FLUXIONS, and it must be so taken, that after being multiplied by \frac{n-m}{n x^{n-1}}, it shall vanish when x=0; for then this hypothesis will make the whole function which expresses the value of s vanish, except its first term 1, as it ought to do.

Example. Let us suppose n=2, then

\int \frac{x^2 \dot{x}}{(1-x)^2} = x + \frac{x}{1-x} + 2 \log. (1-x),
\frac{2-m}{2x} \int \frac{x^2 \dot{x}}{(1-x)^2} = \frac{(2-m)x}{2(1-x)} \\ + \frac{2-m}{x} \log. (1-x),

the fluent being here taken as directed. In this case then, after collecting the terms, we get s, or

1 + \frac{m}{2}x + \frac{m+1}{3}x^2 + \frac{m+2}{4}x^3 +, \&c. \\ = \frac{1}{1-x} + \frac{(2-m)}{x} \log. (1-x).

(24.) There is a branch of the doctrine of series which is of considerable importance in pure mathematics as well as in many physical inquiries, and in the science of astronomy; it is called the Interpolation of series.

To interpolate a series is to interpose among its terms others which shall be subject to the same law, or which shall be formed in the same manner as the original terms of the series; or in other words, it is to find the

Series. value of one or more terms by means of others which are given, and which may be either at equal or unequal intervals from one another, the places of the given terms as well as of those sought being supposed known.

It is easy to see that this problem may be applied to the construction of logarithmic tables; for we may regard the logarithms of the natural numbers 1, 2, 3, 4, &c. ad infinitum as the terms of a particular series of which the numbers themselves are then the indices. Having given the logarithms of some numbers, we may by interpolating deduce from them the logarithms of others.

Again, in astronomy we may consider the numbers which express the successive observed positions of a celestial body as the terms of a series, their indices being the intervals of time between the observations, and some assumed epoch, and the problem we are considering will enable us to determine the position at any instant different from the times of actual observation, provided the intervals between the observations be small, and the instant for which the position is sought not very remote from those at which the observations were made.

(25.) With a view to illustrate the nature of the problem to be resolved, let us consider some particular case, as for example the arithmetical series

a, a+d, a+2d, a+3d, a+4d, \&c.

Let t and t' be two given terms of the series, which are at any distance from one another, and let n and n' be their indices, or numbers which denote their places in the series. Also let y be any term whatever, and x its index. Then by the nature of an arithmetical series,

t = a + (n-1)d, \quad t' = a + (n'-1)d, \\ y = a + (x-1)d,

Now, as there are here three equations, each involving the quantities a and d, we may eliminate both these quantities by the common rules (ALGEBRA, Sect. VII.), and this being done, we get

(x-n')(t'-t) = (n'-n)(y-t');

and hence we find this expression,

y = \frac{x-n'}{n-n'}t + \frac{x-n}{n'-n}t',

which is a general formula for interpolating any arithmetical series, and it is observable, that it is entirely independent both of the first term and common difference.

Example. The 7th term of an arithmetical series is 15, and the 12th term is 25: It is required to find the 10th term.

\text{Here} \quad n=7, \quad n'=12, \quad x=10; \\ t=15, \quad t'=25, \quad y \text{ is sought.}

Therefore by the formula,

y = \frac{2}{5} \times 15 + \frac{3}{5} \times 25 = 21, \text{ the answer.}

(26.) The mode of investigation by which we have found a formula for the interpolation of an arithmetical series will apply also to others, if the law according to which the terms are formed be known; in general, however, the law of a series to be interpolated is either

