aA 2.19 Lse illllll $C Ibfi 123 IN MEMORIAM FLORIAN CAJORI tti^ii /^. e^ei^^'—^^^'^i y" zz '-^^fe<^-^ y/^ A^:f'e^ rT^ W.^tjL » 4 ^ BY JAMES\L0CKHART, F. R. A. S. -J OXFORD : PRINTED FOR THE AUTHOR, BY D. A. TALBOYS. M DCCC XXXIX. CAJora PREFACE. On the day of commemoration, held in June last, at Oxford, to celebrate the memory of the Founders and Benefactors of the University, the writer published, as a homage on the occasion, the resolution of two celebrated equations according to the method of Newton, He asserted, at the conclu- sion, that by means of some properties derivable from the theorem of Sturm, unnoticed by that author, the separation of the roots of equations from each other might be effected, and a few instances of such separations were given. Being now engaged on the second part of a work, relating to approxima- tion, he has thought it advisable to preface it by the examples now placed before the public, the demonstration of the universal accuracy of the separa- tions being reserved for the intended publication, which indeed is so plain that it will readily occur to the scientific reader. Cowley, near Oxford, 1839. Note. — The writer's work, " On the Properties of Equations and the Detection of Imagin- ary Roots," the " Homage," above mentioned, and the present work, are sold by Messrs. T. & C. LocKHAET, 156, Cheapside, London. Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/extensionofcelebOOIockrich SEPARATION OF THE ROOTS OF NUMERAL EQUATIONS FROM EACH OTHER. Our purpose is to shew, by suitable examples, that all the real roots of numeral equations may be separated from each other by means of the func- tions belonging to the theorem of Sturm, using these functions as auxiliary equations, whereby a proposed equation may be fitted for resolution by the method of Newton. Let the equation, having two positive roots, be x^ — ax-\-b = the functions, agreeably to Sturm's theorem, are X = x^ — a x + b Xi =2 X — a Xi =a^ — ^b This is the only case in which the limiting equation is employed. If Xi =0,x= — , being a limit between the two roots if they are un- equal; but if they are equal, — is the value of each. If Xa = 0,^ = 4,which is the case when the two roots are equal, otherwise -r- > 4. There is another o remainder or function not alluded to by Sturm, namely, a x — 2 b, which 2 b made = 0, gives x = — ; this also designates equal roots, or separates them if unequal; in the latter case the expression is less than—; so that between a .2b -— and — no root can exist. 2 EXTENSION OF THE As an example, let a;2 — 6 x + 9 = 0, _ = — . Again, let a; 2 - 19 a; + 90 = where a: = 9 and 10, a T. 9'5, — = 947 .... nearest to the least root. a Let the equation, having three real roots, be x^ + Ox'^-bx + c^O the functions are X = x^ ~ b x + c Xi =3x' - b X2=2bx-3c X, =46'-27c^ If X2 = 0, a: = — , which gives a root if the two positive roots are equal, and separates them if unequal, tending to the least. If Xs = 0,— . = 6'75, in which case the positive roots are equal; but if — ^ > 6-75, the roots are unequal. If — < 6-75, two roots are imaginary. Thus, the equation a;" — 3 a: + 2 = 0, has equal roots, x^ — Sx + 1 =0, has unequal roots, x^ — 3 X + 3 = 0, has two imaginary roots, — = — , separates the roots of the middle equation, tending to the least 2b 6 root, in all such cases. Proceeding in a similar manner with the equation x" — bx + c = 0, n n c being integral and > 1, a derived function gives x = , and another gives n — 1" " ^ 6" = n" c" " *, where the roots are equal, so that if b" « = 4, the equation cannot have two unequal positive roots except ^_^ 256 27" CELEBRATED THEOREM OF STURM. 7 Thus, let a;* — 15 a; + 14 = 0, where a; = 1, 2 . ~ -rr = 1-24 ,. . separates the roots, being nearest to the least. n — lb *^ Let a;* - 4 a; + 3 = here x = I, 1 and l! = ?