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the powers *-, -, &c.; that is, of the powers (z–a)-',(2-a)-, &c.; therefore, by transposition, the proposed equation in z, viz.
Az" +Bz"-!+C -2+.. +L%--N=0, is equivalent to the transformed equation in v or 2-a; viz.
A(2-a)*+,B(2-a)*-'+n-,C(-a)*-+..+,L(2-a)+N=0. Whence the values of ,B, r«,C, r-D, &c. will be found by the following table, viz.
,B= B+ Aal C= C+ Ba ,D= D+ ,Cal&c.
---D=n-,D+r-,Ca&c. n-,C=n-Ctr-Bal B= B+ Aa in which table the value of ,N, found in the last column, will be
N=Aa" + Ban-' + Can-+.. La-N, which is the same as the original equation, excepting that we have a instead of z.
Cor. Hence the last term of a transformed equation, found by diminishing the root of the original equation by a given quantity a, is identical to the expression found by substituting that quantity u in the original equativn.
Def. The limits of the roots of an equation are two numbers, between which all the roots aie contained ; that is, the one is less than the least root, and the other is greater than the greatest root.
Proposition II. Theorem. If any quantity be substituted in an equation, and if the original equation be transformed to another, of which the ruot shall be less than the root of the original equation by the quantity substituted in that original equation, the absolute number of the transformed equation will be equal to the result produced in the original equation.
For, let z Brn-s+ Crn-2--..+Lr-N=0 be the equation proposed ; then, if a be substituted for x, the result will be
a"-Ban- + Cak--..+La-N; but this quantity now exhibited is the absolute number of the transformed equation found by diminishing the root of the proposed equation by the quantity a*. Hence the proposition is manifest.
Thus, suppose that 2 is substituted for x in the equation x3-52? + ir-7=0, the result is -1. For, let the root of the proposed equation be dimninished by 2; then, by the operation of transformation following, viz.
1-3 5 712
we obtain the equation 13+ Su2+52-1=0, where v is =r-2, and the absolute number 1 is eqnal to the result -1.
Hence the result may be fond much more expeditiously by only the first row of figures in the transformed equation, than by substituting --—2 in we original equation ; thus in
1)-3 5 -712
3 the last number being the result,
Proposition III. Theorem. If the results obtained by substituting two affirmative numbers in an equation have the same sign, either none or some even number of afirmative roots lie between them ; but if the results have different signs, some odd number of roots lie between them.
For, let the original equation be fr-N=0, where fr is a function of I.
It is evident that if a be substituted for x, and be increased continually from zero till fa becomes equal to N, a will that instant be a root of the equation ; and if a be still increased, the result fa—N will change its sign, and become affirmative ; therefore, between two reresults with contrary signs there is one root.
Now if a be increased to any magnitude, and its function fa also increase, the equation cannot have any more affirmative roots; for in this case the result fa-N will always contince to have the same sign ; viz. affirınative.
But if, when a still increases, the function fa should decrease, so as to be less than N, the result fa-N will again become negative, and some intermediate value of a will have made fa–N=0: this value of a must also be a root of the equation ; therefore these two adjacent roots lie between results which have like signs.
The same suppositions being continued, it is evident that the signs of the results, when the roots are all possible, are alternately affirmative and negative; and, consequently, if two numbers, when substituted for the unknown quantity, give results which have the same sign, either none or an even number of roots lie between these two numbers; or if the two results have unlike signs, some odd number of roots must lie between these numbers.
Cor. 1, As fa, in the maximum of its increase, or in the minimum of its decrease, may not become equal to N, the equation may have impossible roots, which lie between two numbers producing results affected will like signs.
Cor. 2. If as many numbers, and one more, as there are units in the expo. nent of an equation be substituted successively for x in that equation, and produce in the results an alternate change of signs, a root of the equation will lie between every two adjacent numbers.
Proposition IV. Theorem. Every equation may be transformed to another, which shall have all its terms affirmative, by diminishing the root of the original equation.
For, let I"_Bxn-+ Cxn--...tLx+N=o be the equation proposed.
Let the quantity a bę that by which the root of the transformed equation is less than the root of the original one, and let X=vta; then will the transformed equation be represented by ***(pa-Bon-14.1922qBa + C)e-+ (mas_rBa++C-Den—3...+86N=0 See the co-efficients to No. 1, Prop. I. Here the nth order of figurates, with two terms, is represented by 1, p; the (0-1)th order with
three terms, by 1, 9,4'; the (n-2)th order, with four terms, by 1,
, N, q', and so on; and do represents an—Ban-' +Can-- to.-La, which is the last term of the transformed equation, and is the same as the original equation, with a in place of x.
