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The Collected Works and Correspondence of Chauncey Wright
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Collected Works of Chauncey Wright, Volume 1
Essays and Reviews
Extension of the Prismoidal Formula.

Extension of the Prismoidal Formula.6

BY CHAUNCEY WRIGHT

We have shown that the Prismoidal Formula is an exact definite integral for algebraic expressions of positive integer powers not exceeding the third degree, and that it is sufficiently accurate in all cases where the fourth order of differences may be neglected. We propose now to determine similar formulas, which shall include higher orders of differences.

If we represent by
A mathematical expression showing a sequence of function values: f(x), f(x + h₁), f(x − h₁), f(x + h₂), f(x − h₂), and so on, indicating function evaluations at points incremented and decremented by different step sizes h₁, h₂, etc.

a series of sections made perpendicular to the axis of X, through a solid or plane figure, and at the distances x, x + h1, x - h1, &c. from the origin of x, then the expression
Mathematical expression labeled (A) showing a weighted sum of function values f evaluated at points x offset by -hn through -h1, a central term afx, and positive offsets h1 through hn, with corresponding coefficients hn, ..., h2, h1, a, a1, a2, ..., an.

which contains 2 n + 1 terms, n + 1 coefficients a, a1, a2, &c., and n values h1, h2, &c., may be made to represent the contents of any portion of the solid or plane figure for which f(x) is an algebraic expression of positive integer powers not exceeding the (4 n — l)th degree.

Formula (A) may be written thus: —
A mathematical expression involving sums and functions: \( h_n \sum_{m} a_m \big[ f(x + h_m) + f(x - h_m) \big] + h_n a f(x) \). The terms include coefficients \( h_n \) and \( a_m \), a summation indexed by \( m \), and a function \( f \) evaluated at points \( x + h_m \), \( x - h_m \), and \( x \).
and developed by Taylor’s Theorem, it becomes
The figure shows a mathematical equation involving summations and derivatives of the function \( f(x) \). It contains terms with coefficients \( a_n \), \( a \), and constants such as \( h \) and powers of \( h \) with subscripts, as well as factorial-like denominators (e.g., \( 1.2 \), \( 2.3.4 \)). The equation includes several derivative notations: \( f(x) \), \( f'(x) \), \( f''(x) \), \( f^{III}(x) \), and \( f^{IV}(x) \). The expression is divided into two large bracketed parts, with the left part involving a sum and multiples of \( f(x) \) and its derivatives, and the right part labeled as \( (B) \) containing a similar structure but different coefficients and combination of terms. The equation appears to represent a finite difference or series expansion approximation involving derivatives of \( f \).

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Now, the definite integral
An integral expression featuring the integral symbol with respect to \(x\), integrating the difference between \(f(x + h_n)\) and \(f(x - h_n)\) inside the integrand.
becomes, when developed,
A mathematical expression showing an equality involving integrals of functions with terms for derivatives and coefficients. The left side contains twice the constant hâ‚™ multiplied by the integral of a sum of terms with f(x) and its derivatives, each term scaled by fractional coefficients involving powers of hâ‚™ and factorial-like denominators. The right side shows twice hâ‚™ multiplied by the integral of a similar sum, with terms involving f(x) and its derivatives, and additional terms indicated by "&c." on both sides. The equation is enclosed by parentheses labeled (C').
and, if we make the corresponding terms of (B) and (C) equal, we have the following equations
The figure presents a set of linear equations involving sums of terms with coefficients \(a_1, a_2, \ldots, a_n\) and variables \(h_n\) raised to different powers. Each equation equates a fraction (e.g., \(\frac{h_n^1}{3}, \frac{h_n^2}{5}, \frac{h_n^3}{7}\)) to a weighted sum of powers of \(h_n\). The sums are expressed in sigma notation. A transformation is introduced by defining ratios \(r_1 = \frac{h_1}{h_n}, r_0 = \frac{h_2}{h_n}\), etc., which rewrite the original equations into a simpler normalized form with powers of the ratios \(r_i\) and unknown coefficients \(a_i\). The equations are labeled as (D).
and so on, in which a, a1, a2, &c., are the coefficients of the sections, and r1, r2, &c., the ratios of their distances from the middle section to the distance A, of the extreme sections from the middle.

If by l we denote the whole distance between the extreme sections, then hn = l/2.

The number of the equations (D) which can be satisfied is equal to the number of undetermined quantities a, a1, a2, ...an; r1, r2, ... rn-1, that is, to (n + 1) + (n - 1) = 2 n. But the (2 n + 1)th

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terms of the expressions (B) and (C) contain the coefficients of the 4nth derivative of f(x), which are therefore the first not included in the equations (D); hence (-4) may be made accurate for (4 n — 1) derivatives of f(x). When the sections are made at equal intervals, the distances h1, h2, &c., and the ratios r1, r2, &c., are determined by their number, and (A) can be made accurate only for as many derivatives of f(x) as there are sections, that is, for 2 n + 1. Since there are 2 n equations (D) the last will be of the form
An equation expressing one over (4n minus 1) as the sum of terms, each term having a coefficient a_i multiplied by r_i raised to the power of 4n minus 2. The sum includes terms from i = 1 to i = s.
and the error of (A) for functions of two higher orders of derivatives, that is for functions which have 4 n, or 4 n + 1 derivatives, is
Mathematical expression representing a term in a sequence or series: the fraction with numerator 2 raised to the power (h sub n plus 1) and denominator the product 1·2·3···4^n, multiplied by a sum of terms a1 r1^(4^n) + a2 r2^(4^n) + ... + an, minus the fraction 1/(4^n + 1), all multiplied by the function f^(4^n) applied to x.

EXAMPLES.

