-54 UC-NRLF III III 1 ill $B SM3 b^h IN MEMORIAM FLORIAN CAJORl BEECROFT'S GENERAL METHOD FINDING ALL. THE ROOTS, BOTH REAL AND IMAGINARY, ALGEBRAICAL EQUATIONS, THE Ain f'f" AT^XTTJAT^V EOTTATIONS OF HIGHER DEGREES. HYDE: PRINTED BY ELIZABETH SLATER, MARKET-STREET. 1854. Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/beecroftsgeneralOObeecrich BEECROFT'S GENERAL METHOD FINDING ALL THE ROOTS, BOTH KEAL AND IMAGINAEY, ALGEBRAICAL EQUATIONS, THE AID OF AUXILIARY EQUATIONS OF HIGHER DEGREES. HYDE: PRINTED BY ELIZABETH SLATER, MARKET-STREET. 1854. CAJORI QA214' B^ LIST OF SUBSCRIBERS. W. C. Atkinson, Esq., Bradford. Dr. Bums, Kochester. D. Barker, Esq., Surgeon, Stalybridge 2 copies. T. Barker, Esq., Bowforth. Mr. Broadbent, Stalybridge. J. Burbridge, Esq., Secretaiy to the *• Times Assurance Company," Manchester. R. B. Benson, Esq., Stalybridge. Dr. Boole, Cork College. S. Beecroft, Esq., Surgeon, Hyde 6 copies. G. 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PREFACE The publication of the following researches, comprehending a new and general method of solving numerical equations, &c., has been hastened by the surprise, not unmixed with the appearance of doubt, expressed by any gentleman to whom the author happened to mention that he possessed a method of simultaneously obtaining all the coefficients of a quadratic factor of any equation, in the same form; as all the writers on this subject, and to no branch of mathe- matics has more attention been paid, have only succeeded, up to the present time, in giving methods for obtaining them as functions of n(n + 1) each other, eliciting, by elimination, an equation of the — - — th order for their separate determination when the proposed equation is of the nth. This uncertainty is however easily removed by pub- lishing a small work containing a portion of my investigations on equations, reserving the rest to appear in a larger work, which, if I meet with sufficient encouragement, I hope to produce at a future period; and it was for this reason that I was led to issue a few circulars amongst my friends and those gentlemen with whose correspondence on mathematical subjects I had been previously favoured, and afterwards to such of their friends and other gentlemen distinguished for scientific attainments as they thought likely to take an interest in the subject, in order to ascertain nearly the number of copies likely to be required, so as not to incur the risk of any pecuniary loss by publishing my first work at the price announced in the circular. The establishment of a universal method for finding the roots of air^jeg[uatiQ]3Lijade2end^ their depression, as this method may be made, will fully disprove the opinion that seems to have been generally held by analysts that depression and continuous approxi- mation are one and the same ; the idea of which must have originated Horner's method of solution, which indeed is nothing more than an elegant mode of depressing the roots of an equation leading to their determination, when real, to any degree of exactness. It was, how- ever, a most valuable improvement in the theory of equations. The following method also distinguishes itself from every other yet given, in its process being always the same for any equation, and in the values of the coefficients of the quadratic factors, &c., being directly approximated to, without a previous knowledge of the nature of the roots, whether real or imaginary. The other method following the general one in this work, though not of so great efficacy for finding the quadratic factors of aja. equation as the general method, is yet better adapted than either it, or than any other known method for finding the value of any particular root to a great degree of exactness. It may also be in- teresting as enabling us to express the roots of an equation in a form of series similar to the common trigonometrical expressions for the signs and cosines of an arc in teims of its length. The short chapter on the solution of equations by general alge- braical formulas, contains only a few of the least extensive of the formnluj which I have obtained for the reduction of the higher I equations, as the discussion of the more extensive ones would have interfered too much with the design of the present work, which is intended to establish, chiefly, a new method of approximation, that will be of much greater practical utility, and may probably prove more interesting, than any formulae that might possibly be obtained for the solution of equations by combinations of algebraic symbols. The method of solving equations of the fifth degree, given at the conclusion, appears, to me, to controvert the opinion generally en- tertainedj^ofjthe impossibility of solyinig the general equation of the fifth degree_by_the Jielp of an auxiliary equation below the sixth. Professor Young, after remarking, in page 469 of his work on the higher equations, that no such solution has yet been devised, adds, that "It has indeed been the object of analysts of late to demonstrate the utter impossibility of such a solution by any combination of algebraic symbols; on which subject the student is referred to the profound paper of Sir William Hamilton in the Transactions of the Royal Irish Academy, Vol. XVI." The method, however, which I have added, solves completely the general equations of the fifth degree, without using any auxiliary equation of a higher order, and that by a process equally as definite as the general method of solving a cubic equation complete in all its terms, though of course more laborious, the expressions iov p, q, &c., in the solution, being merely assumed for greater convenience in forming the cubic by help of which four of the roots are determined, after first determining one real root, as in Dr. Rutherford's method of solving an equation of the fifth degree deprived of its second term by help of an auxiliary equation of the tenth order. — ''Ruth. Eq., pages 20, 21." This method of solving equations of the fifth order is not likely, however, to be considered very interesting, after a method depending on so neat an algebraic form, and so general in its application as that contained in this work for determining the quadratic factors of any equation, but it may aid in directing attention to new methods of algebraical reduction. Philip Beecroft. Hyde, August 15th, 1854. NEW MET H D RESOLVING NUMERICAL EQUATIONS, 1. The following general method is founded upon the principle that imaginary roots greater than their reciprocals and real roots greater than unity have identical properties, and are capahle of being determined by the same common process, and, it is therefore evident, that, it will always be necessary first to determine the number of roots in an equation exceeding their reciprocals in magnitude; for this purpose Ave may make use of the following simple rule. Substitute a-^ -f-j;, and — , respectively for the root (a-) of the proposed equation, then, if the equation resulting for a- contains vi variations of sign from -j- to — and — to + amongst the signs of its coefFicients, and the equation for a* contains ii variations amongst tlie signs of its coefficients, the equation proposed will contain 7)i — n roots greater than their reciprocals ; and, by decreasing the value of 2>, and taking the equation for both positive and negative values of cc, ^\■e can thus determine all the roots greater than their reciprocals, and also the limits between which they are placed. This method depends upon an extension of the well-known rule of signs, and \nll be at once comprehended by the practised analyst, but, as other methods will be hereafter given, it may be omitted by the reader until he has perused the examples commencing in the third page of this work, the operations in which are so simple that, with a previous knowledge of the first four rules of common algebra only, he will (iud no difiieulty in understanding them. 2 Every algebraical equation may be made to take the form x''+ax^''-\-hx'^^-^cx^Jf&<^ =0 (1) and if this equation be multiplied by the polynomial m — 1 a; +p^x +2Y + ^m-l^ + A we get, putting m-\-n=r, ^'+(i'i + «)/~' + (i^,+ «i^i + %'~'+&c =0 (2) Multiplying (1) by x , and substracting the product from (2), thei^e remains and, if we take the coefficients it then becomes p/~'+i^P^+^P^l + 'P,^-.^ + >'^^ + {^P,, + %n-l + ^'^'-^ &C.=::0,. (4) or, dividing by the coefficient (j^j) of the first term, and then denoting the coefficients of the other terms by a , b , c , &c..., it will be ^ m' m^ mr- ' r— 1 , n— 1 , , n— 2 , n— 3 , „ -, ,^. X -{-ax -\-b X 4-c X +&c = (5) Now, if we form three equations from this, by substituting w^l, m — 2, m — 3, successively for m, and then eliminate the terms containing x^^ , x^~ , from these three e;^ations, we get, ^tor a little reducing, an equation of the form CRx''-{-Sx+T:)x"~^-{-u/^+u/'~^+&g =0 (6) in which x ^ will always be the highest of the powers of x, since r=w+w, and, by dividing by this power, we immediately perceive that when x is greater than unity or than its reciprocal, the farther we extend the calculation, or the higher we take m, the nearer will the part within the parenthesis approximate in value to zero ; and hence we may take for an approximation the quadratic Il^'^^S^+T = 0, (7) in which the coefficients obtained by the above mentioned operation, will be found to have the values, R = rt h ^-a h S = fl' h .