 
 it is probable that with -4 they will be restored, which is found to be the 
 case as above. 
 
 The equations (a) and (b) will determine the nature of two roots. 
 
 From(«), a;>^ = -13.... 
 
 From (b)y x < -- (1 — -1) = a negative quantity. 
 
 The superior limit being less than the inferior, the two roots are imaginary. 
 The equations (b) and (c) determine the nature of the two remaining roots : 
 ^ 1-4062 5 , . ^2 1 ^ 
 ^>4r90625 = +'^^^^<T~-2=^- 
 
 .*. four roots are imaginary. 
 
 Operating on the equation (c) for the sake of showing that any equation 
 resulting from a depression may be used, we change the alternate signs : 
 
 g r 
 
 x^ +2(x:^ — 2x^ + 2a?2 — 2a? + 2=0 
 
 in(ar — -5) 1 + 4-5 -j- 4-3 + 3*25 — •1875 + 1*40625 («), 
 
 two signs are lost; therefore the two roots related to those signs are 
 
 imaginary ; for (D), the least of them, if real, must be greater than — ; 
 
 but the signs have disappeared by this depressor. 
 
 Two other roots are also unreal, for (a) x > -——.—: = *8 but a 
 
 depression by -1 leaves no change of sign, the 5th term being '6158. This 
 last proof requires nothing more than regarding the smaU term — '1875 
 in contrast with the others. 
 
of numerical equations. 9 
 
 Example 4. 
 
 x^ H- 3a?*-f-2a^ — 3a?2 — 2a: — 2 = 0, 
 
 one root is seen to be positive. 
 Alternating the signs 
 
 1—3 + 2 + 3 — 2+2, 
 
 2 4 
 if another positive root exists it is > — ^^ ~X' ^^* ^ depression by "3 
 
 causes two changes of sign to disappear ; 
 
 .*. the two roots are imaginary 
 
 in (ar--5) 1--5-1-5 + 275 + 1-315 + 1-84375; 
 
 . . . 1-84375 
 
 if a real root exist it is > = -55 
 
 3-34375 
 
 but with '5 1 as depressor the terms are all positive ; 
 .-. there are four imaginary roots. 
 
 Example 5. 
 
 q r 
 ^6 _|. <p5 _^ _ ar^ + a?2 — a? + 1 = 
 
 in(a?--5) I +4 + 5-25 + 2 + -1875— -75 + -609375 (a), 
 
 there are only two changes of sign in this depression ; but as the same 
 
 1 r 
 
 number is lost by that of - which is equal to , the contradiction 
 
 2 ^ q + r 
 
 implies two imaginary roots. 
 Depressing {a) by -2 
 
 m {a)-2^ _^5.2^ g.gs + 7.95 + 2-9915 - -23308 + -492619, 
 or x — '7] 
 
 here = -6 
 
 q + r 
 
 but depressing this last by '3, all the terms are positive, viz. : 
 1 + 7 + 19 + 25 + 17 + 5 + 1 ; 
 
 .*. four imaginary roots exist, as far as this equation has been examined. 
 
 The methods of Sturm and Fourier are compared in the 234th page of 
 Professor Young's work by means of the preceding equation ; and at the 
 same time he remarks, " that certain steps cannot be made without cautious 
 deliberation, and that sort of tact which experience alone can impart to 
 the analyst.'* 
 
 If it were allowed to frame equations having the absolute term decimal, 
 
10 NATURE OF THE ROOTS 
 
 some of low degree might be formed, requiring long attention ; for 
 instance, the following : 
 
 a^—lOx^+ 10a?-2-64163101 =0; 
 
 but examples of this kind, obeying the general laws of reduction, take up 
 the time of a student unnecessarily. 
 
 Example 6. 
 
 In the general equation, 
 
 as^ — ax^ -j- bx — c = 
 if «252 ^ 1 8^5^ ig legg tijan 4ca^ + 463 + 27c^ 
 two roots are imaginary, as in ar^— lOa?^ -f 12a? — 4 = 0. 
 
