Problems concerning Interpolations. By Edward Waring, M. D. F. R. S. and of the Institute of Bononia, Lucasian Professor of Mathematics in the University of Cambridge

VII. Problems concerning Interpolations. By Edward Waring, M.D. F.R.S. and of the Institute of Bononia, Lucasian Professor of Mathematics in the University of Cambridge. Read Jan. 9, 1779. Mr. Briggs was the first person, I believe, that invented a method of differences for interpolating logarithms at small intervals from each other: his principles were followed by Reginald and Movton in France. Sir Isaac Newton, from the same principles, discovered a general and elegant solution of the abovementioned problem: perhaps a still more elegant one on some accounts has been since discovered by Messrs. Nichole and Stirling. In the following theorems the same problem is resolved and rendered somewhat more general, without having any recourse to finding the successive differences. THEOREM I. Assume an equation $a + bx + cx^2 + dx^3 \ldots x^{n-1} = y$, in which the co-efficients $a, b, c, d, e, \&c.$ are invariable; let \( \alpha, \beta, \gamma, \delta, \varepsilon, \&c. \) denote \( n \) values of the unknown quantity \( x \), whose correspondent values of \( y \) let be represented by \( s^\alpha, s^\beta, s^\gamma, s^\delta, s^\varepsilon, \&c. \). Then will the equation \[ a + bx + cx^2 + dx^3 + ex^4 \ldots x^{n-1} = y = \] \[ \frac{x-\beta \times x-\gamma \times x-\delta \times x-\varepsilon \times \&c.}{x-\alpha \times x-\gamma \times x-\delta \times x-\varepsilon \times \&c.} \times s^\alpha + \frac{x-\alpha \times x-\beta \times x-\delta \times x-\varepsilon \times \&c.}{\beta-\alpha \times \beta-\gamma \times \beta-\delta \times \beta-\varepsilon \times \&c.} \times s^\beta + \frac{x-\alpha \times x-\beta \times x-\gamma \times x-\varepsilon \times \&c.}{\gamma-\alpha \times \gamma-\beta \times \gamma-\delta \times \gamma-\varepsilon \times \&c.} \times s^\gamma + \frac{x-\alpha \times x-\beta \times x-\gamma \times x-\delta \times \&c.}{\delta-\alpha \times \delta-\beta \times \delta-\gamma \times \delta-\varepsilon \times \&c.} \times s^\delta + \frac{x-\alpha \times x-\beta \times x-\gamma \times x-\delta \times \&c.}{\varepsilon-\alpha \times \varepsilon-\beta \times \varepsilon-\gamma \times \varepsilon-\delta \times \&c.} \times s^\varepsilon + \&c. \] **Demonstration.** Write \( \alpha \) for \( x \) in the equation \( y = \) \[ \frac{x-\beta \times x-\gamma \times x-\delta \times x-\varepsilon \times \&c.}{x-\alpha \times x-\gamma \times x-\delta \times x-\varepsilon \times \&c.} \times s^\alpha + \frac{x-\alpha \times x-\beta \times x-\delta \times x-\varepsilon \times \&c.}{\beta-\alpha \times \beta-\gamma \times \beta-\delta \times \beta-\varepsilon \times \&c.} \times s^\beta + \&c.; \text{ and all the terms but the first in the resulting equation will vanish, for each of them contains in its numerator a factor } x-\alpha=x-\alpha=0; \text{ and the equation will become } \] \[ y=\frac{\alpha-\beta \times \alpha-\gamma \times \alpha-\delta \times \alpha-\varepsilon \times \&c.