^-a h ,, «t— 3 m—2 7rt— 2 HI— 3 j/i— 1 7H— 3 j/i— 3 »t— 1 T=^ h -a h ^ 3 Putting m — 1, for m, in (5), and then multiplying hj x, we get ^'~'+«.-/+V-i^""'+^.-i^""'+&« --0 (8) and, eliminating x" from tins equation, bj means of (1), we get Making the last equation identical with (5) we shall have a =h —aa ., or & ^—a -\-aa ^.flO) similarly^ ^ — a .-\-aa „, h ^ = a ^ + aa '' m—2 m—l * TO— 2' m— 3 m— 2 ^ wtr— 3 Substituting the three last values in the above expressions for Us S, T, we readily get R = rt a „ — rt „,S = « « „ — « a ,T = rt « — a^ . fft'i m—l m— 3 m—2' m—l m—2 to m— 3' m m—2 m—l ^ / Put^^=^^_^j9^, for any value of m, then equations (a) become a^+a = 0, a^^-{-aa^-}-h = 0, a^+aa^ + ba^-i-c = 0, &c (c) and since these relations would result by taking (l + ax+hx -i-cx +&c...) (1 + a^x + a^x^+a^x H-&c. ..) = !, we shall have from (7) and the last values of R, S, T, the following simple method of finding the quadratic factors of an equation. Rg. Let the proposed\quation be transformed into one containing two roots greater than their reciprocals, and let the transformed equation be denoted by X ■\-ax -{-hx -\-cx 4-A:c = 0. Then, if we ■^\\tf[x) = l-\-ax-\-hx +c.v^+&c , and —. — l-\-ax-\.aar^a^x-\-&LG^.....x cT. J [X) I 4 Q TO thfe quadratic factor containing the two roots eJsceeding their re- ciprocals, may be found to any degree of exactness, by calculating the values of a , by common algebraical division, to any extent, and substituting them in the equation On 2 (11) EXAMPLE I. To find a quadratic factor of x —Sx -^-Qx -|-1 = 0. This equation needs no transformation, as it already contains two roots exceeding their reciprocals, and dividing unity by 45^* - 82/+ 2l/+ 552/+ 1539/+(Src., and taking the cocflicients of /, /, /, /, we shall have, w=8, a^=: -82, a^=n, a^=6h2, ^,^=1539, aa-al = -45705, a .a -a a =1371 QO, ^tn-al = -272385, o/G O/OM bat ITcnce *'(11)" we have the approximate quadratic factor 45705a;2_137790a;+272385=0, or /-3-014768^ + 5-95903 = 0. Similarly, by advancing the division another step, and taking ?/? = 0, we get 30265/-91242^+180369 = 0, or /-3-0147695a!+5-959656=0. AVhich is the same as the preceding quadratic for fi^^ or six places of figures, showing the former quadratic to be correct to about that extent, and the latter being obtained at a step more in advance of the operation is probably correct to the last figure. EXAMPLE II. 3 Solve the equation /- 18/+ 85/ + 1 = 0. This equation has two roots greater than their reciprocals, and we have ■j=l + 18a^+244a;^+2952./+33615/+ l-18.^•+80.^'"+-.r 3rj8892^'"+ctc. Here « =244, r^, = 2952, « =33615, r/,=368892; 4 o i «^ _f/^rr^ = 512244, «3«^-rt^«.^=9221832, al -«^«^=40999041, and . .-. 512244/-9221832a;+40999041=0, nearly, or 56916/-1024048.r+4555449 = 0. Solving this quadratic we get the two real roots 8-007829..., and 9-9949823..., the greater of which is correct to the last figure inclusive, and the lesser value is nearly as exact, the true root being 8*007828 The two roots less than their reciprocals may be found in the same manner by taking the reciprocal equation, those two roots being, in it, greater than their reciprocals, and we shall then have l-[-SOx-lSa}^ + x' = l-80/ + 18./' + C399;t'^-2880.i;'^-ct'C..., Here ^/ =-80, ^/ =18, « =6399, r/ =-2880, or, dividing these by 9 we may take ^^=-8|, ^3 = 2, a^ = 711, r/^=-320; thena/i^-al=-(jSU,a^a^-a^/i.= -1422|, r^/.-r^^^ -500101. Hence, reversing the coetHcients, since we have been finding the roots exceeding their reciprocals for the reciprocal equation, we ha^^ 506161a;Vli22Aa?+6324 = 0, Which contains two roots of the equation proposed correct to six decimals. Obs. In the above solution it will be seen that the approximation advances the most rapidly for the greater of the two roots exceeding their reciprocals, which will always be the case whether the roots sought be real or imaginary ; in the latter case, when the equation contains more than one pair of imaginary roots, the approximation w^U be most rapid for that pair the greatest in proportion to their reciprocals. EXAMPLE HI. To determine the quadratic factors of the equation .r^ + 5/-10./ + 5a;-l = 0. ''Yowig's Math. Dis., 2^. 154." Since the positive coefficients have the same aggregate as the negative ones, one root is unity, and dividing by ^ — 1, we get x^+x-\-QJ-4.x+1 = o (12) This equation has two roots greater than their reciprocals, and here o 3 4 we shall have/(.r)=l-f .^'^-0.^ —ix -\-.t , and = l_^^_5/4-15^'^ + 10^*-119^'^ + 124/+015/-1845/- Here « =615, ^=-1845, fA=-1230, «• =14036, 7 o . y iu or, we may put a=u, a^=^3u, a=—^u, (7^^=14630, and substituting these values in the expressions (h) we get R=-lln^ S = (6?^- 14636)?/, T= -(43908 + 4?^)?/, ll?a'^+(14636-6if)a;4-43908H-4M = 0, nearly, or, restoiing the value of it = 615, this becomes 6765ii?Vl0946.r+46368 = 0, or / + l-01803398.r+6-85410199 = 0. 6 The reciprocal equation of (12) has also two roots exceeding their reciprocals, and putting /(a?) = 1 — 4^+0^ -{-x^+x , and proceeding as before, we get ^,,= -1930, «q=-1845, ^^ = 615, a^^ = U261, and substituting these in the equation (11) when m= 10, and dividing by 015, we get the approximate quadratic 46368^-28667^ + 6765 = 0, or x^ - •618033989a;+-145898033 =0. Advancing the process a few more steps we obtain the quadratic 832040/'-514229a;+191393 = 0. or ^^-•6180339887505a;+-145898033748=0. This may be proved by means of the following finite factor of the equation proposed, 2 V5-I , 7-3V5 _ " 2-"+— 2— = '- EXA^IPLE IV. Solve the equation x — Ax^ -\-Sx —lC)x-{-^0 = 0. ''L. dG. Diary for 1846, p. 96." Substituting x=y-\-2, we get / + 4/ + 8/+4 = (13) which has two roots greater and two less than their reciprocals, and to find the lesser roots, we may take the reciprocal equation, since the corresponding roots in it will be greater than their reciprocals, 234 and putting f(y) = A-\-8y +4?/ +«/ , we get, by division 4 - ^2 3 15 4 , . 5 p 6 46 7 113 8 , ^^ 9 9 10 j^=l-^y -y +^^ +% -6^ -~y +—2/ +281/ —^f- &c 23 113 9 Here a^= — —, a^ = -~-^ «9 = 28, a^^= —■-, and substituting these in the equation Vo-% +(Vo-^r%)2/ + (Vio-«o )/ = 0' (1^) we get 95201 -47312i/+202738/ = 0, or dividing by the coefficient of y^, we get ^'^-•23336523?/+-469576 = 0. By proceeding another step, we get 43174R/-1007552/+202738 = 0, or, /-•233366202/ + -46957485=O. Substituting y=x — 2, in the last quadratic, we get a;^-4-23336599^+4-93630543 = 0. Which, I believe, will give two roots of the proposed equation correctly to the eighth decimal. When we make use of the reciprocal equation, to find the roots of the proposed one, less than their reciprocals, as in the last ex- amples, we must then reverse the coefficients in equation (11) and take for the approximate quadratic factors containing two roots of the proposed equation less than their reciprocals, the equation (a^ -_« a ]cc^-}-(a a ^ — a .a ^a)-\-a —a a „ = 0,(15) \ m— 1 m m-2j ' \ m m— 3 iiir-l m— 2/ ' jht— 2 Tra— 1 Wr-S ^y\.-^'^.f Thus, taking m = 10, we have {% - Vio)^'+(Vio- V9)^+«8 - Va=0' the same as (14), when a; is substituted for y. The two roots greater than their reciprocals in the last example may be approximated to with great ease and rapidity by taking /(i/) = l + 4i/-f-8?/ +47/ , from (13), and proceeding as before. EXAMPLE V. To find a (quadratic factor of the equation x^ — 6x-\-G = 0. Putting x=x^ + l, this equation becomes a;^ + Qxl + 15^^ + 20a.-J + Uxl +1 = 0. The coefficient of x is ±0, and, taking the lower sign, it is im- mediately seen that the equation has two roots less than their reciprocals, and putting f(x^) = 1 + 15a.' + 20x1 + Uxl + 6x1 + 4 and dividing unity by this, as before, we find the coefficients of 7 8 9 10 ^ , a^= -12720, ag = 32995, ^^^^ = 231170, «jq=-5C409. Taking m = 10, in (15), and substituting these values, we get 10947338771/ - 919646334^:^+ 693850485 = 0, or x^ - •08400G38a7^+ -06388074 = 0, which I have proved to be exact to the seventh place of decimals at least, and by restoring the value of x, we immediately obtain a quadratic factor of the equation proposed. EXAMPLE VI. To find two roots of tlie equation /-G/ + 40/+00/-a?-l = 0. ''Yoiuif^ Eq., p. 200." This equation has two roots less than their reciprocals, and we have ;: = l-a? + 0l/~81^''^+3r01^^ -G12T/ + '^24594/-44490l/-fti'c. Here a^ = 3701, rtg=-6197, ^^ = 224594, «^= -444901, and substituting those values in (15) for m = 7, we get 47878393689/-268285443a;- 795014655 = 0, or /-•005603476a?--010604879= (10) By extending the above division another tenn, we find the same as the last equation to the extent here given, showing its correctness to the ninth place of decimals ; and if the division had been carried only to the seventh term, we should have got the equation 795014625/-4454753.'