not known, or it is not taken into account, and we only consider the absolute magnitudes of certain terms, and the numbers expressing their places in the series. To resolve the problem generally with these data, it is usual to proceed as follows: Let a straight line, AB, and a point A in it, be assumed as given in position, and let there be taken the segments AD, AD', AD'', AD''', &c. proportional to the numbers denoting the places of the terms of a series reckoned from any term assumed as a fixed origin, and at the points D, D', D'', D''', let there be erected perpendiculars proportional to the terms themselves. Let us now suppose a curve to pass through C, C', C'', C''', &c. then, if it be so chosen that its curvature may vary gradually in its progress from point to point, without any very abrupt changes of inflexion, and moreover, if the terms (which we may suppose to be either at equal or unequal distances) are pretty near to one another, it is easy to conceive, that if AP be taken equal to the number expressing the place of a term between C''D'', C'''D''', any two others, the term itself will, if not exactly, at least be nearly expressed by PQ, the ordinate to the curve.

As an infinite variety of curves may be found that shall pass through the same given points; in this respect the problem is unlimited; it is, however, convenient to assume such as are simple and tractable. The parabolic class possesses these properties, and accordingly they are commonly employed.

Let us then express the ordinates CD, C'D', C''D'', C'''D''', &c. which are the given terms of the series by

t, t', t'', t''', \&c.

and the abscissae AD, AD', AD'', AD''', or the numbers denoting the order of the terms by

n, n', n'', n''', \&c.

Put y for PQ, a term to be interpolated, and x for AP its place. Then, considering x and y as indefinite co-ordinates, a parabolic curve that shall pass through the points C, C', C'', C''', &c. will have for its equation

y = A + Bx + Cx^2 + Dx^3, \&c.

the number of terms on the right-hand side being supposed equal to that of the given points, and A, B, C, &c. being put to denote constant quantities. To determine these we must consider that when x=n, then y=t, and that when x=n', then y=t' and so on, therefore, substituting the successive corresponding values of x and y we get

\begin{aligned} t &= A + Bn + Cn^2 + Dn^3 + \dots, \&c. \\ t' &= A + Bn' + Cn'^2 + Dn'^3 + \dots, \&c. \\ t'' &= A + Bn'' + Cn''^2 + Dn''^3 + \dots, \&c. \\ t''' &= A + Bn''' + Cn'''^2 + Dn'''^3 + \dots, \&c. \end{aligned}

this series of equations must be continued until their number be the same as that of the coefficient, A, B, C, D, &c. If we now consider t, t', t'', \&c. and n, n', n'', \&c. as known, and A, B, C, &c. as unknown quantities, we may determine these last by eliminating them one after another from the above equations, as is taught in ALGEBRA, Sect. XVII. And the values of A, B, C, &c. being thus determined and substituted in the general equation, we shall have a general expression for y in terms of x the number denoting its place and known

quantities; and this is in substance the solution originally given of the problem by Sir Isaac Newton, who proposed it in the third book of his Principia with a view to its application in astronomy.

A celebrated foreign mathematician (Lagrange) has, in the Cahiers de l'Ecole Normale, given a different form to the expression for y. He has observed that since, when x becomes n, n', n'', n''', \&c. successively, then y becomes t, t', t'', t''', \&c. It follows that the expression for y must have this form.

y = \alpha t + \beta t' + \gamma t'' + \delta t''' + \dots, \&c.

where the quantities \alpha, \beta, \gamma, \&c. must be such functions of x, that if we put x=n, then \alpha=1 and \beta=0, \gamma=0, \&c. and if we put x=n', then \alpha=0, \beta=1, \gamma=0, \&c.; and again, if we make x=n'', then \alpha=0, \beta=0, \gamma=1, \&c. and so on. Hence it is easy to conclude that the values of \alpha, \beta, \gamma, \&c. must have the form

\begin{aligned} \alpha &= \frac{(x-n')(x-n'')(x-n''')}{(n-n')(n-n'')(n-n''')}, \&c. \\ \beta &= \frac{(x-n)(x-n'')(x-n''')}{(n'-n)(n'-n'')(n'-n''')}, \&c. \\ \gamma &= \frac{(x-n)(x-n')(x-n''')}{(n''-n)(n''-n')(n''-n''')}, \&c. \\ \delta &= \frac{(x-n)(x-n')(x-n'')}{(n'''-n)(n'''-n')(n'''-n'')}, \&c. \end{aligned}

and here the number of factors in the numerator and denominator must be each equal to the number of given points in the curve. This formula would be found to be identical with that which may be obtained by the method indicated in last article, if we were to take the actual product of the factors and arrange the whole expression according to powers of x. It possesses however one advantage over the other, viz. that of admitting of the application of logarithms.