}{\alpha-\beta \times \alpha-\gamma \times \alpha-\delta \times \alpha-\varepsilon \times \&c.} \times s^\alpha=s^\alpha. \text{ In the same manner, by writing } \beta, \gamma, \delta, \varepsilon, \&c. \text{ successively for } x \text{ in the given equation it may be proved, that when } x \text{ is equal to } \beta, \gamma, \delta, \varepsilon, \&c. \text{ then will } y \text{ become respectively } s^\beta, s^\gamma, s^\delta, s^\varepsilon, \text{ which was to be demonstrated.} \] 2. Assume \( y=ax^r+bx^{r+1}+cx^{r+2}+dx^{r+3} \ldots x^{r+n-1}; \) and when \( x \) becomes \( \alpha, \beta, \gamma, \delta, \varepsilon, \&c. \) let \( y \) become respectively Dr. Waring on Interpolations. respectively $s^\alpha$, $s^\beta$, $s^\gamma$, $s^\delta$, $s^\epsilon$, &c.; then will $y =$ $$\frac{x' \times x' - \beta' \times x' - \gamma' \times x' - \delta' \times x' - \epsilon' \times \text{&c.}}{\alpha' \times \alpha' - \beta' \times \alpha' - \gamma' \times \alpha' - \delta' \times \alpha' - \epsilon' \times \text{&c.}} \times s^\alpha$$ $$+ \frac{x' \times x' - \alpha' \times x' - \gamma' \times x' - \delta' \times x' - \epsilon' \times \text{&c.}}{\beta' \times \beta' - \alpha' \times \beta' - \gamma' \times \beta' - \delta' \times \beta' - \epsilon' \times \text{&c.}} \times s^\beta$$ $$+ \frac{x' \times x' - \alpha' \times x' - \beta' \times x' - \gamma' \times x' - \epsilon' \times \text{&c.}}{\gamma' \times \gamma' - \alpha' \times \gamma' - \beta' \times \gamma' - \delta' \times \gamma' - \epsilon' \times \text{&c.}} \times s^\gamma + \text{&c.}$$ This may be demonstrated in the same manner as the preceding theorem, by writing $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, &c. successively for $x$. **Problem.** Let there be $n$ values $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, &c. of the quantity $x$, to which the $n$ values $s^\alpha$, $s^\beta$, $s^\gamma$, $s^\delta$, $s^\epsilon$, &c. of the quantity $y$ correspond; suppose these quantities to be found by any function $X$ of the quantity $x$; let $\pi$, $\rho$, $\sigma$, $\tau$, &c. be values of the quantities $x$, to which $s^\pi$, $s^\rho$, $s^\sigma$, $s^\tau$, &c. values of the quantity $y$ correspond: for $x$ substitute its abovementioned values $\pi$, $\rho$, $\sigma$, $\tau$, &c. in the function $X$, and let the quantities resulting be $s^\pi$, $s^\rho$, $s^\sigma$, $s^\tau$, &c. not equal to the preceding $s^\alpha$, $s^\beta$, $s^\gamma$, $s^\delta$, &c. respectively; to find a quantity which added to the function $X$ shall not only give the true values of the quantity $y$ corresponding to the values $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, &c. of the quantity $x$, but also cor- corresponding to the values \( \pi, \rho, \sigma, \tau, \&c. \) of the above- mentioned quantity \( x \). Assume \( s^\pi - s^\pi = T^\pi, s^\rho - s^\rho = T^\rho, s^\sigma - s^\sigma = T^\sigma, s^\tau - s^\tau = T^\tau, \&c.; \) then the errors of the function \( X \) will be respec- tively \( T^\pi, T^\rho, T^\sigma, T^\tau, \&c.; \) and