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become more simple and geneļal, and are more easily deduced.

In this general method, all the terms of any equation are brought to one side, and the equation is expressed by making them equal to o. Therefore, if a root of the equation be inserted instead of (x) the unknown quantity, the positive terms will be equal to the negative, and the whole must be equal to o.

Dev. When any equation is put into this form, the term in which (x) the unknown quantity is of the highest power, is called the First, that in which the index of x is less by 1, is the Second, and so till the last into which the unknown quantity does not enter, and which is called the Absolute Term. Prop. I. If

If any number of equations be multiplied together, an equation will be

produced, of which the dimension * is equal to

the

on,

* The terin dimension, in this treatise, is used in senses somewhat different, but so as not to create any ambiguity. In this chapter it means either the order of an equation, or the number denoting that order,

which

any dimen

the sum of the dimensions of the equations multiplied.

If any number of simple equations be multiplied together, as xmaro, 4-6=0, x=o, &c. it is obvious, that the product will be an equation of a dimension containing as many units as there are simple equations. In like manner, if higher equations are multiplied together, as a cubic and a quadratic, one of the fifth order is produced; and fo on.

Conversely. An equation of fion is considered as compounded either of fimple equations, or of others, such that the sum of their dimensions is equal to the dimension of the given one. By the resolution of equations these inferior equations are difcovered, and by investigating the component simple equations, the roots of any higher equation are found.

Cor. 1. Any equation admits of as many solutions, or has as many roots, as there are fimple equations which compose it, that is, as there are units in the dimension of it.

Cor.

which was formerly defined to be the highest exponent of the unknown quantity in any term of the equation.

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Cor. 2. And conversely, no equation can have more roots than the units in its dimension.

Cor. 3. Imaginary or impossible roots must enter an equation by pairs ; for they arise from quadratics, in which both the roots are such.

Hence also, an equation of an even dimension

may

have all its roots, or any even number of them impossible, but an equation of an odd dimension must at least have one possible root.

Cor. 4. The roots are either positive or negative, according as the roots of the fimple equations, from which they are produced, are positive or negative.

When one root of an equation is discovered, one of the simple equations is found, from which the given one is compounded. The given equation, therefore, being divided by this fimple equation, will give an equation of a dimension lower by 1. Thus, any equation may be depressed as many degrees as there are roots found by any method whatever.

Prop:

Cor. 5.

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Prop. II. To explain the general properties of the signs and coefficients of the terms of an equation.

Let x-a=0, x=0, x=0, x-d =o, &c. be simple equations, of which the roots are any positive quantities ta, tb, tc, +d, &c. and let x+m=0, x=n=o, &c. be simple equations, of which the roots are any negative quantities --M, -n, &c. and let any number of these equations be multiplied together, as in the following table,

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From this table it is plain,

1. That in a complete equation the numa ber of terms is always greater by unit than the dimension of the equation.

2. The coefficient of the first term is I.

The coefficient of the second term is the sum of all the roots a, b, c, m, &c. with their signs changed.

The coefficient of the third term is the sum of all the products that can be made by multiplying any two of the roots together.

The coefficient of the fourth term is the sum of all the products which can be made by multiplying together any three of the roots with their signs changed; and so of others.

The last term is the product of all the roots, with their signs changed.

3. From induction it appears, that in any equation (the terms being regularly arranged as in the preceding example) there are as many positive roots as there are changes in the signs of the terms from'+ to —, Аа

and

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