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III. On the Resolution of Equations in Finite Differences. By the Rev. R. MURPHY, M.A. F.R.S. Honorary Member of the Royal Cork Institution, Fellow of Caius College, and of the Cambridge

Philosophical Society.

[Read Nov. 15, 1835.]

WHEN the degree of equations in Finite Differences does not exceed the first, whatever may be their order, methods for their solution in most cases have been furnished by analysts. With respect to those of higher degrees, scarcely any thing has been done to assist in obtaining explicitly an algebraical expression for the unknown quantity*. The utility of solutions for such equations, occurring, as they do, in the theory of chances, is more apparent by the proof which they afford of the expansibility of various kinds of successive functions on which

some doubt has hitherto existed. The difficulties which those have encountered who attempted to

obtain expansions in an algebraical form, for functions which from their nature may be denominated repeated functions, are known, such

are for instance
(a times)

log, log. log. .........(a times) {a}
sin. sin. sin..........(a times) {a},

* In the great work of Lacroix this subject is entirely passed over.

in which w is to be integer or fractional, positive or negative, real or imaginary: so anomalous have they appeared as to induce a belief in some, that they did not admit of an algebraical expansion, and therefore might be supposed to affect some of the first principles of the Differential Calculus.

In fact, the application of Maclaurin's Theorem requires the knowledge of the differential coefficients, which can only be deduced ā priori, in forms which leave them still unknown, while the application of Taylor's Theorem in Finite Differences introduces impracticable coefficients of a nature more complicated to value than the proposed functions themselves.

As an illustration, suppose we denote by u, the successive function, (r times) e

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e being the base of Napier's Logarithms, then, to find its differential coefficient, we have the equation

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which contains an arbitrary multiplier; but it is plain that the expansion of u, being unknown, Xu, is also unknown, and the successive differential coefficients would similarly be expressed in forms of unknown functions; and therefore the expansion by the immediate application of Maclaurin's Theorem would be impracticable.

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now to find A"u, in this case, we must have recourse to the theorem

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the coefficients thus found being obviously more complicated than the function itself.

Try again to find u, by a series arranged according to the powers of r, and containing indeterminate coefficients, that is, put

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which clearly show that the coefficients cannot be found but by the resolution of equations of infinitely high degrees.

Now similar difficulties opposing the expansion of other kinds of successive functions, these can only be removed by attending carefully to the equations in finite differences, by which the law of the formation of the functions is expressed. I therefore here propose a means for resolving such equations, of whatever order or degree, in an algebraical form.

FIRST CLASS OF EQUATIONS.

All successive functions such as those above mentioned, are represented by the equation u, , , = p(u.), for this manifestly expresses the law of the successive functions,

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Put u, - f'(y) the form of the function f, and the quantity y remaining at present unknown, and also let y' = x.

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and it remains to expand the latter function, in order to compare like powers of x, and thus obtain the assumed coefficients and the quantity y.

Now by Taylor's Theorem the development will be of the following nature, viz.

q}(A) + Z. p'(A) + Z. p"(A) + Z. p"(A) + &c, where Z, Z, Z., &c. are functions of x completely independent of the form of the function p, and p(A), p"(A), &c. represent the successive differential coefficients of p(A).

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