 To this we shall add another example : 
 
 Let x^ + 24x^-30x + 10 = 
 in {x-'5) 1 +25-5-5-25 + 1-125 (a) 
 
 1-125 
 if a real root exist in (a), it is greater than ■ = '17 
 
 O'O/ o 
 
 a depression by -17 renders all the signs positive. 
 
 .*. two roots are imaginary. 
 They are also imaginary by (A) depressing by unity. 
 
 Budan's correlative is 10 + — 6 + 5, designating two real roots, which 
 renders necessary a resort to the method contained in the second or third 
 part of his treatise. 
 
 When this case of Budan's exception occurs in equations of hiffh degree, 
 we know of no means of being aware of it, and have, therefore, lost all 
 confidence in his method on trying occasions, such as when the existence 
 of an even number of real roots is announced. Newton's rule is often 
 more satisfactory. 
 
 EXAMPLE 7. 
 
 a;6— lla;5 + iOx'^—llGx^ + 164a?2-. 132a? + 52 = 
 
 depressed by uniti/, 
 
 afi — 5x^ + 9x^—lOa^ + I5x^ — 5x + 7 = (c), 
 
 this last having, like the original, six changes of sign, two roots at least 
 are imaginary by (E). But we shall find the nature of all the roots from 
 the equation Ca), for the nature of roots is not altered by depression, 
 
 in {x'--5) 1-2 + -25—2 + 8-1875 + 5-625 + 7-421875 (b) 
 
OF NUMERICAL EaUATIONS. 11 
 
 On referring to (C), the only term changed to minus is 5*625, 
 
 '''^ < 13-0468 75 — ' ^ ^ega^ive quantity; 
 
 .'. these two roots are imaginary, 
 
 q r 
 
 in (6 — -8) 1 + 2-8 + 1-85 -3740 + -2835 + 13-27988 + 15-848059, 
 
 ^ 13-279 
 
 ^ < i3-279... + 15-848... '''^^ ^^^^*^"^' 
 
 .-. two roots more are unreal. 
 The two remaining roots are unreal, because -6, which is less than 
 r 
 in this last, causes all the signs to be plus, which being manifest to 
 
 the eye, needs no operation. 
 
 Example 8. 
 «5 — 5a?2 + 16 = 0. 
 This example is taken from the 'Mem. de Turin,' vol. vi, p. 171. 
 In (a? — 1) 1 + 5 + 10 -1- 5 — 5 H- 12, 
 
 if the two changes of sign denote two positive roots, the least is greater 
 
 12 
 than — = '7 . . . But if this last equation be depressed by '5, all the 
 
 signs are positive ; 
 
 .*. these two roots are unreal. 
 Changing the alternate signs of the original equation, 
 
 1+0+0+5+0— 16 (a) 
 
 in (x' — 1) 1 +5_+ 10 + 15_+ 15 — 10 (6). 
 
 There are necessarily three changes of signs in (a), and but one in (b). 
 If the two roots in (a) are real, 
 
 the least > — = 76 . . . 
 
 that in (b) < — = '6. 
 
 The contradiction shows that the two roots are unreal. 
 
 This example was selected by the learned society in order to clear up 
 some doubts respecting the resolution of the higher orders of equations. 
 It shall now be treated in a different manner, 
 
 4i/^ — 5i/2 — 16 = is one limiting equation, 
 5?/^ — 1 Oy is the usual limit. 
 
 The root of the former is nearly 1 -4, 
 
 „ of the latter, nearly 1-25. 
 But as the first value exceeds the second, the four roots are imaginary. 
 
 This method we published in 1837. It is only convenient in binomial 
 and trinomial equations. 
 
12 NATURE OF THE ROOTS 
 
 Example 9. 
 
 r 8 
 
 a^ + 312^3 ^ 23337:»2 _ i4874a? + 2360 = 0. 
 