?- 13201114 = 0, or /-•00560336a?--*01660487 = 0, which quadratic is the next in degree of con-ectness to (IG). Solving the quadratic (16) we obtain the two roots a7 = -131G93087, ir= -'126088011. EXAMPLE Vir. Solve the equation x' -{■J' —x^ —x^ -\-x^ —x-^l=-0. Decreasing the roots by unity we get the equation x\ + ^x\ + l^x\ + 25a?^ + llx^ + 5ar^ + 1 = (ITj where x^=^x — l. The last equation has two roots less than \, and since its reciprocal equation contains none so small, the equation (17) contains two roots less than their reciprocals, and putting -i- = l-5a;+8/+20a;^-130a;*+198/+602/-.tc... .-. rty = 20, rt^=-130, rtj^ = 198, «^. = 602, or dividing by 2, since they all contain this common factor, we may take ^^3 = 10; a^=^m, a. = 99, ^^. = 301, ^ - «f-,«, = 29366, «3«g- «^^g = 9445, a\ - «3^^=3935, and .-. 29366d?j + 94450? +3235 = 0, nearly. And restoring the value of x by substituting x ^x — \^ we get 29366a;^-49287^+93156 = 0> or a;^-l-67837i»+-78855=0. Carrying the process as far as «g, we obtain the quadratic 96886030?^ + 3116068^^+ 1066690 = 0, and restoring the value of x, this gives 9688603^^-16261138a? -1-76392^5 = 0, or ;?;^-l-6783779a;+-788475 = (18) The two last quadratic factors for x, show the former of them to be correct to about the fifth decimal, and the latter, being obtained at a more advanced stage of the same process, is probably exact to the sixth decimal or more. It will however be otherwise tested hereafter. For two oth-er roots take the reciprocal equation and x negative, t^^^ x'^x'^^x^^i-^^-x^X^^ (19) and putting x—x^-\-\, this becomes x\ + lx\ + 2^ + 35^^ + 33^;^ -f- 15a?^+B =0 (20) This equation has two negative roots less than their reciprocals, and putting /(^a;j) = 3-|-15^^ + 33^^ -t-35a;j +2^ ■\-^x^ +^i , then 3 2 80 3 92 4 28 5 1436 e . ■^^=1-5^^+14^^ - y^i +y^i +y^i — y-^i -^'^- and multiplying these coefficients by \ to clear them of fractions, the coefficients of the four last terms exhibited, will then be «^=-60, a=m, «^, = 21, a,= -359, and a^ -aa=m%l, aa-au^^'^^mi, al -« a,:t=25212, 4o0 oo40 546 .-. 25212a;j +20091^^ + 6021 = 0, nearly "(15)/' Substituting x^=x — l, in this equation, we get 25212a;^-30333;»+11142 = 0, which is an approximate quadratic of equation (19), and therefore, for an approximate quadratic factor of the proposed equation, we 10 shaUhave 11142a;V30333a;-f 25212 = 0, or a;V2-723a;+2-262 = 0. By proceeding a step farther in the last division for finding the values of a^, we get, for the next approximation, 186886a;'^+508385a;+422347 = 0, or a?^+2-7203;r+2-2599 = (21) Extending the last division by f{oc^, we may easily obtain the quadratic factor to a much greater degree of correctness, but all the factors of the equation proposed will be given more exactly hereafter, in illustrating a subsequent method of transformation. 2. In the preceding examples the operation for the division of unity by f{x) has not been exhibited, as the labour would not be very great, for these examples, by the common method of algebraical division, but, it will generally be found most convenient to develop the coefficients a^ successively, in the following manner: — a —b —c —d —e — &c — aa —ah —ac —a^d — ^2 -y -%b -a^c - «3 - V -%b - a, —au — 4 4 % &C (h) in which the first line contains the coefficients, with their signs changed, of the divisor, and the second line is the product of the first by a (= — a); then a is the sum of the quantities above it in the same column, and the third line is formed by multiplying the first by a ; then a is the sum of the quantities above it in the same column, and the fourth line is formed by multiplying the first by a ; a 19 the sum of the quantities above it in the same column, and the fifth line is obtained by multiplying the first by a^; and so on, we may continue determining the coefficients a^, a^, a^, &c... , to any extent. The above process is at once suggested by the equation (c), but, another method of obtaining the coefficients a , when the coefficient 11 of the first term of the equation is not unity, or when some of the others a, b, c, d, &c. contain several places of figures, will be subsequently given in the present work. EXAMPLE VIII. To solve the equation Sx —4:X +10a; -\-x+l = 0. This equation has two roots less and two roots greater than their reciprocals, and to find the two lesser roots, we must take the re- ciprocal equation and the operation, according to the above method of division, will be -1 -10 +4 -3 1 +10 -4 +3 _9 • 9 4 -90 -36 +27 23 -23 - 230 +92 -69 60 -60 - 600 +240 -180 323 +3230 -1292 +&c. ■158 158 +1580 -... 3559 -3559 - ... -3451 which gives ag= —323, a^— —158, «y=3559, a^= —3451, and substituting these values in the equation we get 12121223a;Vl676995a;+1174521 = 0, or a;^ + -13835196a;+-09689789 = (22) By extending the process another step, we get ag = 31802, and substituting as before, we have 125092719aj^+17306825a;+12121223 = 0, or a;^ + -138351978a;+-096897901 = (23) Comparing the equations (22), (23), the former is evidently correct to the seventh place of decimals, and the latter will be found to be so to the last figure. 12 EXAMPLE IX. Take the equation x —12a; +12^^ — 3 = 0. '' Bonnycastles Alg. Biq. Eq. Ex. 7." This equation contains two roots greater than their reciprocals, and applying the process (b) we have + 12 -12 +3 +144 -144 +36 -12 -144 +144 -36 147 +1764 -1764 +441 -288 0_-3456 +3456 -864 rtg = 1944 ___0+23328-23328-&c. ay=--5256 0- 63072 + .... a =27225 8 , «9= -87264 and, dividing the last four values of a hy 9, we may take rt =216, a =-584, Of. =3025, a„=-9696, and, substituting these in equation (11), when m = 9, we get 312344/+327736a;-3488161 = 0, which contains a positive and a negative root of the proposed equation correct to about six places of figures. The degree of approximation is not so rapid in this example as in some of the preceding ones, from the roots not being so much in excess of their reciprocals, but the operation may be extended with little trouble, or we may transform the equation into others better adapted for rapidly approximating to the roots by this method, which can be easily done by applying some of the followiixg paethods of transformation. ON TRANSFORMATION. All the methods of approximation apprehended by other writers on equations have been made, I believe, to depend upon the restricted principle, that the rapidity of approximation is increased in proportion as the root is decreased; but this is only a simple consequence and part of a general principle, which may be thus enunciated : — The rapidity of approocimation is increased in proportion as the quotients of the roots divided hy their reciprocals are made to decrease or increase according as they are less or greater than their reciprocals, 3. To increase the greatest roots of an equation. Find m such a quantity that when , , are m m — \ respectively substituted for a; in the proposed equation, the equations for X and x^, shall each contain the same number of variations of sign from + to — and from — to + amongst their respective sets of coefficients, then the greatest roots will be increased in the equation for x^. For, by these assumptions mx=x^-\-l, {m — l)x =zx -\-l, and, by subtraction, x=x^ — x^ (24) Therefore the greatest roots of the equation for x^, exceed those of the proposed one, by the corresponding greatest roots of the equation for x . 4. To decrease the least roots of an equation. Find m such a quantity that when the above substitutions are made for x, the equation for x^ shall contain p more variations amongst the signs of its coefficients than shall subsist amongst the signs of the coefficients of the equation for x , then the p least roots of the equation for x shall be decreased in the equations for x^ and x^. For, in this case the least roots will be positive quantities in the equation for x,, and negative ones in that for x , and when x is 14 negative, (24) becomes ar=aj^+a?g (25) Hence, the least roots of the proposed equation will then be respectively equal to the sum of their corresponding roots in the equations for x^ and x^, and consequently these roots will be diminished in both these equations. Ohs. The equation for x^ may be easily obtained by multiplying the coefficients of x'^ , aP'~ , x^~ , &c... , hj m, m , rn , &c... , respectively, and then decreasing the roots of the equation, with enlarged coefficients, by unity. Cor. If we take m the least integer that will cause no change in the number of variations of sign amongst the coefficients of the equation for x , from the number of variations amongst the coefficients of the proposed equation, the separation of the least from the greatest roots will be widened in the equations for x^ and x , and these equations will be better adapted for rapid approximation by this method than the proposed one. EXAMPLE X. To find a quadratic factor of the equation x^-{-x^-{-4.x^-Ax-{-l = 0. ^'Buth. Eq. p. 16.'