the correcting quantity sought may be \[ \begin{align*} &x - \alpha \times x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c. \\ &\pi - \alpha \times \pi - \beta \times \pi - \gamma \times \pi - \delta \times \pi - \varepsilon \times \&c. \\ &+ x - \alpha \times x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c. \\ &\varepsilon - \alpha \times \varepsilon - \beta \times \varepsilon - \gamma \times \varepsilon - \delta \times \varepsilon - \varepsilon \times \&c. \\ &+ x - \alpha \times x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c. \\ &\sigma - \alpha \times \sigma - \beta \times \sigma - \gamma \times \sigma - \delta \times \sigma - \varepsilon \times \&c. \\ &+ x - \alpha \times x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c. \\ &\tau - \alpha \times \tau - \beta \times \tau - \gamma \times \tau - \delta \times \tau - \varepsilon \times \&c. \\ &+ \&c. \end{align*} \] \[ \text{Aliter.} \] Let \( x - \alpha \times x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c. \times x - \pi \) \( \times x - \rho \times x - \sigma \times x - \tau \times \&c. = N; \) \( \pi - \alpha \times \pi - \beta \times \pi - \gamma \times \pi - \delta \times \pi - \varepsilon \times \&c. \times \pi - \rho \times \pi - \sigma \times \pi - \tau \times \&c. = \Pi; \) \( \varepsilon - \alpha \times \varepsilon - \beta \times \varepsilon - \gamma \times \varepsilon - \delta \times \varepsilon - \varepsilon \times \&c. \times \varepsilon - \rho \times \varepsilon - \sigma \times \varepsilon - \tau \times \&c. = P; \) \( \sigma - \alpha \times \sigma - \beta \times \sigma - \gamma \times \sigma - \delta \times \sigma - \varepsilon \times \&c. \times \sigma - \pi \times \sigma - \rho \times \sigma - \tau \times \&c. = \Sigma; \) \( \tau - \alpha \times \tau - \beta \times \tau - \gamma \times \tau - \delta \times \tau - \varepsilon \times \&c. \times \tau - \pi \times \tau - \rho \times \tau - \sigma \times \&c. = T, \&c.; \) then may the correcting quantity sought be \( N \left( \frac{T^\pi}{\pi \times x - \pi} + \frac{T^\rho}{\rho \times x - \rho} + \frac{T^\sigma}{\sigma \times x - \sigma} + \frac{T^\tau}{\tau \times x - \tau} + \&c. \right). \) This Dr. Waring on Interpolations. This problem may be demonstrated in the same manner as the preceding theorems, by writing for \( x \) in the correcting quantity successively its values \( \pi, \varrho, \sigma, \tau, \&c. \). 2. For the correcting quantity sought may be assumed \[ \frac{x^i - \alpha^i \times x^i - \beta^i \times x^i - \gamma^i \times x^i - \delta^i \times \&c. \times x^i \times x^i - \varepsilon^i \times x^i - \sigma^i}{x^i - \alpha^i \times x^i - \beta^i \times x^i - \gamma^i \times x^i - \delta^i \times \&c. \times x^i \times x^i - \varepsilon^i \times x^i - \sigma^i} \times T^\pi + \frac{x^i - \alpha^i \times x^i - \beta^i \times x^i - \gamma^i \times x^i - \delta^i \times \&c. \times x^i \times x^i - \varepsilon^i \times x^i - \sigma^i}{x^i - \tau^i \times \&c.