REVERSION OF SERIES. WHEN the powers of an unknown quantity are contained in the terms of a series, the finding the value of the unknown quantity in another series, which involves the powers of the quantity to which the given series is equal, and known quantities only, is called reverting the series *. RULE 1. Assume a series for the value of the unknown quantity, of the same form with the series which is required to be reverted. 2. Substitute this series and its powers, for the unknown quantity and its powers, in the given series. 3. Make the resulting coefficients equal to the corresponding coefficients of the given series, whence the values of the assumed coefficients will be obtained. 4. When the series is expressed by means of another, as ax + bx2 + cx3 + . . . = ay + by2 + cy3 + .... the value is to be obtained in the same manner, by assuming x =A y + By2 + Cy3 + ... EXAMPLES. Ex. 1. Let z = ax + bx2 + cx3 + dx1 + be given, to find the value of æ in terms of z and known quantities. Assume x = Az + Bz2 + Cz3 + . . ., and substitute for the powers of x in the given series, the powers of this assumed series. Then we shall have z = aAz + aB + bA2 + ac By equating the coefficients of the homologous terms of z, we shall have This conclusion forms a general theorem for every similar series, involving the like powers of the unknown quantity. *Other methods of reversion are given by different mathematicians. The above is selected for its simplicity, most of the others depending for their evidence on principles more recondite than have yet been laid before the student, or being more difficult of application, or more confined as to generality. This is, evidently, only an application of the Method of Indeterminate Coefficients. Ex. 2. Let the series z = x + x2 + x3 ± 24 + reversion. Here a = 1, b +1, c = 1, d= ± 1, and so on; these values being substituted in the theorem derived from the preceding example, we shall obtain x = 2 + 22 + 23 = 24 + the answer required. =- -1, and so on. Substituting as before, we have a = 1, b == − 1, c = }, d These values being substituted, we shall have x = y + y2 y3 y1 2 6 24 from which if y be given, and sufficiently small for the series to converge, the value of x will be known. Ex. 4. Given the series y = x − } x3 + } x3 of a in terms of y. to find the value Ans. xy+} y3 + 1⁄2 y3 + 375 y2 + . • • x2 x3 x4 2.3 2.3.4 + to find a in ... Ex. 5. Given the series y = 1 + x + + + + 4 Ex. 6. It is required to find x and y from the two following equations: THE METHOD OF FINITE DIFFERENCES. I. Definitions, notation, and principles. 1. Ir the successive integers, 1, 2, 3, 4, ... be given to a in any expression, a series of numbers will be produced, which are called the successive values of that expression and, conversely, that expression is called the general term of the series of numbers thus produced. 2. The general term or function of x from which the series of numbers is formed, is sometimes written, as in algebraic equations, f(x); but more commonly u, or v., the letter u or v being called the characteristic of the function. Thus, if x3 + 6x2 5x + 10 were the general term, it would be written u: and the values of this function, when 1, 2, 3, 4, .. are written for æ, (viz. 12, 32, 76, 150, .) are written u1, U2, U3, U49 · Also, if - 1, 2, 3,... were put for a, the several results, 20, 36, 52, ... are written U —1, U—29 U—39 ·· ... * In this equation y expresses the meridional parts corresponding to the latitude a: and it is remarkable that the numeral coefficients are the same in the direct and reverted series, and differ only in the signs of the even-numbered terms. Leybourn's Rep. II. 44. 3. If a series of n terms be given, the (n + 1)th is called the increment of the series, or the increment of the sum of the series. The increment is the same function of n + 1 that the last term of the series is of n; or in symbols, if u, be the is the increment. last term, un + 1 4. If the first term of a series be subtracted from the second, the second from the third, the third from the fourth, and so on; the several remainders constitute a new series, which is called the first order of differences. Taking the differences of the successive terms of this series, we obtain the second order of differences; and from this again, the third order of differences; and so on, as long as remainders result from such operations. 5. The general expression for the difference of the two consecutive terms, the (x + 1)th and the ath, will, according to the preceding notation, be written Ux+1 u but it is often convenient to write it simply Au,, where ▲ is called the sign of differencing. The second difference, or A(u,+1 u), is written ▲2u,, the third A3u, and so on, as long as any differences exist. 