 An acquaintance with the manner of separating the figures which concur 
 in the real roots of equations is implied in the process of discovering the 
 existence of imaginary roots. ♦ 
 
 The reader will observe that the following is not given as a rule appli- 
 cable under all circumstances, nevertheless it is of such frequent occurrence 
 in success, that it is thought advisable to insert it : 
 
 2s 4720 
 7=lT874=^^^--- 
 
 Depress the roots of the equation by '316, and in the resulting equation 
 take 
 
 2s' -02089 ^^^,^, 
 7=3-R245 = -«^^^^^- 
 
 The sum of these results is -316664, which is so far correct for the least 
 root of the original equation; the greater root being '316665 . . . 
 
 This equation is ably managed in Dr. Rutherford's recent work, and in 
 Prof. Young's * Analysis and Solution of Cubic and Biquadratic Equations.' 
 
 Example 10. 
 a?7 Oafi — 2a?5 Oa;^ _ 3-^ ^. 4^2 _ 5^ ^ g _ 0. 
 
 The roots of the equation are increased by -2 in order to avoid the 
 ambiguity of signs preceding the meros, 
 
 in (a;+-2) 1--1-4 — 1-16+1-720 — 3-7440-J-5-95328 — 6-975552+7-1846272 ... (a) 
 
 depressing the roots in a? by '1, 
 
 c d 
 
 in {x — -1) 1 +-7 — 1-79 — -965 — 3-1965 + 3-08021 — 4-290993 + 5-5369801 ... (A) 
 
 - + - + — 
 
 Here two changes of sign are lost, and it is to be remarked that («) is 
 
 depressed '3, so that |) = -3 c) . 
 
 7* 18 
 
 If a positive root exist in (a) it is >• -^ = -5 . . . 
 
 ^ V y ^ 7-18... +6-975... 
 
 but in {b) the greatest root > -— -7 = a negative quantity ; 
 
 D<jO • « • 
 
 .*. the two roots in this interval are imaginary. 
 
 If the equation in {b) have a positive root it is > '5 ... = ^ 
 
 5' + r' 
 
OF NUMERICAL EQUATIONS. 13 
 
 Depressing it by '6, the result is 
 
 c d 
 
 l+4-9 + 8'29+5-005-4-3965— 5-63053— 5-387457+3-1772143 .. (/). 
 
 Here are only two changes of sign, 
 
 8*29 
 by (C) the root in (b) < — — - -4 = 7 . . . -'4 . . .= -3 . . . which, 
 
 being less than 'S^i^he two roots are imaginary. 
 
 The remaining roots of the equation are real, two positive and one 
 negative ; for if the equation in x be depressed by unity, and again by '5, 
 the last term of the resulting equation will be — 6* 2734375. 
 
 The long route we have taken is for the sake of showing how, in higher 
 equations, the signs belonging to the zeros may be found. 
 
 In the work mentioned in the Preface, the imaginary roots of this 
 equation are found in a compendious way. See the 147th page of 
 Prof. Young's 'Theory of Equations,' where this equation of Fourier 
 occurs. 
 
 Example 11. 
 a?8 4- Ax^ — a?6 — 10^5 ^ 5^4 __ 5^ _ i0a?2 - lOa? — 5 = 0, 
 taken from the * Arithmetica Universalis.' 
 In (a?- 1) 1 + 12 + 55 + 124+ 150 + 91 +2—45—31. 
 
 Here two changes of sign are lost ; if a positive root exist in the interval, 
 5 1 
 
 ^'">I0 = 2' 
 
 150 
 but by (C) it is <-— - — -5 = -82 - -5 = -32 . . . 
 
 181 
 
 .'. the two roots are imaginary. 
 Alternating the signs of the equation 
 
 l-.4_l + 10+5 + 5-10 + l-5, 
 5 
 the least positive root > — = -333 . . . 
 15 
 
 But depressing by '3, in a very early part of the process, two changes 
 of sign are lost, the coefficient related to — 10 becoming + 
 .*. the two roots in this interval are imaginary. 
 
 In the whole process it is readily discovered that the equation contains 
 one positive and three negative roots. 
 
 It was with some surprise that we found, on searching for the surd 
 divisors, to show which seems to be the object of the renowned author, 
 that the equation has the following factors : 
 
 a?6 + 5^5 _|_ 5^4 ^_{. i0a?2 + 5a? +5 = 
 
 a?2 — la? - 1 = 
 
 therefore the surd divisors are confined to the equation of the 6th degree. 
 