1. When there are only three sections, n = 1, and the equations (D) become 1 = 1/2 a + a1 and 1/3 = a1, hence a = 4/3 and (A)
The equation shows a weighted average expression on the left side, involving the function f evaluated at three points: x minus h sub 1, x, and x plus h sub 1, multiplied by h sub a over 3. This is equal to the right side, which is u over 6 times the sum of B, four times B double prime, and B double prime prime, with B and its derivatives indicated as B, B'', and B''.

which is the common prismoidal formula.

2. When n = 2, there are five sections, and the equations (D) become
Four algebraic equations are shown, involving the variables \(a_1\), \(a_2\), and \(a\), with fractional coefficients on the left-hand side. The equations are: 1 equals \(\frac{1}{2}a + a_1 + a_2\), \(\frac{1}{8} = a_1 r_1^2 + a_2\), \(\frac{1}{6} = a_1 r_1^4 + a_2\), \(\frac{1}{7} = a_1 r_1^6 + a_2\). Each equation forms a linear combination of terms involving powers of \(r_1\) and the constants \(a_1\) and \(a_2\), presumably for solving the values of these variables.

If these sections are made at equal intervals, then r1 is determined, and only three of these equations can be satisfied; that is, only five orders of derivatives can be included by them. As the second and fourth sections will in this case bisect the intervals between the middle and the extreme sections, r1 = 1/2 and the equations
A handwritten set of three algebraic expressions showing equalities involving fractions and variables a, a₁, and a₂. The first equation is "l = ⅓ a + a₁ + a₂". The second is "⅓ = a₁ ⅓ + a₂," and the third is "⅓ = a₁ ⅓⅓ + a₂". The notation is in cursive style with some overlapping and joined characters.

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The figure presents the values of three variables expressed as fractions: a equals one and one-sixth, a subscript one equals two and one-sixths, and a subscript two equals one and five-sixths.
so that the formula (A) becomes
Equation labeled (1) showing a mathematical expression: the fraction l divided by 90, multiplied by the sum of terms \(7B + 32B' + 12B'' + 32B''' + 7B^{IV}\).
and its error for functions of six or seven derivatives is
Mathematical notation displaying the expression f^(vi)(x)·l^l divided by the number 1,935,360. The expression f^(vi)(x) indicates the sixth derivative of the function f with respect to x, and l^l denotes l raised to the power of l.

If r1 be taken so that the coefficient a of the middle section shall disappear, then
The image displays a mathematical expression defining three variables: \( r_1 = \sqrt{\frac{1}{3}} \), \( a_1 = \frac{5}{6} \), and \( a_2 = \frac{1}{6} \).
and we obtain a formula of four sections,
A mathematical expression labeled as equation (2) showing a fraction with numerator 1 and denominator 12, multiplied by a sum inside parentheses: B plus 5 times B prime plus 5 times B double prime plus B triple prime. Below the equation, text states that the interval between B prime and B double prime is the fraction 1 over the square root of 15.

This formula is accurate for five derivatives, and its error for functions of six or seven derivatives is
Mathematical notation showing the fraction with numerator "f^(vi)(x), l'" and denominator "1512000."

If r1 is left indeterminate, then the solution of the four equations above will give
The figure presents a mathematical expression defining variables \(a\), \(a_1\), \(a_2\), and \(r_1\). The variable \(a\) is assigned the value \(\frac{9}{4}\), \(a_1\) is defined as \(\frac{3}{2}\), \(a_2\) as \(\frac{5}{4}\), and \(r_1\) as the square root of 7 (\(\sqrt{7}\)).
and we obtain the formula of five sections

The figure presents a mathematical expression labeled (3), which is a weighted sum involving terms 9B, 49B', 64B'', 49B''', and 9B'''' multiplied by the factor l/180. The text explains that the sections B' and B''' are located at a distance of (l/2)√(7/3) from the middle section B''. It notes that the formula is accurate for seven derivatives and provides an error estimate for functions with eight or nine derivatives expressed as a fraction involving the 8th derivative of the function, f^(viii)(x), multiplied by l^8 and divided by 237081600.

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This error and those of formulas (1) and (2) may be estimated in terms of the finite differences of f(x) by the following transformations. If the length l be divided by sections into so many parts m that the corresponding differences of f(x) of a higher order than the ith, may be neglected, then
The figure presents a set of mathematical expressions defining an operator Δ* applied to a function f(x). It shows Δ* f(x) as equal to the mth derivative of f(x), denoted f^(m)(x), or equivalently the logarithmic operator lm* applied to Δ* f(x) is equal to the (i+1)th derivative of f(x), denoted f^(i+1)(x). It further states that the error of formula (3) can be expressed as (lm* Δ* f(x)) divided by 237081600.
Hence, if the number of parts m into which l is divided be about the eighth root of 237081600, or about 11, and if the corresponding value of Δ8f(x) be inappreciable, then formula (3) is sufficiently accurate. In the same way the accuracy of formulas (1) and (2) may be estimated in practice.

In the application of formulas (1), (2), and (3) to symmetrical figures, as for instance for determining the contents of casks, since the sections at equal distances from the middle section are equal, these formulas may be written thus: —
Three mathematical expressions are shown, each representing a weighted sum of terms involving B, B', and B''. The first expression is (1/45) multiplied by (7B + 32B' + 6B''), the second expression is (1/6) multiplied by (B + 5B'), and the third expression is (1/90) multiplied by (9B + 49B' + 32B'').

Cases in practice might arise in which sections at equal intervals could not be obtained. For such cases special formulas might be easily obtained from the equations (D).

In some future number of this journal we shall apply the formulas of the preceding discussion to a variety of problems in Engineering, Tonnage of Vessels, Cask Gauging, &c.