▲'u。 +▲'u ̧ + &c. =▲ ̃1 (Au ̧ + Au ̧ + &c.) = ▲r1 (u„ — u ̧). This formula has the disadvantage of containing the differences of un, which cannot be calculated from the values u, u... u. We may remedy this in the following way: Removing the first two terms from each side since they are obviously equal, and writing u, for Aw, we get In the above investigation we have in reality twice per 1 formed the operation on both sides of an equation. We shall see that Au2 = Av, only enables us to say ux = vx + C and not u=v; hence we should have added an arbitrary constant. But the slightest consideration is sufficient to shew that this constant will in each case be zero. 14. The problems of Interpolation and Mechanical Quadrature are of the greatest practical importance, the formulæ deduced therefrom being used in all extended calculations in order to shorten the labour without affecting greatly the accuracy of the result. This they are well capable of doing; indeed Olivier maintains (Crelle, 11. 252) that calculations proceeding by Differences will probably give a closer approximation to the exact result than corresponding ones that proceed by Differential Coefficients. In consequence of this practical value many Interpolation-formulæ have been arrived at by mathematicians who have had to do with actual calculations, each being particularly suited to some particular calculation. All the most celebrated of these formule will be found in the accompanying examples. Examples of calculations based upon them can usually be found through the references; the papers by Grunert (Archiv, XIV. 225 and xx. 361), which contain a full inquiry into the subject, may also be consulted for this purpose. Numerical examples of the application of several Interpolation-formulæ may also be found in a paper by Hansen (Relationen zwischen Summen und Differenzen, Abhandlungen der Kön. Sächs Gesellschaft, 1865), in which also he gives a very detailed inquiry into the various methods in use, with numerical calculation of coefficients, &c. We must warn the reader against the notation, which is unscientific and wholly in defiance of convention, e. g. Ayx+1 and Ay are used to represent the Ay and Ayx-1 of the ordinary notation. A good paper on the subject by Encke (Berlin. Astron. Jahrbuch, 1830), from which Ex. 7 is taken, labours under the same disadvantage; and Stirling's formula (Ex. 9) is seldom found stated in the correct notation. In speaking of the developments which the theory has received we must mention an important Memoire by Jacobi (Crelle, xxx. 127) on the Cauchy Interpolation-formula of Art. 8. In it the author points out the advantages that it possesses over others, and subjects it to a very full investigation, representing the numerator and denominator in various forms as determinants, and considering especially the case when two or more of the values of the independent variable approach equality. A paper by Rosenhain which follows immediately after it treats also of the above formula in representing the condition that two equations (x)=0 and f(x)=0 should have a common root, in terms of the values of the expression for different values of x. (x) f(x) But the most important researches in the theory of Interpolation have had reference to the Gauss-formula of Art. 12. Minding (Crelle, vi. 91) extends it to the approximate evaluation of double integrals between constant limits. Christoffel (Crelle, LV. 61) investigates the more general problem of determining the ordinates we should choose for observation when certain ordinates are already given, so that the approximation may be as close as possible. Mehler (Crelle, LXIII. 152) shews that a closely analogous method enables us to calculate integrals of the form -1 (1-x)(1+x) f (x) dx with great accuracy, the position of the ordinates chosen being in this case determined by the roots of the equation of the nth degree 1 Jacobi had previously examined the case in which λ=μ=-2; in other words, he had shewn that in the positions of the co-ordinates to be chosen after the analogy of the Gaussformula are given by the roots of which is equivalent to cos (n cos-1x)=0. Hence x=cos 2m+1 Π. In this case the coefficients Aa, A., ... (see (26), page 51) are all equal, each being and the formula becomes π n In most of the above papers the magnitude of the error caused by using the approximate formula instead of the exact value of the function is investigated. The special importance of the method becomes evident when we consider the close relation between it and the celebrated Laplace's functions. This is seen by comparing the expression for the nth Laplace's coefficient of one variable, with the value of M in Art. 12; and the similarity of the corresponding expressions for two variables is equally great. In fact the Gauss-method may be represented as follows: n, n Let us be a rational integral function of the (2n-1)th degree, and Y2 be the nth Laplace's coefficient. Divide u by Y, and let N be the quotient and f(x) the remainder which is of the (n - 1)th degree. Thus u=f(x) + Yn • N. Integrate between the limits 1 and -1, and since N is of a lower degree than Yn' f(x) dx which is accurately found by the Lagrange-formula from the n observed values of u 1 YNdx=0, and we are left with In consequence of this close connexion the method is of great importance in the investigation of Laplace's Functions and of the kindred subject of Hypergeometrical Series. Heine's Handbuch der Kugelfunctionen will supply the reader with materials for discovering the exact relation in which they stand to one another, or he may compare a paper by Bauer on Laplace's functions (Crelle, LVI. 101) with that by Christoffel given above. For instances of numerical calculation he may consult Bertrand (Int. Cal. 339), where, however, the limits 1 and 0 are taken. EXERCISES. 1. Required, an approximate value of log 212 from the following data: == log 210 = 2.3222193, log 213 = 2·3283796, = 2. Find a rational and integral function of x of as low a degree as possible that shall assume the values 3, 12, 15, and -21, when x is equal to 3, 2, 1, and -1 respectively. 19 3. Express v12 and v, approximately, in terms of v., V1, v1, and v, both by Lagrange's formula and the method of (7) Art. 4. 4. The logarithms in Tables of n decimal places differ 5 from the true values by ± at most. Hence shew that 10+1 the errors of logarithms of n places obtained from the Tables by interpolating to first and second differences cannot exceed 1 9 1 + +e and ± x+e' respectively, e and e' being the 10" 10" 8 errors due exclusively to interpolation. (Smith's Prize.) 5. The values of a function of the time are a1, ɑğ, ɑz, α şı at epochs separated by the common interval h; the first differences are d,, d',, d', the second differences are d,, d',, and the third difference d. Hence obtain the following formula of interpolation to third differences: t being reckoned in the first case from the epoch of a,, and in the second from that of a,. 6. If P, Q, R, S, ... be the values of X, an unknown function of x, corresponding to x = p, q, r, s, ..., shew that (under the same hypothesis as in the case of Lagrange's formula), X=P+ (x−p) {p, q} + (x − p) (x − q) {p, q, r} +&c., 7. Shew that, in the notation of the last question, if q=p=r-q=s-r=&c. = 1, |