* This equation has two roots greater than their reciprocals, and we find that by taldng m = 3, two variations of sign are lost in the ar, + l equation for x , and none in that for x , and putting x— — ^— — we get x\ +7^j +5^ -23;rj + 13 = (26) And, applying the preceding method of division, we get 1 l + 7a;+51a;^-23a;^+13a;* = l_7^«2a;'*+894/-2830a;*-239^Vl5509l/-1143660/-&c. Here 6i^=-2830, ^^^=-239, ^^=155091, 0^= -1143660. and substituting these values in the equation {a\ -a^a^)x\ +(«^«^-a^a^).r^ + a^ - a^a^^O, -(11)." we get the approximate quadratic factor 438964651a?j +3273624549a;j + 23779883541 = 0, 15 and, substituting a; =3i» — 1, this gives 3950681859/+7187085741^+90945293643 = 0, or a;^+l-81920134a;+5-30167307 = (27) which will give the values of two roots of the equation proposed to eight places of decimals, and, by extending the above process, we may easily obtain these roots to a much greater degree of exactness. If we had substituted x = — - — , we should then have had ^2 +60.' +28a;^ +10^2+3=0, (28) and, proceeding with this equation in the same manner as we have above made use of in finding the quadratic factor for the equation x^, we easily get, at the same step 3125506a.^ +I'^622861a72 + 80778996 = 0, and substituting x^=2x — l, in this equation, we get 12502024^'^+22743698a;+66281641 = 0, or a;^ + l-81920127a;+5-30167283 = (29) Comparing (27) with (29), they are seen to agree to nearly the last figures of the coefficients. In the last example, the other biquadratic factor may be directly found by operating at once upon the reciprocal equation of that proposed, that is, by taking f{x)=z\ — 4tX'\-4:X -\-x -{-x , and proceeding as before, which will give the approximate factors with tolerable rapidity of approximation ; but the degree of approxi- mation wiU be much more rapid if we take the reciprocal equations of (26) or (28), of which the last will involve less trouble than the other in the division part of the operation. EXAMPLE XI. To find a quadratic factor of the equation 4/-6/-7/+8a;^+7a;^-23aj^-22a7-5=0. ''Young's Eq.p.^OV This equation contains two positive roots less than their reciprocals, x, + l and decreasing the negative roots by putting ~x=: — ~ — , we get the 16 equation /+10irj + 3^+2^-l^+168/+26a;j + l = 0...(% and dividing unity hy this function of x, we find the coefficients of X , X , X , X , in the result to be, = 2-2748284178, c = 2-9678714421. Therefore a?^-2-2748284178^+2-9078714421 = 0. It will be seen that the two roots given by this quadratic coincide with the values of the same roots obtained by Dr. Eutherford to seven places of decimals only, and this is owing to the roots he obtains for the equation of the tenth degree, used as an auxiliary in his so- lution, being correct for only seven places of figures, as may be seen by examining and proceeding with the determination of the root alluded to of his equation of the tenth order. EXAMPLE XIV. To solve the equation x -{-x —x —x-}-x — a7 4-l==0. " Youngs Operating in the manner of process (t) we have 1+1+1+1+1+1+1 • 2-2-2 ' -2 1+3+5-1-3-1+1 Therefore the equation for the squares of the roots of the proposed equation is /_3/ + 5/+/-3/+2/+l = 0.. (33). Which contains two roots exceeding their reciprocals, and perform- ing the division of unity by /(?/) = 1 — 3^+5?/ -\-y —3?/ +t/' +y > according to the process (b), we shall have 20 3-5 -1 +3 ~1 -1 9- 15 -3 +9 -3 -3 ~T" 12-20 -9 +12 -4 -4 -4-12+20 +4 -12 +4 +4 -32 -96 + 160 +32 -96 +32 +32 -72 -216 + 360 +72 -216 +72 -44 -132 + 220 +44 -132 ay = 241 723 -1205 -241 919 2757- 4595 1416 4248 -616 Here « =241, « =919, ^^ = 1416, ^ =— 616, and substituting these in eq. (11), when m = 10, we get 10066l/-289952?/+ 514232 = ^..(34) Extending the operation to a , and reducing by 8, we get «g = 177, rtjQ=-77, ^^^=-1126, «tj2=-2850. and substituting the last four values in (11) we get 20523l/-591152i/+1048426 = 0. Or, dividing by the first fcoefficient /-2-88042252/+5-1085167 = (35) If we divide the preceding quadratic by the coefficient of its first term, we get the same equation as the last for five places of figures in the coefficients, and since (35) was obtained by advancing the operation two more steps, it will evidently be correct to a still greater extent, probably to the last figure expressed. Take y negative then (33) becomes . ♦ /+3/+5/-/-3/-2/ + l = (36) This equation contains two roots less than their reciprocals, and decreasing these roots, according to the first method of transformation, by substituting y= — - — , we get the equation yl + Uyl + Q6y\ + 152i/J + 123yJ - 36^/^+ 3 = 0, and denoting this equation by/(yj) = 0, we get 21 ^ =14-121/. + 1032/' + 69311/' + sm^,y[ + 7698|i/' 1 1 •»• A A 1 -87185|2/i -&c. Multiplying these coefficients by 3, we may take ^ =309, « =2080, « =10402, « =28096, 2 o i and substituting these values in the general equation (15) form = 5, we get 601619242/j -14499496y^ + 1112182 = 0. Similarly by proceeding another step we get 9762412988^^ - 2352847376^^ + 180485772 = 0,. or, y] -•2410108i/j+'0184878 = (37) The preceding equation gives, when divided by its first coefficient, y\ -•24100782/^+-0184864 = 0, which will give the same roots as (37) to five places of figures, and I believe (37) to be exact to the last figure ; it can.however be easily tested by extending the above division another term. The value of y in the proposed equation may be obtained by substituting — — - — for y,or y^=—l — ^y in (37) and then w^e get /+1-12050542/+-3148746 = (38). 2 By substituting 2/= a; , and applying the expressions {d) for finding the quadratic factors for x, the equations (35), (38), will give, respectively a;'4.2-7204463aj+2-2602028 = 0. a;'--04205211a;+-56113689 = 0. By another method of transformation we get, for an approximate quadratic for the other two roots, the equation / - l-6783942a;+ -78846745 = 0. The correctness of the three last factors can easily be tested by equating the product of all three with the proposed equation for a>. R . When an algebraical equation contains only one root greater than its reciprocal, this root may be found to any degree of exactness by taking cc=—!^ (40) m — 1 For, if X be less than unity or less than its reciprocal, the first term of eq. (5) may, by taking m sufficiently large, be always made so small in comparison with the other terms that for an approximation it may be omitted, and we may take a X +6 X +c X +&C =0...» (41) Hence, if the equation contains » — 1 roots less than their reciprocals they will be contained approximately in the equation (41), and since the sum of the roots of this equation is, by the theoiy of equations, h = !!^, and the sum of the roots of the proposed equation (1), is <*« m = —a, the other root of the proposed equation, or that exceeding its reciprocal, will be equal to the difference between these two sums, and will therefore be h h —aa «, , , = -a+-=-^^^ '^^^'-±1 "(lO.r a a a ^ ' mm m This theorem, though neat, is not so valuable as the preceding theorem for finding the quadratic factors of an equation, nor is it, generally, so convenient for finding the real or single roots of an equation to a great extent of figures, as Horner's or another method that J shall hereafter give; but it possesses one advantage over Horner's, in not being tentative but always certain in its operation for obtaining fractions successively approximating more nearly in value to a root of the equation, previously transformed so as to contain either one root less or one root greater than its reciprocal. g Ex. 1. Take the equation a; — Co? + 7= 0. " Ruth. Eq. Ex. I." Putting a; =^ + 1, we get the equation / + 8/-32/4-l = ^.(42.) This equation has only one root greater than its reciprocal, and by division which will, in this case, be most easily performed according to the process [h) wc get 33 -— -i—3= 1 - 3y + 1%^- 46/+ 177/- 681/ + 2620/ 1 + Sy-By +y 7 8 9 -100801/ +38781/-149203/+&C 8 9 Taking the coefficients oi y , y , y^e have a =38781, rtg= —149203. a -149203 and ♦'(40)" y=—= = -3-8473221.- ^ ^ -^ ^8 38781 which, hy proceeding another step, may be shewn to be correct to the seventh decimal inclusive. Hence a;=y+l= -^'SA7S221 To obtain more convenient equations for rapid approximation, we may apply the method of transformation given in the preceding part of this work, thus — cc^ + 1 x^^ + 1 Substitute y= , and y= , respectively, in (42), then we get x^ + 9x^ + Sx+S = 0, x^+12x^-Qx-{-10 = 0, either of which will be found to give the value of the root greater than its reciprocal much more rapidly by this method than the equation above used. Ex. 2. Take the equation a;^-17/+54ir-350 = 0. "Ruth. Eq. Ex, II." Putting x=y-\-16, we get from the proposed equation /+28/+2192/+10 = 0. This equation has one negative root less than its reciprocal, and for this root we shall have — 2 ^ = 1 - 21-97/ + 476-81/ - 10380-919/ 10+2192/+28/+/ 4 +226028-9581/-&C. And taking the coefficients of i/ , y , we shall have, inverting the fractional expression (40), since the root is less than its reciprocal a -10380-919 y^ — ^ = --04593139037 ^ «4 226028-9581 .-. iC = 15 + ?/ = 14-95406860963. 