14 NATURE OF THE ROOTS 
 
 A method for surd divisors was published in 1658 by Johannes Hudde, 
 Burgomaster of Amsterdam, who gave the following instance : 
 
 a^ + 4a;3 _ 3^2 _ 8a? + 4 = 0, 
 whose factors are 
 
 a?2 + (2 - V3) a? - 2 = 
 
 a?2 + (2 + V3) a? - 2 = 
 
 Example 12. 
 a?6 — 7^5 + 19^ __ 22a?3 + l ia?2 — I5x + 25 = 0, 
 
 25 
 the least root > — = '59 . . . 
 ^ 37 
 
 depressing by its inferior limit '5 . 
 
 In (^--5) 1 — 4 +5-25 + 1 — 1-3125 — 13 + 18-484375 («) 
 
 .-. two roots in this interval are imaginary, because two signs are lost. 
 Now depress (a) by -8, in order to render the second term positive, 
 there results 
 
 1 + -8 — 1-15 + 2-44 + 6-9115 — 8-65392 + 8-858199 {b) 
 
 The least root in (S) > '5, by which depress 
 
 in (a?'— -5) 1 + 3-8+4-6+4-64+10-784 — -04992 + 6-532864 (c). 
 
 Two signs being lost, the roots are imaginary in the interval of (a) and (6). 
 
 6*53 
 
 The least root in (c) > '-^ = '9 . . . 
 
 ^ ^ ^ 6-53 . . . +-049 
 
 It is unnecessary to operate further, for it is evident that a very low 
 decimal will occasion the loss of two signs, if (c) is depressed by it. 
 
 The six roots are imaginary. The equation has two pairs of equal roots. 
 They form no obstacle to this method of discovering their character. Nor 
 can any other similar equation be proposed that can defeat the means used 
 in respect of the present equation. 
 
 The factors of the equation are : 
 
 «2 + la? + 1 = 
 
 x^ — 4a? + 5 = twice. 
 
 Example 13. 
 
 a;io - 196830a' + 531442 =0. 
 By (F) all the roots are imaginary. 
 
or NUMERICAL EQUATIONS. 15 
 
 Example 14. 
 x^ - 16a;5 + 85a?4 — 144a^ — 57a?2 + 126^ + 54 = 0. 
 
 This equation was proposed many years since by a learned society in 
 Italy, doubtless with the view that the character of its roots should be 
 ascertained on some general principle. No such object could have been 
 attained. We alliifed to this equation in a former work. It is but lately 
 that we have reverted to this equation, and found that it had the following 
 factors : 
 
 a;2 _ 6a? + 6 = 
 
 a?2 — 5^ ■— 3 = twice. 
 
 Consequently it has six real roots, or four positive and two negative roots, 
 of which two positive roots are equal as well as two negative roots equal, 
 and these are incommensurable roots. 
 
 It is easily seen that an equation of the fourth degree, with integral 
 cofficients, the first being unity, cannot be constituted with only a single 
 pair of equal incommensurable roots. Two pairs of such roots must 
 simultaneously exist. Thence every equation of a higher degree having 
 one pair of such roots, must have another pair, and such equation must 
 necessarily have a biquadratic factor. 
 
 The difficulty of discovering the character of roots that are supposed to 
 be equal by taking the figures of a root to a great extent, and endeavouring, 
 without success, to separate it from another root, is discouraging to the 
 analyst. 
 
 If the root on which he is operating belongs to a pair of equal roots, 
 he will find the same obstacle on examining another root belonging to 
 another pair, as well as that connexion between the decimals of the former 
 and the latter, which amounts to all but certainty that two pairs have, 
 separately, equal roots. 
 
 We refer to Prof. Young's * Theory of Equations,* page 116, where the 
 subject of equal roots is amply discussed. 
 
 The theorem of Sturm will, undoubtedly, discover the presence of equal 
 roots, but it is impracticable in equations of high degree, notwithstanding 
 the great improvement it has received from Prof. Young. 
 
