To determine Z, Z., &c. put p(A) = e”, a being an arbitrary quantity. Now the value of the left-hand member of this equation is also And if we equate the coefficients of x" in each we get 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. and making m = 1, 2, 3, &c. successively, we get the following identities, by which A, y, c, c, c, &c. are completely known. The general law of which equations is thus expressed : 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 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 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'. 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. 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 Immediately applying the general formulae above found, they give A = b + Vb Fa’, ‘y = 1 + V(1 +%), 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 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 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 + &c. for B write — #3. and this series becomes * Trans. of Camb. Phil. Soc. Vol Iv. p. 144. |