62)

OF

EQUATIONS OF ALL DEGREES.

THE following is the translation of a letter received by the Editor of this Course, Jul" .8th, 1820, from the French Institute, on account of an Essay on Involution and Evolution, then published ; which letter will show the degree of approbation attached to the following general method of depressing and augmenting the roots of equations, and the subsequent extraction of their roots, see Nicholson's Analytical and Arithmetical Essays, published November, 1820, where this interesting branch on equations is.

FRENCH INSTITUTE, ROYAL ACADEMY OF SCIENCES.
Paris, July 10, 1820.

The Perpetual Secretary of the Academy to Mr. Nicholson. SIR,-The Academy has received with a lively degree of interest, the Essay that you obligingly addressed to it on Involution and Evolution, or a Method of determining the Numerical Value of any Function of an unknown Quantity. I am desired, in its name, to thank you for sending this interesting work, which has been honourably placed in the Library of the Academy; and to express the sense of obligation which the Institution entertains for your attention. Receive, Sir, I beg, the assurance of the most distinguished respect with which I am, Your's, &c.

Definitions.

B. G. CUVIER.

1. TRANSFORMATION of an equation, is the method of finding another equation which shall have all its roots greater or less than the roots of the original equation by a given quantity.

2. Extraction is the method of finding such a value of the unknown quantity in an equation as will make the numerical value of both sides equal when that value is substituted for the unknown quantity representing it.

3. The Root of an Equation is the value of the unknown quantity, 4. When an equation contains one or more powers of an unknown quantity, it is said to be of such a degree or order as is indicated by the exponent of the highest power of the unknown quantity.

5. A Simple Equation is that in which the exponent of the unknown quantity is unity.

6. A Quadratic Equation is that in which the exponent of the highest power of the unknown quantity is 2

Az

7. A Cubic Equation is that in which the exponent of the highest power of the unknown quantity is 3.

8. A Biquadratic Equation is that in which the exponent of the highest power of the unknown quantity is 4.

9. An Equation of the nth Degree is that in which the exponent of the highest power of the unknown quantity is n, where n may be any assigned value whatever.

Scholium.-The solution of simple equations is not the object of this tract, as the root is found by plain division. Here I shall only treat of the higher equations, and show a general method of extracting their roots. For those who are curious in the history of this important subject, I shall refer them to my Essay on Involution and Evolution.

Proposition I. Problem.

To transform the equation
Az+B+Cx”−2. + Lx=N.

...

into another, of which the root shall be less or greater than the root of the equation proposed by a given quantity a.

Letv be the remaining part of the root; then will z=v+a; also let u the exponent of any power of z or of v+a; therefore, by the binomial theorem,

and this equation will be equivalent to the following:

By substituting n, n−1, n−2, &c. for u, and multiplying the respective powers and their values by the co-efficients, we have the following value of each of the terms of the given equation, viz.

But the values of „B, C, D, &c. exhibited in Prop. 3, Figurate numbers, are identical to the co-efficients now exhibited of

the powers -, -, &c.; that is, of the powers (z—a) "—3, ' (z—a)*~3, &c.; therefore, by transposition, the proposed equation in z, viz. A+B+Cz"¬2+.. + Lx-N=0,

is equivalent to the transformed equation in v or x-a; viz.
A(x-a)"+B(x-a)n—2+-1C(≈—a)n—2 + .. +‚L(≈—a)±‚N=0.
Whence the values of B, C, -2D, &c. will be found by the fol-
lowing table, viz.

B B+ Aa C= C+ Ba D= D+ Ca&c.
,B=B+Aa,C,C+,Ba

,B=,B+Aa

B=B+Aa

&c.

.n which table the value of N, found in the last column, will be N=Aa" + Ba"−1 +Can~~2+.... 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 a in the original equation.

Def. The limits of the roots of an equation are two numbers, between which all the roots are 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 root 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 Br"-1+ Cx-2.. +Lr-N=0 be the equation proposed; then, if a be substituted for x, the result will be

a"-Ba-Ca--..+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 x3-3x2+5x−7=0, the result is -1. For, let the root of the proposed equation be diminished by 2; then, by the operation of transformation following, viz.

-7(2
-1

we obtain the equation w3+Sv2+5c−1 =0, where u is x-2, and the absolute number-1 is equal to the result-1.

Hence the result may be found much more expeditiously by only the first row of figures in the transformed equation, than by substituting -2 in the original equation; thus in

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 affirmative 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 x.

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 continue to have the same sign; viz. affirmative.

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 with like signs.

Cor. 2. If as many numbers, and one more, as there are units in the exponent 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 x-Bx"¬3 +Сx”―2—··· +L+N=0 be the equation proposed.

Let the quantity a be that by which the root of the transformed equation is less than the root of the original one, and let z=v+a; then will the transformed equation be represented by

+(pa-B)+(gʻa2-q Ba+C)in~2+(r°q3~r°Baa+rC—D)x−3...+8a-N0 See the co-efficients to No. 1, Prop. I. Here the nth order of figurates, with two terms, is represented by 1, p; the (n-1)th order with *E

three terms, by 1, q, q'; the (n-2)th order, with four terms, by 1, r, r', '', and so on; and da represents a"-Ba"-+Caa¬2 +..—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 da 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, must 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.