 A thought has occurred to us of finding a substitute for the common 
 measure, in respect of an equation having two pairs of equal incommen- 
 surable roots, and no more. 
 
 The foundation was a biquadratic, formed from the two equations 
 
 a;2 + 2«a? — 1 = 
 
 a?2-f (2« + I)^— 1 =0 
 taking a = 998. 
 
 The quadratics are a;^ + 998a? — 1 = 
 
 a?2 + 999a? —1 = 
 whose roots are — 499 ± \/249002 
 
 -499 5 ±n/249501-25. 
 
16 NATURE OF THE ROOTS 
 
 The decimals in these roots separate almost immediately. 
 The biquadratic is 
 
 x^-\- \997a^+ 997000^2 _ 1997^+ 1 =0. 
 
 As the remarks we are about to make are peculiarly connected with the 
 separation of decimals, it may be of use to state that the decimals following 
 very high numbers, whose difference is unity, separate for the most part 
 at the 5th or 6th place ; for instance, ^ 
 
 V7896547852144 = 2810079-687636 
 
 V7896547852145 = 28100/9-687639 
 
 Supposing that the biquadratic had been proposed from these numbers, 
 we should have soon discovered, by the early separation of decimals, that 
 the roots were not equal. 
 
 The cause of this early separation is the fewness of the figures employed 
 in its construction, connected with their magnitude, and the law of sepa- 
 ration is easily traced. It will be remarked that the greatest number of 
 figures in any of the coefficients is six. 
 
 It is much easier to find eight figures of a root of an equation of high 
 degree than to find what is generally known by the name of the "common 
 measure." 
 
 If assent is given, it may be stated that an equation of any degree, 
 having no coefficient, consisting of more than six figures, and which has, 
 or may be supposed to have, two pairs of equal incommensurable roots, 
 and no more, can be freed from the necessity of our finding its common 
 measure. 
 
 Let it be supposed that such equation is of the 10th degree, having no 
 more than six figures in any one coefficient. If it have two pairs of equal 
 roots, a biquadratic factor must exist. This cannot be avoided. If it have 
 several concurring figures, a separation must take place in not later than 
 the 6th decimal place, even supposing that the equation has no factors. 
 But we are confident that an equation of the 10th or any higher degree 
 cannot be formed without factors, and with such limit in the figures of 
 one coefficient containing six concurrent decimals in two of its roots, the 
 coefficients being integral, as before mentioned. 
 
 This opinion is derived from long study and experience on the subject. 
 We venture, then, to subjoin the following rule in respect of equations of 
 all degrees, above the third of course, whose highest coefficient does not 
 consist of more than six figures : 
 
 Find eight correct decimals of a root : increase the last decimal by a 
 unit. Depress the equation resulting from the previous work by the 
 decimal so changed. If two changes of sign are lost, two roots are equal. 
 
 For example: let the derived decimals be -31479305; depress by 
 •00000006. 
 
 This may cause three changes of signs to be lost, provided another root, 
 such as '314793058 ... be passed over. This, however, would be no 
 obstacle ; it would only show the existence of another positive root. If, 
 then, we have succeeded in the attempt to dispense with the "common 
 measure" when the highest coefficient contains six figures only, we can 
 
if. 
 
 OF NUMERICAL EQUATIONS. 17 
 
 equally do so when the highest coefficient contains any number of figures, 
 because a biquadratic could be easily found accommodated to the require- 
 ments of the case. 
 
 Finally, we would remark, that all idea of perplexity, arising from 
 taking the real part of an imaginary root and proceeding with it as if it 
 were one of the real roots, is totally excluded. See Prof. Young's * Theory 
 of Equations,' page 309, where a constant is immediately preceptible. See 
 also the 1 2th Example, which has been expressly constructed for the pur- 
 pose of showing tnat we are enabled to decide that the equal roots are 
 imaginary, if such be the case, in any proposed interval ; and our previous 
 reasoning is subject to the preceding remark if a case should occur requiring 
 the interference ; but this, we are of opinion, can never happen, under 
 the limited magnitude of the highest coefficient. 
 
