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To determine Z, Z., &c. put p(A) = e”, a being an arbitrary quantity.

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Now the value of the left-hand member of this equation is also

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And if we equate the coefficients of x" in each we get

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where it must be observed, that b, c, d, &c. having previously satisfied

the equation b + 2 c + 3d, &c. = m, the quantity n is then found by summing b, c, d, &c.

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and making m = 1, 2, 3, &c. successively, we get the following identities, by which A, y, c, c, c, &c. are completely known.

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The general law of which equations is thus expressed :

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b, c, d, &c. being regulated by the two conditions before mentioned.

From hence we obtain the complete integral of the proposed equation u, , , = q (u.); for since

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by the reversion of which series B is found in a series arranged according to the powers of u,-4.

Before proceeding to any particular applications of this general solution, a few observations will be useful.

I. When p(u.) is of the form u. 4 const., then A becomes generally infinite, and y becomes 1, the solution therefore fails in this case, but more generally it may be remarked, that it also fails when the value of A deduced from the equation A = p(A) satisfies the equation p'(A) = 1; for this, by making y = 1, renders infinite the coefficients cl, c., &c. Such cases of failure will shortly be separately considered.

II. When the equation u, , , = q (u,) is of the n" degree, the equation for finding A is of the same degree, and therefore A has n values; then y = p (A) has also n corresponding values, which, being represented by y, y, ... y, and putting for abridgment F(Boy') for the series above found for u., we have

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and the complete solution is found by taking the product of the members on the left side and equating to zero: the result will only contain one arbitrary constant, since B, B, &c. are all found in terms of u, as before shewn.

III. u, is a known function of B, u, is the same function of Boy'.

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Let p-' be the function which is inverse to p, that is, such that p-' p (a) = a, then it follows that u-. = p^' p' op '...... {a times; (u). The same formula therefore which represents the ar" successive direct

function of us, will also give the a” successive inverse function by merely writing — a in place of ar.

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Thus the number of times it is necessary to take the successive

- - - log. B functions (p of a, to arrive at u, as a result, determines ; y ' and

since y is known, B may be thus also determined. Vol. VI. PART I. N

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Immediately applying the general formulae above found, they give A = b + Vb Fa’, ‘y = 1 + V(1 +%),

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To facilitate the determination of B in every case of the general solution, viz., u, - A + Boy’ + c, (Boy)" + c, (Boy)", &c, we may apply the general theorem

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where the indeterminate indices are subject to the two conditions

b, + bi + b + &c. = n 2b, + 3b, + &c. = n – 1, which I have given in my Memoir on the Resolution of Algebraic Equations.”

With respect to the inverse function in this particular example, we have u. = V (a + 2bu.. ): ... u_. = v_{a' + 2b u -e-, }; and putting for a successively 1, 2, 3 ... a, we get u_. = V a + 2b V} a + 2b V} a + ... + 2b V(a + 2bu)}, the number of roots being a ; and for the value of this successive function we have

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and B is very readily found by putting r = 1, and thence the required value of u -, is obtained in a series extremely convergent. As another numerical example, let u, , , = 2 u.” 1. Here y = 4, by = 1, and therefore

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+ &c.

for B write — #3. and this series becomes

* Trans. of Camb. Phil. Soc. Vol Iv. p. 144.

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