and so on alternately; thus,

(2) Original sign or signs. Reverse. Original. Reverse. . . . Reverse. (2) ORORORORO... R.

Let n + 1 denote the number of left hand junctions of the reverse with original signs, in the order (Ø R), at each of which a variation is always gained; then will n evidently denote the number of right hand junctions in the order (R O), at each of which a variation may be either lost or gained; which results are thus proved:

Let A and (B- aA) respectively denote the signs of the last coefficient of an original and the first of the next reverse compartment. In order that the part or product aXA may have a sign opposite to that of B, as well as exceed B, it is evident that A and B must have like signs, and so give a permanence in (1), while the corresponding terms in (2) give a variation. Thus at the left hand junction of every reverse compartment (O R), a variation is gained.

Again let — C and (DaC) respectively denote the signs of the last of a reverse, and the first of the next original compartment. At this right hand junction, the signs in (1) will evidently be those of C, D; the signs in (2) will be those of C, D. If the former denote a permanence, the latter will denote a variation, and conversely.

Thus at the n + 1 left hand junctions of reverse compartments, n + 1 variations are always gained in (2) over (1), while at the n right hand junctions only n variations can be lost. Hence the introduction of a positive root is always attended by the net gain of at least one variation in (2) over the previous number in (1).

The nature of the results may also be exhibited as follows:

At n + 1 left hand in (1) A, B, permanence only.
junctions, (OR)........ Jin (2) A, B, variation only.

At n right hand in (1) C, D, variation or permanence. junctions, (RO)... Šin (2) C, D, permanence or variation.

The result is entirely conclusive, that the number of real positive roots cannot exceed the variations of sign. And in like manner, the multiplication by x + b to introduce a negative root, would necessarily give at least one more permanence of signs; whence a similar conclusion follows.

In the next place, let us suppose the signs of the first and last terms of equation (1) to be alike; then must the number of variations along its terms, be zero or an even number; while the first and last terms of (2) being then unlike, the number of variations in (2) must be an odd number. Again, let us suppose the first and last terms of (1) to have unlike signs, which must indicate in total, an odd number of variations; while the first and last terms

of (2) being then alike, must indicate an even number in (2). Thus in every case, the gain of variations in (2) over the previous number in (1), being the difference between an odd and an even number, is always an odd number

IV. Incomplete Equations. We are at liberty to supply the place of deficient terms with zero terms or infinitesimals, having such signs as to render the number of variations a minimum; and so of the permanences. Now if the inserted zeros have the same sign as the preceding term, as in + x3 +0 022, they can evidently give no new variation; and their insertion is superfluous, in respect to positive roots.

In respect to negative roots, an even number of zero terms whose signs alternate with the preceding, as in - x2 + 0—0 + x2, can give no new permanence; since the last zero has the same sign as the preceding real term Consequently the insertion of zero terms is superfluous, when the difference of the including exponents is an odd number, or the number of vacant terms is even.

But when the difference is an even number, between the two including exponents, let us first suppose the two including signs to be unlike, as in

3x600-0-5x2. Here the alternating signs admit of no permanence except at the junction of the last zero with the next real term; which invariably gives one permanence. As the terms + 3x6 5x2 have previously indicated one variation, and now are made by the inserted zero to indicate one permanence, the total result will be rendered by simply copying both the given signs + and -, into the scale of roots, as described in the new Rule.

Lastly, the difference of the two including exponents being still an even number, let us secondly suppose the signs of the two including terms to be alike, as in + x1 — 0 + 2x2. In this, and all such cases, the alternate signs admit not of any permanence. And the previous like signs gave no variation. Hence the two corresponding roots are not real but imaginary; and the correct result will be rendered by temporarily omitting either of the like signs, or by remembering to regard the succession in + x + 2x2, and such terms, as a permanence of imaginary roots only, and omitting it from the scale of the new Rule.

Thus the effect of zero or vacant terms can be generalized into the Rule, so that their further insertion is no longer needed. Nor does it appear necessary here to give separate precepts for the corresponding imaginary roots; since the number of signs or roots of the scale, positive and negative, subtracted from the degree of the equation will give the even excess of imaginary roots, beyond any pair or pairs of real roots noted in the scale.

The invention of Fluxions or the Calculus has opened the way to the next simple criterion of imaginary roots; to which let us now proceed.

V. The Quadratic Type. This name is here suggested for any three terms of an equation, where the three exponents are in the series of natural numbers, and the first and third terms, or extremes, have like signs; whatever may be the sign of the mean or middle term; such as, 13x2+10x -49, or Pa± Qx1 + Ra3.

