to exceed the other; there must, therefore, be a point at which the two magnitudes are equal.

The value of x, whatever it be, which renders

x37x13x2 + 1,

and such a value has been proved to exist, gives

and must necessarily, therefore, be the root of the equation proposed.

What has been shown with respect to the particular equation x3 13x27x- 1 = 0,

may be affirmed of any equation whatever, the positive terms of which I shall designate by P, and the negative by N. Let a be the value of x, which leads to a negative result, and b that which leads to a positive one; these consequences can take place only upon the supposition, that by substituting the first value, we have PN, and by substituting the second, P >N; P, therefore, from being less, having become greater than N, we conclude as above, that there exists a value of x between a and b, which gives P = N.*

*The above reasoning, though it may be regarded as sufficiently evident, when considered in a general view, has been developed by M. Encontre in a manner, that will be found to be useful to those, who may wish to see the proofs given more in detail.

1. It is evident, that the increments of the polynomials P and N may be made as small as we please.

Pax +6 x2

Let

....

+ dx9,

m being the highest exponent of x; if we put a + y in the place of x, this polynomial takes the form

the coefficients, A, B, C, . . . . T, being finite in number and having a finite value; the first term A will be the value the polynomial P assumes, when x = a; the remainder,

By Cy2 .... + Tym = y (B+ Cy.... Tym−1), will be the quantity, by which the same polynomial is increased when we augment by y the value x = a. This being admitted, if S designate the greatest of the coefficients, B, C,.... T, we have

B+ Cy....+ Tym-1 < S (1+ y.... +ym−1);

The statement here given seems to require, that the values assigned to a should be both positive or both negative, for if they have different signs, that which is negative produces a change in the signs of those terms of the proposed equation, which contain odd powers of the unknown quantity, and, consequently, the expressions P and N are not formed in the same manner, when we substitute one value, as when we substitute the other. This difficulty vanishes, if we make x = 0; in this case, the proposed equation reduces itself to its last term, which has necessarily a sign contrary to that of the result arising from the substitution of one or the other of the above mentioned values. Let there be, for example, the equation

the first member of which, when we put

x=1 and x = 2,

becomes + 12 and . 45. If we suppose x = 0, this member is reduced to 3; substituting, therefore,

therefore,

y (B + C y . . . . + Tym−1) < S y (1—ym),

1-y

and, consequently, the quantity by which the polynomial P is increased, will be less than any given quantity m, if we make

less than this last quantity; this is effected by making

because, in this case, y being <1, the quantity

S+m

Sym+1

1-y'

Sy (1-ym)

1

1

Sy

y

=m,

Sy (1—ym),

1-y

will necessarily be less than the quantity

m, which is indefinitely small.

2. If we designate by h the increment of the polynomial P, and by k that of the polynomial N, the change, which will be produced in the value of their difference, will be h-k, and may be rendered smaller than a given quantity, by making smaller than this same quantity the increment, which is the greater of the two; we may, therefore, in the interval between x = a and z = b, take values, which shall make the difference of the polynomials P and N change by quantities as small as we please, and since this difference passes in this interval from positive to negative, it may be made to approach as near to zero as we choose. See Annales de Mathématiques pures et appliquées, published by M. Gergonne, vol. iv. p. 210.

-

we arrive at two results with contrary signs; but putting — y in the place of x, the proposed equation is changed to

and we have

whence

y+2y3-3y2+15y-3=0,

P = y + 2y3 +15y, N = 3y2 + 3,

P<N, when y = 0,

P>N, when y = 1.

Reasoning as before, we may conclude, that the equation in y has a real root, found between 0 and + 1; whence it follows, that the root of the equation in x lies between 0 and — 1, and, consequently, between + 2 and

1.

As every case the proposition enunciated can present, may be reduced to one or the other of those which have been examined, the truth of this proposition is sufficiently established.

212. Before proceeding further, I shall observe, that whatever be the degree of an equation, and whatever its coefficients, we may always assign a number, which, substituted for the unknown quantity, will render the first term greater than the sum of all the others. The truth of this proposition will be immediately apparent from what has been intimated of the rapidity, with which the several powers of a number greater than unity increase (126); since the highest of these powers exceeds those below it more and more in proportion to the increased magnitude of the number employed, so that there is no limit to the excess of the first above each of the others. Observe, moreover, the method by which we may find a number that fulfils the condition required by the enunciation.

It is evident, that the case most unfavourable to the supposition, is that, in which we make all the coefficients of the equation negative, and each equal to the greatest, that is, when instead of xm + Pxm-1+Q xm-2 + Tx+U= 0,

....

S representing the greatest of the coefficients, P, Q, ........ T, U. Giving to the first member of this equation the form

a quantity, which evidently becomes positive, if we make

Now if we divide each member of this equation by Mm, we have

By substituting therefore for a the greatest of the coefficients found in the equation, augmented by unity, we render the first term greater than the sum of all the others.

A smaller number may be taken for M, if we wish simply to render the positive part of the equation greater than the negative; for to do this, it is only necessary to render the first term greater than the sum arising from all the others, when their coefficients are each equal, not to the greatest among all the coefficients, but to the greatest of those which are negative; we have, therefore, merely to take for M this coefficient augmented by unity.*

Hence it follows, that the positive roots of the proposed equation are necessarily comprehended within 0 and S + 1.

In the same way we may discover a limit to the negative roots; for this purpose we must substitutey for x, in the proposed equation, and render the first term positive, if it becomes negative (178). It is evident, that by a transformation of this kind, the positive values of y answer to the negative values of x, and the reverse. If R be the greatest negative coefficient after this change, R+1 will form a limit to the positive values of y; consequently R-1 will form that of the negative values of x. Lastly, if we would find for the smallest of the roots a limit approaching as near to zero as possible, we may arrive at it by

* In the Résolution des équations numériques, by Lagrange, there are formulas, which reduce this number to narrower limits, but what has been said above is sufficient to render the fundamental propositions for the resolution of numerical equations independent of the consideration of infinity.

1

substituting for a in the proposed equation, and preparing the

y

equation in y, which is thus obtained, according to the directions given in art. 178. As the values of y are the reverse of those of x, the greatest of the first will correspond to the least of the second, and, reciprocally, the greatest of the second to the least of the first. If, therefore, S'+ 1 represent the highest limit to the values of y, that is, if

Indeed, it is very evident, that we may, without altering the relative magnitude of two quantities separated by the signor >, multiply or divide them by the same quantity, and that we may also add the same quantity to or subtract it from each side of the signs and >, which possess, in this respect, the same properties as the sign of equality.

213. It follows from what precedes, that every equation of a degree denoted by an odd number has necessarily a real root affected with a sign contrary to that of its last term; for if we take the number M such, that the sign of the quantity

...

Mm + PMm-1 + QMm−2 . . . . . + TM ± U, depends solely on that of its first term Mm, the exponent m being an odd number, the term Mm will have the same sign as the number M (128). This being admitted, if the last term U has the sign+, and we make x = — - M, we shall arrive at a result affected with a sign contrary to that, which the supposition of x=0 would give; from which it is evident, that the proposed equation has a root between 0 and M, that is, a negative root. If the last term U has the sign we make x + M; the result will then have a sign contrary to that given by the supposition of x = 0, and in this case, the root will be found between 0 and + M, that is, it will be positive.

214. When the proposed equation is of a degree denoted by an even number, as the first term M remains positive, whatever sign we give to M, we are not, by the preceding observations, furnished with the means of proving the existence of a real root,