We shall now show the application of this formula.

Ex. 1. Having given the logarithms of 101, 102, 104, and 105, it is required to find the logarithm of 103.

In this case we may reckon the terms of the series forward from the first given term, viz. log. 101, so that we have

\begin{aligned} t &= \log. 101 = 2.0043214, & n &= 0, \\ t' &= \log. 102 = 2.0086002, & n' &= 1, \\ y &= \log. 103 = \text{term sought}, & n'' &= 2, \\ t'' &= \log. 104 = 2.0170333, & n''' &= 3, \\ t''' &= \log. 105 = 2.0211893, & n'''' &= 4. \end{aligned}

Substituting now in the general formula we get

\begin{aligned} \alpha &= \frac{1 \times -1 \times -2}{-1 \times -3 \times -4} = -\frac{1}{6}, & \gamma &= \frac{2 \times 1 \times -2}{3 \times 2 \times -1} = \frac{2}{3}, \\ \beta &= \frac{2 \times -1 \times -2}{1 \times -2 \times -3} = \frac{2}{3}, & \delta &= \frac{2 \times 1 \times -1}{4 \times 3 \times 1} = -\frac{1}{6}. \end{aligned}
\begin{aligned} \text{Therefore } y &= -\frac{t}{6} + \frac{2t'}{3} + \frac{2t''}{3} - \frac{t'''}{6} \\ &= \frac{1}{3}(t' + t'') - \frac{1}{6}(t + t''') \\ &= 2.0128372 \text{ the answer.} \end{aligned}

Ex. 2. Given a comet's distance from the sun on the following days at 12 at night, to find its distance December 20th.

December 12. distance 301, Dec. 24. distance 715,
21. 629, 26. 772.

Here we shall estimate the places of the terms from the time of the first position, viz. December 12. Therefore

\begin{aligned} t &= 301, & n &= 0, \\ y & \text{ is fought, } & x &= 8, \\ t' &= 629, & n' &= 9, \\ t'' &= 715, & n'' &= 12, \\ t''' &= 772, & n''' &= 14. \end{aligned}

In this case the general formula gives us

a = \frac{1}{63}, \quad b = \frac{6}{45}, \quad \gamma = -\frac{2}{3}, \quad \delta = \frac{8}{35},
y = \frac{t}{63} + \frac{64t'}{45} - \frac{2t''}{3} + \frac{8t'''}{35}.

= 586.3 the answer.

We shall conclude this article with a brief enumeration of the best works on the subject which we have been treating of.

Ars Conjectandi, (Jac. Bernoulli). Methodus Differentialis, (Newton). Methodus Incrementorum, (Taylor). Methodus Differentialis, five Tractatus de Summatione et Interpolatione Serierum, (Stirling). Institutiones Calculi Diff. (Euler). Emerson's Method of Increments. The differential method, (same author). Miscellanea Analytica, (De Moivre). The various writings of Landen and Simpson. Theorie des Fonctions Analytiques, (Lagrange). Du Calcul des Derivation, (Arbogast). Traité des différences et des Series, (a sequel to Lacroix's work on the Calcul Differential, &c.). Dr Hutton's Mathematical and Philosophical Tracts. An Essay on the Theory of the various orders of Logarithmic Transcendents, with an Inquiry into their applications to the Integral Calculus, and the Summation of Series, by W. Spence, &c. &c.