24 » Ex. 3. Take x^-lx-l^O. *'Ruth.Eq. Ex. III." Putting a;=?/+3, we get the equation This equation contains one root less than unity, and the' determination of the coefficient a^ according to the process of division (b) will be 20 +9 +1 400 +180 4-20 409 8180 +3681 +409 a3 = 8361 167220 +75849 rt^ = 170921 3418420+ &C..,. a =3494078 &c. a 170921 And here y= —= =-04891733954 «5 3494078 Hence i»=?/+3 = 3-04891733954. The Theorem R is expressed in 'a form best adapted for general application, but it will be most easily committed to memorj^in the following more elegant form, which will also be more convenient for obtaining the quadratic factors with reduced coefficients when several of the coefficients a have a common factor. m Rj 2^ Let the proposed equation be transformed into one containing two roots greater than their reciprocals, and let the transformed equation be denoted by a;"+«^^^+6a;"^2_^ca;**~^+&c...=0 (1) Then, if we put 7-2 3 — : — = l+«ia?+« a? +a a?^+&c...a a?**, the quadratic factor containing these two roots may be found to any degree of exactness from the equation Ex. 1. Take the equation a7*-5a;^+10a;^+5 = 0. By the common method, or the more convenient process (b) of division, we get 25 ■=l + 6x+16x^+^6x^ - 30ic*-425a;^-1900a;^ 2 4 -5375:^; -7725/+17250a; -&c... Hence, taking the coefficients of i» , a? , a? , a; , we have "(m)" (1900a;-5375)(7725a^+17250) = (5375a;-7725f, or, di\dding by 5 , this becomes [76X-216) {S09x+mO) = {216x-Q90f, which, by multiplying out and transposing, becomes 2274l/-118875a! 4-243831-0, or a;^-5-227342a;-f 10-72208 = 0. The correctness of this quadratic factor may be proved by carrying the above division another step further, but the degree of approxi- mation is not very rapid for this example since the reciprocals of the roots considered are not very small ; other equations may however be obtained by transforming the proposed one, as directed in the first method of transformation, for which this method will approximate to the values of the quadratic factors with great rapidity. Ex. 2. T:akex^-{-6x^-10x^ + 6x-l = 0. ''YoungsDis.p.Ui:' Dividing this equation by a; — 1, since one root is unity, we get a;^4.^^ + 6a;^-4a^+l = 0, and dividing unity by this equation, we get as in Ex. Ill to Theorem R^, which is here repeated to show the advantage gained in elimination by using the expression (m), «y=615, a^=-1845, ag=-1230, ^^^=14636. Hence "(m)" (615a;+1845) (-1230a;-14636) = (1845a7-1230f, or, dividing by 615 , this becomes (^+3) (-2ar-23|^) = (3a^-2f, or lla?^+17|^|a?-f75|4-f.=0, which contains two roots of the proposed equation correct to eight places of decimals, as shown in Ex. Ill to Theorem U^. The equation (m) will apply, generally, whether the equation con- tains one or two roots exceeding their reciprocals. For, when the equation contains only one root greater than its reciprocal, we shall have, from (40), as approximations, a .x = a , a x = a ,, a x — a ^, wi— 1 to' m— 2 TO— r m— 3 m— 2' by which values the equation (m) will be fulfilled. E ANOTHER METHOD OF FINDING THE REAL ROOTS OF ANY NUMERICAL EQUATION. Let r be any quantity approximating to a root of the proposed equation for x, and substitute r-\-rx^ for a;; then find r^ any quantity approximating to a root of the equation for x^, and substitute ^i"'"^j^2' ^^^ ^1 ^^ *^^^^ equation, then find r an approximation to a root of the equation for x^, and substitute in this equation X =r -\-r x^, and continue in this manner to find the values, r , r , r^, &c., and substituting irg=rg-f-rg.r^, ^^=^^+^^^5, &c...; then, we shall have ^^ _ ^. _|_ ^.^, __ ^ _|_ ^^ + rr a? ^ ^ ^ ^^^ _|. ^^.^^^ ^ ^^^^^a?g = &c ^r-\-rr^-\-rr^r^-\-rrr^r^J^rr^r^r^r^+kc (t) The equation for x may be found from the equation for x by multiplying the coefficients for a;", x'~ , o;"^^, &c...,by r, r~ ,r~ ' &c..., respectively, and then decreasing the roots of the equation, with its coefficients thus enlarged, by unity; and in the same manner the equation for x , may be found from that for x , by multiplying the first, second, third, &c... , coefficients of that equation by the descending powers of r , and then decreasing the roots of the equation, with its coefficients thus enlarged or diminished, by unity. J[n the follg^ung examples, the common process (Budan's] is em- ployed in decreasing the roots by unity, but in Simple Equations a slight modification is made in the arrangement, which, though at once obvious to those accustomed to the common process, is fully explained in the first example. Ex. Take the equation 8*14159265a:=3, to, express x in the form (i). 27 The operation will stand thus : 3-14159265-3 (l===r 3-14159265 + -14159265(-04 = r^ -•12566371 + 1592894 (-12 = ;-^ -1507964 + 84930 (-05 =r J -75398 + 9532 (-12=9'^ -9047 + 485 (-05: -452 -33(-07 = r^ Of which the first three lines are like common division, and by dividing, mentally, the third line by the second, the fifth by the fourth, the seventh by the sixth, and so on, we find the leading figures of the quotients to be -04, -12, -05, -12, -05, and -07; also the fourth line is got by multiplying the second by -04, the sixth by multiplying the fourth by -12, the eighth by multiplying the sixth by -05, and so on. The values of r being in this case negative with respect to r, we have a; = l--04--04X-12--04X-12X-05-&c , which series is very easily summed, for six terms thus : 1 • - -04 - 048 240 2880 - 14400 - 1008 a;=l--0450703408 = -9549296592. In which if we denote the first, second, third, fourth, &c... , lines by (1), (2), (3), (4), &c , then (2) = (l)X-04, (3)^(2)X-12, (4) = (3)X-05, (5) = (4)X-12, &c... , the ciphers being omitted at the commencement of each line after the second. 28 The above process may be materially shortened by occasionsdly expressing the values of the convergent factors r as reciprocals of integral numbers, that is of the form r = — , each value of u being TO 2< ' m " m then obtained by mental division of the 2m thjline of the process by the following one. Ex. 2. Take 3-14159265358979323846a;-3 = 0. - 3-14159265358789323846-3 (l=r 3-14159265358979323846 14159265358979323846 (•04=rj -12566370614359172954 1592894744620150892 (^ = ^3 •1570796326794896619 22098417825254273 ( -19634954084936207 ■STJ- •04 05 00625 78125 26041666666666 2893518518518 11574074074 1653439153 206679894 8267196 330688 8267 918 22 1 - -04507034144862798540 2463463740318066 (I =r^ -2454369260617026 9094479701040 {■^ = r^ -8181230868723 913248832317 {\=^r^ -909025652080 4223180237 (-004=/^ 3636102608 587077629 (^=rg 519443229 67634400 (i=r^ -64930404 2703996 (-04 = rj^ -2597216 106780 (-04=/^^ -103888 2597 iv= -95492965855137201460 295(^=r^3 -288 29 In the above operation the values 8, 80, 8, 300, &c... , of u, u^, u^^ &c..., or the reciprocals of r^, r^, r^, &c..., are found mentally, by the division of the leading digits in the fourth, sixth, eighth, &c... , lines, by the first digits in the fifth, seventh, ninth, &c... , lines, respectively; these values being taken the next integers, suitable for short division, above the quotients thus mentally ob- tained; and the sixth, eighth, tenth, &c..., lines are obtained by dividing the fourth, sixth, eighth, &c... , lines as they are successively obtained by the quantities 8, 80, 8, &c..., respectively. And the m th Hne in the summation of the series (t), is obtained by multi- plication of the preceding line the same as in the preceding example when r occurs, and by division of the preceding line by w .or m in — 1 when u takes place. 'n^l The last value of x gives the diameter of a circle whose circum- ference is 3, true to twenty places of figures, and wiU serve to show the average amount of labour for division by this method when the divisor shall contain that number of figures, as I have not met with another instance in which the approximation was not more rapid. The rapidity with which the left hand digits vanish at each step of subtraction by this method of division, will more than compensate for the additional labour involved in the summation of the series (e), and each convergent factor is obtained with just the same ease as a digit of the quotient by the common method of division ; but a greater advantage in the application of this method to simple equations exists in the use which may be made of it in forming rules for ready computations, as in the following instance. To find the diameter of a circle of a given circumference, EuLE. — Divide the given circumference by -4, the result by 8, the last result by 60, then multiply the last result by '11, and divide the result thus produced by 20. The sum of the last four results will give the diameter correctly to eight places of figures. Ex. 3. Find the diameter of a circle whose circumference is 876495820. 