} \times T^\varrho + \&c. \] 3. In general, let \( z \) be any quantity which is = \( a \), when \( x \) becomes either \( \alpha, \beta, \gamma, \delta, \varepsilon, \&c. \); let \( z \) become successively \( A, B, C, D, \&c. \) when \( x \) becomes \( \pi, \varrho, \sigma, \tau, \&c. \) respectively. When \( x \) either = \( \varrho, \sigma, \tau, \&c. \), let \( \Pi = o \); but if \( x = \pi \), let \( \Pi = p \): in the same manner when \( x \) either = \( \pi, \sigma, \tau, \&c. \), let \( P = o \); but when \( x = \varrho \) let \( P = r \): and similarly, let \( \Sigma = o \) when \( x \) is either \( \pi, \varrho, \tau, \&c. \); but when \( x = \sigma \) let \( \Sigma = s \): and likewise, when \( x \) is either \( \pi, \varrho, \sigma, \&c. \), let \( T = o \); but when \( x = \tau \) let \( T = t \): &c. then for the correcting quantity sought may be assumed \[ \frac{z}{A} \times \frac{\pi}{p} \times T^\pi + \frac{z}{B} \times \frac{\varrho}{r} \times T^\varrho + \frac{z}{C} \times \frac{\sigma}{s} \times T^\sigma + \frac{z}{D} \times \frac{\tau}{t} \times T^\tau + \&c. \] THEOREM. Assume (n) quantities \( \alpha, \beta, \gamma, \delta, \varepsilon, \&c. \) then will the sum of all the (n) quantities of the following kind \[ \frac{\alpha^m}{\alpha - \beta \times \alpha - \gamma \times \alpha - \delta \times \alpha - \&c.} + \frac{\beta^m}{\beta - \alpha \times \beta - \gamma \times \beta - \delta \times \beta - \&c.} \] \[ + \frac{\gamma^m}{\gamma - \alpha \times \gamma - \beta \times \gamma - \delta \times \gamma - \&c.} + \frac{\delta^m}{\delta - \alpha \times \delta - \beta \times \delta - \gamma \times \delta - \&c.} \] \[ + \frac{\varepsilon^m}{\varepsilon - \alpha \times \varepsilon - \beta \times \varepsilon - \gamma \times \varepsilon - \&c.} + \&c. = 0, \text{ if } m \text{ be any whole number less than } n - 1; \text{ but if } m = n - 1, \text{ then will the above mentioned sum } = 1. \text{ In general, the sum of the } \] \( n \) terms \[ \frac{\alpha^m (\beta \gamma \delta \&c. + \beta \gamma \varepsilon \&c. + \beta \delta \varepsilon \&c. + \&c.)}{\alpha - \beta \times \alpha - \gamma \times \alpha - \delta \times \alpha - \&c.} \] \[ + \frac{\beta^m (\alpha \gamma \delta \&c. + \alpha \gamma \varepsilon \&c. + \alpha \delta \varepsilon \&c. + \&c.)}{\beta - \alpha \times \beta - \gamma \times \beta - \delta \times \beta - \&c.} \] \[ + \frac{\gamma^m (\alpha \beta \delta \&c. + \alpha \beta \varepsilon \&c. + \alpha \delta \varepsilon \&c. + \&c.)}{\gamma - \alpha \times \gamma - \beta \times \gamma - \delta \times \gamma - \&c.} \] \[ + \frac{\delta^m (\alpha \beta \gamma \&c. + \alpha \beta \varepsilon \&c. + \alpha \gamma \varepsilon \&c. + \&c.)}{\delta - \alpha \times \delta - \beta \times \delta - \gamma \times \delta - \&c.} \] \[ + \frac{\varepsilon^m (\alpha \beta \gamma \&c. + \alpha \beta \delta \&c. + \alpha \gamma \delta \&c. + \beta \gamma \delta \&c. + \&c.)}{\varepsilon - \alpha \times \varepsilon - \beta \times \varepsilon - \gamma \times \varepsilon - \&c.