^, 33 27 We proceed to equations having three positive roots. Xi =Sx^ — 2ax + b X2 ={2a^-6b)x -{ab-9c) X., = + The last is not given symbolically, being very complicated, and rising to the fifth degree. It gives, however, an expression whereby three equal roots are designated. If Xo = 0, the expression - — - — — answers the same purpose, for it be- comes - if the three roots are equal. If two of the roots are equal the expression gives one of them, which was noticed by Kinckhuysen, in 1661 ; if the three roots are unequal (and not in arithmetical progression, one then being obtained) the expression separates the two roots nearest in magnitude to each other, but always tends to the middle root. Thus, let a;^ — 13 a;2 + 54 ar - 72 = 0, where ar = 3, 4, 6, the separator is ^ = 3-8 . . . 14 let x^ - 49x2 + 658a: - 1379 = . 19831 the separator is -gcT- = 23*221 . . . Sturm's equation is x" + 11a;*— 102a; + 181 =0; the preceding is connected with it by adding 20 to each root. We also learn from the expression that if a 6 < 9 c, or 2 a * < Gb, the equation cannot have three positive roots. The reader who is acquainted with the practical part of Sturnis method will have remarked that there is a first and second remainder in the opera- tions. The first ones have extensive uses. In the present case there is 8 EXTENSION OF THE ax- — ^bx + 3c, which made = separates the three roots from each other. Thus 49 x^ - 13l6x + 4137 = 0, has the roots 3-636 . . . 23*2205 and the equation has the roots 2"557351 23-213112 23-2295 and thus in the equation x^ — 6x" + 11a; — 6 = 0, where ^ = 1,2,3 a;2 — -a-a: + 3 = has roots which separate I from 2, and 2 from 3. It now remains to shew the consequences of Xi = 0, in respect of cubic equations, having two positive roots and one negative. Let x' — ax^ — bx + c = 9c-ab the expression g^z + 6b ^^P*'^^'^^ "^^ ^°°^^ nearest to each other. thus x^ — 5x^ — 34a; + 80 = has the roots _ 5, + 2, + 8 .550 ^, the separator is agl =2-1 if a;' — 7a;2 — 40 a; + 100 = the roots are — 5, + 2, + 10 . 620 , „ the separator is aaa = I'o . . . . Let x^ — ax^ + bx + c =■ ab + 9e , , , the expression « — s _ ai . separates the roots nearest to each other thus x^ - 9x'' + 6x + 16 = has the roots _ 1, + 2, + 8 the separator is -=- = 1 -5 . . . between the negative and positive roots and let x^ — i>x^ + 1 a; + 6 = the roots are — 1, + 2, + 3 29 the separator is Tg" = 2-2 between the two positive roots. CELEBRATED THEOREM OF STURM. 9 Let x^ + a x^ — b X + c =z 0, in this case the two positive roots are of necessity nearest to each other ; the expression — — separates these roots. Sturm has selected this form of equation to exemplify his method, and his own example is now given. Let x^ + Ux^- I02x + 181 = where x = 3-213 .... 3-229 .... ,, , . 2751 393 the separator is -^^ = -^ = 3-221 Let x^ + 0-179«2 _ 36-11549 or + 81-42057 = 0, where x = 3-411, 3-41, - 7 739-24980271 the separator is — ^ = 3-4104 nearest to the least positive root. 216'7570;^2 Let x^ — ax^ + c = — — separates the roots nearest to each other. Thus let a;3 - 7a;2 + 36 = where a; = — 2, + 3, + 6 324 the separator is -— - = 3-3 ... . and let x^ — 9 x^ ■\- \0 = where x = — \,5 +_ ^15 the separator is -— = -55 between the negative and least positive root. In all the preceding cubic equations if the roots are in arithmetical pro- gression the middle one is obtained, or if two of them are equal, one even root is derived. Let X* + ox^ + \x^ — 70 J- + 120 = where x = 2, 3 ^ ; -—r-, separates the two positive roots This gives = 2-3 .... nearest to the least root. 10 EXTENSION OF THE Let a;* + ox3 — bx~ + cx — d = Ohave three positive roots, the function •S 6 y^ — 3cy + 4c? = contains two separations. and — -^ — ~ . . ^ — separates two roots, or exhibits equal roots. Let X" — ax'' + b X — c = 0, have three positive roots, n being a whole number, greater than 2, the function n — 2 a x^ — n — I. b x + n c = separates the roots. let a;« - 45 x2 + 50 