30 The operation will be 4 )876495 820 8)219123955 60) 27390494-3 X-11 456508-2 20) 50215-9 2510-r 27899729-1 Ans. Here the first line is divided by 4, the second by 8, the third by 60, the fourth is multiplied by -11, and the fifth is divided by 20. The answer in the last line is the sum of the last four lines above it. This rule was immediately found by the above method applied to the equation 3-14159265a;=-4, and there seems no doubt that others stiU better may be found by this method with a little more trouble. An equally convenient rule may be given for finding the circum- ference from the diameter, and an exceedingly convenient one for finding common from hyperbolic logarithms will be given with a method for finding these logarithms without the aid of any tables, which I hope to publish, with some other researches, next after another small work on equations intended to follow the present one, as an appendix to it. Ex. 4. To find a root of the equation 87315.»^-174544a;V354632aj-698336 = 0. ^. , 2 354632 698336 Dividing by x\ 87315^-174544+ ir- = ^' ^ X from which we immediately see that a; =2 nearly, and taking r=2, the operation for finding the equation for x^ will be 349260-349088 + 354632-349168 172 + 354804+ 5636 849432 + 704286 698693 the first line containing the products of the coefficients of x^, x, x, x, -o 1 by r , r, r , r , respectively, or the products of the coefficients of 31 the proposed equation by 4, 2, 1, I, respectively; and each of the other hnes is obtained, from that preceding it, by simple addition, each coefficient in any of them being equal to that preceding it in the same line added to the one placed immediately above it, the first coefficient being understood in every line. Hence, 349260ar^ +69869^ +704236a.'^+ 5636 = 0, the coefficients of the equation for x^ being the last in each column of the preceding operation. Since the last coefficient but one of this equation is the greatest, we may obtain an approximate value of x^ by omitting the two first terms, and then we get, by division, 0?^= —-008 nearly, and taking r =— -008, and proceeding as before, we shall have 22-35264-5589-536 +704236 -704500 -5567-18336 + 698668-81664- 5831-18336 -5544-83072 + 693123-98592 -5522-47808 the first line of which is obtained by multiplying the coefficients of the preceding equation for x^, by (--008), —-008, 1, _ respectively, the same as in the preceding operation, and the co- efficients in the succeeding lines are obtained by addition as before. The bottom coefficients being those of the equation for x , we have 22-352640^^ -5522-47808a;^ +693123-98592a;2- 5831-18336 = 0. Dividing the last coefficient of this equation by the last but one, we find that i»2 = -008, nearly, and putting r^='OOS, and proceeding as before, we have, carrying the operations to five places of decimals only •00143-44-17982 + 693123-98592-728897-92000 -44-17839 + 698079-80753- 35818-11247 -44-17696 + 693035-63057 -44-17553 the first line being obtained as before, by multiplying the coefficients of the equation for x^, by (.008)^, -008, 1, tttttt^ respectively, and the rest by addition as before. The bottom coefficients give the equation for x.^, that is, 32 'OOUSxl -44-17553a;^ +693035-63057a; -35818-11247 = 0. Dividing the last coefficient by the last hut one, we get x^ = -0^ nearly, and taking r^ = -06, and proceeding as before, the first term will vanish as its first five decimal digits will he ciphers, and the operation is -2-208 + 693035-630 -716362-249 693033-422- 23328-827 .693031-214 the first line of which is obtained by multiplying the coefficients of * 1 w^ , X , X , by -05, 1, and -;Trr— respectively, and the others -are obtained by addition as before. The last results give the equation for x^, -2-208a^^ +693031-214^^-23328-827 = 0. Proceeding as before we have a?^ = -03 nearly, and, if we multiply the coefficients of the last equation by the quantities -03, 1, , res- pectively, the last result will only be correct to one place of decimals, and the first becomes -06624, which may therefore be omitted in the succeeding part of the operation, and we may then proceed as in Ex. 1, thus 693031-214-23328-827 ( .03 20790937 2537890 (-12 2494912 42978 (-01 24949 18029 (-7 17464 565 (-03 624 41 (-08 Collecting the preceding processes, the entire work may be arranged as follows: 33 87315 -174544 +354639 -698336(2=^ 349260 -349088 + 354632 -349168 + 172 + 354804 + . 5636 (-•008 = r +349432 + 704236 -704500 +698692 . + 698668'81664 — 5831-18336(-008 = r2 22-35264-5589-536 + 693123-98^92 - -728897-92 -5567-18336 + 693079-80753 -35818-11247 (-05 -5544-83072 + 693035-63057 -716362-249 -5522-47808 + 693033-422 -23328-827(-03 •00143-44-17982 + 693031-214 20790936 -44-17839 2537891(-12 -44-17696 2494912 -44-17553 42979(-01 0-2-208 24949 18039(-7 17464 566(-03 524 42('08 And summing the series (je) with the last values of r , we shaU have r=2 rr^=--016 rr^r^= - 0128 ^V2^3=- 64 &c. =- 192 2304 2304 16128 484 39 a;=2--01613461543691 = i*9^386538456308i In which the first line contains r^ the second its product by r l the third the preceding line multiplied by n ; the fourth the third mul- tiplied by r^i and so on to the tenth, which is produced by multiplying the ninth by ?- =-06. The last line is the sum of all the others tttid gives the numerical value of Xi 84 Ex. 5. Take the equation x^ — ^x +Ax —8x —1 = 0. By proceeding as in the last solution, we immediately see that a; =2 nearly, and the operation will stand l_2-j. ,4_ 8 - - -1 (2=r 8-8+ 8-8-0 - - •125 + 8-0-0 -0 - •125 (--008 = ^ 8-f 16+ 16+16 + 16 244140-625 16 + 32+ 48 +64 250000 -2037-329(-008 ==^2 24+ 56 + 104 - 8000 242103-296 -2037-329 32 + 88 104 - 7896-704+ 234309-184 -163-348(-08 = =^ 40--704 103-296- 7794-112 + 1874-473 149-879 102-592- 101-888- 7692-224 1873-981 13-469(-08 = 11-990 =^ -•492 + 1873-489 + 101-184 . 149-879 1479(12 = = ^5 0+0 1439 ■ ' 40(-02 = 29 = ^6 ll(-4=r^ In this example we first multiply the coefficients of the given equation by 2^ 2^ 2, 2^ 2~\ 2~^ 2~^ or 8, 4, 2, 1, ^ h i, respectively, and then decreasing by unity as before, we obtain the coefficients 8, 40, 88, 104, 64, 16, — -125, of the equation for x^; from the two last of which we get by division r = — -008. The equation for x^ is then found in the same manner, by multiplying the last set of coefficients by the descending powers of r^, and applying the common process for decreasing the roots of an equation by unity, neglecting in the mul- tiplication the two first coefficients, since* the operation is here only 3 2. carried to three places of decimals, and the values of 8r , 40r , will 1 be less than TTs : the coefficients of the equation for x , tiius obtained, carried onlyto three places of decimals being —7692-224,234309-184, — 2037-329, we get from the two last, as before by division, r = -008. Proceeding in the same manner with r , we find that two other co- efficients will vanish for three places of decimals, and we get for x a quadratic whose coefficients are -^492, 1873^489, — 163^848, and 35 dividing the last by the preceding one, we get rg = -08. Proceeding in the same manner with /„ we get the simple equation 149-879aj -13-469 = 0, 4 which gives ?'^=-08, and the rest of the operation is performed in the same manner as Ex. 1. The summation of (t) with these values of r will stand thus _2 -•016 0128 1024 8192 98304 1966 786 aT=2 -•016139160256=1-93860839743. This root agrees with that found in the preceding example for six places of figures, and the roots of the cubic equation forming the preceding example will be found to agree to about the same extent with three of the roots of the last equation of the sixth order. 10. In the preceding solutions the convergent factors r have been taken any values approximating to a root of the equation for sc , but if this equation be represented by A/''+A^/''+A^^x'-''+&c A^«+A=0 (g) and we take u any value that will nearly fulfil the condition A+—+-Y+ ^+&c.... = 0, (45), u u u mm m that is take u any value approximating to a root of the reciprocal equation of (g), the root may, by operating in every other respect as before, be developed in the form r r r r '""''"^ ^"^ v^"^ vv*^"^ vw^"^ < When A is greater than the other coefficients, which will generally be the case when the roots are decreased below unity, it will be sufficient to take the two first terras of (45) only, that is 30 \ \ A+ — =0, or « =— — , nearly, (/»). u "' A This process will immediately be seen to be identical with my method of decreasing the roots of an equation, given in the preceding general method of finding both the real and unreal roots; and the equation for x ^ may be formed from that for x , by multiplying 2 3 its first, second, third, &c..., coefficients by 1, r , r , r , &c..., respectively, and then decreasing the roots of the equation with enlarged coefficients by unity, as before, or we may divide them by the descending powers of r , commencing with A , or the ascending powers commencing with A , as in the expression (45), and then apply the process for decreasing the roots of an equation by unity. In assuming the values of r , or t* , it is not necessary that they shall always be positive quantities, and in some cases the integral fulfilling most nearly the condition (45) or (k), will exceed the root of the equation for x , and then the value of the next convergent factor u must be taken a negative quantity. This assumption of both positive and negative values in (f) will add very little to the trouble of its summation, while it will materially shorten the process for obtaining a sufficient number of convergent factors, for obtaining the root correctly to a great extent. 