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Multiply (5) by A, (6) by B, and subtract the sum from (4), making the coefficients of each differential equal to zero. We thus obtain

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x2 + y2 + z2 = A* (l2 + m2 + n2) + 2 AB (++)

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a2

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v2 - c2

putting for a2+y+, and for A its value v, this becomes

B=v (r2 - v2).

Substituting these values of A and B, equation (7) becomes

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1

{a° (r2 — v2) + v2 (a2 - r2)} for a2-v2, this becomes

Putting = { a° (r2 — v2) + v2 (a° − r°) }

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Multiplying these by x, y, z, respectively, and adding

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This is the simplest form which the equation to the wave surface can

assume.

If we clear it of fractions, and replace by x2+y2+x2, we obtain the equation given by Fresnel, viz.:

(x2+y2+z2) (a2x2+b2y2+c2x2)−a2 (b2+c2) x2 −b2 (a2+c2) y2 −c2 (a2 +b2) z2 +a2b2 c2 = 0.

TRINITY COLLEGE,

May 8, 1835.

VOL. VI. PART I.

M

ARCH". SMITH.

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

a

α

(x times)

a

log. log. log.......... (x times) {a}

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

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

in which 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 à 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,

(x times)

€ €

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

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