a; - 2 = 0, here x is nearly -03, M17, 2'29 180 a;2 - 250 a; + 12 = has the roots 0-408 ... and 1'239 . . . Taking the function containing the simple equation, it gives = "7004 .... which separates the lesser roots, shewing also that 30601211 ^ ^ no root exists between •? and 0*498. Hitherto the functions have been represented symbolically ; but as they become very complex in equations of a high degree, having several positive roots, they will now be introduced under such form as they would receive on determining the nature of the roots of a particular equation, agreeably to the directions of Sturm; and these will, in almost every case, be sufficient without employing the first remainders before mentioned. The equation x* - 14 a:^ + 67 a;^ - 126 a; + 72 = has the roots 1, 3, 4, 6. Xi = 2 a:^ _ 21 a:* + 67 a; - 63 X» = 13 a;2 - 91 a; + 153 Xs = 26 a; -91 X* = + Xs = 0, gives the roots 2'8 . . ., 4-19 . . ., Xs =0 gives SS, being the arithmetical mean, between the middle roots, which is occasioned by the dif- ference of the exterior roots and their adjacents being the same. CELEBRATED THEOREM OF STURM. ] I Let X* - 14 x3 + 6.5 x^ — 112 x + 60 = 0, the roots are 1, 2, 5, 6. Xi = 2 a;3 - 21 x2 + 65 a: - 56 Xi = 17x2— 119 a; + 152 Xs =34 a; — 119 X* = + Xi = 0, separates 1 and 2, and 5 and 6. Xx gives 3'5, being the arith- metical mean, as before mentioned. Let X* — 17x3 + 99 ^^ -.22Sx + 140 = 0, the roots are 1, 4, 5, 7. Xi = 4 a:3 _ 51 x"- + 198 a; — 223 Xe = 25 a;2 — 230 x + 517 Xs =a; — 4-6 X4 = + X2 = 0, gives 3-9071 . . . and 5-292 ... Xs = gives 4-6, the arith- metical mean not being applicable in this case. Leta;4 — 14 x^ + 66 j;^ — 123a; -f 90 = X2 = 10 a:^ — 62 a; -h 47 Xs = — 82 a; + 527 X4 = — This equation, from the signs of the first terms of the functions, has two imaginary roots, the other two being evidently positive. The last are 5, and 6. Xe = 0, separates by one of its roots the two positive roots. We cannot avail ourselves of X3, the function being prefaced by a negative sign. Had an equation been selected with roots containing interminable decimals, the succeeding process of Sturm, about to be noticed, would have enabled us to determine which root in X2 = 0, ought to be adopted as a separator. It may here be remarked that equations of the third and fourth degree do not admit of more than two roots between and 1, or between any two con- secutive numbers, the first coefficient being unity and the rest integral. 12 EXTENSION OF THE Let x' — 80 a;3 + 1998 x^ — 14937 x + 5000 = Xo = 201 x^ — 8777-25 x + 6968 5 X3 = 163973-5625 x — 5192733-3 X, = + The separations are left to the industry of the reader. They will be found very efficient. One of the four positive roots is discovered in a-compendious manner by Newton's rule, and the operation will be found in the work men- tioned in the Preface. The root is 12-75644179448074402 Hitherto we have not considered equations having negative as well as positive roots. This will require a detail of some length, but it will be serviceable in every other equation of the kind. Much labour, and not unfrequently attendant confusion, will be spared by rendering the roots of all such equations positive. This can be accomplished with great ease by the means pointed out by Professor Young, in the 24th page of his Treatise on " Algebraical Equa- tions." Supposing the most negative root to lie between — 9 and — 10; by increasing each root by + 10, it is evident that all the roots Mill be positive. To elucidate the operations we recommend to be adopted in these cases, the Professor's own example, taken from the 150th page of his work, shall be employed ; but, in order to work with small coefficients, the alternate signs shall be changed. Let x* + 2 a;3 — 7 a:2 — 10 .r + 10 = the two negative roots are found between — 2 and — 3. Increasing each root of the Iquation by + 3, there is derived y4 _ 10 2,3 + 29 3^* — 22 y + 4 = Yi =^y^—\5y^ + ^y—n Y^ = 17y2 — 79y + 39 Y, = 152 y — 151 y* = + having added 3 to each root in x, whose values have been found to lie CELEBRATED THEOREM OF STURM. 