11, Substitute rii for u in (f) then we get Hence if we decrease the root by r and then operate as in the preceding examples, we may express it in the form (g). Ex. Take the equation x —^x=6. One root is nearly 2, and substituting ajj 4-2, for a:, this equation becomes ftnd proceeding as in the p-eceding ex9,mpl€, the operation will be 37 1 +6 + 10 -1 (11 1 +66 + 1210 -1331 67 1277 - 54 (25 68 1345 -843750 -1399 69 840625 (600 1725 842351 -302184000000 1726 844078 1685116^ 16576605 301 (-200 1727 303868080000 8-905075(10 1728 303869116801 1383098 7-546791(11 1036800 + 303870153603 18708-358107(800 1036801 -1519350768-015000 1443-009124(12 ' 1036802 -1519350742^094925 -1438-779082 + 1036803 -1519350716-174850 ^ 1-230042(400 ^•000000+25-92007 5-151935071-617485 3596947 •259201 -151935071^358284 633095(6 -151935071^099083 ^13812279-190826 599491 •002142 33604( 20 -13812279-188684 29974 -13812279^186542 3630(9 •000000 - 17265^348983 3330 And summing (g) for these values we have r=2 •090909090909090909090909 300( 12 278 22(13 3636363636363636363636 6060606060606060606 -30303030303030303 -3030303030303030 -275482093663911 -344352617079 -28696051423 -71740128 -119*56688 -597834 -66426 -5535 •^425 a;=2'094551481542326591482369 38 This value of the root agrees to the twenty-third decimal with that given by Professor de Morgan, ''Math. p. 390, Vol. Ill," and it may be easily obtained to a much greater extent, by proceeding a few more steps in the equation u , before commencing with the abbre- viation ; which, in the above solution, commences with u . by carrying the division of the preceding set of coefficients, l-j- 1036803 + 303870153603 + 1685116801, of the equation for x^ by (-200)3, ( — 200)2, —200, 1, respectively, to only six places of decimals. The subsequent factors may be obtained much larger, and therefore the approximation be made more rapidly, by taking u =11, but the above, in which it will be seen that many figures are imnecessarily repeated, is sufficiently rapid in approximation. 12. The last method of expressing the real roots of an equation may be very conveniently employed in determining, by elimination, whether any two equations of the same degree contain a common factor. My method is first to eliminate the first terms and then the last from the proposed equations, thus obtaining two equations one degree lower than those proposed ; then, by treating the resulting equations in the same manner, to obtain two other equations another degree lower; then, by proceeding in the same manner with the two last equations and continuing this process of elimination until we arrive at two equations of which the coefficients of any two terms in one equation, shall have the same ratio, to each other as the co- efficients of the corresponding terms in the other, the two equations consequently containing the same roots. Ex. To determine the common factor, if any, of the two following equations : 17821a;*-21719/+4349a?^+1852a;-1982=0, 3898aj*+156040/-37494a;'^+1982a;+17821 = 0. Taking 17821t/=3898, and proceeding as in the preceding methods we find that w = — + + + +&C... , ^ 5 5.11 ^5.11.40 5.11.40.5^5.11.40.5.30^ and dividing all the coefficients in the first of the two above equations omitting the leading one, by 5, all the results by 11, the last set of a9 results by 110, then the last set by 5, and so on to any extent, and then adding the results in each column, the process will stand 5) -81719 +4349 +1852 -1982 11) -4343-8 +869-8 + 370-4 -396-4 40) 394-8909 79-07272 33-67272 36-03636 5) 9-8722 1-97681 •84181 •90090 30) 1-9744 -39536 •16836 • -18018 8) 658 1317 561 606 12) 82 164 70 75 6 13 5 6 -4750.6121 +951-25983 +405-08925 -433-52433 and since the first coefficients of each equation have the same signs, we must subtract from these the coefficients of all the terms but the first in the second equation, that is the coefficients 156040 -37494 +1982 +1782 and we get the coefficients of the cubic -160790-6121a;^+38445-25983a?^-1576-91075aj-18254-52433 = 0, (46) resulting from th^ elimination of the first terms of the proposed equations. In the same manner, by eliminating the two last terms of the same equations, we obtain another cubic equation, and it will be found that the coefficients of the two cubics have the same ratios to one another in each equation, for nine places of figures. Hence the three roots of the cubic (46), wiU also be the roots of the two proposed equations for nine places of figures correctly, or it is a common factor of both equations to that extent. The two equations forming the last example, respectively contain the same three roots of the equation to about ten or more places of figures correctly, and the factor con- taining the two other roots may be found by the same method of elimination, repeated on each pair of resulting equations successively diminishing in degree, from the equations 40 Q366a?*-3505a;^+22966a;^-15C46a;+7298 = 0, 7298/-9664a;^+69817a;^-37562a;+15646 = 0, which will be found to contain two of the imaginary roots of the preceding equation of the fifth order to a greater degree of exactness than the three nearly common roots of the two equations in the preceding example, afe found to coincide. 13. The same method of elimination may also be applied in de- termining? the values of the coefficients a , b , &c... , in my general method for finding the quadratic factors of an equation in the following manner. Ex. Take the equation ru^V^a?^- 5 = 0. ''Math.Vollll.p.UQer Putting 00 4-1, for x this becomes x^-i-lOxl +4.^0x1 -fl^OoTj +2100;' +25^ +210^;* +12^ -\-61xl +16^^^-2 = (47) And putting x =y, vie get by the process (t) /^ + l0/ + 45/+120/+182/ + 318/+12/+2368/-2143/ + 4007/ + 4=:0. . We find, by my first method of transformation, that two of these w +1 roots may be decreased by putting y=~ , and then we get o y^i+^of^ +720i/' +76802/J +51492./' +260478/^ + 701988^/^ + 6420108t/^ +26030881/^ -29938041/^+513409=0, and decreasing the roots of this equation in the same manner by y +1 ' . putting ?/j = _i , which I perform, in this case, by dividing the 4 Coefficients of the last equation by 4 , 4 , 4®, &c... , in their order respectively, and then decreasing the roots by unity as in the previous examples, first omitting the four leading terms, since they become less than unity by this division, and the determination of the quadratic factor containing the two roots to about six or seven places of figures^ will be sufficient to show the method of elimination ^ we then get the equation ,41 1^1 +3322/^' +44412/J +115018i/' +4842962/' -1088712/^+ 31208 = (48) And the operation for finding the two roots less than their reciprocals will be -108871+484226+ 115018+ 4441+&C... -326613 + 1452678+ 345054+ — 54435+ 242113+ 57509 + 1088- 4842- 1150 + 155- 691- 164 + 3- 13- 3 104424 + 1804263+ 405687 326613-1452678 36290- 161408 1209- .5382 172- 768 10- 48 2168557-1214597 &c... Here, we find, by the preceding method of obtaining the real roots . ,. ., . 108871 ^33 3 3 01 an equation, that =3+-^ 31208 ' 6 6.50 6.50.7 6.50.7.50 — &c. . . , and, arranging the coefficients of the equation for y , omitting the last, in their reverse order, the second line consists of the co- efficients in the first multiplied by 3 ; the third line of those in the second divided by 6; the fourth of the third divided by — 50 ; the fifth of the fourth divided by 7; the sixth of the fifth divided by 50; and the next line contains the sums of the coefficients of the preceding lines in each column. Then we find that oiono =3+7^+- 31208 ' 9 ' 9.30 3 3 "^ 9 SO 7 "^ 9 SO 7 ir +&C..., and the eighth line in the above operation is obtained by multiplying the first by — 3, the ninth by dividing the eighth by 9, the tenth by dividing the ninth by 30, the the eleventh by dividing the tenth by 7, the next by dividing the preceding one by 16, and the next by adding the columns in the last six lines. In this manner we might continue the process by taking 42 ^ 2168557 ^ ,. 1 ^ , . . , , the yalue of , and proceeding as before, but it is already sufficiently advanced, for obtaining the two roots in consideration, to the extent of about six or seven places of decimals. The first lines of coefficients may be taken to represent the values of a , fc^, c , d , &c..., in eq. (5) at the commencement of this work, for the equation (48), the seventh the values a^, h , c , &c..., the thirteenth the values a , 6^, c , &c... , and we may therefore take a^=_-108871, 6^ = -484226, a =-104424, 6 =1-804263, a =2-168557, b =-1-214597. 2 '2 '3 '3 And substituting these values in the equation 2/2 ^2 we get 4-039480?