CHA P. X. OF THE METHOD OF SERIES BY WHICH YOU MAY APPROXIMATE TO THE ROOTS OF LITERAL EQUATIONS. § 97. I F there be only two letters, x and a, in the proposed equation, suppose a equal to unit, and find the root of the numeral equation that arifes from the substitution, by the rules of the last chapter. Multiply thefe roots by a, and the products will give the roots of the propofed equation. Thus the roots of the equation x2 – 16x + 55 are found, in § 84, to be 5 and 11. And therefore the roots of the equation x2 - 16ax + 55a2 = o, will be 5a and 11a. The roots of the equation x3 + a3x — 2a3 = 0 are found by enquiring what are the roots of the numeral equation x3 + x 2 = 0, and fince one of these is 1, it follows that one of the roots of the proposed equation is a; the other two are imaginary. § 98. If the equation to be more than two letters, as x3 + -y3o, then the value of x refolved involves a2x — 2 a3 + ayx may be exhibited in a feries having its terms compofed of the powers of a and y with their refpective coef ficients; which will "converge the fooner the lefs y is in respect of a, if the terms are continually multiplied by the powers of y, and divided by thofe of a.". Or, "will converge the fooner the greater y is in respect of a, if the terms be continually multiplied by the powers of a, and divided by thofe of y." Since when y is very little in respect of a, the terms y, 1, 1, 1, 1, &c. decrease very quickly. If y vanish in respect of a, the fecond term will vanish in respect of the first, fince 2 fame manner yy: a. And after the a vanishes in respect of the term immediately preceding it. But when y is vaftly great in refpect of a, then a is vastly great in respect of —, and a3 respect of ~; fo that the terms a, a2 y in e &c. in this cafe decrease very swiftly. In either cafe, the feries converge fwiftly that confift of fuch terms; and a few of the firft terms will give a near value of the root required. $99. If a feries for x is required from the propofed equation that fhall converge the fooner, the lefs y is in refpect of a; to find the first term of this feries, we fhall fuppofe y to vanish; and extracting the root of the equation No13 + a3x — 2a3 = o, confifting of the remaining parts of the equation that do not vanish with y, we find, by § 97, that x = a; which is the true value of x when y vanishes, but is only near its value when y does not vanish, but only is very little. To get a value ftill nearer the true value of x, fuppofe the difference of a from the true value to be p, or that x = a + p. And fubftituting a +p in the given equation for x, you will find, x23 = a3 + 3a2p + зap2 + p + a2x = a3 + a2p But fince, by fuppofition, y and p are very little in respect of a, it follows that the terms 4ap, ay, where y and p are feparately of the leaft dimenfions, are vaftly great in respect of the reft; fo that, in determining a near value of p, the reft may be neglected: and from 4a2p + a2y = 0, we find p = y. So that x = a + p = a y, nearly. - Then to find a nearer value of p, and confequently of x, fuppofe p = ty + q, and subftituting this value for it in the laft equation, you will find, p3 3ap2 = • y2 + y2 q = £y q2 + q3] 4a2p = a2y + 4a2q ajp = ay2 + ayg azy = a1y And fince, by the fuppofition, q is = 0. very little in refpect of p, which is nearly = – y, therefore q will be very little in respect of y; and confequently all the terms of the laft equation will be very little in refpect of these two, viz. - Tay', + 4 a'q, where y and q are of leaft dimensions feparately: particularly the term ayq is little in refpect of 4a'q, becaufe y is very little in refpect of a; and it is little in respect of ―ay, because q is little in respect of y. Neglect therefore the other terms, and fup x2zay2 + 4a2q = 0, you will have § 100. When it is required to find a feries for x that fhall converge fooner, the greater y is in refpect of any quantity a, you need only fuppofe a to be very little in respect of y, and proceed by the fame reasoning as in the last example on the fuppofition of y being very little. Thus, to find a value for x in the equation a2x + ayx — y3 = 0 that shall converge the fooner the greater y is in respect of a, suppose a to vanish, and the, remaining terms will give ×3 — y3 = 0, or x=y. So that when y is vaftly great, it appears that xy nearly. But to have the value of x more accurately, put x = y + p, then = + 3y2p + 3YP2 + p3 — a3y — a3p + ay2 + ayp: = where the terms 3y'p + ay become vaftly greater than the reft, y being vaftly greater than a or p; and consequently p = - + a nearly. Again, by supposing p = ➡a + 9, you will transform the laft equation into 24a3 + 3y2q + 3yq* + q' a'y ayq- aq2 — — a2q where the two terms 3932 a'y must be vaftly greater than any of the reft, a being vaftly less R 3 than |