13 between known integers, two positive roots in y lie between and 1 ; one between 3 and \, and one between 5 and 6. It is not necessary to examine the following Table in respect of the integral boundaries of the roots, that having been done in respect of x; and it is only given for the sake of those who wish to prosecute the enquiry. H H 1- 4 variations 1 + + — + + 2 var. s -\ h + 2 var. 4 + + 1 var. 5 1- + + + 1 var. 6 + + + + + The function Fa = gives y, -S^ and 4*08 nearly, as separators for the two lesser and two greater roots. 151 The function Fs = gives — - = "9934 . . . separating the middle roots, for it will be found that the greater of the two lesser roots is less than •9934 That '56 separates the lesser roots will become evident by diminishing the roots of the equation by '5 each. But an approximator may wish for and even need exterior limits and other confirmations. These can always be given when the previous boundaries are ascertained. The equation in z/ is 1 — 10 + 29 — 22 + 4 in(^— 1) 1—6 + 5 + 10+2 It is seen by the decrease of signs that there are two positive roots be- tween and 1. 4 By known rules the least > = '153 . . . ^ 22 + 4 and the greater < = "8 . . . and this confirms Ys = and Fs = The equation in (2/ — 3) is 1 + 2 — 7 — 10 + 10 and in (y — 4) is 1 + 6 + 5—14 — 4 14 EXTENSION OF THE One root is contained between 3 and 4, and this is greater than 3 + — = 3*5, and less than 3 + 10 + 10 14 + 4 3-777 and this confirms Tz = 0. The equation in (y — 5) is 1 + 10 + 29+18 — 6 in (y _ 6) is 1 + 14 + 65 + 110 + 52 the greatest root is contained between 5 and 6, and is greater than 5 + -^ = 5-25; but less than 5 + — !i^ = 5-679 24 110 + 52 The exterior and interior limits furnishing boundaries to the several roots of the equation in y, it is now adapted for resolution by the method of Newton, whence the correct figures of each root may be derived to any extent; any oscillation to different roots being immediately detected or coun- teracted. These values being found, and each diminished by 3, give, of course, the several values of x. When only one negative root is found in an equation, it will be preferable to discover the separators without altering the nature of the roots. Letx^— 12a;* + 4€a;3 — 48a;2 — 47 a; + 60 = 0, the roots are 1, 3, 4, 5, — 1 Xi =5a;* — 48a;3 + 138a;= — 96ar — 47 X^ = 29a;3 — 234^;^ + 523 a; — 234 Xi =mdx^ — 2772 X + 3659 X* = 111853 a; — 388542 Xs = + by Xs =i 0, the separators 3-7 .. . and 2-1 . . are obtained, shewing that one root at least is less than 2-1 .. . and that another root exists between 2*1 ... and 3-7 .. . by X* =0 there is obtained 3-47 . . . which separating the middle roots, there can be no root between 3-7 . . and 3*47 . . . CELEBRATED THEORY OF STURM. 15 The exterior limit derived as before from the equation itself is — = '555 . . . separating the least positive root from the negative. As all the roots are exactly obtained by Sturm's table, further investiga- tion is unnecessary. The last equation we shall notice has the united roots of the two following equations : x^ + Ux^'—imx + 181 = x^ — ix + 2 = The former one is chosen by Sturm to exemplify his theorem. It would seem that he had been considering the equation y^ — 7y + 7 = 0, and de- sirous of obtaining another whose roots should be nearer to each other, by putting __ = y, the wished-for equation arose. As the writings of this gentleman command respect and attention, this short digression will be excused. Letx^ + Ix*— 144x3 ^ 611 a;2_ 928a; + 362 = Xi = 5a;* + 28a;=' — 432a;2 + \22%x — 928 Xa = 1636a;'— 12189 a;2 + 27114 a;— 15546 Xs = 3073059 a;« — 20133742 a; + 32968246 Xi = 20969919426169 a; — 67576648427974 Xa = + For a; = 0, a;= 