/^ -•91783742/2+-2469967 = (49) And restoring y , from y^ = Ay — 1, we get 4-039480i/j -2-249199i/^ + -325269 = 0, And substituting y^ = Sy — l, we get 4-039482/^-3-442719i/+-7a4883 = 0. or /--8522679i/+-1819255 = 0. And substituting y = x^, applying eqs. (d) we get x\ --0280314^j + -4265272 = 0. Restoring the value of x from x^=x—l, this gives ^^-2-0280314a;+l-4545586 = (50). Which agrees very nearly with one of the approximating factors given by the late Mr. Weddle, by means of his ingenious method of approximating, trionometrically, to the imaginary roots of a trinomial equation. ''See Math. Vol. III. p. 288." Several of the other factors may be obtained by means of my first method of transformation and the above method of elimination, with' out taking the equation for the squares of the roots of any of the transformed equations, and some of them with more rapidity than (50) was obtained; but the approximation to the roots considered in eq. (48), is very rapid, and they may be found to ten or more places 43 of decimals with but little more trouble, by the above method, and retaining the whole of its terms as derived from the substitution of 2/2 = — J— , in the equation for y^ One of the real roots may be easily obtained at once from eq. (47), by means of eq. (40), but I intend to reconsider this, and some of the other examples in this work, in a small work that I am preparing to publish, to be considered, though separate, in the light of an Appendix to the present one. The determination of the above factor by any other algebraical method yet published, would have involved the finding of one root of an equation of the forty-Jlfth order. ON THE SOLUTION OE EQUATIONS GENERAL FORMULA 1. The following Theorem is connected with the general principle of division, partially considered in the introductory investigations at the commencement of my general method for finding the approximate quadratic factors of any equation, upon which that method depends ; and it is not entirely unconnected with the ''General Theory of Reciprocity for points referred to co-ordinate axes,'' commenced by me in the Lady's and Gentleman s Diary for 1849. It admits of different enunciations, but the following one seems to be sufficiently explicit, ^ l-{-ax-\-bx -\-cx -\-dx -\-ex -{-&c Put 2 3 4 b l-\-a^x-]-a^x -\-a^x -\-a^x +a^x -\-&c = l + [b^ + a)x-]-(^b^+b^a-^b)x'+(b^ + ba^-^b^h + c)x' + &c (^), then, this equation will be fulfilled by taking for every value of m, a =—b —b ,a^—b ^a^ — b a — &c (A,) m m m— 1 1 m— 2 2 m— 3 3 VI' or b =—a —a ,b.—a ^b^ — a „b^ — &c TO TO TO— 1 1 TO— 2 2 TOr-3 3 the latter of which will always subsist with the former, that is the quantities a , 6^, are interchangeable in {h) for any value of m. 45 Taking m=:l, =2, =3, ==:4, =&c , s.uccessively, and subr stituting, we readily get the values or b^=z-a^, ^2=-^2+ai» ^3=-S+^V2~4' ^^ W In biquadratic equations, we take "(^)" = <'(hyi^l-b^x+(hl -by][l + [h^ + a)x+{b^ + b^a + b)x'] Multiplying out these factors and equating the coefficients of the same powers of x on both sides of the last equation, we get h\a+^b^)=.bl -.bb^-c...(k'), (PI -b^){b^ + ab^-^b) = d...{\\ If we eliminate 6^, by substituting in (k ), its value from (k ), we shall obtain an equation whose roots are double those of the equation (7), in page 15, of Dr. Rutherford's small tract on the complete so- lution of numerical equations, but, I think, the equation for a affords a more convenient equation for the general solution of biquadratics and it may be easily found from the last two equations by substituting b —a, for b^, and then eliminating by The resulting equation, putting y for a will be / - bif+(ac - d)y^+(2bd - ad - c^)y^+(ac - d)dy^ - bd^y+d^ = 0. . (5 1) which is the equation for the products of each pair of the roots of the equation 00 -\-ax -\-bx^-{-cx-\-d = 0, and can easily be resolved into three factors of the form {y'+Piy+^{y'+p<^'\-d)(y^+p^y+d)=o (52), Of which the coefficients p^, p^, p^, will be the roots of the equation p^-{-bp^+{ac-M)p + ad+e^-Ud = 0......{l). In the same manner we may proceed with the general equation of the sixth order. Resolving it by means of (Ij) into two cubic factors involving only three unknown quantities b ,b^,b., or a , a , a , 1 <4 3 12 3 for the determination of which equations maybe formed by multiplying the two factors, involving these quantities, together, and equating, as above, the coefficients of the same powers of x. One of the auxiliary 46 equations resulting from the elimination of two of these quantities from the equations involving all three will admit of reduction, but the low price of the present work will hardly permit its greater extension, and I also feel anxious for its completion, as it has already been unwillingly delayed beyond the time advertised for its appearance. 2. By another method, too extensive to be discussed in this ^tnall work after the investigations it now contains, I have succeeded in obtaining greater improvements in the solution of equations by general formulae than by means of equation (1^), froin which I take the following as the first instance of the general equation of the fifth order being solved, algebraically, without the help of an auxiliary equation of a higher order. To solve the general equation of the fifth order 5 4 . 7 3 2,7 rv ♦ oc —ax -\-ox —ex -\-dx — e = K). Every equation of the fifth order contains one real root at least, that may be determined without the help of any auxiliary equation, which denote by r, and put e — dr « 2 (r — ay r ^-i, /. = .<- 4. (,) S = 2 Then if we solve the cubic equation 3 /16j9 4g+sA 2 (Qp \ ps. ^' +(-H +-^J^ + (x ^t^-i^^' ^''^ hz and moreover put k = 5' — g^ , ^^ , we shall have a;=— - ;5 ± vL^-N j (54) Ex. Solve the equation x^-^Sx^ — Sx + 5a; — 1=0. This equation contains four imaginary roots and one real one =1, and, here, wc have 47 a = 0, fc = 3, c = 8, d=6, e = l, and r=l, and substituting these values in equations (r), we get and substituting these values in the cubic (53), we get 47/-52/-68;^-15=0. The positive root of this equation is veiy easily found by the preceding method for finding the real root of an equation, to be ;£;=: 1-93793493 and substituting this and the preceding values in the above expressions for k, M, N, we get k=2-74514601, M = 3-37586986+ 'v/1-740292202, which gives for M the two values M = 4-69507121, or M = 2-05666851, according as we take the surd quantity, positive or negative, also N=/+-i-^s^+k = 3-75559179— ^^+k = 5-53177043... Substituting this value of N, and the first of the above values of M in (54) we get * a?=-409600675± '/-•02084701. And if we take the last value of M and the rest as before, then, the same equation (54) gives a?= -•909600675± >/- 4-47429909. Ex. 2. Take the equation x^-6x^-\-x^-\-lQx^-20x+16 = 0. " Young's Eqs. p. 327." We easily find 2 to be an integral root of this equation, and to find the others by the above method, we shall have a = 5, 6 = 1, c=-16, ^=-20, 6= -16, r=2. And substituting these values in the equations (r), we get gr=-2, p=-12, s^ = S, s=-l, /i = 13, and substituting these values in the cubic (53), it becomes 39/-309/-326;5;- 108 = 0, one root of which we find to be 9, and substituting z=9, and the preceding values in the above expressions for M and N, we easily get M = 19 or M=:20, and N = 91. 48 Hence, taking the first value of M, we get "(54)" x=l±: V — ^, and taking M = 20, then (54) gives ic=l±3 = 4or —2. The investigation of this method of solving equations of the fifth degree will be given, with many others, in a small work intended to appear, in the course of the present year, as an appendix to this. By applying the above method to the solution of the quintic, solved in pages SI, 22, of Dr. Rutherford's tract, it will be seen that the advantage gained by this over the method employed in that work, consists in the same being here effected by means of the cubic (53), as is effected by means of an equation of the tenth order in the tract referred to ; one root being previously determined in both methods. In the appendix, to appear hereafter, will also be added several other methods of increasing the rapidity of approximation in applying my general theorems, including an elegant mode of finding the coefficients of the different powers of x in the result of unity divided by f{x-{-p), from the coefficients [a ), resulting from the division of unity by f{x), as in the first part of this work. The theorem R^, in this work, will also be generalized in the appendix, for any number of roots exceeding their reciprocals, and it will also contain some important applications of the general theorem to the summation of series, rationalising of surd equations, &c. NOTE TO THEOREM R. 2 As cases will frequently occur of equations containing one root nearly equal to unity, it may be remarked, before concluding, that, in place of -— — , we must then take . . /(^) ■j— = l+a^x-\-a^x -\ra^x -\-a^x +&c. . as i their reciprocals. and proceed as before for finding the values of the roots exceeding Elizabeth Slater, Printer, Market-street, Hyde. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 3l0ct'50f(j, ,0l»n'58ESi REC'D LD MAY 2 7 ^g-^ ) 21-100m-ll,'49(B71468l6)476 I