 Any number of equal roots, mutatis mutandis^ can be discovered to 
 exist in an equation of any degree. 
 
 Example 15. 
 
 a?4 __ 72a?3 4- 1344a?2 — 1728:r -}- 576 = 0. . 
 
 Two roots of some kind appear to take place between -6 and 7, and a 
 
 2o 
 separation of the decimals not being yet attainable, we take — , the result 
 
 is '66. 
 
 In {x --m) 1 — 69-36 -M204-0536 - 46*859616 + -45643536, 
 
 taking -f = -0194, and adding it to -66, there is derived -6794. 
 
 Depressing the last equation by -0194, found as before, and taking -\, 
 
 It 
 
 and adding, as before, x = -679491924 . . . 
 
 This is so far the value of two equal roots ; for, depressing again by 
 •000000005, two changes of sign are lost. (See the last Example). 
 
 Observation on the Limiting Equations. 
 
 Let a?4 — 10a?3 + 31a?2- 30a? + 9 = 0, 
 
 ^^^^ 3a^ — 20a?3-F31a?2--9 
 
 4aj3 _ 30a;2 + 62a? _ 30 "" O' 
 
 There are two pairs of equal roots, which in this case are 
 
 2-5 + V3-25 . 
 
 2-5--V3-25'^^^^- 
 (See the 8th Example.) 
 
18 ON THE PECULIAR EttUATION X^ — bx==l. 
 
 Example 16. 
 afi + 370x^ + 33736ar* — 90160a^ + 116333^^ - 50858a^ + 6962 = 0. 
 
 No means are yet known of obtaining, without trial, the first figure of 
 the root of an equation. 
 
 This example is given to exemplify the fact, for it obtains in the first 
 and second figures ; therefore recourse has been had to trial for the first 
 three figures. 
 
 There are also some peculiarities in the. roots, the development of which 
 is left to the student. The three following steps are given, unabridged, 
 as well for his assistance, as to give him confidence in the accuracy of the 
 calculations in this treatise. 
 
 Depressed by '3, 
 
 1 + 371-8 + 34292-35 - 49343-260 -f 53.506-4615 — 
 
 1742-91242 + 14-411429. 
 This by -01, 
 
 1 + 371-86 + 34310-9415 — 47971-194180 + 52046-74282875 — 
 
 687-4489800094 +2-283950650681. 
 This by -006, 
 
 1+371-896 + 34322-097840-47147-597709080 + 51 190673299529040 
 
 -68-039307979776544 + -022622204367150016. 
 
 ADDENDA. 
 
 ON THE PECULIAR EQUATION a^-~bx=\, 
 
 1 . Let p^ — rp =she & cubic equation having three real roots, and let 
 it be represented by 
 
 . /b*-6ix 2«» 2i'^, ... 
 
 ^-(— 3-)^=27--T + * (*) 
 
 then will 
 
 i.,.,-.^.. <B, 
 
 where b is consequently greater than (6*75)^. If 6 = 9, the quotient is 
 6*750000 . . ., and the greater b is assumed, the number of zero* following 
 6-75 increases in a known ratio, for such number is obvious when the 
 values of b and s are given. The number of zeros would be unlimited if 
 the magnitude of b were unlimited, because s is greater than b. 
 
ON THE PECULIAR EQUATION SC^ ■— bx =^ \ . 19 
 
 2. The equation a?^ — 6a? = I is brought into relation with p^—rp = s, 
 by assuming 
 
 X = i, (C). 
 
 Let a^ -— 30^ = 1; then b =■ 30, and by (A) we get the equation 
 p^ - 269940^) = 53982001 ; 
 and by (B) the quotient 
 
 ^ = 675000000000 . . . 
 
 b^ 
 When the roots of the equation x^ — bx = c are equal, then — =6*75 
 
 exactly, and the positive root = y • 
 
 3* 
 The equation in p is so closely allied to this case that — can be safely 
 
 taken as the positive root in p to a great extent of figures. The result 
 is 599*933329, which is so far correct, and had the division been further 
 extended, it would probably have afforded a still closer approximation, 
 
 Applying this value of^ to the formula (C), it gives x=: — -033334568038, 
 which is one of the roots of the equation a^ — 30^= 1, and correct to the 
 last decimal figure. 
 