1, a; = 3, a; = 4, The Table. . -I 1 1- 4 variations + + — + 3 var. — + + + — + 3 var. + + + + + + thus there is one positive root between and 1 , and three positive roots be- tween 3 and 4, being the greatest number of roots that can exist between 16 EXTENSION OF THE two consecutive integers in an equation of the fifth degree, the first coeffi- cient being unity and the rest whole numbers. It is desirable, in the first place, to narrow as much as can be conveniently done the exterior limits of all these roots. Since a root exists between and 1, this root is greater than oact = -2806 (1) by the equation. 928 + 6&2 And since there are three roots between 3 and 4, the roots of the function Xz = are diminished by 4 ; in {X — 4) 1636 + 7443 + 8130 + 2590 changing the alternate signs, the functional root is less than 3 + __!H2 = 3-758 (2) 8130 + 2590 ^ ' the exterior limits of the four roots are found by (1) and (2). The following process serves to determine a limit less than either of the three greater roots. Xo = in (a; — 3) gives 1636 + 2535 _ 1848 + 267 267 hence the least of the three greater roots > 3 + -— --— - = 3-1262 — (3) lo4o + 2d7 the three greater roots are thus contained between (2) and (3). Now let a superior limit be found for the root between and 1. For this purpose the roots of the original equation are diminished by unity, and with the alternate signs changed, the coefficients are 1 — 12 — 106 — 231 — 105 + 91 231 hence the root between and 1 < ^„^ ^^ = "717 ... (4) and thus the least root hes between -2806 . . and -717 ... (1) (4) We now proceed to separate the three greater roots from each other, and for this purpose the quadratic function Xs = is employed, and its roots found to a convenient number of decimals ; the roots are 3-2758469 + -0549609 = 3-3308078 ... and 3-220886 ... (5) root root 3-229. .. 3-414. . . exterior . 3-3308. 3-758.. CELEBRATED THEOREM OF STURM. 17 the simple equation or function X+ = separates the middle roots of the equation, and gives x = 3-2225. We give a summary of these roots and separations : root root •5857... 3-213... exterior -717... 3-2208. •2806... 3-12(J2... 3-2225... No trial has been employed in any of the processes which have now fitted the equation for resolution by the method of Newton. The negative root of the equation has been passed over, as we trust it is evident that the limits for such root can be obtained with facility. We find it now our duty to state, from experience, that Sturm's theorem ceases to be practicable with complete equations of the fifth degree ; for, at the sixth degree, the coefficients of the functions contain oftentimes sixty or more figures each, which seems destructive of the "mental anticipation" alluded to by Professor Young, although indeed it may be exerted on other occasions. It will then still remain necessary in higher equations to resort to other rules to determine the nature of their roots. Perhaps the following one may in most cases be effective. In Budan's work he points out an anomaly occa- sioned by imaginary roots of the form + a + V — b. An equation very easy of solution will shew his meaning. Let^c^ + 2x^ — 3a;+ 1=0 in (a; — 1) 1+5 +4 +1 in this last there is no change of sign, and it is uncertain whether two roots are imaginary or not. The equation — is 1 — 3 + 2 + 1 a- .1+0-1 + 1 if the equation have two positive roots, the equation in f 1 j must have two changes of signs, and such changes appear. But the two roots are imaginary ; and this is the anomaly. As a is necessarily less than 1, if we diminish the roots of the equation by 18 EXTENSION OF THE the tenths between and 1 , there is scarcely an equation containing the ano- maly, the nature of whose roots cannot be determined. ' Decreasing the roots by '5 in {x — -5) 1 + 3-5 — 0-25 + 0-125 in ( ^^ .g — 1^ -125 + -125 + 3-375 + 4-375 which being all positive proves that two roots