the series (x) after each substitution indicates the number of roots comprehended between the numbers substituted. A few trials may suffice to effect the complete separation of the roots; that is, to assign to each root two limits between which it alone is comprehended. But when two or more of the roots are very nearly equal, it becomes necessary to multiply the number of substitutions in order to separate the roots; and as the equation has no equal roots, it is certain that the separation can always be effected. The tediousness of the calculations is compensated by the advantage of determining these roots with a close degree of approximation. The rate of increase of the numbers 0, a, ß, y, . . . . is arbitrary. But it is convenient to commence with the numbers 0, 1, 10, 100, &c., for thus to the advantage of having only calculations easily performed is joined that of determining how many roots there are between 0 and 1, 1 and 10, 10 and 100, &c. If there are roots between 1 and 10, the integer part of each is determined by substituting the numbers 1, 2, 3, 9; if there are roots between 10 and 100, the figure of the tens of each is obtained by the substitution of the numbers 10, 20, 30, . . . . . 90, &c. In like manner, for roots less than 1, the highest decimal figure is obtained by substituting the numbers from 1 to 9, or from '01 to 09, &c. By proceeding in this manner the highest figure contained in each root is obtained. To determine the next figure, suppose, for the sake of perspicuity, that it is required to find certain roots between 100 and 1000, and that it has been found that one of these roots is between 300 and 400; it is evident that the substitution of the numbers 300, 310, 320, . . . . in the series (r) will lead to the discovery of the figure of the tens. But, to facilitate the calculation, let r be changed into 300+; the result is a new series (x) in which it is only necessary to substitute the numbers 0, 10, 20, .... Admitting, next, that after the determination of the tens it has been found that there are roots between 350 and 360, the substitution of the numbers 350, 351, 352, . in the series (r) will lead to the discovery of the figure of the units. But in this instance also the calculation may be simplified. By the change of x into 300+ the series (r) has been already changed into another series (x). Now this may be in like manner transformed into a series ('') if r' is replaced by 50+x'; and it is evident, that if in this last series the numbers 0, 1, 2, 3, are substituted, the same results must be obtained as by the substitution of the numbers 50, 51, 52, . . . . in the series (x), or the numbers 350, 351, 352 . . . . in the series (x). These calculations may be continued so as to discover the tenths, the hundredths, &c.; and thus the roots are not only separated, but also calculated with any required accuracy of approximation. 286. As an application of Sturm's Theorem, let it be required to determine the roots of the equation x+4x3—4x2-11x+4=0 ? 1st. To determine the functions X X X X Since X=x+4x3—4x2—11x+4, X‚=4x3+12x2—8x−11. Dividing X by X, as follows, x+4x3-4x2-11x+4 4 4x3+12x2—8x—11)4x2+16x3—16x2—44x+16(x+1 Changing the signs of this remainder, X,,=20x2+25x-27. Changing the signs of this remainder, X,,,=227x+31: Changing the sign of this remainder, X=+1547988: X=x++4x3-4x2—11x+4, X=4x+12x2-8x-11, X=20x2+25x—27, X=227x+31, 2d. To determine how many real roots are contained in the proposed equation, let, first, and, second,+ ∞, be substituted for z in the first term of each function of the series (x), viz. in x1, 4x3, 20x2, &c.; then, since for x= ∞ the series of signs is +, −, +, −, +, four variations, and for the series of signs is +, +, +, +, +, no variation, the four roots of the proposed equation are real. To ascertain how many of the four roots are positive, and how many negative, make x=0. (By this substitution each of the functions of the series (x) is reduced to its last term.) Now for x=0 the series of signs is +, therefore 2 roots are comprehended between ∞ and; that is, 2 are negative and 2 positive. +, +, two variations; and 0, and 2 between 0 Before attempting the complete separation of the roots, it is convenient to determine closer limits than ∞ and +∞. Making x=-10, the series of signs is +, −, +, −, +, four variations. The number of variations for x=-10 being the same as for x=-— ∞, it follows that -10 is a limit of the negative roots. Next, making x=+10, the series of signs is +, +, +,+, +, no variation; whence +10 is a superior limit of the positive roots. The separation of the positive roots is effected by substituting 1, 2, 3 ... successively for x in the series (x). For x=+1 the series of signs is -, x=+2, +, +, +, +, +, no variation. +, +, +, one variation; and for One variation being lost between 0 and +1, and a second between +1 and +2, one of the positive roots is comprehended between 0 and 1, and the other between 1 and 2. The separation of the negative roots is, in like manner, made by successively substituting -1, -2, -3, &c. in the series (x), until the numbers between which the two negative roots are comprehended have been determined. − − +, −, +, three variations. +, −, +, −, +, four variations. One variation being lost between-2 and -1, and another between -5 and -4, one of the negative roots is comprehended between 1 and -2, and the other between 4 and 5. The complete separation of the roots is consequently accomplished. When different numbers are successively substituted for x, the signs of the functions X, X, X,,, &c. are found as in the following instances for x=+2 and for x=-2: Xa positive number +1547, &c. ; Whence (as above) for x=2 the signs are +, +, t, t, t, For x=-2, Xa positive number +1547, &c., and the series of signs for x=-2 is −, +, +, − +. Let it, next, be required to find an approximate value of the root which is comprehended between 1 and 2. Since 1 is the integer part of the root, it is necessary to transform the equation X=0 into another equation, whose roots shall be less by 1 than the roots of the equation X=0. Making x=x+1, and employing the method of Article 236, the transformation is made as follows: 1, +4, −4, −11, +4 5, +1, -10, -6 6, +7, -3 7, +14 8 and the transformed equation in x' is X'=0, or ́x2+8x'3 +14x^2-3x′-6=0. The tenths of the root are to be determined from this equation; to obtain the limits between which the figure expressing these tenths is comprehended, it is necessary to form the series (r). The functions X, X, &c. may be formed from X', as X, X,, . . . . from X; or x, in each of the functions X,, X,,.. ...., may be replaced by x'+1, and the function transformed into another in x', in the same manner as the function X has been transformed into X'. If in these functions 1 is substituted for r, the signs are one variation. And if 1 is substituted for x, the signs are +, +, +, +, +, no variation. Therefore one root of the equation in x' is comprehended between ·1 and 1. Making x= =2, the signs of the series (x ́) are —, +, +, +, +, one variation. 0.3 0.4 0.5 0.6 -,+,+,+,+, one variation. −,+,+,+,+, one variation. −,+,+,+,+, one variation. −, +,+,+,+, one variation. +, +, +, +, +, no variation. Consequently a root of the equation in r' is comprehended between 0-6 and 0.7, and therefore a root of the equation in ≈ between 1·6 and 1·7. 0.7 To obtain the figure expressing the hundredths of the root, it is necessary to diminish the roots of the equation in r' by 6; making x'='+6, and transforming the equation in z' into an equation in x', as follows; 1, +8, +14, -3, 9.8, +30.56 10.4 -6 the transformed equation in x' is, x"4+10'4x′′3+30′56x^2+23·304x”— 0.9024=0; And the functions of the series (x') are, X"=x+10·4x3+30·56x'"+23·304x′′-0·9024, X"=20x^2+89x”+64·2, X"=227x"+394.2, Xa positive number. By substituting 01 and 1 successively for " in the series ('), it is found that one root of the equation X"=0 is comprehended between these limits; and by similar substitutions of 02, 03, 04, it is ascertained that this root falls between 03 and 04; therefore the required root of the equation in x falls between 1.63 and 1.64. Making "="+03, effecting the transformation, and forming the series (x), it is found that the figure which expresses the thousandths of the root is 6; and, similarly, by making "=""'+0.006, effecting the transformation, and forming the series (''), that the figure which expresses the ten thousandths of the root is 9, &c. &c. =x Consequently the required root of the equation in x is 1.6369+. 2d Example. It is required to separate the positive roots of the equation, x+11x-102x+181=0? X=x+11x-102x+181, X,,,+1566114. Making x=- ∞, the series of signs is -, +, -, +, three variations. +, -, -, +, two variations. Whence the three roots are real; and of the three, two are positive and one negative. To effect the separation of the positive roots let x+10, the series of signs is +, +, +, +, no variation. Therefore 10 is a superior limit of the positive roots. +, two variations. 十,一, -,+, two variations. +, +, +, +, no variation. The two positive roots are, consequently, both comprehended between 3 and 4. Making x=x+3, and transforming the functions of the series (x), X'=x3+20x'2—9x′+1, X'=3x^2+40x′-9, X'=122x-27, X′ =+ an absolute number. Making =0, the series (x) gives the signs +, x=1 x=2 x=3 ,,+, two variations. +, -, -, +, two variations. +, −, −, +, two variations. +, +, +, +, no variation. Therefore the figure which expresses the tenths of both the positive roots is 2 (these roots being comprehended between 2 and 3). Making =x+2, and transforming the functions of the series (x) into functions of the series (x'), X"=x+20·6x''2—·88x''+'008, X" =3x2+41·2x''—·88, X" =122x"−2·6, X" an absolute number. = Consequently the hundredths of one root are comprehended between '01 and 02, and the hundredths of the other root between 02 and '03. Therefore the figure which expresses the hundredths of the first root is 1; the figure which expresses the hundredths of the other root is 2, and the roots 3.21 and 3.22. To obtain the figures which express the thousandths of the two roots, it is necessary to proceed as if for two distinct roots; the figure of the root 3.21 is obtained by making "=""+01, and forming the series (x"); and the figure of the root 3.22 by x=x'"+02, and forming another series (x'"). Calculating by the known method, it is found that the figure which expresses the thousandths of the root 3.21 is 3, and the figure which expresses the thousandths of the root 3.22 is 9; so that the roots approximated to thousandths are 3·213 and 3.229. HORNER'S METHOD FOR THE RESOLUTION OF NUMERICAL EQUATIONS. 287. If the absolute term of an equation is transposed to the right of the sign, and x is replaced by an exact numerical value, the equation is reduced to an identity; but if the number substituted for x is less than a root of the equation, the numerical value of the left member is less than the absolute term (the signs, if negative, being left out of consideration); and if the number substituted for x is greater than a root, the numerical value of the left member is greater than the absolute term. In determining the approximate value of a positive root by finding the successive figures of that root, it is evident that both members of the equation are successively diminished by a quantity less than that which corresponds to an exact root; and that the absolute number after each operation expresses the numerical value of the transformed equation which has the undetermined figures of the root x for one of its roots. |