6. The symbol prefixed to an expression u,, signifies the operation of finding another expression v such that v1+1 ・vx = Ux• In other words, it expresses an operation directly the reverse of taking the difference of a function; and hence ▲ and Σ are indicative of operations each of which neutralises the effect of the other. The symbol Σ is called the sign of integration. 7. A factorial is an expression composed of factors in arithmetical progression, as x(x ± a) (x ± 2a) (x+na); where every factor differs from the prea. .... ceding by the common quantity In respect of notation, the following has the advantage of concentrated writing, viz. ++" where the first factor of the series is written down; then the index of the number of factors in the factorial, in the manner of the binomial index; and lastly, separated from the index by a vertical line, the common difference of the several successive factors *. II. To find the general term of the successive orders of differences of a given function u 1. The expression for the ath term of the nth order of differences is the coefficients being those of the expanded binomial (1 — 1)". - {Us+1 A2u = {u,+1-u,} = {Us+2 — Us+1} — { Us+, — u,} Proceeding thus we find the fourth, fifth, and subsequent differences, to any extent, to retain the assigned form: but to complete the proof, we must establish the necessary continuity of the law. * This very elegant notation was invented by M. Kramp, Professor of Mathematics at Strasburgh. See his Elémens d'Arith. Un. p. 347, a work of great originality and value. Now this will be done, if, supposing it true for the mth difference, we prove that it will be so for the (m + 1)th difference. In this case we have And writing x + 1 for x, and subtracting A"u, from the result, we have +.. which establishes the general and necessary continuity of the law of the series. · зu1+2 + зu2+1 — u ̧ = a(x + 3)3 — 3a(x + 2)3 + 3a(x + 1)3 — ax3 and so on, to any required extent. 2. The preceding formula applies to any form of the function u, but when that function is one of an algebraic form, the actual process is simpler when the reductions are made, pari passu, from one step to another of the differencing. Thus, if the function were u ̧ = ax3 bx2 + cx +d, we should have a {(x + 1)3 — x3 } − b {(x + 1)2 — x2} +c {(x + 1) − x} + {dd} =3ax2 + (3a — 2b) x + a − -b+c ▲3u, = 3a {(x+1)2x2} + (3a−2b) {(x+1)−x} + {(a−b+c) — (a−b+c)} =6ax + (6a - 2b) ▲3u ̧ = 6a {(x + 1) − x} + {(6a — 2b) — (6a — 2b)} = 6α A1u, 0, and so on for all higher orders of differences. = It may also be remarked, that though d — d, (a − b + c) — (a − b + e), and so on, are written in the expressions above, they are unnecessary in practice, as the absolute term is always cancelled in differencing. Scholium. It is very clear, since (x + 1)* " is an expression of the (n - 1)th degree, that the difference of an algebraic function is one degree lower than the function itself. In like manner, the second difference is one degree lower than the first difference, or two degrees lower than the given function. Proceeding thus, we shall find the nth difference constant, and the (n + 1)th, (n + 2)th, and all subsequent differences, severally equal to 0. EXAMPLES. Ex. 1. Find the general term of the several orders of differences of (1). x10 + 6x9 + 3x2 The solution of the first of these examples is subjoined. [To facilitate the student's comprehension of the signification of these results, let him apply them to the following particular examples : ▲ (Ux Vx) = 2. Hence, (3x3-2x) (2x-4) + (x2-4x+6) (9x2—2) + (2x−4) (9x2—2). Reduce this, and then actually multiply the values of u, and v, together, and take the difference: they will be found identical. .... Ex. 4. Let a series of factors in arithmetical progression be given, or Aux x(x + a) ... (x + na), (x + a) (x + 2a).... (x + na) { x + (n + 1) a = (n + 1)(x + a) (x + 2a) .... (x + na) a * That is, the difference sought is the product of all the factors except the first, by the common difference and the number of factors. That is, the difference is found by multiplying the denominator, the preceding value of the function, and the numerator, by the common difference, into the number of factors in the denominator so increased. Ex. 6. It is required to determine the first differences of xa, xa*, and xa ̄”. Ex. 7. Find the numerical values of the first ten terms of the first order of differences in the functions whose coefficients are numerical in Ex. 1. Ex. 8. Find the second, third, and fourth differences of the functions given in Ex. 6. * Since a = Ax, this latter expression may be written for it in the answers above. The factors are also often written xx1 x2 x3 ; and hence the results obtained in the text may be |