are imaginary. This is of easy application to equations of any degree whatever. In order that the reader may pass his own judgment on two several means of detecting the existence of imaginary roots in an equation of the fifth degree, he is first requested to refer to the 250th page of the Latin edition, in 1722, of the "Arithmetica universalis." The equation x^ — 4j:* — 4.r^ — 2x^ — bx — 4 = is there asserted to have one positive, two negative, and two imaginary roots. The name of Newton is not prefixed to the work, but it is known that it was published by Mr. Whiston, and translated partly by Mr. Raphson and partly by Mr. Cunn, and thus sent into the world as the original perform- ance oi Newton. The equation alluded to is found in the 198th page of the English trans- lation, pubUshed and revised in 1728. It is apparent that Newton was not concerned in either of these publica- tions, much less with the faulty additions and corrections of Mr. Whiston, copied by Mr. Cunn, beginning with " hinc etiam cognosci" and ending with the word " eveniat." Nearly the whole of the sentence is erroneous, the attempted reasoning being incompatible with the precepts given by Newton in the first instance. The " sequitur" of Mr. Whiston has probably led many a student into error. Changing the alternate signs of the above-mentioned equation it becomes x^ + 4a;* — 4a;» + 2x^ — 5x + 4 Xi = 5x* + mx'—l2x^ + 4a; X^ = 52a:^-39x^-h58.r-60 X., = 299a;2 + 1326x^1924 x* = - 2958176 X + 3342989 X, = + 12183750969204152 CELEBRATED THEOREM OF STURM. 19. there being two variations in the leading terms of the functions, the equation has four imaginary roots, contrary to Mr. Whiston's assertion. By the other process, the equation vnth. alternate signs changed is X* + 4 a;* — 4 a;' + 2x^- — 5 a; + 4 in (a; — 1) 1+9 +22 +24 +8 +2 if positive roots exist they must lie between and I » ip — (or reciprocal equation,) (^0 5 + 2 — 4 + 4 + 1 . . all positive /. this equation has no positive root ; but its roots are one negative and four imaginary. And consequently the roots of Whiston's equation are, one positive and four imaginary. This last process does not require the work of three minutes' time. However, had the anomaly taken place, it might have been very tedious. This statement is due to both processes. ADDENDA. The nature of the roots of every cubic equation may be thus known : .o each Cher, .,„ ,„„,., a. le.^, Zf^T,. each Xr""" '" ""'" thus let x» — 7-4a;2 + 7'ix — 2 = ^' — 7-5x^ + 7-5x — 2 = two roots of the former are imaginary, but those of the last are positive, manner" " '' ^'^ '^^'^ ^' ^" ^'^"^'''^"^ ^^ ^- '^"-" ^" « simila.. manner. Kl^^tjc^) 18 EXTENSION OF THE the tenths between and 1 , there is scarcely an equation containing the ano- maly, the nature of whose roots cannot be determined. * Decreasing the roots by "5 in {x — -5) I + 3-5 — 0-25 + 0*125 in ( _ .g — l) -125 + -125 + 3-375 + 4-375 which being all positive proves that two roots are imaginary. This is of easy application to equations of any degree whatever. In order that the reader may pass his own judgment on two several means of detecting the existence of imaginary roots in an equation of the fifth degree, he is first requested to refer to the 250th page of the Latin edition, in 1722, of the "Arithmetica universalis." The equation x^ — ^x* — ^x^ — 2x" — 5x — 4 =Ois there asserted to have one positive, two negative, and two imaginary roots. The name of Newton is not prefixed to the work, but it is known that it was published by Mr. Whiston, and translated partly by Mr. Raphson and partly by Mr. Cunn, and thus sent into the world as the original perform- CELEBRATED THEOREM OF STURM. 19. there being two variations in the leading terms of the functions, the equation has four imaginary roots, contrary to Mr. Whiston's assertion. By the other process, the equation with alternate signs changed is xs + 4