 When one of the roots is known, the others are derived by applying 
 
 X C 3x^^ - 
 such root to the formula — — ±:<b ~r ( - The remaining roots 
 
 p are — 299-9664599 and— 299*966869, which, if somewhat more ex- 
 tended, would give the other roots in x. 
 
 3. The equation a^ -{■ bx=^\ may be treated in the same manner as the 
 foregoing equation, for the three formulas remain unaltered, but b and its 
 odd powers are here negative. 
 
 Let, for example, x^ + i 2a? = 1 be the proposed equation ; then 
 
 p^ -6936 p = 222337; 
 .-. p = 96-1665224, and thence x = -083285191 . . . 
 
 There are some very remarkable features connected with the resolution 
 of all cubic equations arising from the present case. 
 
 A few small values of b are, in the present state of this problem, not 
 suited to the resolution. The efiectiveness of 6 = 6 may be shown in 
 the following manner, and as this number answers the purpose, all superior 
 integers and mixed fractions will be equally effective. 
 
 Let x^ — 6a? = 1, 
 
 a?'3 + 6a?' = 1, 
 
 and as the equations in p were introduced only to render the subject more 
 intelligible, they will now be dispensed with. 
 
 m 
 
20 ON THE PECULIAR EQUATION 0^ ^ bx =■ 1. 
 
 ^ , 3* 252 2 3 
 
 Now ^and^=-=----^j— ^^; 
 
 .-. JO = 23-664, / = 24-331. 
 
 ^5*664 
 and by (C) ^ =- -2;^^^ = - -16744919 
 
 ^=+^s=+->«^««^^« 
 
 It is apparent that these approximating roots may be derived from each 
 other, an unusual circumstance in ordinary operations. 
 
 By means of the resolution of the proposed equation, the following are 
 resolved : 
 
 1. The equation y^ + hy'^=. 1, whose roots are — (a^-j-a^), _-(52^^5)^ 
 + ab when the roots of the proposed are a -\- b, — a, —b. 
 
 2. The equation y^ — b^y = b^, since bx ^-y. 
 
 3. The equation y^ — I'y =■ s, when -^ = b^, for — = y, reference 
 
 s s 
 
 being had to the equation a^ — b^x = b^. 
 
 Another class comprehends all cubic equations having roots real or 
 imaginary. The same formulas are used as in the preceding resolutions, 
 with the exception that a variable quantity is employed, which in the 
 former was a constant, and that constant was — b. 
 
 Keeping to the equation x^ — bx = I, having three real roots, and 
 where the value of b is not less than 6, let it be connected with 
 
 p^ ^ (^-n^ + 3n + b)p = (^l - —y + bn^ -]- —n+ 1, 
 
 where n may be any number or fraction whatever, positive or negative. 
 
 Having found the approximate values of the roots of o?^ — bx = l, equations 
 
 are obtained in p, which are brought into relation with those in x by 
 
 assuming 
 
 2bn 
 i> + — + w2 
 
 ^ = * (D.) 
 
 np — --n^+l 
 
 Thus let the equation be x^ — 6x — I, one of whose roots has been 
 found above, and if 
 
 w = + 1, then we have p^ -- 2lp = 16, 
 
 n= — l „ „ y _ 15^ = _ 2, 
 
 w = + 2 „ „ p3__60i) = -47, 
 
 71 =-2 „ „ p^-4Sp= 97, &c.,;) = 7776. 
 And the roots of these equations can readily be obtained from those of the 
 equation a:^—6a?=l. Selecting the equationjp^ — 15p^=— 2, we have, by(D), 
 
 P~^ . = _ -16744919 = Xi 
 -i>- 1 
 hence jp = +3*8045115. 
 
 C. and J. Adlard, Printers, Bartholomew Close. 
 
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