m the series Zamp will be expressed in ascending factorials of the above form. But if in expressing π and p we assume ▲x=-1, then since the series will be expressed in factorials of the latter form. Ex. 1. Given (x − a) ux − (2x — a − 1) ux-1 + (1 − q3) (x − 1) u‰-2 = 0 ; required the value of u, in descending factorials. Multiplying by x, and assuming T = x Ax-1, we have x (x − a) u ̧ — (2x − a − 1) pu ̧+ (1 − q3) p3u„ = 0), whence, substituting + p for x, developing reducing, by (13), and .(a). the initial values of a corresponding to m = 0 and m = a being arbitrary, and the succeeding ones determined by the law It may be observed that the above difference-equation might be so prepared that the complete solution should admit of expression in finite series. For assuming u2 = μ*v; and then transforming as before, we find μ3π (π − α) vx + (μ2 — μ) (2π — a –— - 1) pvx + {(μ − 1)2 — q2} p2v2 =.0......................... (c), which becomes binomial if μ=1+q, thus giving the initial value of m being 0 or a, and all succeeding values determined by the law It follows from this that the series in which the initial value of m is 0 terminates when a is a positive odd number, and the series in which the initial value of m is a terminates when a is a negative odd number. Inasmuch however as there are two values of μ, either series, by giving to μ both values in succession, puts us in possession of the complete integral. Thus in the particular case in which a is a positive odd number we find The above results may be compared with those of p. 454 of Differential Equations. Finite solution of Difference-Equations. 6. The simplest case which presents itself is when the symbolical equation (8) is monomial, i.e. of the form Resolving then {ƒ (7)} as if it were a rational algebraic fraction, the complete value of u will be presented in a series of terms of the form (π − a) X = p2 (π) ̃ ̄pTMa X.......... .(28). ... It will suffice to examine in detail the case in which Ax=1 in the expression of π and p. To interpret the second member of (28) we have then paÞ (x) = x (x + 1) ... (x + a − 1) $ (x+a), · (x + 1) (x + 2) ... (x+a) ' π*† (x) = (xA) ̄*¤ (x) = × 1 × 1- ... • (∞) ; the complex operation Σ, denoting division of the subject by x and subsequent integration, being repeated i times. Should X however be rational and integral it suffices to express it in factorials of the forms X, x(x+1), x(x+1)(x+2), &c. to replace these by p, p3, p3, &c. and then interpret (27) at once by the theorem {ƒ, (π)} ̄1 pTM = {ƒ。 (m)} ̄1 pTM = {fo {ƒ。 (m)} ̄1 x (x + 1) ... (x + m − 1)...(29). As to the complementary function it is apparent from (28) that we have This method enables us to solve any equation of the form x (x + 1) ... (x+ n − 1) ▲”u + A ̧x (x + 1) ... n .. (x + n − 2) ▲” ̄1u... + A2u = X......(31). For symbolically expressed any such equation leads to the monomial form or {π (π − 1) ... (π − n + 1) +  ̧π (π −1)... ... (π − n + 2) ... + A2} u = X................(32). Ex. 2. Given x (x + 1) A3u - 2x▲u + 2u = x (x + 1) (x + 2). The symbolical form of this equation is Hence π (π — 1) и — 2πи + 2u = x (x + 1) (x+2)..............(a), since the factors of 2-3π +2 are π· -2 and π-1. Thus we have 7. Let us next suppose the given equation binomial and therefore susceptible of reduction to the form in which U is a known, u the unknown and sought function of x. The possibility of finite solution will depend upon the form of the function (7), and its theory will consist of two parts, the first relating to the conditions under which the equation is directly resolvable into equations of the first order, the second to the laws of the transformations by which equations not obeying those conditions may when possible be reduced to equations obeying those conditions. As to the first point it may be observed that if the equation be u+ 1 аптЂри= it will, on reduction to the ordinary form, be integrable as an equation of the first order. Again, if in (33) we have $ (π) = f (π) if (π − 1) ... † (π − n + 1), a system of equations of the first order. This depends upon the general theorem that the equation u +α ̧§ (π) pu + a‚§ (π) $ (π − 1) p3u ... + α„$ (π) $ (π − 1) ... † (π − n + 1) pˆu = U |