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lefs than the roots of the given equation by i=e, Cor. 4. Prop. 2. When x=ò the result is the absolute term of the given equation. When for x is inserted --I =-e, the result is the absolute term of an equation whose roots exceed the roots of the given equation by 1=e. Hence, if the terms of the series, 1, 0, -1, -2, &c. be inserted successively for x, the results will be the absolute terms of so many equations, of which the roots form an increasing arithmetical series with the difference I. But, as the commenfurate roots of these equations must be among the divisors of their abfolute terms, they must also be among the arithmetical progressions found by this rule. The roots of the given equation, therefore, are to be sought for among the terms of these progressions which are divisors of the result, upon the supposition of x=0, because that result is its absolute


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It is plain that the progression must always be increasing, only it is to be observed, that a decreasing series, with the sign t, becomes increasing with the sign Cc


Thus, in the preceding example, -4, -3, --2, is an increasing series, of which -3 is to be tried, and it succeeds.

If, from the substitution of three terms of the progression, 1,0,-1, &c. these arise a number of arithmetical serieses, by substituting more terms of that progression, some of the serieses will break off, and, of course, fewer trials will be necessary. III. Examples of Questions producing the

higher Equations.

E X A M P L E I.

It is required to divide 16 l. between two

perfons, so that the cube of the one's fbare

may exceed the cube of the other's by 386. Let the greater share be * pounds, And the less will be 16-x;

By the question, a--16--1 =386. And by Inv. 2x:-48x2 +-768x---4096=386 Trans, and

x—24x2 divide Suppos. Results. Divisors. If x=-1; - 1880 1, 2, 4, 5, 8, 10, 20,

o; - 2241 1, 3, 9, 27, 83. x=-I; - 2650 - 'I, 2, 5, 10, 25, 53..


are 9

Where 8, 9, 10, differ by i ; therefore +9 is to be tried; and being inserted for x, the equation is =o.

The two shares then
and which succeed. Since x=9;

*—9=o is one of the simple equations from
which this cubic is produced; therefore
23—24x2 +384x==2241

=%2—15*+ 249 = 0.
And the two roots of this quadratic are im-



What two numbers are those, whose product

multiplied by the greater will produce
405, and their difference multiplied by the

lefs 20 ?
Let the greater number be x, and the less y.


xy Xy=xy-- y2=20

j? + 20


g+ +40y? 4400 And




y4+40y' +400_405



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Mult, and tranf. y4+40y2–40537400=o.

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This biquadratic resolved by divifors, gives y=5; and therefore x=9. Also, y4+4072-405y+400

-=y?+592 +65y—80=0. This cubic equation has one positive incommensurate root, viz. 1.114, &c. which may be found by the rule in the next fection, and two impoflible. The incommenfurate root y=1,114, &c. gives x = 19.067, &c. and these two answer the conditions very nearly.

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The sum of the squares of two numbers 208,

and the sum of their cubes 2240 being given, to find them.

Let the greater be xty, and the less x-) Then x+31 +x-3=2x2+2y2 = 208.

Hence y2 = 104—*2. Also x+y+xm-y] =2x--6xy2 = 2240. Substitute for gits value and 2x3 +624x—

6x3 = 2240.

This reduced gives x?-156x+560=0.


The roots of this equation are +10, +4, _14

If x=10 then y=2, and the numbers sought are 12 and 8, which give the only just solution. If x=4, then yż=88 and y=/88. The numbers fought are therefore 4+88 and 4–88. The last is negative, but they answer the conditions. Lastly, if x=-14, then y=-92, hence y=v=92, is impossible; but still the two numbers –14 +92, -14-7392, being inserted, would answer the conditions. But it has been frequently observed, that such solutions are both useless, and without meaning.

IV. Solution of Equations by Approximation.

may be

By the former rules, the roots of equations when they are commensurate obtained: These, however, more rarely.occur; and when they are incommensurate, we can find only an approximate value of them, but to any degree of exactness required. There are various rules for this


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