} + \&c. = 0, \] if \( m \) be less than \( n \), and \( m + r \) not equal to \( n - 1 \), where \( r \) is equal to the number of letters contained in each of the contents above mentioned \( \beta \gamma \delta, \&c. \beta \gamma \varepsilon, \&c. \beta \delta \varepsilon, \&c. \gamma \delta \varepsilon, \&c. \&c. \&c. \&c. \) respectively: but if \( m + r = n - 1 \), then will the above mentioned sum \( = \pm 1 \); it will be \( +1 \) if \( r \) be an even number, otherwise \( -1 \). DEMON- Dr. Waring on Interpolations. Demonstration. Suppose \(a + b\alpha + c\alpha^2 + d\alpha^3 + e\alpha^4 + \&c.\) = \(s^\alpha\), \(a + b\beta + c\beta^2 + d\beta^3 + e\beta^4 + \&c.\) = \(s^\beta\), \(a + b\gamma + c\gamma^2 + d\gamma^3 + e\gamma^4 + \&c.\) = \(s^\gamma\), \(a + b\delta + c\delta^2 + d\delta^3 + e\delta^4 + \&c.\) = \(s^\delta\), \(a + b\varepsilon + c\varepsilon^2 + d\varepsilon^3 + e\varepsilon^4 + \&c.\) = \(s^\varepsilon\), multiply these equations into \(A, B, C, D, E, \&c.\) unknown coefficients to be investigated, and there result \[ A \times s^\alpha = Aa + Ab\alpha + Ac\alpha^2 + Ad\alpha^3 + Ae\alpha^4 + \&c. \] \[ B \times s^\beta = Ba + Bb\beta + Bc\beta^2 + Bd\beta^3 + Be\beta^4 + \&c. \] \[ C \times s^\gamma = Ca + Cb\gamma + Cc\gamma^2 + Cd\gamma^3 + Ce\gamma^4 + \&c. \] \[ D \times s^\delta = Da + Db\delta + Dc\delta^2 + Dd\delta^3 + De\delta^4 + \&c. \] \[ E \times s^\varepsilon = Ea + Eb\varepsilon + Ec\varepsilon^2 + Ed\varepsilon^3 + Ee\varepsilon^4 + \&c. \&c. \&c. \] Now suppose \(As^\alpha + Bs^\beta + Cs^\gamma + Ds^\delta + Es^\varepsilon + \&c.\) = \(a + bx + cx^2 + dx^3 + ex^4 + \&c.\) and the correspondent parts respectively equal to each other; that is, \(a(A + B + C + D + E + \&c.) = a;\) \(b(A\alpha + B\beta + C\gamma + D\delta + E\varepsilon + \&c.) = bx;\) \(A\alpha^2 + B\beta^2 + C\gamma^2 + D\delta^2 + E\varepsilon^2 + \&c. = x^2;\) \(A\alpha^3 + B\beta^3 + C\gamma^3 + D\delta^3 + E\varepsilon^3 + \&c. = x^3;\) \(A\alpha^4 + B\beta^4 + C\gamma^4 + D\delta^4 + E\varepsilon^4 + \&c. = x^4, \&c.\). But it follows from Theorem I. that (if \(As^\alpha + Bs^\beta + Cs^\gamma + Ds^\delta + Es^\varepsilon + \&c.\) \(= a + bx + cx^2 + dx^3 + ex^4 + \&c.\)) \(A = \frac{x - \beta \times x - \gamma \times x - \delta \times x - \varepsilon \times \&c.}{a - \beta \times a - \gamma \times a - \delta \times a - \varepsilon \times \&c.}\), Vol. LXIX. Dr. Waring on Interpolations. \[ B = \frac{x - \alpha}{\beta - \alpha} \times \frac{x - \gamma}{\beta - \gamma} \times \frac{x - \delta}{\beta - \delta} \times \cdots \times \text{&c.}, \quad C = \frac{x - \alpha}{\gamma - \alpha} \times \frac{x - \beta}{\gamma - \beta} \times \frac{x - \delta}{\gamma - \delta} \times \cdots \times \text{&c.}, \] \[ D = \frac{x - \alpha}{\delta - \alpha} \times \frac{x - \beta}{\delta - \beta} \times \frac{x - \gamma}{\delta - \gamma} \times \cdots \times \text{&c.}, \quad E = \frac{x - \alpha}{\epsilon - \alpha} \times \frac{x - \beta}{\epsilon - \beta} \times \frac{x - \gamma}{\epsilon - \gamma} \times \cdots \times \text{&c.}, \] &c.: substitute these values for \( A, B, C, D, E, \) &c. respectively in the preceding equations (\( A + B + C + D + E + \text{&c.