The leading idea is, that, when the final differential coefficient, placed equal to zero, has imaginay roots, the original equation must have at least an equal number of them. Let it here be observed, that the types to give independent indications of imaginary roots, can only be contiguous; that is, the middle term of one must not occur in the next, but the last term of one type may be taken as the first of the next type. Let it also be remembered, to guard against a common mistake, that if all the types in an aquation give real roots, we cannot infer conversely that the roots of the original equation are all real; for imaginary roots are still possible, with certain exceptions. Let the original equation be denoted by,

u = AxTM +...+ Px" + Qx”−1 + Rx"-2 + ... = 0.

Differentiating n-2 times, we then substitute the reciprocal of y in place of x. Multiplying by ym-"+2, and differentiating m―n times with respect to y, then restoring the value of y, and omitting common factors, we obtain the isolated type,

n(n−1)Px2+2(n-1) (m—n+1) Qx+(m—n + 2) (m—n+1)R = 0. Resolving this quadratic,

If the quantity under the radical sign be negative, the two roots of the last derivative are imaginary, and consequently two roots of the original equation. are imaginary. This is rendered very evident by tracing the curve of any equation and its derivatives, after the manner of Rolle, of which some elegant examples are given in Montucla.

It will be noticed that the value of ƒ is the ratio of two consecutive factors of the general binomial coefficient. When the degree of the equation m is an odd number, the middle or minimum value of ƒ is

middle, the values of ƒ are equal to each other. Thus,

When the quadratic type begins with the first term, or ends with the last

term of an equation, we have nm, and ƒ =

2m

f 1(m — 1)*

When the type begins with the regular second term, or ends with the

When the type begins with the normal third term, counted in from either

extremity of an equation, n = m2, and f

=

4(m 2)
3(m3)

To recapitulate,-In the common quadratic equation two imaginary roots exist when Q2 -4PR is negative.

In any equation of the third degree, two roots are imaginary, if Q-3PR is negative.

In any equation of the fourth degree, the negative type of two imaginary roots, is QPR; that is, if P denotes the first term or R the last term of the equation. And in the middle part the type is Q2- PR.

In any equation of the fifth degree the values of ƒ are, f= 5, 2, 5. Yet the general formula may be preferable; thus, to find f;

The highest exponent of the type is one factor off, and the difference between the lowest exponent of the type and the degree of the whole equation, is the other factor of the numerator. Subtracting unity from each, gives the two factors of the denominator.

The quadratic type occurs, for example, in the last three terms of, 23 13x2+16x48

= 0.

Here f, and Q2-ƒPR = (16) × 13 × 48-1408. This result being negative, it follows that two roots are imaginary.

The quadratic type occurs in the first three terms of the equation, x3 +3x2 + 7x+4=0.

Here 32-3X1X7 being negative, two roots are imaginary.

It may be remarked that if the first and last terms of the type had unlike signs, no calculation were needed to show that since — PR is then positive, the criterion would be positive and indecisive.

VI. The Cubic Type. By this term we designate any three or four terms of an equation, which terms divided by their lowest power of x would give the form of the common cubic equation. For exa nple,

Let us suppose the given equation to be first divided by P, and let Q÷ PQ', etc., then,

3

A'xTM +...+ x2 + Q'x′′-1 + R'x"-2 + S'x"-3 + ...=0. Differentiating n-3 times, and dividing by (n-3)...3.2.1,

Dividing through by am-+3, then substituting y in place of the reciprocal of x, differentiating mn times with respect to y, then restoring x and dividing by the left hand coefficient,

If this final derivative has two imaginary roots, to be ascertained presently by solution or by Sturm's theorem, then must the preceding derivative, and consequently the original equation, have two such roots, as previously described. Causing the second term to disappear by making

The general equation z + qz + r=0, is proved in Algebra to have two imaginary roots whenever (39)3-(r) is negative. Substituting and di—(1r)2 viding by Q", which cannot alter the sign, the criterion becomes,

Or multiplying and dividing the last term by (fR'÷Q'2)2 denoted by f2, and introducing g' to be presently described, the criterion is,

(1 — ƒ′)3 — (1 — }ƒ' + 1⁄2ƒ"'g')2 = negative.

Expanding and multiplying by 4÷f12, which cannot alter the sign, the criterion becomes symmetrical; thus,

(C)

3—4(f' +g') + 6f'g' — (f'g')2 = neg.

Now to facilitate the practical application, let us suppose this criterion to be the ordinate, and g' the abscissa of a continuous curve. When g' is infinite positive or negative, the criterion is evidently negative in both branches of the curve; while the intermediate portion between the two roots of g', is positive. Thus when the criterion is placed equal to zero, the two roots g1, 92 will be limits; outside of, and not between which, the actual value of g' must fall to indicate two imaginary roots. We find,