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Which values of A, B, C, &c, being fubftituted for them in the affumed fluent or area, will give the area as in the propofition. 2. E. D.

And much after the fame manner we may find the area of the curve whofe abfcifs is z, and ordinate (a+ßz”+yz2”+&c)Xz2-1×(a+ßz”+yz2′′ +dz3′′ +&c)? ́1 × (A+B≈"+cz2" + Dz3" + &c)"1× &c, what

ever be the number of the feries.

3

When, after fome of the first terms, the numerators of each of the following terms of the feries

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z" + &c, become equal to nothing,

a (r + 1) 2"

the series will break off, and terminate; and then the curve is faid to be quadrable. If otherwife, it is said to be non-quadrable.—If r be either nothing. or a negative integer number, it is evident that the denominator of one of the terms of the above feries will become equal to nothing, and then that term will be infinite; and if this happen before the feries terminate, by means of the numerators becoming equal to nothing, the value of the area will come out infinite; in which cafe the feries is faid to fail.

The curve is denominated from the number of terms contained in the quantity

(a + Bz" +yz2” +dz3”+ &c)2. So, if it contain only one term a, it is a fimple nomial; if it contain two terms (a + Bz"), it is a binomial; if three (a + Bz" + yz22), it is a trinomial; and fo on. Hh 2

But

But in what follows I fhall confider none beyond the trinomial.

The area might be expreffed by a defcending feries, and from thence other cafes of the termination of the feries might be pointed out; but this I fhall do in the particular forms, as in them it will be done with greater eafe.

Nor fhould it be wondered at that the area admits of two different values; for when an ordinate flows, the area on one fide of it will increase as fast as that on the other decreafes; confequently the fluxions of thofe two areas will be equal to each other, that is, the fluxions of the areas are both exprcffed by the fame quantity; and it is therefore but right that the fluxion fhould admit of two different fluents, anfwering to the two areas on the oppofite fides of the ordinate. When the expreffion comes out affirmative, it denotes the area lying on that fide of the ordinate from which it is fuppofed to move; but when it comes out negative, it denotes the area on the other fide of the ordinate. When it comes out infinite, it denotes the area lying along the abfcifs infinitely produced.

If all the terms of the ferics a + bz" + cz3 " + &c, after the first, vanifh, and by that means the expreffion for the ordinate become only

azt × (a + Bz" + yz2" + &c), the curve or area is faid to be fimple; but if there be more terms than one, it is faid to be compound.

In what follows I fhall confider chiefly those cafes in which the curve is faid to be fimple; not only because thofe are the cafes that commonly

happen

happen in practice, but because every compound cafe may be refolved into as many fimple ones, as there are terms in the feries a + bz" + cz22 + &c); and then, by finding the area for every fimple cafe, the aggregate of thofe will be the area for the whole compound one.

In the quantity expreffing any area, write that particular value of the abfcifs which it is fuppofed to have where the arca commences, or when it is equal to nothing, and the value of the area refulting from that fubftitution, will be equal to nothing, if the first area be rightly affigned, and then it needs no correction; but if the area, by this fubftitution, come out of fome value, then by just so much will the first area differ from the truth, and it must be corrected by fubtracting the faid value from it.

It may alfo be obferved, that when the ordinate is oblique to the abfcifs, the area, as found by the feries, must be drawn into the fine of the angle of inclination of the ordinate to the abfcifs.

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To have this in the fame form with the general

feries, we must exprefs it thus

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y = ̧ ̈1 ̈× (3a — bzz) × (a — bz2 + cz3);

'or it

may be expreffed thus

× ( − b + 3az ̈3) × (c− b x ̄ ̈1 + az ̃3 ̧1⁄2• ̧

Now by comparing the first of these forms with the general expreffion of the area, we obtain

a = 3a, b = 0, c = − b ; a = a, ß = 0, y = —b,

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Hence r =

= 221 = − 3, s = r + q = − 1, t = 5 +

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2

q = − 1, v = t + q = o, w = v + q = 1, &c.

Then, by fubftituting these values in the general feries, we have

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all the reft of the terms after the firft vanishing. And because this expreffion is negative, it denotes the area on the other fide of the ordinate.

Again, by comparing the latter form with the general feries, we obtain a = b, b = 0, c = 3a; a = c,B==b, y = o, d = a; p = −1, n = −1, q = 1.

9

Hencer == 1,5=r+q=2,t=s+q=2,&c.

n

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for the area in this cafe; where the law of the pro

greffion is manifeft, and where A, B, C, &c, denote the whole coefficients of the firft, fecond, third, &c,

terms,

COROLLARY

1.

When b, c, d, &c, are each equal to nothing, the curve will be fimple, and the general expreffion for fim

ple

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az

ple areas becomes ×(«+ßz”+yz2”+dz3” +&c)?×

an

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WE A+ (V+1)♪ в + (t + 2) y c + (s+3)BD_

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If the curve be only a trinomial, as yaz

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(a + Bx" + yz2”;1a, that is, d, e, &c, each equal to nothing, the area in the last corollary will become

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But to find another expreffion of the area in a defcending feries, we have the ordinate y or az x (a+ßz”+yz2”)? ̈1— azp-1+2nq-2′′ × (y+ßz” +az ̈2”)1 ́1; then, by comparing this with the original feries,

az

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n

2n

the area will come out · × ( a + B = " + y z 2 " ) ? × ( — —

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(s—3)BB+(r−2) A (s−4)Bc + (r−3) × B

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