Because the signs of the quantities in the co-efficients of the transformed equation follow one another in the same order as the signs of the terms of the original equation, and since a is only variable, and its highest power is affirmative, such a value may be given to it that every one of the co-efficients may become affirmative, and that the quantity represented by <x may be made to exceed N; in this case da-N will be affirmative ; therefore the quantity a, by which the root of the original equation is diminished, may be so great, that all the terms of the transformed equation will be affirmative.
Cor. Hence it is evident that, if the terms of an equation be all affirmative, none of the terms of a transformed equation can ever become negative by diminishing the root of the proposed equations.
Proposition V. Theorem, If an equation be transformed to another which has all its terms affirmative, the number by which the root of the original equation is diminished, is greater than the greatest root of the equation.
For, in such a transformed equation, the quantity fa-N, which is the result or absolute number, will be affirmative, and can never afterwards become negative by any increase of a, and therefore the proposed equation cannot have any more roots; the quantity a, by which the root is increased, nust therefore exceed the greatest root.
Proposition Vi. Problem. To diminish or increase the roots of an equation by any given num. ber.
Place the co-efficients of a given equation in a row in the same order as in that equation.
Add the number by which it is intended that the root shall be diminished or increased to the co-efficient of the second term, under which write this sum.
Proceed with the member now found as with the co-efficient above, and so on from one member to another, until the number of members found in the column are just one less than the number of units contained in the exponent of the power, of which the number at the head is the co-efficient.
Multiply the first number of the column now constructed, by the diminishing or increasing figure, and add the product to the co-efficient of the second term, under which write the sum. Proceed with the number now found as done with the co-efficient above, and so on, until the number of members found in the column are just one less than the number of units contained in the exponent of the power, of which the number at the head is the co-efficient.
Proceed from column to column exactly in the same manner as with the last column, until the whole is constructed; then the last number of every column is the co-efficient of the corresponding term of the transformed equation to that of the original equation, of which the pumber at the head is the co-efficient.
N.B.-The addition which is here spoken of is that of algebraic addition, and is the sign of tbe number by which the root is to be increased, must be considered negative.
Er. 1. Diminish the root of the cubic equation x + 5.xx® +71-47 =0 by the number 2.
Whence the equation (1-2)' +11(1-2)! +39(x-2)-5=0, has its roots less by 2 than the roots of the proposed equation.
Er. 2. Diminish the root of the biquadric equation -47+819165+20=0 by the number 3.
Whence (1-3)*+8(1-3)' +26(1-3) +32(0-3) +17=0 is the equation required.
Er. 3. Diminish the root of the cubic equation X* +3.034x + 1 by the number to='3.
1/3 Observe in this process, that 3.3 -3.01 0.097 the numbers are placed accord3.6 -1.93 ing to the rules of decimal frac3.9
Whence (1-3)* +3.96.2--3) -1°93(2-3)+097=0 is the equation required.
Er. 4. Increase the roots of the cubic equation 24-71+7=0 by the number 3.
0 -7 7(-3 3
2 6 20
-9 Whence (x+3)3-9(x+3)+20(x+3)+1=0 is the equation required, or substituting v for x+3 the transformed equation will be v-9vo +200+1=0. To ascertain the first digit of every possible root of an equation.
Proposition VII. Problem. Find such a digit a, either at once or by trial, that when the original equation is diminished separately by a and a +1, the transformed equation in x-a-1 may have one or more changes of signs less than those of the equation in x-a; then for every change that disappears so many affirmative roots will be contained between a and a ti; and, if the number of changes that disappear be odd, the equation will at least have one possible root; therefore, if only one change disappears, a will either be the entire root, or the first figure of the root, according as the value of the absolute number of the equation which arises by diminishing the root of the original equation by a is zero, or a real magnitude; but if more changes disappear, a will be the first digit of each root contained between a and atl.
If we now ascertain every two transformed equations, of which the root of the one is less than that of the other by unity, and of which the changes of signs, in that which has its root diminished, are fewer than in that other, we shall have all the affirmative roots.
In order to discover the negative roots, we may either substitute --x for x in the original equation, and proceed as before, or we may increase the roots of the original equation, and consider the permanences of signs that disappear instead of the changes.
N.B. One certain sign of two or some even numbers of impossible roots, is, wlien any co-efficient beconies zero between two adjacent co-efficients which have like signs, viz. either both + or both
Er. 1. Find the first digit of each of the roots of the cubic equatica --111° +5.7--14.
This equation having no permanencies of signs has no negative roots, that is, all its roots are affirmative.
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