} = 1, \) \( A \alpha + B \beta + C \gamma + D \delta + E \epsilon + \text{&c.} = x, \) \( A \alpha^2 + B \beta^2 + C \gamma^2 + D \delta^2 + E \epsilon^2 + \text{&c.} = x^2, \) \( A \alpha^3 + B \beta^3 + C \gamma^3 + D \delta^3 + E \epsilon^3 + \text{&c.} = x^3, \) &c.) and there result the equations (1) \[ \frac{x - \beta}{\alpha - \beta} \times \frac{x - \gamma}{\alpha - \gamma} \times \frac{x - \delta}{\alpha - \delta} \times \cdots \times \text{&c.} + \frac{x - \alpha}{\beta - \alpha} \times \frac{x - \gamma}{\beta - \gamma} \times \frac{x - \delta}{\beta - \delta} \times \cdots \times \text{&c.} + \text{&c.} = 1; \] (2) \( \alpha \times \frac{x - \beta}{\alpha - \beta} \times \frac{x - \gamma}{\alpha - \gamma} \times \frac{x - \delta}{\alpha - \delta} \times \cdots \times \text{&c.} + \beta \times \frac{x - \alpha}{\beta - \alpha} \times \frac{x - \gamma}{\beta - \gamma} \times \frac{x - \delta}{\beta - \delta} \times \cdots \times \text{&c.} + \gamma \times \frac{x - \alpha}{\gamma - \alpha} \times \frac{x - \beta}{\gamma - \beta} \times \frac{x - \delta}{\gamma - \delta} \times \cdots \times \text{&c.} + \text{&c.} = x; \) (3) \( \alpha^2 \times \frac{x - \beta}{\alpha - \beta} \times \frac{x - \gamma}{\alpha - \gamma} \times \frac{x - \delta}{\alpha - \delta} \times \cdots \times \text{&c.} + \beta^2 \times \frac{x - \alpha}{\beta - \alpha} \times \frac{x - \gamma}{\beta - \gamma} \times \frac{x - \delta}{\beta - \delta} \times \cdots \times \text{&c.} + \gamma^2 \times \frac{x - \alpha}{\gamma - \alpha} \times \frac{x - \beta}{\gamma - \beta} \times \frac{x - \delta}{\gamma - \delta} \times \cdots \times \text{&c.} = x^2; \) and in general, \[ \alpha^m \times \frac{x - \beta}{\alpha - \beta} \times \frac{x - \gamma}{\alpha - \gamma} \times \frac{x - \delta}{\alpha - \delta} \times \cdots \times \text{&c.} + \beta^m \times \frac{x - \alpha}{\beta - \alpha} \times \frac{x - \gamma}{\beta - \gamma} \times \frac{x - \delta}{\beta - \delta} \times \cdots \times \text{&c.} + \gamma^m \times \frac{x - \alpha}{\gamma - \alpha} \times \frac{x - \beta}{\gamma - \beta} \times \frac{x - \delta}{\gamma - \delta} \times \cdots \times \text{&c.} + \delta^m \times \frac{x - \alpha}{\delta - \alpha} \times \frac{x - \beta}{\delta - \beta} \times \frac{x - \gamma}{\delta - \gamma} \times \cdots \times \text{&c.} + \text{&c.} = x^m, \] whatever may be the values of the quantities \( x; \alpha, \beta, \gamma, \delta, \epsilon, \) &c.: reduce all these fractions into terms, proceeding according to the dimensions of the quantity \( x, \) and it is evident, that the sum of all the fractions multiplied tiplied into any dimension of \( x \) not equal to \( m \) will be \( = 0 \); but the sum of all the fractions multiplied into \( x^m \) will be \( = 1 \): from this proposition the theorem is easily deduced. I have invented and demonstrated from different principles to the preceding the first part of this theorem, a particular case of which was published by me many years ago. From this theorem may easily be deduced several others of a similar nature.